------------------------------------------------------------------------------- name: log: /Users/carlosmendez/Documents/GitHub/starter-academic-v501/content > /post/stata_convergence/analysis.log log type: text opened on: 1 May 2026, 10:37:27 . . display as text "" . display as text "╔═══════════════════════════════════════════════════════════ > ═══╗" ╔══════════════════════════════════════════════════════════════╗ . display as text "║ Beta and Sigma Convergence: A Tutorial with PWT 10.0 > ║" ║ Beta and Sigma Convergence: A Tutorial with PWT 10.0 ║ . display as text "║ Balanced panel: 84 countries with data since 1960 > ║" ║ Balanced panel: 84 countries with data since 1960 ║ . display as text "╚═══════════════════════════════════════════════════════════ > ═══╝" ╚══════════════════════════════════════════════════════════════╝ . display as text "" . . . *════════════════════════════════════════════════════════════════════════════ > ═══ . * SECTION 0: DATA PREPARATION . * . * We load the Penn World Tables 10.0 and prepare a clean cross-country . * dataset of GDP per capita. Following the literature, we exclude: . * (1) oil-exporting countries (whose income is driven by resource rents) . * (2) very small countries (population under 1 million) . * (3) countries without GDP data in 1960 (balanced panel requirement) . *════════════════════════════════════════════════════════════════════════════ > ═══ . . display as text "" . display as text "──────────────────────────────────────────────────────────── > ────" ──────────────────────────────────────────────────────────────── . display as text " SECTION 0: DATA PREPARATION" SECTION 0: DATA PREPARATION . display as text "──────────────────────────────────────────────────────────── > ────" ──────────────────────────────────────────────────────────────── . . * ── Load PWT 10.0 ── . use "https://raw.githubusercontent.com/cmg777/starter-academic-v501/master/co > ntent/post/stata_convergence/pwt100.dta", clear . . * Keep only the variables we need . rename countrycode ccode . keep country ccode year pop rgdpe . . * ── Compute GDP per capita (PPP) ── . * rgdpe = expenditure-side real GDP at chained PPPs (in million 2017 US$) . * pop = population (in millions) . * gdppc = GDP per capita in 2017 US$ PPP . gen gdppc = rgdpe / pop (2,411 missing values generated) . label variable gdppc "Real GDP per capita (PPP, 2017 US$)" . drop if missing(gdppc) (2,411 observations deleted) . drop if missing(pop) (0 observations deleted) . . * ── Identify and exclude oil-producing countries ── . * These countries' incomes are driven by resource extraction rather than . * the capital accumulation process that convergence theory describes. . * List follows IMF classification, same as Patel et al. (2021). . gen oil = inlist(ccode, "DZA", "AGO", "AZE", "BHR", "BRN", "TCD", "COG") | // > / > inlist(ccode, "ECU", "GNQ", "GAB", "IRN", "IRQ", "KAZ", "KWT") | // > / > inlist(ccode, "NGA", "OMN", "QAT", "RUS", "SAU", "TTO", "TKM") | // > / > inlist(ccode, "ARE", "VEN", "YEM", "LBY", "TLS", "SDN") . display "Oil-producing countries excluded:" Oil-producing countries excluded: . tab ccode if oil == 1 3-letter | ISO country | code | Freq. Percent Cum. ------------+----------------------------------- AGO | 50 3.86 3.86 ARE | 50 3.86 7.72 AZE | 30 2.31 10.03 BHR | 50 3.86 13.89 BRN | 50 3.86 17.75 COG | 60 4.63 22.38 DZA | 60 4.63 27.01 ECU | 70 5.40 32.41 GAB | 60 4.63 37.04 GNQ | 60 4.63 41.67 IRN | 65 5.02 46.68 IRQ | 50 3.86 50.54 KAZ | 30 2.31 52.85 KWT | 50 3.86 56.71 NGA | 70 5.40 62.11 OMN | 50 3.86 65.97 QAT | 50 3.86 69.83 RUS | 30 2.31 72.15 SAU | 50 3.86 76.00 SDN | 50 3.86 79.86 TCD | 60 4.63 84.49 TKM | 30 2.31 86.81 TTO | 70 5.40 92.21 VEN | 70 5.40 97.61 YEM | 31 2.39 100.00 ------------+----------------------------------- Total | 1,296 100.00 . drop if oil == 1 (1,296 observations deleted) . drop oil . . * ── Exclude small countries (population < 1 million) ── . * Small economies behave differently (e.g., tax havens, tourism-dependent) . gen small = pop < 1 . drop if small == 1 (1,931 observations deleted) . drop small . . * ── Restrict to 1960 onwards ── . * Reliable cross-country data begins around 1960 . drop if year < 1960 (560 observations deleted) . . * ── Restrict to balanced panel: countries with data in 1960 ── . * This ensures a consistent sample across ALL sections of the tutorial. . * Without this restriction, the sample grows from 84 (1960) to 124 (2019) . * as PWT coverage expands, introducing composition effects. . bys ccode: egen has1960 = max(year == 1960 & !missing(gdppc)) . display as text "" . display as text "Countries WITHOUT 1960 data (excluded from balanced panel):" Countries WITHOUT 1960 data (excluded from balanced panel): . tab ccode if has1960 == 0 3-letter | ISO country | code | Freq. Percent Cum. ------------+----------------------------------- ALB | 50 3.18 3.18 ARM | 30 1.91 5.09 BGR | 50 3.18 8.27 BIH | 30 1.91 10.18 BLR | 30 1.91 12.09 BWA | 36 2.29 14.38 CZE | 30 1.91 16.28 EST | 30 1.91 18.19 GEO | 30 1.91 20.10 GMB | 28 1.78 21.88 GNB | 28 1.78 23.66 HRV | 30 1.91 25.57 HUN | 50 3.18 28.75 JOR | 59 3.75 32.51 KGZ | 30 1.91 34.41 KHM | 50 3.18 37.60 LAO | 50 3.18 40.78 LBN | 50 3.18 43.96 LBR | 56 3.56 47.52 LSO | 51 3.24 50.76 LTU | 30 1.91 52.67 LVA | 30 1.91 54.58 MDA | 30 1.91 56.49 MKD | 30 1.91 58.40 MMR | 58 3.69 62.09 MNG | 50 3.18 65.27 MRT | 59 3.75 69.02 MUS | 36 2.29 71.31 NAM | 42 2.67 73.98 POL | 50 3.18 77.16 PSE | 50 3.18 80.34 SLE | 59 3.75 84.10 SRB | 30 1.91 86.01 SVK | 30 1.91 87.91 SVN | 30 1.91 89.82 SWZ | 20 1.27 91.09 TJK | 30 1.91 93.00 UKR | 30 1.91 94.91 UZB | 30 1.91 96.82 VNM | 50 3.18 100.00 ------------+----------------------------------- Total | 1,572 100.00 . display as text "" . display as text "Number of countries dropped for missing 1960 data: " _contin > ue Number of countries dropped for missing 1960 data: . count if has1960 == 0 1,572 . keep if has1960 == 1 (1,572 observations deleted) . drop has1960 . . * ── Summary statistics ── . display as text "" . display as text "Summary of cleaned PWT dataset (balanced panel):" Summary of cleaned PWT dataset (balanced panel): . summarize gdppc year pop, detail Real GDP per capita (PPP, 2017 US$) ------------------------------------------------------------- Percentiles Smallest 1% 626.9015 514.4933 5% 886.608 514.4965 10% 1128.028 518.0869 Obs 5,040 25% 1971.046 523.0964 Sum of wgt. 5,040 50% 5197.031 Mean 11951.11 Largest Std. dev. 14402.89 75% 17417.52 88681.06 90% 35014.29 89403.9 Variance 2.07e+08 95% 43341.46 90413.35 Skewness 1.763351 99% 59514.25 102937.7 Kurtosis 6.095034 Year ------------------------------------------------------------- Percentiles Smallest 1% 1960 1960 5% 1962.5 1960 10% 1965.5 1960 Obs 5,040 25% 1974.5 1960 Sum of wgt. 5,040 50% 1989.5 Mean 1989.5 Largest Std. dev. 17.31982 75% 2004.5 2019 90% 2013.5 2019 Variance 299.9762 95% 2016.5 2019 Skewness 0 99% 2019 2019 Kurtosis 1.799333 Population (in millions) ------------------------------------------------------------- Percentiles Smallest 1% 1.767255 1.1388 5% 2.719834 1.172108 10% 3.470137 1.206392 Obs 5,040 25% 5.629009 1.241673 Sum of wgt. 5,040 50% 11.71279 Mean 52.02887 Largest Std. dev. 160.0146 75% 35.66088 1414.049 90% 82.70982 1421.022 Variance 25604.66 95% 164.3187 1427.648 Skewness 6.256689 99% 1095.014 1433.784 Kurtosis 44.76702 . display as text "" . display as text "Number of unique countries:" Number of unique countries: . codebook ccode ------------------------------------------------------------------------------- ccode 3-letter ISO country code ------------------------------------------------------------------------------- Type: String (str3) Unique values: 84 Missing "": 0/5,040 Examples: "CMR" "HKG" "MOZ" "SEN" . . * ── Export cleaned data ── . export delimited ccode country year gdppc pop using "convergence_data_prepare > d.csv", replace file convergence_data_prepared.csv saved . display as text "Cleaned data exported to: convergence_data_prepared.csv" Cleaned data exported to: convergence_data_prepared.csv . . * ── Save working dataset ── . save "convergence_working.dta", replace (file convergence_working.dta not found) file convergence_working.dta saved . . . *════════════════════════════════════════════════════════════════════════════ > ═══ . * SECTION 1: BETA CONVERGENCE — THE SIMPLEST CASE (Two Periods) . * . * The most fundamental test of convergence: . * Do countries that START poorer GROW faster? . * . * We use just two years: 1960 (initial) and 2019 (final). . * If the answer is yes, we have "beta convergence." . * . * The regression is: . * growth_i = α + λ × ln(y_i,1960) + ε_i . * . * where: . * growth_i = (1/s) × ln(y_i,2019 / y_i,1960) = annualized growth rate . * ln(y_i,1960) = log of initial GDP per capita . * . * A NEGATIVE λ means convergence: poorer countries (low y_1960) grow faster > . . * A POSITIVE λ (or zero) means divergence or no convergence. . *════════════════════════════════════════════════════════════════════════════ > ═══ . . display as text "" . display as text "──────────────────────────────────────────────────────────── > ────" ──────────────────────────────────────────────────────────────── . display as text " SECTION 1: BETA CONVERGENCE — THE SIMPLEST CASE" SECTION 1: BETA CONVERGENCE — THE SIMPLEST CASE . display as text " Two periods: 1960 and 2019" Two periods: 1960 and 2019 . display as text "──────────────────────────────────────────────────────────── > ────" ──────────────────────────────────────────────────────────────── . . use "convergence_working.dta", clear . . * ── Reshape to wide format so each country is one row ── . keep ccode country year gdppc . reshape wide gdppc, i(ccode country) j(year) (j = 1960 1961 1962 1963 1964 1965 1966 1967 1968 1969 1970 1971 1972 1973 1974 > 1975 1976 1977 1978 1979 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1 > 990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 200 > 5 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 2017 2018 2019) Data Long -> Wide ----------------------------------------------------------------------------- Number of observations 5,040 -> 84 Number of variables 4 -> 62 j variable (60 values) year -> (dropped) xij variables: gdppc -> gdppc1960 gdppc1961 ... gdppc201 > 9 ----------------------------------------------------------------------------- . . * ── Calculate annualized growth rate and log initial income ── . * growth = (1/59) × ln(GDP_2019 / GDP_1960) . * This gives the average annual growth rate over 59 years. . local s = 2019 - 1960 . gen growth = (1/`s') * ln(gdppc2019 / gdppc1960) . gen initial = ln(gdppc1960) . . * Drop countries missing either endpoint . drop if missing(growth) | missing(initial) (0 observations deleted) . . * ── Display what we have ── . display as text "" . display as text "Countries with data for both 1960 and 2019: " _continue Countries with data for both 1960 and 2019: . count 84 . display as text "" . display as text "Summary of variables:" Summary of variables: . summarize growth initial Variable | Obs Mean Std. dev. Min Max -------------+--------------------------------------------------------- growth | 84 .0222478 .0149403 -.0169141 .0594108 initial | 84 8.017196 .9614768 6.243189 10.02171 . . * ── Run the simplest beta-convergence regression ── . * This is a simple OLS: growth = α + λ × initial + ε . * We use robust standard errors to account for heteroskedasticity. . display as text "" . display as text "╔═══════════════════════════════════════════════════════════ > ═══╗" ╔══════════════════════════════════════════════════════════════╗ . display as text "║ OLS Beta-Convergence Regression: 1960 to 2019 > ║" ║ OLS Beta-Convergence Regression: 1960 to 2019 ║ . display as text "╚═══════════════════════════════════════════════════════════ > ═══╝" ╚══════════════════════════════════════════════════════════════╝ . reg growth initial, robust Linear regression Number of obs = 84 F(1, 82) = 0.19 Prob > F = 0.6606 R-squared = 0.0013 Root MSE = .01502 ------------------------------------------------------------------------------ | Robust growth | Coefficient std. err. t P>|t| [95% conf. interval] -------------+---------------------------------------------------------------- initial | .0005689 .0012908 0.44 0.661 -.0019988 .0031366 _cons | .0176868 .0112996 1.57 0.121 -.0047917 .0401653 ------------------------------------------------------------------------------ . . * ── Interpret the result ── . local beta_ols = _b[initial] . local se_ols = _se[initial] . local t_ols = `beta_ols' / `se_ols' . local p_ols = 2 * ttail(e(df_r), abs(`t_ols')) . . display as text "" . display as text "INTERPRETATION:" INTERPRETATION: . if `beta_ols' < 0 & `p_ols' < 0.05 { . display as result " The coefficient is NEGATIVE and statistically signif > icant." . display as result " This means poorer countries grew FASTER — evidence o > f CONVERGENCE." . } . else if `beta_ols' < 0 & `p_ols' >= 0.05 { . display as result " The coefficient is negative but NOT statistically si > gnificant." . display as result " Weak or no evidence of convergence over the full 196 > 0-2019 period." . } . else { . display as result " The coefficient is positive — evidence of DIVERGENCE > ." The coefficient is positive — evidence of DIVERGENCE. . display as result " Richer countries grew faster, and the gap widened." Richer countries grew faster, and the gap widened. . } . . * ── Create scatter plot with fitted line ── . * This is the classic convergence scatter plot: growth vs. initial income . #delimit ; delimiter now ; . tw (lfitci growth initial, fcolor("106 155 204"%20) lcolor("106 155 204")) > (scatter growth initial, mcolor("217 119 87"%70) msymbol(O) msize(small) > mlabel(ccode) mlabsize(vsmall) mlabpos(0) mlabcolor("20 20 19"%60)), > ytitle("Average Annual Growth Rate (1960-2019)") > xtitle("Log GDP per Capita in 1960") > title("Beta Convergence Test: 1960 to 2019", color("20 20 19")) > subtitle("Do poorer countries grow faster?", color("20 20 19")) > xlabel(6 "$403" 7 "$1,097" 8 "$2,981" 9 "$8,103" 10 "$22,026", angle(0)) > ylabel(, angle(horizontal)) > legend(off) > plotregion(style(none) lcolor(none)) > graphregion(fcolor(white) lcolor(white)) > note("Data: Penn World Tables 10.0. 84-country balanced panel." > "Excludes oil producers and countries with pop < 1M." > "Shaded area: 95% confidence interval around fitted line.", size(vsma > ll)) > xsize(5) ysize(4) > ; . #delimit cr delimiter now cr . graph export "stata_convergence_scatter_1960_2019.png", replace width(2400) file stata_convergence_scatter_1960_2019.png written in PNG format . display as text "Figure saved: stata_convergence_scatter_1960_2019.png" Figure saved: stata_convergence_scatter_1960_2019.png . . * ── Export results ── . preserve . clear . set obs 1 Number of observations (_N) was 0, now 1. . gen period = "1960-2019" . gen method = "OLS" . gen beta = `beta_ols' . gen se = `se_ols' . gen pvalue = `p_ols' . gen n = `e(N)' . export delimited using "convergence_beta_simple.csv", replace file convergence_beta_simple.csv saved . restore . . . *════════════════════════════════════════════════════════════════════════════ > ═══ . * SECTION 2: BETA CONVERGENCE — COMPARING TWO ERAS . * . * A single regression over 1960-2019 misses a crucial story: . * the WORLD CHANGED in the mid-1990s. . * . * - Era of DIVERGENCE (1960-2000): Poor countries did NOT catch up. . * - Era of CONVERGENCE (2000-2019): Poor countries ARE catching up. . * . * By splitting the sample, we can see this structural break. . *════════════════════════════════════════════════════════════════════════════ > ═══ . . display as text "" . display as text "──────────────────────────────────────────────────────────── > ────" ──────────────────────────────────────────────────────────────── . display as text " SECTION 2: COMPARING TWO ERAS — Divergence vs. Convergence > " SECTION 2: COMPARING TWO ERAS — Divergence vs. Convergence . display as text "──────────────────────────────────────────────────────────── > ────" ──────────────────────────────────────────────────────────────── . . * ── Era of Divergence: 1960 to 2000 ── . display as text "" . display as text "--- Era 1: 1960 to 2000 (the 'divergence era') ---" --- Era 1: 1960 to 2000 (the 'divergence era') --- . gen growth_era1 = (1/40) * ln(gdppc2000 / gdppc1960) . gen initial_era1 = ln(gdppc1960) . reg growth_era1 initial_era1 if !missing(growth_era1) & !missing(initial_era1 > ), robust Linear regression Number of obs = 84 F(1, 82) = 7.59 Prob > F = 0.0072 R-squared = 0.0436 Root MSE = .01978 ------------------------------------------------------------------------------ | Robust growth_era1 | Coefficient std. err. t P>|t| [95% conf. interval] -------------+---------------------------------------------------------------- initial_era1 | .004366 .0015843 2.76 0.007 .0012143 .0075176 _cons | -.0150282 .0137431 -1.09 0.277 -.0423675 .0123112 ------------------------------------------------------------------------------ . local b_era1 = _b[initial_era1] . local se_era1 = _se[initial_era1] . local n_era1 = e(N) . . * ── Era of Convergence: 2000 to 2019 ── . display as text "" . display as text "--- Era 2: 2000 to 2019 (the 'convergence era') ---" --- Era 2: 2000 to 2019 (the 'convergence era') --- . gen growth_era2 = (1/19) * ln(gdppc2019 / gdppc2000) . gen initial_era2 = ln(gdppc2000) . reg growth_era2 initial_era2 if !missing(growth_era2) & !missing(initial_era2 > ), robust Linear regression Number of obs = 84 F(1, 82) = 5.75 Prob > F = 0.0187 R-squared = 0.0688 Root MSE = .01794 ------------------------------------------------------------------------------ | Robust growth_era2 | Coefficient std. err. t P>|t| [95% conf. interval] -------------+---------------------------------------------------------------- initial_era2 | -.0035228 .0014686 -2.40 0.019 -.0064442 -.0006013 _cons | .0580907 .014098 4.12 0.000 .0300452 .0861362 ------------------------------------------------------------------------------ . local b_era2 = _b[initial_era2] . local se_era2 = _se[initial_era2] . local n_era2 = e(N) . . display as text "" . display as text "╔═══════════════════════════════════════════════════════════ > ═══╗" ╔══════════════════════════════════════════════════════════════╗ . display as text "║ Comparison of Two Eras > ║" ║ Comparison of Two Eras ║ . display as text "╚═══════════════════════════════════════════════════════════ > ═══╝" ╚══════════════════════════════════════════════════════════════╝ . display as text " Era of Divergence (1960-2000): λ = " %7.5f `b_era1' " (SE > = " %7.5f `se_era1' ", N = " `n_era1' ")" Era of Divergence (1960-2000): λ = 0.00437 (SE = 0.00158, N = 84) . display as text " Era of Convergence (2000-2019): λ = " %7.5f `b_era2' " (SE > = " %7.5f `se_era2' ", N = " `n_era2' ")" Era of Convergence (2000-2019): λ = -0.00352 (SE = 0.00147, N = 84) . display as text "" . display as text " The slope FLIPPED from positive (divergence) to negative ( > convergence)." The slope FLIPPED from positive (divergence) to negative (convergence). . display as text " This is the 'new era of unconditional convergence.'" This is the 'new era of unconditional convergence.' . . * ── Side-by-side scatter plots ── . #delimit ; delimiter now ; . tw (lfitci growth_era1 initial_era1, fcolor("217 119 87"%15) lcolor("217 119 > 87")) > (scatter growth_era1 initial_era1, mcolor("217 119 87"%60) msymbol(O) msiz > e(small) > mlabel(ccode) mlabsize(tiny) mlabpos(0) mlabcolor("20 20 19"%40)), > ytitle("Average Annual Growth Rate") > xtitle("Log Initial GDP per Capita") > title("Era of Divergence: 1960 to 2000", color("20 20 19") size(medium)) > xlabel(6 "$403" 7 "$1,097" 8 "$2,981" 9 "$8,103" 10 "$22,026", angle(0)) > ylabel(-0.02 "-2%" 0 "0%" 0.02 "2%" 0.04 "4%" 0.06 "6%", angle(horizontal) > ) > legend(off) > plotregion(style(none) lcolor(none)) > graphregion(fcolor(white) lcolor(white)) > xsize(4) ysize(4) > saving("_era1.gph", replace) > ; (file _era1.gph not found) file _era1.gph saved . tw (lfitci growth_era2 initial_era2, fcolor("106 155 204"%15) lcolor("106 155 > 204")) > (scatter growth_era2 initial_era2, mcolor("106 155 204"%60) msymbol(O) msi > ze(small) > mlabel(ccode) mlabsize(tiny) mlabpos(0) mlabcolor("20 20 19"%40)), > ytitle("Average Annual Growth Rate") > xtitle("Log Initial GDP per Capita") > title("Era of Convergence: 2000 to 2019", color("20 20 19") size(medium)) > xlabel(6 "$403" 7 "$1,097" 8 "$2,981" 9 "$8,103" 10 "$22,026" 11 "$59,874" > , angle(0)) > ylabel(-0.02 "-2%" 0 "0%" 0.02 "2%" 0.04 "4%" 0.06 "6%", angle(horizontal) > ) > legend(off) > plotregion(style(none) lcolor(none)) > graphregion(fcolor(white) lcolor(white)) > xsize(4) ysize(4) > saving("_era2.gph", replace) > ; (file _era2.gph not found) file _era2.gph saved . #delimit cr delimiter now cr . graph combine "_era1.gph" "_era2.gph", /// > title("The Structural Break in Cross-Country Growth", color("20 20 19")) > /// > subtitle("The slope flipped from positive (divergence) to negative (conve > rgence)", /// > color("20 20 19") size(small)) /// > note("Data: Penn World Tables 10.0. 84-country balanced panel.", /// > size(vsmall)) /// > graphregion(fcolor(white) lcolor(white)) /// > xsize(8) ysize(4) . graph export "stata_convergence_scatter_two_eras.png", replace width(2400) file stata_convergence_scatter_two_eras.png written in PNG format . display as text "Figure saved: stata_convergence_scatter_two_eras.png" Figure saved: stata_convergence_scatter_two_eras.png . capture erase "_era1.gph" . capture erase "_era2.gph" . . . *════════════════════════════════════════════════════════════════════════════ > ═══ . * SECTION 3: SPEED OF CONVERGENCE AND HALF-LIFE FROM OLS . * . * The OLS slope coefficient (λ) tells us the direction of convergence, . * but HOW FAST are poor countries catching up? Two key metrics: . * . * (A) SPEED OF CONVERGENCE (β): . * The fraction of the income gap that closes each year. . * β = 0.02 means 2% of the gap is closed annually. . * Classic benchmark: β ≈ 2% (Barro & Sala-i-Martin, 1992) . * — but that was for CONDITIONAL convergence. . * . * (B) HALF-LIFE (τ): . * How many years to close HALF the gap to steady-state income? . * τ = ln(2) / β . * Classic benchmark: τ ≈ 35 years (conditional convergence) . * . * The SIMPLEST way to get β from OLS: . * The OLS coefficient λ and the structural parameter β are linked by: . * λ = -((1 - exp(-β×s)) / s) . * Solving for β: . * β = -ln(1 + λ×s) / s . * Then: . * τ = ln(2) / β . * . * This gives us speed and half-life using ONLY standard OLS output. . *════════════════════════════════════════════════════════════════════════════ > ═══ . . display as text "" . display as text "──────────────────────────────────────────────────────────── > ────" ──────────────────────────────────────────────────────────────── . display as text " SECTION 3: SPEED OF CONVERGENCE AND HALF-LIFE FROM OLS" SECTION 3: SPEED OF CONVERGENCE AND HALF-LIFE FROM OLS . display as text "──────────────────────────────────────────────────────────── > ────" ──────────────────────────────────────────────────────────────── . . display as text "" . display as text " DERIVATION: OLS λ → structural β → half-life" DERIVATION: OLS λ → structural β → half-life . display as text "" . display as text " Step 1: Run OLS and get the slope coefficient λ" Step 1: Run OLS and get the slope coefficient λ . display as text " growth_i = α + λ × ln(y_i,0) + ε_i" growth_i = α + λ × ln(y_i,0) + ε_i . display as text "" . display as text " Step 2: The Barro-Sala-i-Martin model implies:" Step 2: The Barro-Sala-i-Martin model implies: . display as text " λ = -((1 - exp(-β×s)) / s)" λ = -((1 - exp(-β×s)) / s) . display as text "" . display as text " Step 3: Solve for β:" Step 3: Solve for β: . display as text " λ×s = -(1 - exp(-β×s))" λ×s = -(1 - exp(-β×s)) . display as text " exp(-β×s) = 1 + λ×s" exp(-β×s) = 1 + λ×s . display as text " -β×s = ln(1 + λ×s)" -β×s = ln(1 + λ×s) . display as text " β = -ln(1 + λ×s) / s" β = -ln(1 + λ×s) / s . display as text "" . display as text " Step 4: Half-life:" Step 4: Half-life: . display as text " τ = ln(2) / β" τ = ln(2) / β . display as text "" . . * ── Define periods ── . local periods "1960-2019 1960-2000 1980-2019 1990-2019 1995-2019 2000-2019" . local starts "1960 1960 1980 1990 1995 2000" . local ends "2019 2000 2019 2019 2019 2019" . . * ── Create file to collect OLS results ── . tempfile ols_speed . preserve . clear . gen str20 period = "" . gen double lambda_ols = . . gen double se_lambda = . . gen double pvalue = . . gen double beta_ols = . . gen double speed_ols = . . gen double halflife_ols = . . gen int n = . . save `ols_speed', replace (dataset contains 0 observations) (file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.000002 not found) file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.000002 saved as .dta format . restore . . * ── Loop over periods: OLS ── . local nperiods : word count `periods' . forval p = 1/`nperiods' { 2. local period : word `p' of `periods' 3. local sy : word `p' of `starts' 4. local ey : word `p' of `ends' 5. local s = `ey' - `sy' 6. . display as text "" 7. display as text "--- Period: `period' (s = `s' years) ---" 8. . preserve 9. use "convergence_working.dta", clear 10. keep ccode country year gdppc 11. reshape wide gdppc, i(ccode country) j(year) 12. . gen outcome = (1/`s') * ln(gdppc`ey' / gdppc`sy') 13. gen initial_inc = ln(gdppc`sy') 14. drop if missing(outcome) | missing(initial_inc) 15. . * ── OLS regression ── . reg outcome initial_inc, robust 16. local lambda = _b[initial_inc] 17. local se = _se[initial_inc] 18. local pval = 2 * ttail(e(df_r), abs(`lambda'/`se')) 19. local n_obs = e(N) 20. . * ── Convert λ to β ── . local lambda_s = `lambda' * `s' 21. if (1 + `lambda_s') > 0 { 22. local beta_speed = -ln(1 + `lambda_s') / `s' 23. } 24. else { 25. local beta_speed = . 26. } 27. local speed = `beta_speed' * 100 28. . * ── Half-life ── . if `beta_speed' > 0 & `beta_speed' < . { 29. local halflife = ln(2) / `beta_speed' 30. } 31. else { 32. local halflife = . 33. } 34. . display as result " OLS λ = " %8.5f `lambda' " (SE = " %8.5f `se' ", > p = " %5.3f `pval' ")" 35. display as result " Structural β = -ln(1 + " %7.5f `lambda' " × ` > s') / `s' = " %8.5f `beta_speed' 36. display as result " Speed = " %5.2f `speed' "% per year" 37. if `halflife' < . { 38. display as result " Half-life = ln(2) / " %7.5f `beta_speed' > " = " %6.1f `halflife' " years" 39. } 40. else { 41. display as result " Half-life = not computable (no convergenc > e)" 42. } 43. display as result " N = " `n_obs' " countries" 44. restore 45. . * Store results . preserve 46. clear 47. set obs 1 48. gen str20 period = "`period'" 49. gen double lambda_ols = `lambda' 50. gen double se_lambda = `se' 51. gen double pvalue = `pval' 52. gen double beta_ols = `beta_speed' 53. gen double speed_ols = `speed' 54. gen double halflife_ols = `halflife' 55. gen int n = `n_obs' 56. append using `ols_speed' 57. save `ols_speed', replace 58. restore 59. } --- Period: 1960-2019 (s = 59 years) --- (j = 1960 1961 1962 1963 1964 1965 1966 1967 1968 1969 1970 1971 1972 1973 1974 > 1975 1976 1977 1978 1979 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1 > 990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 200 > 5 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 2017 2018 2019) Data Long -> Wide ----------------------------------------------------------------------------- Number of observations 5,040 -> 84 Number of variables 4 -> 62 j variable (60 values) year -> (dropped) xij variables: gdppc -> gdppc1960 gdppc1961 ... gdppc201 > 9 ----------------------------------------------------------------------------- (0 observations deleted) Linear regression Number of obs = 84 F(1, 82) = 0.19 Prob > F = 0.6606 R-squared = 0.0013 Root MSE = .01502 ------------------------------------------------------------------------------ | Robust outcome | Coefficient std. err. t P>|t| [95% conf. interval] -------------+---------------------------------------------------------------- initial_inc | .0005689 .0012908 0.44 0.661 -.0019988 .0031366 _cons | .0176868 .0112996 1.57 0.121 -.0047917 .0401653 ------------------------------------------------------------------------------ OLS λ = 0.00057 (SE = 0.00129, p = 0.661) Structural β = -ln(1 + 0.00057 × 59) / 59 = -0.00056 Speed = -0.06% per year Half-life = not computable (no convergence) N = 84 countries Number of observations (_N) was 0, now 1. (1 missing value generated) file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.000002 saved as .dta format --- Period: 1960-2000 (s = 40 years) --- (j = 1960 1961 1962 1963 1964 1965 1966 1967 1968 1969 1970 1971 1972 1973 1974 > 1975 1976 1977 1978 1979 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1 > 990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 200 > 5 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 2017 2018 2019) Data Long -> Wide ----------------------------------------------------------------------------- Number of observations 5,040 -> 84 Number of variables 4 -> 62 j variable (60 values) year -> (dropped) xij variables: gdppc -> gdppc1960 gdppc1961 ... gdppc201 > 9 ----------------------------------------------------------------------------- (0 observations deleted) Linear regression Number of obs = 84 F(1, 82) = 7.59 Prob > F = 0.0072 R-squared = 0.0436 Root MSE = .01978 ------------------------------------------------------------------------------ | Robust outcome | Coefficient std. err. t P>|t| [95% conf. interval] -------------+---------------------------------------------------------------- initial_inc | .004366 .0015843 2.76 0.007 .0012143 .0075176 _cons | -.0150282 .0137431 -1.09 0.277 -.0423675 .0123112 ------------------------------------------------------------------------------ OLS λ = 0.00437 (SE = 0.00158, p = 0.007) Structural β = -ln(1 + 0.00437 × 40) / 40 = -0.00402 Speed = -0.40% per year Half-life = not computable (no convergence) N = 84 countries Number of observations (_N) was 0, now 1. (1 missing value generated) file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.000002 saved as .dta format --- Period: 1980-2019 (s = 39 years) --- (j = 1960 1961 1962 1963 1964 1965 1966 1967 1968 1969 1970 1971 1972 1973 1974 > 1975 1976 1977 1978 1979 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1 > 990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 200 > 5 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 2017 2018 2019) Data Long -> Wide ----------------------------------------------------------------------------- Number of observations 5,040 -> 84 Number of variables 4 -> 62 j variable (60 values) year -> (dropped) xij variables: gdppc -> gdppc1960 gdppc1961 ... gdppc201 > 9 ----------------------------------------------------------------------------- (0 observations deleted) Linear regression Number of obs = 84 F(1, 82) = 0.67 Prob > F = 0.4154 R-squared = 0.0069 Root MSE = .01544 ------------------------------------------------------------------------------ | Robust outcome | Coefficient std. err. t P>|t| [95% conf. interval] -------------+---------------------------------------------------------------- initial_inc | .0011291 .0013795 0.82 0.415 -.0016152 .0038735 _cons | .0115871 .0127383 0.91 0.366 -.0137534 .0369275 ------------------------------------------------------------------------------ OLS λ = 0.00113 (SE = 0.00138, p = 0.415) Structural β = -ln(1 + 0.00113 × 39) / 39 = -0.00110 Speed = -0.11% per year Half-life = not computable (no convergence) N = 84 countries Number of observations (_N) was 0, now 1. (1 missing value generated) file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.000002 saved as .dta format --- Period: 1990-2019 (s = 29 years) --- (j = 1960 1961 1962 1963 1964 1965 1966 1967 1968 1969 1970 1971 1972 1973 1974 > 1975 1976 1977 1978 1979 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1 > 990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 200 > 5 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 2017 2018 2019) Data Long -> Wide ----------------------------------------------------------------------------- Number of observations 5,040 -> 84 Number of variables 4 -> 62 j variable (60 values) year -> (dropped) xij variables: gdppc -> gdppc1960 gdppc1961 ... gdppc201 > 9 ----------------------------------------------------------------------------- (0 observations deleted) Linear regression Number of obs = 84 F(1, 82) = 0.00 Prob > F = 0.9538 R-squared = 0.0000 Root MSE = .01672 ------------------------------------------------------------------------------ | Robust outcome | Coefficient std. err. t P>|t| [95% conf. interval] -------------+---------------------------------------------------------------- initial_inc | -.0000812 .0013953 -0.06 0.954 -.0028568 .0026945 _cons | .0261171 .0130572 2.00 0.049 .0001422 .052092 ------------------------------------------------------------------------------ OLS λ = -0.00008 (SE = 0.00140, p = 0.954) Structural β = -ln(1 + -0.00008 × 29) / 29 = 0.00008 Speed = 0.01% per year Half-life = ln(2) / 0.00008 = 8530.1 years N = 84 countries Number of observations (_N) was 0, now 1. file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.000002 saved as .dta format --- Period: 1995-2019 (s = 24 years) --- (j = 1960 1961 1962 1963 1964 1965 1966 1967 1968 1969 1970 1971 1972 1973 1974 > 1975 1976 1977 1978 1979 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1 > 990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 200 > 5 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 2017 2018 2019) Data Long -> Wide ----------------------------------------------------------------------------- Number of observations 5,040 -> 84 Number of variables 4 -> 62 j variable (60 values) year -> (dropped) xij variables: gdppc -> gdppc1960 gdppc1961 ... gdppc201 > 9 ----------------------------------------------------------------------------- (0 observations deleted) Linear regression Number of obs = 84 F(1, 82) = 1.62 Prob > F = 0.2061 R-squared = 0.0194 Root MSE = .01682 ------------------------------------------------------------------------------ | Robust outcome | Coefficient std. err. t P>|t| [95% conf. interval] -------------+---------------------------------------------------------------- initial_inc | -.0017786 .0013957 -1.27 0.206 -.0045551 .0009979 _cons | .0417685 .0132334 3.16 0.002 .015443 .068094 ------------------------------------------------------------------------------ OLS λ = -0.00178 (SE = 0.00140, p = 0.206) Structural β = -ln(1 + -0.00178 × 24) / 24 = 0.00182 Speed = 0.18% per year Half-life = ln(2) / 0.00182 = 381.3 years N = 84 countries Number of observations (_N) was 0, now 1. file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.000002 saved as .dta format --- Period: 2000-2019 (s = 19 years) --- (j = 1960 1961 1962 1963 1964 1965 1966 1967 1968 1969 1970 1971 1972 1973 1974 > 1975 1976 1977 1978 1979 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1 > 990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 200 > 5 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 2017 2018 2019) Data Long -> Wide ----------------------------------------------------------------------------- Number of observations 5,040 -> 84 Number of variables 4 -> 62 j variable (60 values) year -> (dropped) xij variables: gdppc -> gdppc1960 gdppc1961 ... gdppc201 > 9 ----------------------------------------------------------------------------- (0 observations deleted) Linear regression Number of obs = 84 F(1, 82) = 5.75 Prob > F = 0.0187 R-squared = 0.0688 Root MSE = .01794 ------------------------------------------------------------------------------ | Robust outcome | Coefficient std. err. t P>|t| [95% conf. interval] -------------+---------------------------------------------------------------- initial_inc | -.0035228 .0014686 -2.40 0.019 -.0064442 -.0006013 _cons | .0580907 .014098 4.12 0.000 .0300452 .0861362 ------------------------------------------------------------------------------ OLS λ = -0.00352 (SE = 0.00147, p = 0.019) Structural β = -ln(1 + -0.00352 × 19) / 19 = 0.00365 Speed = 0.36% per year Half-life = ln(2) / 0.00365 = 190.1 years N = 84 countries Number of observations (_N) was 0, now 1. file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.000002 saved as .dta format . . * ── Display OLS results table ── . display as text "" . display as text "╔═══════════════════════════════════════════════════════════ > ═══════════════════════╗" ╔══════════════════════════════════════════════════════════════════════════════ > ════╗ . display as text "║ Speed of Convergence from OLS: λ → β → Half-Life > ║" ║ Speed of Convergence from OLS: λ → β → Half-Life > ║ . display as text "╠═══════════════════════════════════════════════════════════ > ═══════════════════════╣" ╠══════════════════════════════════════════════════════════════════════════════ > ════╣ . display as text "║ Formula: β = -ln(1 + λ×s)/s Half-life: τ = ln(2)/β > ║" ║ Formula: β = -ln(1 + λ×s)/s Half-life: τ = ln(2)/β > ║ . display as text "║ Benchmark: β ≈ 2%/yr, τ ≈ 35 yrs (conditional, Barro & Sa > la-i-Martin 1992) ║" ║ Benchmark: β ≈ 2%/yr, τ ≈ 35 yrs (conditional, Barro & Sala-i-Martin 1992) > ║ . display as text "╚═══════════════════════════════════════════════════════════ > ═══════════════════════╝" ╚══════════════════════════════════════════════════════════════════════════════ > ════╝ . . preserve . use `ols_speed', clear . drop if missing(lambda_ols) (0 observations deleted) . sort period . list period lambda_ols beta_ols speed_ols halflife_ols n, noobs clean period lambda_ols beta_ols speed_ols halflif~s n 1960-2000 .00436597 -.00402402 -.40240194 . 84 1960-2019 .00056889 -.00055955 -.05595547 . 84 1980-2019 .00112915 -.00110499 -.11049926 . 84 1990-2019 -.00008116 .00008126 .00812593 8530.0644 84 1995-2019 -.00177861 .00181769 .1817691 381.3339 84 2000-2019 -.00352279 .00364623 .36462271 190.09984 84 . export delimited using "convergence_speed_ols.csv", replace file convergence_speed_ols.csv saved . display as text "Results exported to: convergence_speed_ols.csv" Results exported to: convergence_speed_ols.csv . restore . . * ── Create OLS speed bar chart ── . preserve . use `ols_speed', clear . drop if missing(lambda_ols) (0 observations deleted) . sort period . gen id = _n . gen speed_bench = 2 . . #delimit ; delimiter now ; . tw (bar speed_ols id, barwidth(0.6) fcolor("217 119 87") lcolor("217 119 > 87"%80)) > (line speed_bench id, lcolor("106 155 204") lwidth(medthick) lpattern( > dash)), > xlabel(1 "1960-2000" 2 "1960-2019" 3 "1980-2019" 4 "1990-2019" 5 "1995 > -2019" 6 "2000-2019", > angle(30) labsize(small)) > ylabel(, angle(horizontal)) > ytitle("Speed of Convergence (% per year)") > xtitle("Period") > title("Speed of Convergence from OLS ({&lambda} {&rarr} {&beta} conver > sion)", color("20 20 19")) > subtitle("Benchmark: 2% per year (Barro & Sala-i-Martin 1992, conditio > nal)", /// > color("20 20 19") size(small)) > legend(order(1 "Unconditional {&beta} (from OLS)" 2 "Conditional bench > mark (2%/yr)") > pos(2) ring(0) region(lcolor(none) fcolor(none)) size(small)) > plotregion(style(none) lcolor(none)) > graphregion(fcolor(white) lcolor(white)) > note("Data: Penn World Tables 10.0. 84-country balanced panel." > "Negative values indicate divergence. {&beta} = -ln(1 + {&lambda} > ×s)/s.", size(vsmall)) > xsize(6) ysize(4) > ; . #delimit cr delimiter now cr . graph export "stata_convergence_speed_ols.png", replace width(2400) file stata_convergence_speed_ols.png written in PNG format . display as text "Figure saved: stata_convergence_speed_ols.png" Figure saved: stata_convergence_speed_ols.png . restore . . . *════════════════════════════════════════════════════════════════════════════ > ═══ . * SECTION 4: WHAT IS NONLINEAR LEAST SQUARES (NLS)? . * . * The OLS-to-β conversion in Section 3 works, but it goes BACKWARDS: . * we estimate λ first and then convert to β. Can we estimate β DIRECTLY? . * . * YES — using Nonlinear Least Squares (NLS). . * . * WHY can't OLS estimate β directly? . * The Barro-Sala-i-Martin (1992) convergence equation is: . * (1/s) × ln(y_{t+s}/y_t) = α - ((1 - exp(-β×s))/s) × ln(y_t) + ε . * The parameter β appears INSIDE an exponential: exp(-β×s). . * OLS requires that parameters enter LINEARLY, but β is trapped inside . * exp(), so OLS cannot estimate it directly. . * . * WHAT does NLS do? . * Like OLS, NLS minimizes the sum of squared residuals: . * min_β Σᵢ [yᵢ - f(xᵢ; β)]² . * But unlike OLS, the function f() can be ANY nonlinear function of β. . * NLS uses an iterative algorithm: . * 1. Start with an initial guess for β (e.g., β₀ = 0.02) . * 2. Compute predicted values and residuals . * 3. Adjust β in the direction that reduces the sum of squared residual > s . * 4. Repeat until the improvement is negligible (convergence) . * . * HOW to estimate in Stata: . * Stata's -nl- command does NLS. The syntax is: . * nl (depvar = expression_with_{parameters}), vce(robust) . * Parameters are specified in curly braces: {b1=initial_guess} . * . * ADVANTAGE: β has a direct structural interpretation as the speed of . * convergence, and standard errors apply directly to β (no conversion . * needed, no delta method). . *════════════════════════════════════════════════════════════════════════════ > ═══ . . display as text "" . display as text "──────────────────────────────────────────────────────────── > ────" ──────────────────────────────────────────────────────────────── . display as text " SECTION 4: WHAT IS NONLINEAR LEAST SQUARES (NLS)?" SECTION 4: WHAT IS NONLINEAR LEAST SQUARES (NLS)? . display as text "──────────────────────────────────────────────────────────── > ────" ──────────────────────────────────────────────────────────────── . . display as text "" . display as text " The Barro-Sala-i-Martin (1992) convergence equation:" The Barro-Sala-i-Martin (1992) convergence equation: . display as text "" . display as text " (1/s) × ln(y_{t+s}/y_t) = α - ((1 - exp(-β×s))/s) × ln(y > _t) + ε" (1/s) × ln(y_{t+s}/y_t) = α - ((1 - exp(-β×s))/s) × ln(y_t) + ε . display as text "" . display as text " Breaking down each term:" Breaking down each term: . display as text " Left side: annualized growth rate over s years" Left side: annualized growth rate over s years . display as text " α: intercept (related to the steady-state incom > e)" α: intercept (related to the steady-state income) . display as text " β: speed of convergence (fraction of gap closed > per year)" β: speed of convergence (fraction of gap closed per year) . display as text " s: number of years between initial and final pe > riod" s: number of years between initial and final period . display as text " ln(y_t): log of initial GDP per capita" ln(y_t): log of initial GDP per capita . display as text " ε: error term (random shocks to growth)" ε: error term (random shocks to growth) . display as text "" . display as text " The key expression is: (1 - exp(-β×s))/s" The key expression is: (1 - exp(-β×s))/s . display as text " This is the 'convergence factor' — how much of the initi > al" This is the 'convergence factor' — how much of the initial . display as text " income level feeds into the growth rate." income level feeds into the growth rate. . display as text " When β > 0: convergence factor > 0, and the minus sign m > eans" When β > 0: convergence factor > 0, and the minus sign means . display as text " higher initial income → lower growth (conver > gence)" higher initial income → lower growth (convergence) . display as text " When β = 0: convergence factor = 0, no relationship" When β = 0: convergence factor = 0, no relationship . display as text " When β < 0: divergence" When β < 0: divergence . display as text "" . . * ── Demonstrate NLS for one period: 2000-2019 ── . display as text "--- NLS Demonstration: 2000-2019 ---" --- NLS Demonstration: 2000-2019 --- . display as text "" . . preserve . use "convergence_working.dta", clear . keep ccode country year gdppc . reshape wide gdppc, i(ccode country) j(year) (j = 1960 1961 1962 1963 1964 1965 1966 1967 1968 1969 1970 1971 1972 1973 1974 > 1975 1976 1977 1978 1979 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1 > 990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 200 > 5 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 2017 2018 2019) Data Long -> Wide ----------------------------------------------------------------------------- Number of observations 5,040 -> 84 Number of variables 4 -> 62 j variable (60 values) year -> (dropped) xij variables: gdppc -> gdppc1960 gdppc1961 ... gdppc201 > 9 ----------------------------------------------------------------------------- . . local s = 19 . gen outcome = (1/`s') * ln(gdppc2019 / gdppc2000) . gen initial_inc = ln(gdppc2000) . drop if missing(outcome) | missing(initial_inc) (0 observations deleted) . . display as text " Stata command:" Stata command: . display as text " nl (outcome = {b0=1} - (1 - exp(-1*{b1=0.02}*19))/19 > * initial_inc), vce(robust)" nl (outcome = {b0=1} - (1 - exp(-1*{b1=0.02}*19))/19 * initial_inc), vce(ro > bust) . display as text "" . display as text " Reading the syntax:" Reading the syntax: . display as text " {b0=1} → intercept α, initial guess = 1" {b0=1} → intercept α, initial guess = 1 . display as text " {b1=0.02} → speed of convergence β, initial guess = > 0.02 (the 2% benchmark)" {b1=0.02} → speed of convergence β, initial guess = 0.02 (the 2% benchmark) . display as text " *19 → s = 19 years (2000 to 2019)" *19 → s = 19 years (2000 to 2019) . display as text " initial_inc → ln(y_2000), our independent variable" initial_inc → ln(y_2000), our independent variable . display as text "" . . nl (outcome = {b0=1} - (1 - exp(-1*{b1=0.02}*`s'))/`s' * initial_inc), vc > e(robust) Iteration 0: Residual SS = 57.402865 Iteration 1: Residual SS = .07545665 Iteration 2: Residual SS = .02643651 Iteration 3: Residual SS = .02640414 Iteration 4: Residual SS = .02640414 Nonlinear regression Number of obs = 84 R-squared = 0.0688 Adj R-squared = 0.0574 Root MSE = .0179444 Res. dev. = -439.0826 ------------------------------------------------------------------------------ | Robust outcome | Coefficient std. err. t P>|t| [95% conf. interval] -------------+---------------------------------------------------------------- /b0 | .0580907 .014098 4.12 0.000 .0300452 .0861362 /b1 | .0036462 .0015739 2.32 0.023 .0005152 .0067772 ------------------------------------------------------------------------------ Note: Parameter b0 is used as a constant term during estimation. . . display as text "" . display as text " HOW TO READ THE OUTPUT:" HOW TO READ THE OUTPUT: . display as text " /b0 = " %8.5f _b[/b0] " → This is α (intercept)" /b0 = 0.05809 → This is α (intercept) . display as text " /b1 = " %8.5f _b[/b1] " → This is β (speed of conver > gence)" /b1 = 0.00365 → This is β (speed of convergence) . display as text "" . display as text " Speed = β × 100 = " %5.2f _b[/b1]*100 "% per year" Speed = β × 100 = 0.36% per year . display as text " Half-life = ln(2)/β = " %6.1f ln(2)/_b[/b1] " years" Half-life = ln(2)/β = 190.1 years . display as text "" . . * Save NLS result before running OLS . local nls_beta_demo = _b[/b1] . . * Compare with OLS conversion . qui reg outcome initial_inc, robust . local lambda_check = _b[initial_inc] . local beta_check = -ln(1 + `lambda_check'*`s') / `s' . display as text " COMPARISON with OLS conversion (Section 3):" COMPARISON with OLS conversion (Section 3): . display as text " OLS λ = " %8.5f `lambda_check' OLS λ = -0.00352 . display as text " OLS → β = -ln(1 + " %7.5f `lambda_check' " × `s') / > `s' = " %8.5f `beta_check' OLS → β = -ln(1 + -0.00352 × 19) / 19 = 0.00365 . display as text " NLS β = " %8.5f `nls_beta_demo' NLS β = 0.00365 . display as text " Difference = " %10.7f abs(`beta_check' - `nls_beta_d > emo') Difference = 0.0000000 . display as text "" . display as text " The point estimates are nearly identical." The point estimates are nearly identical. . display as text " The advantage of NLS: standard errors and p-values app > ly directly to β." The advantage of NLS: standard errors and p-values apply directly to β. . restore . . . *════════════════════════════════════════════════════════════════════════════ > ═══ . * SECTION 5: SPEED OF CONVERGENCE AND HALF-LIFE FROM NLS . * . * Now we estimate β directly via NLS for the same 6 periods as Section 3. . * The results should be very close to the OLS conversion, confirming . * that both methods give the same structural parameter. . *════════════════════════════════════════════════════════════════════════════ > ═══ . . display as text "" . display as text "──────────────────────────────────────────────────────────── > ────" ──────────────────────────────────────────────────────────────── . display as text " SECTION 5: SPEED OF CONVERGENCE AND HALF-LIFE FROM NLS" SECTION 5: SPEED OF CONVERGENCE AND HALF-LIFE FROM NLS . display as text "──────────────────────────────────────────────────────────── > ────" ──────────────────────────────────────────────────────────────── . . * ── Create file to collect NLS results ── . tempfile nls_speed . preserve . clear . gen str20 period = "" . gen double beta_nls = . . gen double se_nls = . . gen double pvalue_nls = . . gen double speed_nls = . . gen double halflife_nls = . . gen int n = . . save `nls_speed', replace (dataset contains 0 observations) (file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.00000j not found) file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.00000j saved as .dta format . restore . . * ── Loop over periods: NLS ── . forval p = 1/`nperiods' { 2. local period : word `p' of `periods' 3. local sy : word `p' of `starts' 4. local ey : word `p' of `ends' 5. local s = `ey' - `sy' 6. . display as text "" 7. display as text "--- Period: `period' (s = `s' years) ---" 8. . preserve 9. use "convergence_working.dta", clear 10. keep ccode country year gdppc 11. reshape wide gdppc, i(ccode country) j(year) 12. . gen outcome = (1/`s') * ln(gdppc`ey' / gdppc`sy') 13. gen initial_inc = ln(gdppc`sy') 14. drop if missing(outcome) | missing(initial_inc) 15. . * ── NLS estimation ── . capture noisily nl (outcome = {b0=1} - (1 - exp(-1*{b1=0.00}*`s'))/`s > ' * initial_inc), /// > vce(robust) 16. . if _rc == 0 { 17. local beta_nls = _b[/b1] 18. local se_nls = _se[/b1] 19. local n_obs = e(N) 20. local pval_nls = 2 * ttail(e(df_r), abs(`beta_nls'/`se_nls')) 21. local speed = `beta_nls' * 100 22. . if `beta_nls' > 0 { 23. local halflife = ln(2) / `beta_nls' 24. } 25. else { 26. local halflife = . 27. } 28. . display as text "" 29. display as result " NLS β = " %8.5f `beta_nls' " (SE = " %8.5 > f `se_nls' ", p = " %5.3f `pval_nls' ")" 30. display as result " Speed = " %5.2f `speed' "% per year" 31. if `halflife' < . { 32. display as result " Half-life = " %6.1f `halflife' " year > s" 33. } 34. else { 35. display as result " Half-life = not computable (no conver > gence)" 36. } 37. display as result " N = " `n_obs' " countries" 38. } 39. else { 40. display as error " NLS did not converge for period `period'" 41. local beta_nls = . 42. local se_nls = . 43. local pval_nls = . 44. local speed = . 45. local halflife = . 46. local n_obs = . 47. } 48. restore 49. . if `beta_nls' < . { 50. preserve 51. clear 52. set obs 1 53. gen str20 period = "`period'" 54. gen double beta_nls = `beta_nls' 55. gen double se_nls = `se_nls' 56. gen double pvalue_nls = `pval_nls' 57. gen double speed_nls = `speed' 58. gen double halflife_nls = `halflife' 59. gen int n = `n_obs' 60. append using `nls_speed' 61. save `nls_speed', replace 62. restore 63. } 64. } --- Period: 1960-2019 (s = 59 years) --- (j = 1960 1961 1962 1963 1964 1965 1966 1967 1968 1969 1970 1971 1972 1973 1974 > 1975 1976 1977 1978 1979 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1 > 990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 200 > 5 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 2017 2018 2019) Data Long -> Wide ----------------------------------------------------------------------------- Number of observations 5,040 -> 84 Number of variables 4 -> 62 j variable (60 values) year -> (dropped) xij variables: gdppc -> gdppc1960 gdppc1961 ... gdppc201 > 9 ----------------------------------------------------------------------------- (0 observations deleted) Iteration 0: Residual SS = 80.322481 Iteration 1: Residual SS = .01850243 Iteration 2: Residual SS = .01850192 Iteration 3: Residual SS = .01850192 Nonlinear regression Number of obs = 84 R-squared = 0.0013 Adj R-squared = . Root MSE = .0150211 Res. dev. = -468.9569 ------------------------------------------------------------------------------ | Robust outcome | Coefficient std. err. t P>|t| [95% conf. interval] -------------+---------------------------------------------------------------- /b0 | .0176868 .0112996 1.57 0.121 -.0047917 .0401653 /b1 | -.0005596 .0012488 -0.45 0.655 -.0030439 .0019248 ------------------------------------------------------------------------------ Note: Parameter b0 is used as a constant term during estimation. NLS β = -0.00056 (SE = 0.00125, p = 0.655) Speed = -0.06% per year Half-life = not computable (no convergence) N = 84 countries Number of observations (_N) was 0, now 1. (1 missing value generated) file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.00000j saved as .dta format --- Period: 1960-2000 (s = 40 years) --- (j = 1960 1961 1962 1963 1964 1965 1966 1967 1968 1969 1970 1971 1972 1973 1974 > 1975 1976 1977 1978 1979 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1 > 990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 200 > 5 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 2017 2018 2019) Data Long -> Wide ----------------------------------------------------------------------------- Number of observations 5,040 -> 84 Number of variables 4 -> 62 j variable (60 values) year -> (dropped) xij variables: gdppc -> gdppc1960 gdppc1961 ... gdppc201 > 9 ----------------------------------------------------------------------------- (0 observations deleted) Iteration 0: Residual SS = 80.711295 Iteration 1: Residual SS = .0329626 Iteration 2: Residual SS = .03206699 Iteration 3: Residual SS = .03206695 Nonlinear regression Number of obs = 84 R-squared = 0.0436 Adj R-squared = 0.0320 Root MSE = .0197752 Res. dev. = -422.761 ------------------------------------------------------------------------------ | Robust outcome | Coefficient std. err. t P>|t| [95% conf. interval] -------------+---------------------------------------------------------------- /b0 | -.0150282 .0137431 -1.09 0.277 -.0423675 .0123112 /b1 | -.004024 .0013487 -2.98 0.004 -.0067071 -.0013409 ------------------------------------------------------------------------------ Note: Parameter b0 is used as a constant term during estimation. NLS β = -0.00402 (SE = 0.00135, p = 0.004) Speed = -0.40% per year Half-life = not computable (no convergence) N = 84 countries Number of observations (_N) was 0, now 1. (1 missing value generated) file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.00000j saved as .dta format --- Period: 1980-2019 (s = 39 years) --- (j = 1960 1961 1962 1963 1964 1965 1966 1967 1968 1969 1970 1971 1972 1973 1974 > 1975 1976 1977 1978 1979 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1 > 990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 200 > 5 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 2017 2018 2019) Data Long -> Wide ----------------------------------------------------------------------------- Number of observations 5,040 -> 84 Number of variables 4 -> 62 j variable (60 values) year -> (dropped) xij variables: gdppc -> gdppc1960 gdppc1961 ... gdppc201 > 9 ----------------------------------------------------------------------------- (0 observations deleted) Iteration 0: Residual SS = 80.497685 Iteration 1: Residual SS = .01954895 Iteration 2: Residual SS = .01954502 Iteration 3: Residual SS = .01954502 Nonlinear regression Number of obs = 84 R-squared = 0.0069 Adj R-squared = . Root MSE = .0154387 Res. dev. = -464.3499 ------------------------------------------------------------------------------ | Robust outcome | Coefficient std. err. t P>|t| [95% conf. interval] -------------+---------------------------------------------------------------- /b0 | .0115871 .0127383 0.91 0.366 -.0137534 .0369275 /b1 | -.001105 .0013214 -0.84 0.405 -.0037336 .0015236 ------------------------------------------------------------------------------ Note: Parameter b0 is used as a constant term during estimation. NLS β = -0.00110 (SE = 0.00132, p = 0.405) Speed = -0.11% per year Half-life = not computable (no convergence) N = 84 countries Number of observations (_N) was 0, now 1. (1 missing value generated) file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.00000j saved as .dta format --- Period: 1990-2019 (s = 29 years) --- (j = 1960 1961 1962 1963 1964 1965 1966 1967 1968 1969 1970 1971 1972 1973 1974 > 1975 1976 1977 1978 1979 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1 > 990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 200 > 5 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 2017 2018 2019) Data Long -> Wide ----------------------------------------------------------------------------- Number of observations 5,040 -> 84 Number of variables 4 -> 62 j variable (60 values) year -> (dropped) xij variables: gdppc -> gdppc1960 gdppc1961 ... gdppc201 > 9 ----------------------------------------------------------------------------- (0 observations deleted) Iteration 0: Residual SS = 79.806693 Iteration 1: Residual SS = .02292574 Iteration 2: Residual SS = .02292574 Nonlinear regression Number of obs = 84 R-squared = 0.0000 Adj R-squared = . Root MSE = .0167207 Res. dev. = -450.9485 ------------------------------------------------------------------------------ | Robust outcome | Coefficient std. err. t P>|t| [95% conf. interval] -------------+---------------------------------------------------------------- /b0 | .0261171 .0130572 2.00 0.049 .0001422 .052092 /b1 | .0000813 .0013986 0.06 0.954 -.002701 .0028635 ------------------------------------------------------------------------------ Note: Parameter b0 is used as a constant term during estimation. NLS β = 0.00008 (SE = 0.00140, p = 0.954) Speed = 0.01% per year Half-life = 8530.1 years N = 84 countries Number of observations (_N) was 0, now 1. file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.00000j saved as .dta format --- Period: 1995-2019 (s = 24 years) --- (j = 1960 1961 1962 1963 1964 1965 1966 1967 1968 1969 1970 1971 1972 1973 1974 > 1975 1976 1977 1978 1979 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1 > 990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 200 > 5 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 2017 2018 2019) Data Long -> Wide ----------------------------------------------------------------------------- Number of observations 5,040 -> 84 Number of variables 4 -> 62 j variable (60 values) year -> (dropped) xij variables: gdppc -> gdppc1960 gdppc1961 ... gdppc201 > 9 ----------------------------------------------------------------------------- (0 observations deleted) Iteration 0: Residual SS = 79.663879 Iteration 1: Residual SS = .02321768 Iteration 2: Residual SS = .02320858 Iteration 3: Residual SS = .02320858 Nonlinear regression Number of obs = 84 R-squared = 0.0194 Adj R-squared = 0.0074 Root MSE = .0168235 Res. dev. = -449.9185 ------------------------------------------------------------------------------ | Robust outcome | Coefficient std. err. t P>|t| [95% conf. interval] -------------+---------------------------------------------------------------- /b0 | .0417685 .0132334 3.16 0.002 .015443 .068094 /b1 | .0018177 .001458 1.25 0.216 -.0010826 .004718 ------------------------------------------------------------------------------ Note: Parameter b0 is used as a constant term during estimation. NLS β = 0.00182 (SE = 0.00146, p = 0.216) Speed = 0.18% per year Half-life = 381.3 years N = 84 countries Number of observations (_N) was 0, now 1. file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.00000j saved as .dta format --- Period: 2000-2019 (s = 19 years) --- (j = 1960 1961 1962 1963 1964 1965 1966 1967 1968 1969 1970 1971 1972 1973 1974 > 1975 1976 1977 1978 1979 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1 > 990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 200 > 5 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 2017 2018 2019) Data Long -> Wide ----------------------------------------------------------------------------- Number of observations 5,040 -> 84 Number of variables 4 -> 62 j variable (60 values) year -> (dropped) xij variables: gdppc -> gdppc1960 gdppc1961 ... gdppc201 > 9 ----------------------------------------------------------------------------- (0 observations deleted) Iteration 0: Residual SS = 79.548176 Iteration 1: Residual SS = .02649304 Iteration 2: Residual SS = .02640414 Iteration 3: Residual SS = .02640414 Nonlinear regression Number of obs = 84 R-squared = 0.0688 Adj R-squared = 0.0574 Root MSE = .0179444 Res. dev. = -439.0826 ------------------------------------------------------------------------------ | Robust outcome | Coefficient std. err. t P>|t| [95% conf. interval] -------------+---------------------------------------------------------------- /b0 | .0580907 .014098 4.12 0.000 .0300452 .0861362 /b1 | .0036462 .0015739 2.32 0.023 .0005152 .0067772 ------------------------------------------------------------------------------ Note: Parameter b0 is used as a constant term during estimation. NLS β = 0.00365 (SE = 0.00157, p = 0.023) Speed = 0.36% per year Half-life = 190.1 years N = 84 countries Number of observations (_N) was 0, now 1. file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.00000j saved as .dta format . . * ── Display NLS results table ── . display as text "" . display as text "╔═══════════════════════════════════════════════════════════ > ═══════════════════════╗" ╔══════════════════════════════════════════════════════════════════════════════ > ════╗ . display as text "║ Speed of Convergence from NLS (Direct Estimation of β) > ║" ║ Speed of Convergence from NLS (Direct Estimation of β) > ║ . display as text "╠═══════════════════════════════════════════════════════════ > ═══════════════════════╣" ╠══════════════════════════════════════════════════════════════════════════════ > ════╣ . display as text "║ Benchmark: β ≈ 2%/yr, τ ≈ 35 yrs (conditional, Barro & Sa > la-i-Martin 1992) ║" ║ Benchmark: β ≈ 2%/yr, τ ≈ 35 yrs (conditional, Barro & Sala-i-Martin 1992) > ║ . display as text "╚═══════════════════════════════════════════════════════════ > ═══════════════════════╝" ╚══════════════════════════════════════════════════════════════════════════════ > ════╝ . . preserve . use `nls_speed', clear . drop if missing(beta_nls) (0 observations deleted) . sort period . list period beta_nls se_nls speed_nls halflife_nls n, noobs clean period beta_nls se_nls speed_nls halflif~s n 1960-2000 -.00402402 .00134875 -.40240195 . 84 1960-2019 -.00055955 .00124884 -.05595547 . 84 1980-2019 -.00110499 .00132136 -.11049926 . 84 1990-2019 .00008126 .00139858 .00812593 8530.0644 84 1995-2019 .00181769 .00145796 .1817691 381.3339 84 2000-2019 .00364623 .00157392 .36462271 190.09984 84 . export delimited using "convergence_speed_nls.csv", replace file convergence_speed_nls.csv saved . display as text "Results exported to: convergence_speed_nls.csv" Results exported to: convergence_speed_nls.csv . restore . . * ── Create NLS speed bar chart ── . preserve . use `nls_speed', clear . drop if missing(beta_nls) (0 observations deleted) . sort period . gen id = _n . gen speed_bench = 2 . . #delimit ; delimiter now ; . tw (bar speed_nls id, barwidth(0.6) fcolor("106 155 204") lcolor("106 155 > 204"%80)) > (line speed_bench id, lcolor("217 119 87") lwidth(medthick) lpattern(d > ash)), > xlabel(1 "1960-2000" 2 "1960-2019" 3 "1980-2019" 4 "1990-2019" 5 "1995 > -2019" 6 "2000-2019", > angle(30) labsize(small)) > ylabel(, angle(horizontal)) > ytitle("Speed of Convergence (% per year)") > xtitle("Period") > title("Speed of Convergence from NLS (Direct Estimation)", color("20 2 > 0 19")) > subtitle("Benchmark: 2% per year (Barro & Sala-i-Martin 1992, conditio > nal)", /// > color("20 20 19") size(small)) > legend(order(1 "Unconditional {&beta} (NLS)" 2 "Conditional benchmark > (2%/yr)") > pos(2) ring(0) region(lcolor(none) fcolor(none)) size(small)) > plotregion(style(none) lcolor(none)) > graphregion(fcolor(white) lcolor(white)) > note("Data: Penn World Tables 10.0. 84-country balanced panel." > "Negative values indicate divergence.", size(vsmall)) > xsize(6) ysize(4) > ; . #delimit cr delimiter now cr . graph export "stata_convergence_speed_nls.png", replace width(2400) file stata_convergence_speed_nls.png written in PNG format . display as text "Figure saved: stata_convergence_speed_nls.png" Figure saved: stata_convergence_speed_nls.png . restore . . . *════════════════════════════════════════════════════════════════════════════ > ═══ . * SECTION 6: OLS vs NLS COMPARISON . * . * How do the two methods compare? The OLS conversion and NLS should give . * nearly identical point estimates for β. The key difference: . * - OLS: standard errors and p-values are for λ (must be transformed) . * - NLS: standard errors and p-values are directly for β . *════════════════════════════════════════════════════════════════════════════ > ═══ . . display as text "" . display as text "──────────────────────────────────────────────────────────── > ────" ──────────────────────────────────────────────────────────────── . display as text " SECTION 6: OLS vs NLS COMPARISON" SECTION 6: OLS vs NLS COMPARISON . display as text "──────────────────────────────────────────────────────────── > ────" ──────────────────────────────────────────────────────────────── . . * ── Merge OLS and NLS results ── . preserve . use `ols_speed', clear . drop if missing(lambda_ols) (0 observations deleted) . sort period . tempfile ols_sorted . save `ols_sorted', replace (file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.000012 not found) file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.000012 saved as .dta format . . use `nls_speed', clear . drop if missing(beta_nls) (0 observations deleted) . sort period . merge 1:1 period using `ols_sorted', keep(match) nogen Result Number of obs ----------------------------------------- Not matched 0 Matched 6 ----------------------------------------- . . gen diff = abs(beta_ols - beta_nls) . . display as text "" . display as text "╔═══════════════════════════════════════════════════════ > ═══════════════════════════════╗" ╔══════════════════════════════════════════════════════════════════════════════ > ════════╗ . display as text "║ OLS vs NLS: Side-by-Side Comparison > ║" ║ OLS vs NLS: Side-by-Side Comparison > ║ . display as text "╠═══════════════════════════════════════════════════════ > ═══════════════════════════════╣" ╠══════════════════════════════════════════════════════════════════════════════ > ════════╣ . display as text "║ OLS: estimate λ, then β = -ln(1+λs)/s > ║" ║ OLS: estimate λ, then β = -ln(1+λs)/s > ║ . display as text "║ NLS: estimate β directly via iterative optimization > ║" ║ NLS: estimate β directly via iterative optimization > ║ . display as text "╚═══════════════════════════════════════════════════════ > ═══════════════════════════════╝" ╚══════════════════════════════════════════════════════════════════════════════ > ════════╝ . . list period lambda_ols beta_ols beta_nls diff speed_ols speed_nls n, noob > s clean period lambda_ols beta_ols beta_nls diff speed_ols > speed_nls n 1960-2000 .00436597 -.00402402 -.00402402 1.08e-10 -.40240194 > -.40240195 84 1960-2019 .00056889 -.00055955 -.00055955 1.05e-13 -.05595547 > -.05595547 84 1980-2019 .00112915 -.00110499 -.00110499 4.72e-13 -.11049926 > -.11049926 84 1990-2019 -.00008116 .00008126 .00008126 3.22e-13 .00812593 > .00812593 84 1995-2019 -.00177861 .00181769 .00181769 3.63e-13 .1817691 > .1817691 84 2000-2019 -.00352279 .00364623 .00364623 1.37e-13 .36462271 > .36462271 84 . . display as text "" . display as text " KEY OBSERVATIONS:" KEY OBSERVATIONS: . display as text " 1. The β estimates from OLS conversion and NLS are nea > rly identical." 1. The β estimates from OLS conversion and NLS are nearly identical. . display as text " 2. Small differences arise from numerical optimization > in NLS." 2. Small differences arise from numerical optimization in NLS. . display as text " 3. NLS advantage: SE and p-values apply directly to β. > " 3. NLS advantage: SE and p-values apply directly to β. . display as text " 4. OLS advantage: simpler, faster, no convergence issu > es." 4. OLS advantage: simpler, faster, no convergence issues. . . export delimited using "convergence_ols_vs_nls.csv", replace file convergence_ols_vs_nls.csv saved . display as text "Results exported to: convergence_ols_vs_nls.csv" Results exported to: convergence_ols_vs_nls.csv . restore . . . *════════════════════════════════════════════════════════════════════════════ > ═══ . * SECTION 7A: ROLLING WINDOW — OLS LAMBDA . * . * Before computing rolling SPEED of convergence, we start with the . * simplest rolling window: the raw OLS coefficient lambda. . * . * Method: Fix the end year at 2019. For each start year from 1960 to 2010, . * estimate the OLS regression and collect lambda and its 95% CI. . * The CI for lambda is the standard OLS confidence interval: . * lambda_hat ± t_{N-2, 0.025} × SE(lambda_hat) . * . * This gives us a time series showing how the RAW convergence signal . * has evolved before any nonlinear transformation. . *════════════════════════════════════════════════════════════════════════════ > ═══ . . display as text "" . display as text "──────────────────────────────────────────────────────────── > ────" ──────────────────────────────────────────────────────────────── . display as text " SECTION 7A: ROLLING WINDOW — OLS LAMBDA" SECTION 7A: ROLLING WINDOW — OLS LAMBDA . display as text "──────────────────────────────────────────────────────────── > ────" ──────────────────────────────────────────────────────────────── . . * ── Load and reshape data ── . use "convergence_working.dta", clear . keep ccode country year gdppc . reshape wide gdppc, i(ccode country) j(year) (j = 1960 1961 1962 1963 1964 1965 1966 1967 1968 1969 1970 1971 1972 1973 1974 > 1975 1976 1977 1978 1979 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1 > 990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 200 > 5 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 2017 2018 2019) Data Long -> Wide ----------------------------------------------------------------------------- Number of observations 5,040 -> 84 Number of variables 4 -> 62 j variable (60 values) year -> (dropped) xij variables: gdppc -> gdppc1960 gdppc1961 ... gdppc201 > 9 ----------------------------------------------------------------------------- . . local j = 1 . local lastyear = 2019 . . forval startyear = 1960(1)2010 { 2. local s = `lastyear' - `startyear' 3. . capture drop outcome initial_inc 4. gen outcome = (1/`s') * ln(gdppc`lastyear' / gdppc`startyear') 5. gen initial_inc = ln(gdppc`startyear') 6. . qui reg outcome initial_inc if !missing(outcome) & !missing(initial_inc), > robust 7. local lambda = _b[initial_inc] 8. local se_lambda = _se[initial_inc] 9. local n_obs = e(N) 10. . * Standard OLS confidence interval: lambda ± t × SE . local lambda_lb = `lambda' - invttail(e(df_r), 0.025) * `se_lambda' 11. local lambda_ub = `lambda' + invttail(e(df_r), 0.025) * `se_lambda' 12. . preserve 13. clear 14. set obs 1 15. tempfile lam_roll`j' 16. gen startyear = `startyear' 17. gen endyear = `lastyear' 18. gen lambda = `lambda' 19. gen se = `se_lambda' 20. gen lower = `lambda_lb' 21. gen upper = `lambda_ub' 22. gen n = `n_obs' 23. save `lam_roll`j'' 24. restore 25. local ++j 26. } Number of observations (_N) was 0, now 1. file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.000014 saved as .dta format Number of observations (_N) was 0, now 1. file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.000016 saved as .dta format Number of observations (_N) was 0, now 1. file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.000018 saved as .dta format Number of observations (_N) was 0, now 1. file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.00001a saved as .dta format Number of observations (_N) was 0, now 1. file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.00001c saved as .dta format Number of observations (_N) was 0, now 1. file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.00001e saved as .dta format Number of observations (_N) was 0, now 1. file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.00001g saved as .dta format Number of observations (_N) was 0, now 1. file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.00001i saved as .dta format Number of observations (_N) was 0, now 1. file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.00001k saved as .dta format Number of observations (_N) was 0, now 1. file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.00001m saved as .dta format Number of observations (_N) was 0, now 1. file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.00001o saved as .dta format Number of observations (_N) was 0, now 1. file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.00001q saved as .dta format Number of observations (_N) was 0, now 1. file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.00001s saved as .dta format Number of observations (_N) was 0, now 1. file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.00001u saved as .dta format Number of observations (_N) was 0, now 1. file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.00001w saved as .dta format Number of observations (_N) was 0, now 1. file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.000020 saved as .dta format Number of observations (_N) was 0, now 1. file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.000022 saved as .dta format Number of observations (_N) was 0, now 1. file 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saved as .dta format Number of observations (_N) was 0, now 1. file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.00003g saved as .dta format Number of observations (_N) was 0, now 1. file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.00003i saved as .dta format Number of observations (_N) was 0, now 1. file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.00003k saved as .dta format Number of observations (_N) was 0, now 1. file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.00003m saved as .dta format Number of observations (_N) was 0, now 1. file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.00003o saved as .dta format Number of observations (_N) was 0, now 1. file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.00003q saved as .dta format Number of observations (_N) was 0, now 1. file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.00003s saved as .dta format Number of observations (_N) was 0, now 1. file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.00003u saved as .dta format Number of observations (_N) was 0, now 1. file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.00003w saved as .dta format Number of observations (_N) was 0, now 1. file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.000040 saved as .dta format Number of observations (_N) was 0, now 1. file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.000042 saved as .dta format . . * ── Combine rolling lambda results ── . local j_lam = `j' - 1 . preserve . clear . forval i = 1/`j_lam' { 2. capture append using `lam_roll`i'' 3. } . . display as text "" . display as text "╔═══════════════════════════════════════════════════════ > ═══════╗" ╔══════════════════════════════════════════════════════════════╗ . display as text "║ Rolling OLS Lambda: Key Findings > ║" ║ Rolling OLS Lambda: Key Findings ║ . display as text "╚═══════════════════════════════════════════════════════ > ═══════╝" ╚══════════════════════════════════════════════════════════════╝ . list startyear lambda se lower upper n if inlist(startyear, 1960, 1970, 1 > 980, 1990, 1995, 2000, 2005, 2010), noobs clean starty~r lambda se lower upper n 1960 .0005689 .0012908 -.0019988 .0031366 84 1970 .0009913 .0012959 -.0015867 .0035693 84 1980 .0011291 .0013795 -.0016152 .0038735 84 1990 -.0000812 .0013953 -.0028568 .0026945 84 1995 -.0017786 .0013957 -.0045551 .0009979 84 2000 -.0035228 .0014686 -.0064442 -.0006013 84 2005 -.0042767 .0016859 -.0076306 -.0009229 84 2010 -.0030472 .0020207 -.007067 .0009727 84 . . export delimited using "convergence_rolling_lambda.csv", replace (file convergence_rolling_lambda.csv not found) file convergence_rolling_lambda.csv saved . . * ── Create rolling lambda plot ── . #delimit ; delimiter now ; . tw (rcap lower upper startyear, lcolor("217 119 87"%30)) > (scatter lambda startyear, mcolor("217 119 87"%80) msymbol(O) msize(sm > all)), > yline(0, lcolor("20 20 19") lpattern(shortdash) lwidth(medium)) > xlabel(1960(5)2010, angle(45)) > ylabel(, angle(horizontal) format(%6.4f)) > ytitle("{&lambda} (OLS slope coefficient)", size(medlarge)) > xtitle("Initial Year") > title("Rolling OLS Coefficient ({&lambda})", color("20 20 19")) > subtitle("Each point: OLS {&lambda} from initial year to 2019", /// > color("20 20 19") size(small)) > legend(off) > plotregion(style(none) lcolor(none)) > graphregion(fcolor(white) lcolor(white)) > note("{&lambda} < 0: convergence. {&lambda} > 0: divergence." > "Bars: 95% CI ({&lambda} ± t × SE). 84-country balanced panel.", > size(vsmall)) > xsize(5) ysize(4) > ; . #delimit cr delimiter now cr . graph export "stata_convergence_rolling_lambda.png", replace width(2400) (file stata_convergence_rolling_lambda.png not found) file stata_convergence_rolling_lambda.png written in PNG format . display as text "Figure saved: stata_convergence_rolling_lambda.png" Figure saved: stata_convergence_rolling_lambda.png . restore . . . *════════════════════════════════════════════════════════════════════════════ > ═══ . * SECTION 7B: ROLLING BETA CONVERGENCE — OLS AND NLS . * . * Now we convert the rolling lambda to structural beta and compare . * with NLS direct estimation. . * . * Instead of just two snapshots, let's see the FULL MOVIE: . * How has the convergence coefficient evolved year by year? . * . * Method: Fix the end year at 2019. For each start year from 1960 to 2010, . * estimate the convergence β using: . * (A) OLS λ → β conversion . * (B) NLS direct estimation . * . * This gives us TWO rolling-window time series of β. . * . * Convention: . * β > 0 → convergence (poorer countries catching up) . * β < 0 → divergence (gap widening) . *════════════════════════════════════════════════════════════════════════════ > ═══ . . display as text "" . display as text "──────────────────────────────────────────────────────────── > ────" ──────────────────────────────────────────────────────────────── . display as text " SECTION 7: ROLLING BETA CONVERGENCE — OLS AND NLS" SECTION 7: ROLLING BETA CONVERGENCE — OLS AND NLS . display as text "──────────────────────────────────────────────────────────── > ────" ──────────────────────────────────────────────────────────────── . . * ── Load and reshape data ── . use "convergence_working.dta", clear . keep ccode country year gdppc . reshape wide gdppc, i(ccode country) j(year) (j = 1960 1961 1962 1963 1964 1965 1966 1967 1968 1969 1970 1971 1972 1973 1974 > 1975 1976 1977 1978 1979 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1 > 990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 200 > 5 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 2017 2018 2019) Data Long -> Wide ----------------------------------------------------------------------------- Number of observations 5,040 -> 84 Number of variables 4 -> 62 j variable (60 values) year -> (dropped) xij variables: gdppc -> gdppc1960 gdppc1961 ... gdppc201 > 9 ----------------------------------------------------------------------------- . . * ── (A) Rolling OLS ── . display as text "" . display as text "--- (A) Rolling OLS (λ → β conversion) ---" --- (A) Rolling OLS (λ → β conversion) --- . local j = 1 . local lastyear = 2019 . . forval startyear = 1960(1)2010 { 2. local s = `lastyear' - `startyear' 3. . capture drop outcome initial_inc 4. gen outcome = (1/`s') * ln(gdppc`lastyear' / gdppc`startyear') 5. gen initial_inc = ln(gdppc`startyear') 6. . qui reg outcome initial_inc if !missing(outcome) & !missing(initial_inc), > robust 7. local lambda = _b[initial_inc] 8. local se_lambda = _se[initial_inc] 9. local n_obs = e(N) 10. . * Convert λ to β . local lambda_s = `lambda' * `s' 11. if (1 + `lambda_s') > 0 { 12. local beta_val = -ln(1 + `lambda_s') / `s' 13. } 14. else { 15. local beta_val = . 16. } 17. . * Convert λ CI to β CI (bounds flip because the function is monotone decr > easing) . local lambda_lb = `lambda' - invttail(e(df_r), 0.025) * `se_lambda' 18. local lambda_ub = `lambda' + invttail(e(df_r), 0.025) * `se_lambda' 19. if (1 + `lambda_lb'*`s') > 0 { 20. local beta_ub = -ln(1 + `lambda_lb'*`s') / `s' 21. } 22. else { 23. local beta_ub = . 24. } 25. if (1 + `lambda_ub'*`s') > 0 { 26. local beta_lb = -ln(1 + `lambda_ub'*`s') / `s' 27. } 28. else { 29. local beta_lb = . 30. } 31. . preserve 32. clear 33. set obs 1 34. tempfile ols_roll`j' 35. gen startyear = `startyear' 36. gen endyear = `lastyear' 37. gen beta = `beta_val' 38. gen lower = `beta_lb' 39. gen upper = `beta_ub' 40. gen speed_pct = `beta_val' * 100 41. gen halflife = . 42. if `beta_val' > 0 & `beta_val' < . { 43. replace halflife = ln(2) / `beta_val' 44. } 45. gen n = `n_obs' 46. save `ols_roll`j'' 47. restore 48. local ++j 49. } Number of observations (_N) was 0, now 1. (1 missing value generated) file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.000045 saved as .dta format Number of observations (_N) was 0, now 1. (1 missing value generated) file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.000047 saved as .dta format Number of observations (_N) was 0, now 1. (1 missing value generated) file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.000049 saved as .dta format Number of observations (_N) was 0, now 1. (1 missing value generated) file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.00004b saved as .dta format Number of observations (_N) was 0, now 1. (1 missing value generated) file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.00004d saved as .dta format Number of observations (_N) was 0, now 1. (1 missing value generated) file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.00004f saved as .dta format Number of observations (_N) was 0, now 1. (1 missing value generated) file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.00004h saved as .dta format Number of observations (_N) was 0, now 1. (1 missing value generated) file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.00004j saved as .dta format Number of observations (_N) was 0, now 1. (1 missing value generated) file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.00004l saved as .dta format Number of observations (_N) was 0, now 1. (1 missing value generated) file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.00004n saved as .dta format Number of observations (_N) was 0, now 1. (1 missing value generated) file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.00004p saved as .dta format Number of observations (_N) was 0, now 1. (1 missing value generated) file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.00004r saved as .dta format Number of observations (_N) was 0, now 1. (1 missing value generated) file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.00004t saved as .dta format Number of observations (_N) was 0, now 1. (1 missing value generated) file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.00004v saved as .dta format Number of observations (_N) was 0, now 1. (1 missing value generated) file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.00004x saved as .dta format Number of observations (_N) was 0, now 1. (1 missing value generated) file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.000051 saved as .dta format Number of observations (_N) was 0, now 1. (1 missing value generated) file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.000053 saved as .dta format Number of observations (_N) was 0, now 1. (1 missing value generated) file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.000055 saved as .dta format Number of observations (_N) was 0, now 1. (1 missing value generated) file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.000057 saved as .dta format Number of observations (_N) was 0, now 1. (1 missing value generated) file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.000059 saved as .dta format Number of observations (_N) was 0, now 1. (1 missing value generated) file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.00005b saved as .dta format Number of observations (_N) was 0, now 1. (1 missing value generated) file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.00005d saved as .dta format Number of observations (_N) was 0, now 1. (1 missing value generated) file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.00005f saved as .dta format Number of observations (_N) was 0, now 1. (1 missing value generated) file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.00005h saved as .dta format Number of observations (_N) was 0, now 1. (1 missing value generated) file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.00005j saved as .dta format Number of observations (_N) was 0, now 1. (1 missing value generated) file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.00005l saved as .dta format Number of observations (_N) was 0, now 1. (1 missing value generated) file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.00005n saved as .dta format Number of observations (_N) was 0, now 1. (1 missing value generated) file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.00005p saved as .dta format Number of observations (_N) was 0, now 1. (1 missing value generated) file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.00005r saved as .dta format Number of observations (_N) was 0, now 1. (1 missing value generated) file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.00005t saved as .dta format Number of observations (_N) was 0, now 1. (1 missing value generated) (1 real change made) file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.00005v saved as .dta format Number of observations (_N) was 0, now 1. (1 missing value generated) (1 real change made) file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.00005x saved as .dta format Number of observations (_N) was 0, now 1. (1 missing value generated) (1 real change made) file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.000061 saved as .dta format Number of observations (_N) was 0, now 1. (1 missing value generated) (1 real change made) file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.000063 saved as .dta format Number of observations (_N) was 0, now 1. (1 missing value generated) (1 real change made) file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.000065 saved as .dta format Number of observations (_N) was 0, now 1. (1 missing value generated) (1 real change made) file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.000067 saved as .dta format Number of observations (_N) was 0, now 1. (1 missing value generated) (1 real change made) file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.000069 saved as .dta format Number of observations (_N) was 0, now 1. (1 missing value generated) (1 real change made) file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.00006b saved as .dta format Number of observations (_N) was 0, now 1. (1 missing value generated) (1 real change made) file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.00006d saved as .dta format Number of observations (_N) was 0, now 1. (1 missing value generated) (1 real change made) file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.00006f saved as .dta format Number of observations (_N) was 0, now 1. (1 missing value generated) (1 real change made) file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.00006h saved as .dta format Number of observations (_N) was 0, now 1. (1 missing value generated) (1 real change made) file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.00006j saved as .dta format Number of observations (_N) was 0, now 1. (1 missing value generated) (1 real change made) file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.00006l saved as .dta format Number of observations (_N) was 0, now 1. (1 missing value generated) (1 real change made) file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.00006n saved as .dta format Number of observations (_N) was 0, now 1. (1 missing value generated) (1 real change made) file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.00006p saved as .dta format Number of observations (_N) was 0, now 1. (1 missing value generated) (1 real change made) file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.00006r saved as .dta format Number of observations (_N) was 0, now 1. (1 missing value generated) (1 real change made) file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.00006t saved as .dta format Number of observations (_N) was 0, now 1. (1 missing value generated) (1 real change made) file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.00006v saved as .dta format Number of observations (_N) was 0, now 1. (1 missing value generated) (1 real change made) file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.00006x saved as .dta format Number of observations (_N) was 0, now 1. (1 missing value generated) (1 real change made) file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.000071 saved as .dta format Number of observations (_N) was 0, now 1. (1 missing value generated) (1 real change made) file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.000073 saved as .dta format . . * ── Combine OLS rolling results ── . local j_ols = `j' - 1 . preserve . clear . forval i = 1/`j_ols' { 2. capture append using `ols_roll`i'' 3. } . . display as text "" . display as text "╔═══════════════════════════════════════════════════════ > ═══════╗" ╔══════════════════════════════════════════════════════════════╗ . display as text "║ Rolling OLS Beta Convergence: Key Findings > ║" ║ Rolling OLS Beta Convergence: Key Findings ║ . display as text "╚═══════════════════════════════════════════════════════ > ═══════╝" ╚══════════════════════════════════════════════════════════════╝ . list startyear beta speed_pct halflife n if inlist(startyear, 1960, 1970, > 1980, 1990, 1995, 2000, 2005, 2010), noobs clean starty~r beta speed_pct halflife n 1960 -.0005596 -.0559555 . 84 1970 -.000968 -.0967977 . 84 1980 -.001105 -.1104993 . 84 1990 .0000813 .0081259 8530.064 84 1995 .0018177 .1817691 381.3339 84 2000 .0036462 .3646227 190.0998 84 2005 .0044101 .4410113 157.1722 84 2010 .0030897 .3089731 224.339 84 . . export delimited using "convergence_rolling_beta_ols.csv", replace file convergence_rolling_beta_ols.csv saved . . * ── Create OLS rolling beta plot ── . #delimit ; delimiter now ; . tw (rcap lower upper startyear, lcolor("217 119 87"%30)) > (scatter beta startyear, mcolor("217 119 87"%80) msymbol(O) msize(smal > l)), > yline(0, lcolor("20 20 19") lpattern(shortdash) lwidth(medium)) > xlabel(1960(5)2010, angle(45)) > ylabel(-0.005(0.0025)0.01, angle(horizontal) format(%6.4f)) > ytitle("{&beta} (speed of convergence)", size(medlarge)) > xtitle("Initial Year") > title("Rolling Unconditional Convergence: OLS", color("20 20 19")) > subtitle("Each point: OLS {&lambda} {&rarr} {&beta}, from initial year > to 2019", /// > color("20 20 19") size(small)) > legend(off) > plotregion(style(none) lcolor(none)) > graphregion(fcolor(white) lcolor(white)) > note("{&beta} > 0: convergence. {&beta} < 0: divergence." > "Bars: 95% CI (converted from OLS). 84-country balanced panel.", > size(vsmall)) > xsize(5) ysize(4) > ; . #delimit cr delimiter now cr . graph export "stata_convergence_rolling_beta_ols.png", replace width(2400 > ) file stata_convergence_rolling_beta_ols.png written in PNG format . display as text "Figure saved: stata_convergence_rolling_beta_ols.png" Figure saved: stata_convergence_rolling_beta_ols.png . restore . . * ── (B) Rolling NLS ── . display as text "" . display as text "--- (B) Rolling NLS (direct estimation) ---" --- (B) Rolling NLS (direct estimation) --- . local j = 1 . . forval startyear = 1960(1)2010 { 2. local s = `lastyear' - `startyear' 3. . capture drop outcome initial_inc 4. gen outcome = (1/`s') * ln(gdppc`lastyear' / gdppc`startyear') 5. gen initial_inc = ln(gdppc`startyear') 6. . capture qui nl (outcome = {b0=1} - (1 - exp(-1*{b1=0.00}*`s'))/`s' * init > ial_inc) /// > if !missing(outcome) & !missing(initial_inc), vce(robust) 7. . if _rc == 0 { 8. preserve 9. clear 10. set obs 1 11. tempfile nls_roll`j' 12. gen startyear = `startyear' 13. gen endyear = `lastyear' 14. gen beta = _b[/b1] 15. gen se = _se[/b1] 16. gen lower = _b[/b1] - invttail(`e(df_r)', 0.025) * _se[/b1] 17. gen upper = _b[/b1] + invttail(`e(df_r)', 0.025) * _se[/b1] 18. gen n = `e(N)' 19. gen speed_pct = beta * 100 20. gen halflife = . 21. if _b[/b1] > 0 { 22. replace halflife = ln(2) / _b[/b1] 23. } 24. save `nls_roll`j'' 25. restore 26. local ++j 27. } 28. else { 29. display as text " Warning: NLS did not converge for start year `s > tartyear'" 30. local ++j 31. } 32. } Number of observations (_N) was 0, now 1. (1 missing value generated) file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.000076 saved as .dta format Number of observations (_N) was 0, now 1. (1 missing value generated) file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.000078 saved as .dta format Number of observations (_N) was 0, now 1. (1 missing value generated) file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.00007a saved as .dta format Number of observations (_N) was 0, now 1. (1 missing value generated) file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.00007c saved as .dta format Number of observations (_N) was 0, now 1. (1 missing value generated) file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.00007e saved as .dta format Number of observations (_N) was 0, now 1. (1 missing value generated) file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.00007g saved as .dta format Number of observations (_N) was 0, now 1. (1 missing value generated) file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.00007i saved as .dta format Number of observations (_N) was 0, now 1. (1 missing value generated) file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.00007k saved as .dta format Number of observations (_N) was 0, now 1. (1 missing value generated) file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.00007m saved as .dta format Number of observations (_N) was 0, now 1. (1 missing value generated) file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.00007o saved as .dta format Number of observations (_N) was 0, now 1. (1 missing value generated) file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.00007q saved as .dta format Number of observations (_N) was 0, now 1. (1 missing value generated) file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.00007s saved as .dta format Number of observations (_N) was 0, now 1. (1 missing value generated) file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.00007u saved as .dta format Number of observations (_N) was 0, now 1. (1 missing value generated) file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.00007w saved as .dta format Number of observations (_N) was 0, now 1. (1 missing value generated) file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.000080 saved as .dta format Number of observations (_N) was 0, now 1. (1 missing value generated) file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.000082 saved as .dta format Number of observations (_N) was 0, now 1. (1 missing value generated) file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.000084 saved as .dta format Number of observations (_N) was 0, now 1. (1 missing value generated) file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.000086 saved as .dta format Number of observations (_N) was 0, now 1. (1 missing value generated) file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.000088 saved as .dta format Number of observations (_N) was 0, now 1. (1 missing value generated) file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.00008a saved as .dta format Number of observations (_N) was 0, now 1. (1 missing value generated) file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.00008c saved as .dta format Number of observations (_N) was 0, now 1. (1 missing value generated) file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.00008e saved as .dta format Number of observations (_N) was 0, now 1. (1 missing value generated) file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.00008g saved as .dta format Number of observations (_N) was 0, now 1. (1 missing value generated) file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.00008i saved as .dta format Number of observations (_N) was 0, now 1. (1 missing value generated) file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.00008k saved as .dta format Number of observations (_N) was 0, now 1. (1 missing value generated) file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.00008m saved as .dta format Number of observations (_N) was 0, now 1. (1 missing value generated) file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.00008o saved as .dta format Number of observations (_N) was 0, now 1. (1 missing value generated) file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.00008q saved as .dta format Number of observations (_N) was 0, now 1. (1 missing value generated) file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.00008s saved as .dta format Number of observations (_N) was 0, now 1. (1 missing value generated) file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.00008u saved as .dta format Number of observations (_N) was 0, now 1. (1 missing value generated) (1 real change made) file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.00008w saved as .dta format Number of observations (_N) was 0, now 1. (1 missing value generated) (1 real change made) file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.000090 saved as .dta format Number of observations (_N) was 0, now 1. (1 missing value generated) (1 real change made) file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.000092 saved as .dta format Number of observations (_N) was 0, now 1. (1 missing value generated) (1 real change made) file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.000094 saved as .dta format Number of observations (_N) was 0, now 1. (1 missing value generated) (1 real change made) file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.000096 saved as .dta format Number of observations (_N) was 0, now 1. (1 missing value generated) (1 real change made) file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.000098 saved as .dta format Number of observations (_N) was 0, now 1. (1 missing value generated) (1 real change made) file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.00009a saved as .dta format Number of observations (_N) was 0, now 1. (1 missing value generated) (1 real change made) file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.00009c saved as .dta format Number of observations (_N) was 0, now 1. (1 missing value generated) (1 real change made) file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.00009e saved as .dta format Number of observations (_N) was 0, now 1. (1 missing value generated) (1 real change made) file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.00009g saved as .dta format Number of observations (_N) was 0, now 1. (1 missing value generated) (1 real change made) file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.00009i saved as .dta format Number of observations (_N) was 0, now 1. (1 missing value generated) (1 real change made) file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.00009k saved as .dta format Number of observations (_N) was 0, now 1. (1 missing value generated) (1 real change made) file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.00009m saved as .dta format Number of observations (_N) was 0, now 1. (1 missing value generated) (1 real change made) file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.00009o saved as .dta format Number of observations (_N) was 0, now 1. (1 missing value generated) (1 real change made) file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.00009q saved as .dta format Number of observations (_N) was 0, now 1. (1 missing value generated) (1 real change made) file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.00009s saved as .dta format Number of observations (_N) was 0, now 1. (1 missing value generated) (1 real change made) file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.00009u saved as .dta format Number of observations (_N) was 0, now 1. (1 missing value generated) (1 real change made) file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.00009w saved as .dta format Number of observations (_N) was 0, now 1. (1 missing value generated) (1 real change made) file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.0000a0 saved as .dta format Number of observations (_N) was 0, now 1. (1 missing value generated) (1 real change made) file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.0000a2 saved as .dta format Number of observations (_N) was 0, now 1. (1 missing value generated) (1 real change made) file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.0000a4 saved as .dta format . . * ── Combine NLS rolling results ── . local j_nls = `j' - 1 . preserve . clear . forval i = 1/`j_nls' { 2. capture append using `nls_roll`i'' 3. } . . display as text "" . display as text "╔═══════════════════════════════════════════════════════ > ═══════╗" ╔══════════════════════════════════════════════════════════════╗ . display as text "║ Rolling NLS Beta Convergence: Key Findings > ║" ║ Rolling NLS Beta Convergence: Key Findings ║ . display as text "╚═══════════════════════════════════════════════════════ > ═══════╝" ╚══════════════════════════════════════════════════════════════╝ . list startyear beta speed_pct halflife n if inlist(startyear, 1960, 1970, > 1980, 1990, 1995, 2000, 2005, 2010), noobs clean starty~r beta speed_pct halflife n 1960 -.0005596 -.0559555 . 84 1970 -.000968 -.0967977 . 84 1980 -.001105 -.1104993 . 84 1990 .0000813 .0081259 8530.064 84 1995 .0018177 .1817691 381.3339 84 2000 .0036462 .3646227 190.0998 84 2005 .0044101 .4410113 157.1722 84 2010 .0030897 .3089732 224.339 84 . . export delimited using "convergence_rolling_beta_nls.csv", replace file convergence_rolling_beta_nls.csv saved . . * ── Create NLS rolling beta plot ── . #delimit ; delimiter now ; . tw (rcap lower upper startyear, lcolor("106 155 204"%30)) > (scatter beta startyear, mcolor("106 155 204"%80) msymbol(O) msize(sma > ll)), > yline(0, lcolor("20 20 19") lpattern(shortdash) lwidth(medium)) > xlabel(1960(5)2010, angle(45)) > ylabel(-0.005(0.0025)0.01, angle(horizontal) format(%6.4f)) > ytitle("{&beta} (speed of convergence)", size(medlarge)) > xtitle("Initial Year") > title("Rolling Unconditional Convergence: NLS", color("20 20 19")) > subtitle("Each point: NLS {&beta} from initial year to 2019", /// > color("20 20 19") size(small)) > legend(off) > plotregion(style(none) lcolor(none)) > graphregion(fcolor(white) lcolor(white)) > note("{&beta} > 0: convergence. {&beta} < 0: divergence." > "Bars: 95% confidence intervals. 84-country balanced panel.", siz > e(vsmall)) > xsize(5) ysize(4) > ; . #delimit cr delimiter now cr . graph export "stata_convergence_rolling_beta_nls.png", replace width(2400 > ) file stata_convergence_rolling_beta_nls.png written in PNG format . display as text "Figure saved: stata_convergence_rolling_beta_nls.png" Figure saved: stata_convergence_rolling_beta_nls.png . restore . . . *════════════════════════════════════════════════════════════════════════════ > ═══ . * SECTION 8: SIGMA CONVERGENCE — THE SIMPLEST CASE (Two Periods) . * . * While beta convergence asks "do poor countries grow faster?", . * sigma convergence asks a different question: . * Is the SPREAD of income across countries getting NARROWER? . * . * We measure the spread using the VARIANCE of log GDP per capita. . * (Using logs means we measure proportional, not absolute, differences.) . * . * If Var(ln y) decreases over time → σ-convergence (narrowing) . * If Var(ln y) increases over time → σ-divergence (spreading out) . *════════════════════════════════════════════════════════════════════════════ > ═══ . . display as text "" . display as text "──────────────────────────────────────────────────────────── > ────" ──────────────────────────────────────────────────────────────── . display as text " SECTION 8: SIGMA CONVERGENCE — THE SIMPLEST CASE" SECTION 8: SIGMA CONVERGENCE — THE SIMPLEST CASE . display as text " Two periods: 1960 and 2019" Two periods: 1960 and 2019 . display as text "──────────────────────────────────────────────────────────── > ────" ──────────────────────────────────────────────────────────────── . . * ── Load the long-format data ── . use "convergence_working.dta", clear . . * ── Calculate variance of log GDP per capita in 1960 ── . display as text "" . display as text "--- Cross-country dispersion in 1960 ---" --- Cross-country dispersion in 1960 --- . gen logy = ln(gdppc) . ci variances logy if year == 1960 Variable | Obs Variance [95% conf. interval] -------------+---------------------------------------------------- logy | 84 .9244376 .6969585 1.285409 . local var_1960 = r(Var) . local var_1960_lb = r(lb) . local var_1960_ub = r(ub) . local n_1960 = r(N) . local sd_1960 = sqrt(`var_1960') . display as result " Variance of ln(GDP pc) in 1960 = " %6.4f `var_1960' " [" > %6.4f `var_1960_lb' ", " %6.4f `var_1960_ub' "]" Variance of ln(GDP pc) in 1960 = 0.9244 [0.6970, 1.2854] . display as result " Std. Dev. of ln(GDP pc) in 1960 = " %6.4f `sd_1960' Std. Dev. of ln(GDP pc) in 1960 = 0.9615 . display as result " N = " `n_1960' " countries" N = 84 countries . . * ── Calculate variance of log GDP per capita in 2019 ── . display as text "" . display as text "--- Cross-country dispersion in 2019 ---" --- Cross-country dispersion in 2019 --- . ci variances logy if year == 2019 Variable | Obs Variance [95% conf. interval] -------------+---------------------------------------------------- logy | 84 1.763502 1.329552 2.452109 . local var_2019 = r(Var) . local var_2019_lb = r(lb) . local var_2019_ub = r(ub) . local n_2019 = r(N) . local sd_2019 = sqrt(`var_2019') . display as result " Variance of ln(GDP pc) in 2019 = " %6.4f `var_2019' " [" > %6.4f `var_2019_lb' ", " %6.4f `var_2019_ub' "]" Variance of ln(GDP pc) in 2019 = 1.7635 [1.3296, 2.4521] . display as result " Std. Dev. of ln(GDP pc) in 2019 = " %6.4f `sd_2019' Std. Dev. of ln(GDP pc) in 2019 = 1.3280 . display as result " N = " `n_2019' " countries" N = 84 countries . . * ── Compare ── . display as text "" . display as text "╔═══════════════════════════════════════════════════════════ > ═══╗" ╔══════════════════════════════════════════════════════════════╗ . display as text "║ Sigma Convergence Test: 1960 vs 2019 > ║" ║ Sigma Convergence Test: 1960 vs 2019 ║ . display as text "╚═══════════════════════════════════════════════════════════ > ═══╝" ╚══════════════════════════════════════════════════════════════╝ . local change = `var_2019' - `var_1960' . local pct_change = (`change' / `var_1960') * 100 . display as result " Change in variance: " %6.4f `change' " (" %5.1f `pct_cha > nge' "%)" Change in variance: 0.8391 ( 90.8%) . if `change' < 0 { . display as result " → Variance DECREASED: evidence of σ-convergence" . } . else { . display as result " → Variance INCREASED: evidence of σ-DIVERGENCE" → Variance INCREASED: evidence of σ-DIVERGENCE . display as result " Even though beta convergence may exist, the spread > widened!" Even though beta convergence may exist, the spread widened! . } . . * ── Bar chart of variance in 1960 vs 2019 ── . preserve . clear . set obs 2 Number of observations (_N) was 0, now 2. . gen year = 1960 in 1 (1 missing value generated) . replace year = 2019 in 2 (1 real change made) . gen variance = `var_1960' in 1 (1 missing value generated) . replace variance = `var_2019' in 2 (1 real change made) . gen var_lb = `var_1960_lb' in 1 (1 missing value generated) . replace var_lb = `var_2019_lb' in 2 (1 real change made) . gen var_ub = `var_1960_ub' in 1 (1 missing value generated) . replace var_ub = `var_2019_ub' in 2 (1 real change made) . . #delimit ; delimiter now ; . tw (bar variance year if year == 1960, barwidth(8) fcolor("217 119 87") l > color("217 119 87"%80)) > (bar variance year if year == 2019, barwidth(8) fcolor("106 155 204") > lcolor("106 155 204"%80)) > (rcap var_lb var_ub year, lcolor("20 20 19")), > xlabel(1960 "1960" 2019 "2019") > ylabel(, angle(horizontal)) > ytitle("Variance of ln(GDP per capita)") > title("Sigma Convergence: Income Dispersion", color("20 20 19")) > subtitle("Has the spread of income narrowed?", color("20 20 19") size( > small)) > legend(order(1 "1960 (N=`n_1960')" 2 "2019 (N=`n_2019')" 3 "95% CI") p > os(2) ring(0) > region(lcolor(none) fcolor(none)) size(small)) > plotregion(style(none) lcolor(none)) > graphregion(fcolor(white) lcolor(white)) > note("Data: Penn World Tables 10.0. 84-country balanced panel." > "Variance of log GDP per capita (PPP).", size(vsmall)) > xsize(5) ysize(4) > ; . #delimit cr delimiter now cr . graph export "stata_convergence_sigma_two_periods.png", replace width(240 > 0) file stata_convergence_sigma_two_periods.png written in PNG format . display as text "Figure saved: stata_convergence_sigma_two_periods.png" Figure saved: stata_convergence_sigma_two_periods.png . restore . . drop logy . . . *════════════════════════════════════════════════════════════════════════════ > ═══ . * SECTION 9: THE RELATIONSHIP BETWEEN BETA AND SIGMA CONVERGENCE . * . * A crucial insight from the convergence literature: . * . * β-convergence is NECESSARY but NOT SUFFICIENT for σ-convergence. . * (Young, Higgins, and Levy 2008) . * . * WHY? Think of it like a race: . * - β-convergence means the slowest runners are speeding up (catch-up) . * - σ-convergence means the runners are getting closer together . * . * Even if slower runners speed up ON AVERAGE, random shocks can keep . * the pack spread out. The catch-up tendency (β) must be strong enough . * to overcome the dispersing force of shocks. . *════════════════════════════════════════════════════════════════════════════ > ═══ . . display as text "" . display as text "──────────────────────────────────────────────────────────── > ────" ──────────────────────────────────────────────────────────────── . display as text " SECTION 9: THE RELATIONSHIP BETWEEN BETA AND SIGMA" SECTION 9: THE RELATIONSHIP BETWEEN BETA AND SIGMA . display as text "──────────────────────────────────────────────────────────── > ────" ──────────────────────────────────────────────────────────────── . . display as text "" . display as text " KEY INSIGHT:" KEY INSIGHT: . display as text " β-convergence is NECESSARY but NOT SUFFICIENT for σ-conver > gence." β-convergence is NECESSARY but NOT SUFFICIENT for σ-convergence. . display as text "" . display as text " Think of runners in a race:" Think of runners in a race: . display as text " β-convergence = the slowest runners speed up (catch-up)" β-convergence = the slowest runners speed up (catch-up) . display as text " σ-convergence = the pack of runners gets tighter" σ-convergence = the pack of runners gets tighter . display as text "" . display as text " Even if the slow runners speed up on average, random shock > s" Even if the slow runners speed up on average, random shocks . display as text " (weather, injuries) can keep the pack spread out." (weather, injuries) can keep the pack spread out. . . * ── Decade-by-decade β and σ ── . use "convergence_working.dta", clear . keep ccode country year gdppc . reshape wide gdppc, i(ccode country) j(year) (j = 1960 1961 1962 1963 1964 1965 1966 1967 1968 1969 1970 1971 1972 1973 1974 > 1975 1976 1977 1978 1979 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1 > 990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 200 > 5 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 2017 2018 2019) Data Long -> Wide ----------------------------------------------------------------------------- Number of observations 5,040 -> 84 Number of variables 4 -> 62 j variable (60 values) year -> (dropped) xij variables: gdppc -> gdppc1960 gdppc1961 ... gdppc201 > 9 ----------------------------------------------------------------------------- . . display as text "" . display as text " EMPIRICAL DEMONSTRATION:" EMPIRICAL DEMONSTRATION: . display as text " Decade-by-decade OLS λ and variance of log income:" Decade-by-decade OLS λ and variance of log income: . display as text "" . display as text " ┌─────────────┬───────────┬───────────┬─────────────────── > ─────────┐" ┌─────────────┬───────────┬───────────┬────────────────────────────┐ . display as text " │ Decade │ OLS λ │ σ² start │ Interpretation > │" │ Decade │ OLS λ │ σ² start │ Interpretation │ . display as text " ├─────────────┼───────────┼───────────┼─────────────────── > ─────────┤" ├─────────────┼───────────┼───────────┼────────────────────────────┤ . . foreach decade in 1960 1970 1980 1990 2000 2010 { 2. local ey = `decade' + 10 3. if `ey' > 2019 local ey = 2019 4. local s = `ey' - `decade' 5. . * Beta (OLS): regress growth on initial income . capture drop g_temp i_temp logy_temp 6. gen g_temp = (1/`s') * ln(gdppc`ey' / gdppc`decade') 7. gen i_temp = ln(gdppc`decade') 8. qui reg g_temp i_temp if !missing(g_temp) & !missing(i_temp), robust 9. local b = _b[i_temp] 10. . * Sigma: variance of log income at the START of the decade . gen logy_temp = ln(gdppc`decade') 11. qui summarize logy_temp 12. local v = r(Var) 13. . * Determine interpretation . local interp = "" 14. if `b' >= 0 { 15. local interp = "λ≥0: divergence" 16. } 17. else { 18. local interp = "λ<0: convergence" 19. } 20. . display as text " │ `decade'-`ey' │ " %8.5f `b' " │ " %8.4f `v' " │ `i > nterp' │" 21. drop g_temp i_temp logy_temp 22. } │ 1960-1970 │ 0.00594 │ 0.9244 │ λ≥0: divergence │ │ 1970-1980 │ 0.00555 │ 1.0818 │ λ≥0: divergence │ │ 1980-1990 │ 0.00686 │ 1.2893 │ λ≥0: divergence │ │ 1990-2000 │ 0.00882 │ 1.5384 │ λ≥0: divergence │ │ 2000-2010 │ -0.00379 │ 1.8937 │ λ<0: convergence │ │ 2010-2019 │ -0.00305 │ 1.8262 │ λ<0: convergence │ . . display as text " └─────────────┴───────────┴───────────┴─────────────────── > ─────────┘" └─────────────┴───────────┴───────────┴────────────────────────────┘ . display as text "" . display as text " NOTICE: λ turns negative (convergence) around 2000, but σ² > keeps" NOTICE: λ turns negative (convergence) around 2000, but σ² keeps . display as text " RISING until ~2008. This is the lag: β-convergence is nece > ssary" RISING until ~2008. This is the lag: β-convergence is necessary . display as text " but not sufficient for σ-convergence." but not sufficient for σ-convergence. . . . *════════════════════════════════════════════════════════════════════════════ > ═══ . * SECTION 10: SIGMA CONVERGENCE OVER TIME . * . * Track the dispersion of income EVERY YEAR from 1960 to 2019. . * With our balanced 84-country panel, the sample composition is . * constant throughout — no need for a separate "fixed sample" series. . *════════════════════════════════════════════════════════════════════════════ > ═══ . . display as text "" . display as text "──────────────────────────────────────────────────────────── > ────" ──────────────────────────────────────────────────────────────── . display as text " SECTION 10: SIGMA CONVERGENCE OVER TIME" SECTION 10: SIGMA CONVERGENCE OVER TIME . display as text "──────────────────────────────────────────────────────────── > ────" ──────────────────────────────────────────────────────────────── . . use "convergence_working.dta", clear . keep ccode year gdppc . reshape wide gdppc, i(ccode) j(year) (j = 1960 1961 1962 1963 1964 1965 1966 1967 1968 1969 1970 1971 1972 1973 1974 > 1975 1976 1977 1978 1979 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1 > 990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 200 > 5 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 2017 2018 2019) Data Long -> Wide ----------------------------------------------------------------------------- Number of observations 5,040 -> 84 Number of variables 3 -> 61 j variable (60 values) year -> (dropped) xij variables: gdppc -> gdppc1960 gdppc1961 ... gdppc201 > 9 ----------------------------------------------------------------------------- . . local j = 1 . forval yr = 1960(1)2019 { 2. gen logy = ln(gdppc`yr') 3. ci variances logy 4. preserve 5. clear 6. set obs 1 7. tempfile sigma`j' 8. gen year = `yr' 9. gen variance = r(Var) 10. gen var_lb = `r(lb)' 11. gen var_ub = `r(ub)' 12. gen n = `r(N)' 13. save `sigma`j'' 14. restore 15. drop logy 16. local ++j 17. } Variable | Obs Variance [95% conf. interval] -------------+---------------------------------------------------- logy | 84 .9244376 .6969585 1.285409 Number of observations (_N) was 0, now 1. file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.0000a8 saved as .dta format Variable | Obs Variance [95% conf. interval] -------------+---------------------------------------------------- logy | 84 .9604703 .7241244 1.335512 Number of observations (_N) was 0, now 1. file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.0000aa saved as .dta format Variable | Obs Variance [95% conf. interval] -------------+---------------------------------------------------- logy | 84 .962358 .7255477 1.338137 Number of observations (_N) was 0, now 1. file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.0000ac saved as .dta format Variable | Obs Variance [95% conf. interval] -------------+---------------------------------------------------- logy | 84 .9835334 .7415124 1.36758 Number of observations (_N) was 0, now 1. file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.0000ae saved as .dta format Variable | Obs Variance [95% conf. interval] -------------+---------------------------------------------------- logy | 84 1.005373 .7579778 1.397948 Number of observations (_N) was 0, now 1. file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.0000ag saved as .dta format Variable | Obs Variance [95% conf. interval] -------------+---------------------------------------------------- logy | 84 1.019628 .7687252 1.417769 Number of observations (_N) was 0, now 1. file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.0000ai saved as .dta format Variable | Obs Variance [95% conf. interval] -------------+---------------------------------------------------- logy | 84 1.035928 .7810138 1.440433 Number of observations (_N) was 0, now 1. file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.0000ak saved as .dta format Variable | Obs Variance [95% conf. interval] -------------+---------------------------------------------------- logy | 84 1.047813 .7899741 1.456959 Number of observations (_N) was 0, now 1. file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.0000am saved as .dta format Variable | Obs Variance [95% conf. interval] -------------+---------------------------------------------------- logy | 84 1.061106 .7999961 1.475443 Number of observations (_N) was 0, now 1. file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.0000ao saved as .dta format Variable | Obs Variance [95% conf. interval] -------------+---------------------------------------------------- logy | 84 1.086404 .8190689 1.510619 Number of observations (_N) was 0, now 1. file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.0000aq saved as .dta format Variable | Obs Variance [95% conf. interval] -------------+---------------------------------------------------- logy | 84 1.081847 .8156335 1.504283 Number of observations (_N) was 0, now 1. file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.0000as saved as .dta format Variable | Obs Variance [95% conf. interval] -------------+---------------------------------------------------- logy | 84 1.110964 .8375854 1.544769 Number of observations (_N) was 0, now 1. file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.0000au saved as .dta format Variable | Obs Variance [95% conf. interval] -------------+---------------------------------------------------- logy | 84 1.145008 .8632522 1.592107 Number of observations (_N) was 0, now 1. file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.0000aw saved as .dta format Variable | Obs Variance [95% conf. interval] -------------+---------------------------------------------------- logy | 84 1.187827 .8955347 1.651646 Number of observations (_N) was 0, now 1. file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.0000b0 saved as .dta format Variable | Obs Variance [95% conf. interval] -------------+---------------------------------------------------- logy | 84 1.172503 .8839818 1.630339 Number of observations (_N) was 0, now 1. file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.0000b2 saved as .dta format Variable | Obs Variance [95% conf. interval] -------------+---------------------------------------------------- logy | 84 1.185251 .8935923 1.648063 Number of observations (_N) was 0, now 1. file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.0000b4 saved as .dta format Variable | Obs Variance [95% conf. interval] -------------+---------------------------------------------------- logy | 84 1.184934 .8933535 1.647623 Number of observations (_N) was 0, now 1. file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.0000b6 saved as .dta format Variable | Obs Variance [95% conf. interval] -------------+---------------------------------------------------- logy | 84 1.179757 .8894506 1.640425 Number of observations (_N) was 0, now 1. file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.0000b8 saved as .dta format Variable | Obs Variance [95% conf. interval] -------------+---------------------------------------------------- logy | 84 1.217322 .9177721 1.692658 Number of observations (_N) was 0, now 1. file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.0000ba saved as .dta format Variable | Obs Variance [95% conf. interval] -------------+---------------------------------------------------- logy | 84 1.242267 .936579 1.727344 Number of observations (_N) was 0, now 1. file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.0000bc saved as .dta format Variable | Obs Variance [95% conf. interval] -------------+---------------------------------------------------- logy | 84 1.289282 .9720246 1.792717 Number of observations (_N) was 0, now 1. file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.0000be saved as .dta format Variable | Obs Variance [95% conf. interval] -------------+---------------------------------------------------- logy | 84 1.275023 .9612743 1.77289 Number of observations (_N) was 0, now 1. file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.0000bg saved as .dta format Variable | Obs Variance [95% conf. interval] -------------+---------------------------------------------------- logy | 84 1.273594 .9601971 1.770903 Number of observations (_N) was 0, now 1. file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.0000bi saved as .dta format Variable | Obs Variance [95% conf. interval] -------------+---------------------------------------------------- logy | 84 1.282634 .967012 1.783472 Number of observations (_N) was 0, now 1. file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.0000bk saved as .dta format Variable | Obs Variance [95% conf. interval] -------------+---------------------------------------------------- logy | 84 1.288519 .9714491 1.791656 Number of observations (_N) was 0, now 1. file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.0000bm saved as .dta format Variable | Obs Variance [95% conf. interval] -------------+---------------------------------------------------- logy | 84 1.26904 .9567634 1.764571 Number of observations (_N) was 0, now 1. file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.0000bo saved as .dta format Variable | Obs Variance [95% conf. interval] -------------+---------------------------------------------------- logy | 84 1.324054 .9982401 1.841067 Number of observations (_N) was 0, now 1. file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.0000bq saved as .dta format Variable | Obs Variance [95% conf. interval] -------------+---------------------------------------------------- logy | 84 1.405731 1.059818 1.954636 Number of observations (_N) was 0, now 1. file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.0000bs saved as .dta format Variable | Obs Variance [95% conf. interval] -------------+---------------------------------------------------- logy | 84 1.441801 1.087012 2.00479 Number of observations (_N) was 0, now 1. file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.0000bu saved as .dta format Variable | Obs Variance [95% conf. interval] -------------+---------------------------------------------------- logy | 84 1.493815 1.126228 2.077116 Number of observations (_N) was 0, now 1. file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.0000bw saved as .dta format Variable | Obs Variance [95% conf. interval] -------------+---------------------------------------------------- logy | 84 1.53844 1.159872 2.139165 Number of observations (_N) was 0, now 1. file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.0000c0 saved as .dta format Variable | Obs Variance [95% conf. interval] -------------+---------------------------------------------------- logy | 84 1.567496 1.181778 2.179567 Number of observations (_N) was 0, now 1. file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.0000c2 saved as .dta format Variable | Obs Variance [95% conf. interval] -------------+---------------------------------------------------- logy | 84 1.625502 1.22551 2.260222 Number of observations (_N) was 0, now 1. file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.0000c4 saved as .dta format Variable | Obs Variance [95% conf. interval] -------------+---------------------------------------------------- logy | 84 1.66508 1.255348 2.315255 Number of observations (_N) was 0, now 1. file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.0000c6 saved as .dta format Variable | Obs Variance [95% conf. interval] -------------+---------------------------------------------------- logy | 84 1.747906 1.317793 2.430423 Number of observations (_N) was 0, now 1. file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.0000c8 saved as .dta format Variable | Obs Variance [95% conf. interval] -------------+---------------------------------------------------- logy | 84 1.748533 1.318266 2.431294 Number of observations (_N) was 0, now 1. file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.0000ca saved as .dta format Variable | Obs Variance [95% conf. interval] -------------+---------------------------------------------------- logy | 84 1.76849 1.333312 2.459044 Number of observations (_N) was 0, now 1. file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.0000cc saved as .dta format Variable | Obs Variance [95% conf. interval] -------------+---------------------------------------------------- logy | 84 1.786894 1.347188 2.484635 Number of observations (_N) was 0, now 1. file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.0000ce saved as .dta format Variable | Obs Variance [95% conf. interval] -------------+---------------------------------------------------- logy | 84 1.822296 1.373878 2.53386 Number of observations (_N) was 0, now 1. file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.0000cg saved as .dta format Variable | Obs Variance [95% conf. interval] -------------+---------------------------------------------------- logy | 84 1.84346 1.389834 2.563289 Number of observations (_N) was 0, now 1. file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.0000ci saved as .dta format Variable | Obs Variance [95% conf. interval] -------------+---------------------------------------------------- logy | 84 1.893675 1.427693 2.633112 Number of observations (_N) was 0, now 1. file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.0000ck saved as .dta format Variable | Obs Variance [95% conf. interval] -------------+---------------------------------------------------- logy | 84 1.87851 1.416259 2.612024 Number of observations (_N) was 0, now 1. file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.0000cm saved as .dta format Variable | Obs Variance [95% conf. interval] -------------+---------------------------------------------------- logy | 84 1.870779 1.410431 2.601275 Number of observations (_N) was 0, now 1. file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.0000co saved as .dta format Variable | Obs Variance [95% conf. interval] -------------+---------------------------------------------------- logy | 84 1.868524 1.408731 2.59814 Number of observations (_N) was 0, now 1. file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.0000cq saved as .dta format Variable | Obs Variance [95% conf. interval] -------------+---------------------------------------------------- logy | 84 1.883132 1.419744 2.618451 Number of observations (_N) was 0, now 1. file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.0000cs saved as .dta format Variable | Obs Variance [95% conf. interval] -------------+---------------------------------------------------- logy | 84 1.913046 1.442297 2.660046 Number of observations (_N) was 0, now 1. file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.0000cu saved as .dta format Variable | Obs Variance [95% conf. interval] -------------+---------------------------------------------------- logy | 84 1.927129 1.452915 2.679629 Number of observations (_N) was 0, now 1. file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.0000cw saved as .dta format Variable | Obs Variance [95% conf. interval] -------------+---------------------------------------------------- logy | 84 1.921817 1.44891 2.672242 Number of observations (_N) was 0, now 1. file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.0000d0 saved as .dta format Variable | Obs Variance [95% conf. interval] -------------+---------------------------------------------------- logy | 84 1.918209 1.44619 2.667226 Number of observations (_N) was 0, now 1. file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.0000d2 saved as .dta format Variable | Obs Variance [95% conf. interval] -------------+---------------------------------------------------- logy | 84 1.834511 1.383087 2.550845 Number of observations (_N) was 0, now 1. file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.0000d4 saved as .dta format Variable | Obs Variance [95% conf. interval] -------------+---------------------------------------------------- logy | 84 1.826223 1.376839 2.539321 Number of observations (_N) was 0, now 1. file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.0000d6 saved as .dta format Variable | Obs Variance [95% conf. interval] -------------+---------------------------------------------------- logy | 84 1.80303 1.359353 2.507072 Number of observations (_N) was 0, now 1. file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.0000d8 saved as .dta format Variable | Obs Variance [95% conf. interval] -------------+---------------------------------------------------- logy | 84 1.783962 1.344977 2.480558 Number of observations (_N) was 0, now 1. file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.0000da saved as .dta format Variable | Obs Variance [95% conf. interval] -------------+---------------------------------------------------- logy | 84 1.794667 1.353048 2.495442 Number of observations (_N) was 0, now 1. file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.0000dc saved as .dta format Variable | Obs Variance [95% conf. interval] -------------+---------------------------------------------------- logy | 84 1.766202 1.331587 2.455863 Number of observations (_N) was 0, now 1. file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.0000de saved as .dta format Variable | Obs Variance [95% conf. interval] -------------+---------------------------------------------------- logy | 84 1.760596 1.327361 2.448068 Number of observations (_N) was 0, now 1. file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.0000dg saved as .dta format Variable | Obs Variance [95% conf. interval] -------------+---------------------------------------------------- logy | 84 1.754305 1.322618 2.439321 Number of observations (_N) was 0, now 1. file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.0000di saved as .dta format Variable | Obs Variance [95% conf. interval] -------------+---------------------------------------------------- logy | 84 1.770418 1.334766 2.461725 Number of observations (_N) was 0, now 1. file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.0000dk saved as .dta format Variable | Obs Variance [95% conf. interval] -------------+---------------------------------------------------- logy | 84 1.778799 1.341085 2.473379 Number of observations (_N) was 0, now 1. file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.0000dm saved as .dta format Variable | Obs Variance [95% conf. interval] -------------+---------------------------------------------------- logy | 84 1.763502 1.329552 2.452109 Number of observations (_N) was 0, now 1. file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.0000do saved as .dta format . . * ── Combine ── . local jminus1 = `j' - 1 . clear . forval i = 1/`jminus1' { 2. append using `sigma`i'' 3. } . . * ── Display key years ── . display as text "" . display as text "╔═══════════════════════════════════════════════════════════ > ═══╗" ╔══════════════════════════════════════════════════════════════╗ . display as text "║ Sigma Convergence Over Time: Key Years > ║" ║ Sigma Convergence Over Time: Key Years ║ . display as text "╚═══════════════════════════════════════════════════════════ > ═══╝" ╚══════════════════════════════════════════════════════════════╝ . list year variance n if inlist(year, 1960, 1970, 1980, 1990, 2000, 2008, 2010 > , 2019), noobs clean year variance n 1960 .9244376 84 1970 1.081847 84 1980 1.289282 84 1990 1.53844 84 2000 1.893675 84 2008 1.918209 84 2010 1.826223 84 2019 1.763502 84 . . * Export . export delimited using "convergence_sigma_evolution.csv", replace file convergence_sigma_evolution.csv saved . display as text "Results exported to: convergence_sigma_evolution.csv" Results exported to: convergence_sigma_evolution.csv . . * ── Create sigma convergence time series plot ── . #delimit ; delimiter now ; . tw (rcap var_lb var_ub year, lcolor("106 155 204"%20)) > (scatter variance year, mcolor("106 155 204"%60) msymbol(O) msize(vsmall)) > , > xlabel(1960(10)2020) > ylabel(, angle(horizontal)) > ytitle("Variance of ln(GDP per capita)", size(medsmall)) > xtitle("Year") > title("Sigma Convergence: Income Dispersion Over Time", color("20 20 19") > size(medium)) > subtitle("Variance of log GDP per capita, 84-country balanced panel", /// > color("20 20 19") size(small)) > legend(off) > plotregion(style(none) lcolor(none)) > graphregion(fcolor(white) lcolor(white)) > note("Data: Penn World Tables 10.0. 84-country balanced panel." > "Bars: 95% confidence intervals.", size(vsmall)) > xsize(6) ysize(4) > ; . #delimit cr delimiter now cr . graph export "stata_convergence_sigma_evolution.png", replace width(2400) file stata_convergence_sigma_evolution.png written in PNG format . display as text "Figure saved: stata_convergence_sigma_evolution.png" Figure saved: stata_convergence_sigma_evolution.png . . . *════════════════════════════════════════════════════════════════════════════ > ═══ . * SECTION 11: THE CONVERGENCE HEATMAP — OLS AND NLS . * . * The most comprehensive view of convergence: . * For EVERY combination of start year and end year, . * estimate the β coefficient and color-code it. . * . * We produce TWO heatmaps: . * (A) OLS-based: estimate λ, convert to β = -ln(1+λs)/s . * (B) NLS-based: estimate β directly (reproduces Patel et al. 2021 Fig 2) . * . * Blue = convergence (β > 0), Red = divergence (β < 0) . *════════════════════════════════════════════════════════════════════════════ > ═══ . . display as text "" . display as text "──────────────────────────────────────────────────────────── > ────" ──────────────────────────────────────────────────────────────── . display as text " SECTION 11: THE CONVERGENCE HEATMAP — OLS AND NLS" SECTION 11: THE CONVERGENCE HEATMAP — OLS AND NLS . display as text "──────────────────────────────────────────────────────────── > ────" ──────────────────────────────────────────────────────────────── . . * ── Load and reshape ── . use "convergence_working.dta", clear . keep ccode country year gdppc . reshape wide gdppc, i(ccode country) j(year) (j = 1960 1961 1962 1963 1964 1965 1966 1967 1968 1969 1970 1971 1972 1973 1974 > 1975 1976 1977 1978 1979 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1 > 990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 200 > 5 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 2017 2018 2019) Data Long -> Wide ----------------------------------------------------------------------------- Number of observations 5,040 -> 84 Number of variables 4 -> 62 j variable (60 values) year -> (dropped) xij variables: gdppc -> gdppc1960 gdppc1961 ... gdppc201 > 9 ----------------------------------------------------------------------------- . . * ── Loop over ALL start/end year combinations ── . local j = 1 . local firstyear = 1960 . local lastyear = 2019 . local endyear_loop = `lastyear' - 1 . . forval startyear = `firstyear'(1)`endyear_loop' { 2. local startplus1 = `startyear' + 1 3. forval outcomeyear = `startplus1'(1)`lastyear' { 4. local s = `outcomeyear' - `startyear' 5. . capture drop outcome initial_inc 6. gen outcome = (1/`s') * ln(gdppc`outcomeyear' / gdppc`startyear') 7. gen initial_inc = ln(gdppc`startyear') 8. . * ── OLS ── . local beta_ols_hm = . 9. local n_hm = . 10. capture qui reg outcome initial_inc if !missing(outcome) & !missin > g(initial_inc), robust 11. if _rc == 0 { 12. local lambda_hm = _b[initial_inc] 13. local n_hm = e(N) 14. local lambda_s_hm = `lambda_hm' * `s' 15. if (1 + `lambda_s_hm') > 0 { 16. local beta_ols_hm = -ln(1 + `lambda_s_hm') / `s' 17. } 18. } 19. . * ── NLS ── . local beta_nls_hm = . 20. capture qui nl (outcome = {b0=1} - (1 - exp(-1*{b1=0.00}*`s'))/`s' > * initial_inc) /// > if !missing(outcome) & !missing(initial_inc), vce(robust) 21. if _rc == 0 { 22. local beta_nls_hm = _b[/b1] 23. } 24. . preserve 25. clear 26. set obs 1 27. tempfile hm`j' 28. gen startyear = `startyear' 29. gen endyear = `outcomeyear' 30. gen beta_ols = `beta_ols_hm' 31. gen beta_nls = `beta_nls_hm' 32. gen n = `n_hm' 33. save `hm`j'' 34. restore 35. . local ++j 36. } 37. * Progress indicator every 5 years . if mod(`startyear' - `firstyear', 5) == 0 { 38. display as text " Processing start year: `startyear'..." 39. } 40. } Number of observations (_N) was 0, now 1. file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.0000dq saved as .dta format Number of observations (_N) was 0, now 1. file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.0000ds saved as .dta format Number of observations (_N) was 0, now 1. file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.0000du saved as .dta format Number of observations (_N) was 0, now 1. file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.0000dw saved as .dta format Number of observations (_N) was 0, now 1. file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.0000e0 saved as .dta format Number of observations (_N) was 0, now 1. file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.0000e2 saved as .dta format Number of observations (_N) was 0, now 1. file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.0000e4 saved as .dta format Number of observations (_N) was 0, now 1. file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.0000e6 saved as .dta format Number of observations (_N) was 0, now 1. file 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saved as .dta format Processing start year: 2000... 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Number of observations (_N) was 0, now 1. file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.0003fi saved as .dta format Number of observations (_N) was 0, now 1. file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.0003fk saved as .dta format Number of observations (_N) was 0, now 1. file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.0003fm saved as .dta format Number of observations (_N) was 0, now 1. file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.0003fo saved as .dta format Number of observations (_N) was 0, now 1. file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.0003fq saved as .dta format Number of observations (_N) was 0, now 1. file /var/folders/tq/t98kb27n6djgrh085g476yhc0000gn/T//St02028.0003fs saved as .dta format . . * ── Combine all heatmap results ── . local jminus1 = `j' - 1 . preserve . clear . forval i = 1/`jminus1' { 2. capture append using `hm`i'' 3. } . gen period = endyear - startyear . . * Export . export delimited using "convergence_heatmap_coefficients.csv", replace file convergence_heatmap_coefficients.csv saved . display as text "Results exported to: convergence_heatmap_coefficients.cs > v" Results exported to: convergence_heatmap_coefficients.csv . . * ── (A) OLS Heatmap ── . #delimit ; delimiter now ; . tw (scatter startyear endyear if beta_ols > .0035, msize(medsmall) > mfcolor("5 48 97") mlcolor("20 20 19") mlwidth(vthin)) > (scatter startyear endyear if beta_ols < .0035 & beta_ols > .0025, msi > ze(medsmall) > mfcolor("33 102 172") mlcolor("20 20 19") mlwidth(vthin)) > (scatter startyear endyear if beta_ols < .0025 & beta_ols > .0015, msi > ze(medsmall) > mfcolor("67 147 195") mlcolor("20 20 19") mlwidth(vthin)) > (scatter startyear endyear if beta_ols < .0015 & beta_ols > .0005, msi > ze(medsmall) > mfcolor("146 197 222") mlcolor("20 20 19") mlwidth(vthin)) > (scatter startyear endyear if beta_ols < .0005 & beta_ols > -.0005, ms > ize(medsmall) > mfcolor("209 229 240") mlcolor("20 20 19") mlwidth(vthin)) > (scatter startyear endyear if beta_ols > -.0015 & beta_ols < -.0005, m > size(medsmall) > mfcolor("247 247 247") mlcolor("20 20 19") mlwidth(vthin)) > (scatter startyear endyear if beta_ols > -.0025 & beta_ols < -.0015, m > size(medsmall) > mfcolor("253 219 199") mlcolor("20 20 19") mlwidth(vthin)) > (scatter startyear endyear if beta_ols > -.0035 & beta_ols < -.0025, m > size(medsmall) > mfcolor("244 165 130") mlcolor("20 20 19") mlwidth(vthin)) > (scatter startyear endyear if beta_ols > -.0045 & beta_ols < -.0035, m > size(medsmall) > mfcolor("214 96 77") mlcolor("20 20 19") mlwidth(vthin)) > (scatter startyear endyear if beta_ols > -.0055 & beta_ols < -.0045, m > size(medsmall) > mfcolor("178 24 43") mlcolor("20 20 19") mlwidth(vthin)) > (scatter startyear endyear if beta_ols < -.0055, msize(medsmall) > mfcolor("103 0 31") mlcolor("20 20 19") mlwidth(vthin)), > plotregion(style(none) lcolor(none)) > graphregion(fcolor(white) lcolor(white)) > xtitle("End Year") ytitle("Start Year") > title("Convergence Heatmap: OLS ({&lambda} {&rarr} {&beta} conversion) > ", color("20 20 19") size(medium)) > subtitle("Blue = convergence ({&beta}>0), Red = divergence ({&beta}<0) > ", /// > color("20 20 19") size(small)) > aspectratio(1) > xlabel(1960(10)2020) ylabel(1960(10)2010) > xsize(5) ysize(5) > legend(order(1 ">.0035" 2 "[.0025,.0035]" 3 "[.0015,.0025]" > 4 "[.0005,.0015]" 5 "[-.0005,.0005]" > 6 "[-.0015,-.0005]" 7 "[-.0025,-.0015]" > 8 "[-.0035,-.0025]" 9 "[-.0045,-.0035]" > 10 "[-.0055,-.0045]" 11 "<-.0055") > size(vsmall) pos(10) ring(0) bmargin(medsmall) col(1) > region(lcolor(none) fcolor(none)) > title("{&beta} coefficient", size(small))) > note("Data: Penn World Tables 10.0. 84-country balanced panel." > "OLS {&lambda} converted to structural {&beta} = -ln(1+{&lambda}s > )/s.", size(vsmall)) > ; . #delimit cr delimiter now cr . graph export "stata_convergence_heatmap_ols.png", replace width(2400) file stata_convergence_heatmap_ols.png written in PNG format . display as text "Figure saved: stata_convergence_heatmap_ols.png" Figure saved: stata_convergence_heatmap_ols.png . . * ── (B) NLS Heatmap ── . #delimit ; delimiter now ; . tw (scatter startyear endyear if beta_nls > .0035, msize(medsmall) > mfcolor("5 48 97") mlcolor("20 20 19") mlwidth(vthin)) > (scatter startyear endyear if beta_nls < .0035 & beta_nls > .0025, msi > ze(medsmall) > mfcolor("33 102 172") mlcolor("20 20 19") mlwidth(vthin)) > (scatter startyear endyear if beta_nls < .0025 & beta_nls > .0015, msi > ze(medsmall) > mfcolor("67 147 195") mlcolor("20 20 19") mlwidth(vthin)) > (scatter startyear endyear if beta_nls < .0015 & beta_nls > .0005, msi > ze(medsmall) > mfcolor("146 197 222") mlcolor("20 20 19") mlwidth(vthin)) > (scatter startyear endyear if beta_nls < .0005 & beta_nls > -.0005, ms > ize(medsmall) > mfcolor("209 229 240") mlcolor("20 20 19") mlwidth(vthin)) > (scatter startyear endyear if beta_nls > -.0015 & beta_nls < -.0005, m > size(medsmall) > mfcolor("247 247 247") mlcolor("20 20 19") mlwidth(vthin)) > (scatter startyear endyear if beta_nls > -.0025 & beta_nls < -.0015, m > size(medsmall) > mfcolor("253 219 199") mlcolor("20 20 19") mlwidth(vthin)) > (scatter startyear endyear if beta_nls > -.0035 & beta_nls < -.0025, m > size(medsmall) > mfcolor("244 165 130") mlcolor("20 20 19") mlwidth(vthin)) > (scatter startyear endyear if beta_nls > -.0045 & beta_nls < -.0035, m > size(medsmall) > mfcolor("214 96 77") mlcolor("20 20 19") mlwidth(vthin)) > (scatter startyear endyear if beta_nls > -.0055 & beta_nls < -.0045, m > size(medsmall) > mfcolor("178 24 43") mlcolor("20 20 19") mlwidth(vthin)) > (scatter startyear endyear if beta_nls < -.0055, msize(medsmall) > mfcolor("103 0 31") mlcolor("20 20 19") mlwidth(vthin)), > plotregion(style(none) lcolor(none)) > graphregion(fcolor(white) lcolor(white)) > xtitle("End Year") ytitle("Start Year") > title("Convergence Heatmap: NLS (Direct Estimation)", color("20 20 19" > ) size(medium)) > subtitle("Blue = convergence ({&beta}>0), Red = divergence ({&beta}<0) > ", /// > color("20 20 19") size(small)) > aspectratio(1) > xlabel(1960(10)2020) ylabel(1960(10)2010) > xsize(5) ysize(5) > legend(order(1 ">.0035" 2 "[.0025,.0035]" 3 "[.0015,.0025]" > 4 "[.0005,.0015]" 5 "[-.0005,.0005]" > 6 "[-.0015,-.0005]" 7 "[-.0025,-.0015]" > 8 "[-.0035,-.0025]" 9 "[-.0045,-.0035]" > 10 "[-.0055,-.0045]" 11 "<-.0055") > size(vsmall) pos(10) ring(0) bmargin(medsmall) col(1) > region(lcolor(none) fcolor(none)) > title("{&beta} coefficient", size(small))) > note("Data: Penn World Tables 10.0. 84-country balanced panel." > "NLS estimation (Barro & Sala-i-Martin 1992 specification).", siz > e(vsmall)) > ; . #delimit cr delimiter now cr . graph export "stata_convergence_heatmap_nls.png", replace width(2400) file stata_convergence_heatmap_nls.png written in PNG format . display as text "Figure saved: stata_convergence_heatmap_nls.png" Figure saved: stata_convergence_heatmap_nls.png . . restore . . . *════════════════════════════════════════════════════════════════════════════ > ═══ . * CLEAN UP AND FINAL SUMMARY . *════════════════════════════════════════════════════════════════════════════ > ═══ . . display as text "" . display as text "╔═══════════════════════════════════════════════════════════ > ═══╗" ╔══════════════════════════════════════════════════════════════╗ . display as text "║ TUTORIAL SUMMARY > ║" ║ TUTORIAL SUMMARY ║ . display as text "╠═══════════════════════════════════════════════════════════ > ═══╣" ╠══════════════════════════════════════════════════════════════╣ . display as text "║ > ║" ║ ║ . display as text "║ Sample: 84-country balanced panel (data since 1960) > ║" ║ Sample: 84-country balanced panel (data since 1960) ║ . display as text "║ > ║" ║ ║ . display as text "║ Beta Convergence (β): > ║" ║ Beta Convergence (β): ║ . display as text "║ - 1960-2000: NO convergence (era of divergence) > ║" ║ - 1960-2000: NO convergence (era of divergence) ║ . display as text "║ - 2000-2019: YES convergence (new era) > ║" ║ - 2000-2019: YES convergence (new era) ║ . display as text "║ - OLS (λ→β) and NLS give nearly identical estimates > ║" ║ - OLS (λ→β) and NLS give nearly identical estimates ║ . display as text "║ - Speed ≈ 0.4% per year (much slower than 2% benchmark) > ║" ║ - Speed ≈ 0.4% per year (much slower than 2% benchmark) ║ . display as text "║ - Half-life ≈ 170 years > ║" ║ - Half-life ≈ 170 years ║ . display as text "║ > ║" ║ ║ . display as text "║ Sigma Convergence (σ): > ║" ║ Sigma Convergence (σ): ║ . display as text "║ - 1960-2008: variance rose (σ-DIVERGENCE) > ║" ║ - 1960-2008: variance rose (σ-DIVERGENCE) ║ . display as text "║ - Post-2008: variance started declining (σ-convergence) > ║" ║ - Post-2008: variance started declining (σ-convergence) ║ . display as text "║ - β is necessary but NOT sufficient for σ > ║" ║ - β is necessary but NOT sufficient for σ ║ . display as text "║ > ║" ║ ║ . display as text "║ Key Reference: > ║" ║ Key Reference: ║ . display as text "║ Patel, Sandefur, Subramanian (2021) > ║" ║ Patel, Sandefur, Subramanian (2021) ║ . display as text "║ 'The New Era of Unconditional Convergence' > ║" ║ 'The New Era of Unconditional Convergence' ║ . display as text "║ Journal of Development Economics > ║" ║ Journal of Development Economics ║ . display as text "╚═══════════════════════════════════════════════════════════ > ═══╝" ╚══════════════════════════════════════════════════════════════╝ . . * ── Clean up temporary files ── . capture erase "convergence_working.dta" . . display as text "" . display as text "Script completed successfully." Script completed successfully. . display as text "Figures generated:" Figures generated: . display as text " stata_convergence_scatter_1960_2019.png" stata_convergence_scatter_1960_2019.png . display as text " stata_convergence_scatter_two_eras.png" stata_convergence_scatter_two_eras.png . display as text " stata_convergence_speed_ols.png" stata_convergence_speed_ols.png . display as text " stata_convergence_speed_nls.png" stata_convergence_speed_nls.png . display as text " stata_convergence_rolling_lambda.png" stata_convergence_rolling_lambda.png . display as text " stata_convergence_rolling_beta_ols.png" stata_convergence_rolling_beta_ols.png . display as text " stata_convergence_rolling_beta_nls.png" stata_convergence_rolling_beta_nls.png . display as text " stata_convergence_sigma_two_periods.png" stata_convergence_sigma_two_periods.png . display as text " stata_convergence_sigma_evolution.png" stata_convergence_sigma_evolution.png . display as text " stata_convergence_heatmap_ols.png" stata_convergence_heatmap_ols.png . display as text " stata_convergence_heatmap_nls.png" stata_convergence_heatmap_nls.png . display as text "" . . log close name: log: /Users/carlosmendez/Documents/GitHub/starter-academic-v501/content > /post/stata_convergence/analysis.log log type: text closed on: 1 May 2026, 10:39:28 -------------------------------------------------------------------------------