Shall we take Solow seriously??
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Shall we take Solow seriously?? Empirics of growth Ania Nicińska Agnieszka Postępska Paweł Zaboklicki Original & augmented models: Solow Mankiw, Romer &Weil Assuming neoclassical production function steady state level of capital per capita is determined by saving and population growth rates Yes – predicted directions of influence are consistent with the data… but…we need to add accumulation of human capital to get the right magnitudes Why? For any given rate of human capital accumulation higher saving or lower population growth leads to a higher level of income and thus a higher level of human capital. Omitting human capital accumulation causes bias if correlated with saving and population growth rates. We will prove that… Including a proxy for human capital as an additional explanatory variable in the regression equation leads to magnitudes predicted by Solow. The augmented model accounts for about 80% of the cross country variation in income!!!! So…after all Solow was right – he just forgot about some details… happens!!!! What about convergence?? We will prove that once accounting for the saving and population growth rate we observe convergence at roughly the rate that Solow predicted. Data We used data from Barro and Lee data set. It contains different country set to the one used by authors so our results vary in magnitudes. However, the spirit remains unchanged! Data cont. We have data from years 1960-1985 for 121 countries divided into three groups: 1 – non-oil countries 2 – intermediate countries 3 – OECD countries The dataset includes real income, investments, population growth and proxy for human capital accumulation. Basic model First, we will look at the results obtained from the basic model equation estimation, which takes the following form: Y ln( ) 1 2 ln( s) 3 ln( n g d ) L We want to investigate whether real income is higher in countries with higher saving rate and lower in countries with higher values of n+g+d. We assume that g+d =0.05 is constant across countries, where g reflects the advancement of knowledge, which is not country specific. Results (OLS): gen lny=ln(gdp) gen lns=ln(inv) gen lnngd=ln(gpop +0.05) reg lny lns lnngd if group1==1 Source | SS df MS -------------+-----------------------------Model | 324.412641 2 162.20632 Residual | 273.650411 563 .486057568 -------------+-----------------------------Total | 598.063052 565 1.05851868 Number of obs F( 2, 563) Prob > F R-squared Adj R-squared Root MSE = = = = = = 566 333.72 0.0000 0.5424 0.5408 .69718 -----------------------------------------------------------------------------lny | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------lns | .7575753 .0404936 18.71 0.000 .6780384 .8371122 lnngd | -2.186215 .1768619 -12.36 0.000 -2.533605 -1.838825 _cons | 3.342269 .4937159 6.77 0.000 2.372519 4.312019 ------------------------------------------------------------------------------ Results (panel): tis czas iis kraj xtreg lny lns lnngd if group1==1 Random-effects GLS regression Group variable (i): kraj Number of obs Number of groups = = 566 97 R-sq: Obs per group: min = avg = max = 2 5.8 6 within = 0.0849 between = 0.6024 overall = 0.5421 Random effects u_i ~ Gaussian corr(u_i, X) = 0 (assumed) Wald chi2(2) Prob > chi2 = = 111.65 0.0000 -----------------------------------------------------------------------------lny | Coef. Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------lns | .3205264 .0401647 7.98 0.000 .2418049 .3992478 lnngd | -.9881011 .1455752 -6.79 0.000 -1.273423 -.7027789 _cons | 5.629819 .3984202 14.13 0.000 4.84893 6.410708 -------------+---------------------------------------------------------------sigma_u | .62736692 sigma_e | .2624178 rho | .85109147 (fraction of variance due to u_i) Human capital 1 Yij Kij H ij ( Aij Lij ) 0 1 ln yij 1 2 ln sij 3 ln(nij d g ) 4 ln SCHOOLij ij SCHOOLij dg the fraction of the eligible population (aged 12-17) enrolled in secondary school multiplied by the fraction of population at working-age that is of school age (15-17) for country j and time period i. assumed to be constant over time and equal to 0.05 for all countries 1 const 2 1 3 1 4 1 2 3 4 0 Introducing human capital xtreg lny lns lnngd lnSCHOOL if group1==1 4.17918 5.703551 -------------+---------------------------------------------------------------sigma_u | .48950053 sigma_e | .23470675 rho | .81307219 (fraction of variance due Random-effects GLS regression Number of obs = 497 Group variable (i): kraj Number of groups = 86 R-sq: within = 0.3090 between = 0.7432 overall = 0.7029 Random effects u_i ~ Gaussian corr(u_i, X) = 0 (assumed) Obs per group: min = avg = max = Wald chi2(3) Prob > chi2 = = 1 5.8 6 363.77 0.0000 -----------------------------------------------------------------------------lny | Coef. Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------lns | .3002022 .0389558 7.71 0.000 .2238503 .3765542 lnngd | -1.038666 .1443974 -7.19 0.000 -1.32168 -.7556523 lnSCHOOL | .2845598 .021147 13.46 0.000 .2431125 .3260071 _cons | 4.941366 .3888772 12.71 0.000 to u_i) ------------------------------------------------------------------------------ Hausman test Hausman specification test ---- Coefficients ---| Fixed Random lny | Effects Effects Difference -------------+----------------------------------------lns | .2198654 .3002022 -.0803368 lnngd | -.8024757 -1.038666 .2361903 lnSCHOOL | .2400482 .2845598 -.0445116 Test: Ho: difference in coefficients not systematic chi2( 3) = (b-B)'[S^(-1)](b-B), S = (S_fe - S_re) = Prob>chi2 = 165.07 0.0000 We reject Ho hypothesis and run fixed effect regression Fixed effect regression xtreg lny lns lnngd lnSCHOOL if group1==1, fe Fixed-effects (within) regression Group variable (i): kraj Number of obs Number of groups = = 497 86 R-sq: Obs per group: min = avg = max = 1 5.8 6 within = 0.3099 between = 0.7441 overall = 0.7026 corr(u_i, Xb) = 0.5936 F(3,408) Prob > F = = 61.07 0.0000 -----------------------------------------------------------------------------lny | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------lns | .2198654 .0420508 5.23 0.000 .1372021 .3025288 lnngd | -.8024757 .1530299 -5.24 0.000 -1.103301 -.5016503 lnSCHOOL | .2400482 .0222506 10.79 0.000 .1963082 .2837883 _cons | 5.511256 .4062469 13.57 0.000 4.712658 6.309854 -------------+---------------------------------------------------------------sigma_u | .6270438 sigma_e | .23470675 rho | .87711176 (fraction of variance due to u_i) -----------------------------------------------------------------------------F test that all u_i=0: F(85, 408) = 27.03 Prob > F = 0.0000 ln yij 1 2 ln sij 3 ln(nij d g ) 4 ln SCHOOLij ij 2 3 4 0 ln yij 1 3 4 ln sij 3 ln( nij d g ) 4 ln SCHOOLij ij ln yij 1 3 ln sij 4 ln sij 3 ln(nij d g ) 4 ln SCHOOLij ij ln yij 1 3 ln(nij d g ) ln sij 4 ln SCHOOLij ln sij ij ln yij 1 3 rhs1ij 4 rhs 2ij ij gen rhs1= lnngd-lns gen rhs2= lnSCHOOL-lns 1 4 1 3 Restricted fixed effect regression xtreg lny rhs1 rhs2 if group1==1, fe Fixed-effects (within) regression Group variable (i): kraj Number of obs Number of groups = = 497 86 R-sq: Obs per group: min = avg = max = 1 5.8 6 within = 0.3023 between = 0.7273 overall = 0.6850 corr(u_i, Xb) = 0.5756 F(2,409) Prob > F = = 88.61 0.0000 -----------------------------------------------------------------------------lny | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------rhs1 | -.4935334 .0456002 -10.82 0.000 -.5831734 -.4038934 rhs2 | .2524585 .021553 11.71 0.000 .2100899 .294827 _cons | 6.339631 .1076709 58.88 0.000 6.127974 6.551289 -------------+---------------------------------------------------------------sigma_u | .63733151 sigma_e | .23570005 rho | .87968598 (fraction of variance due to u_i) -----------------------------------------------------------------------------F test that all u_i=0: F(85, 409) = 28.91 Prob > F = 0.0000 Convergence . reg lny lnY60 if group1==1 Source | SS df MS -------------+-----------------------------Model | 294.766441 1 294.766441 Residual | 307.152329 544 .564618253 -------------+-----------------------------Total | 601.918771 545 1.10443811 Number of obs F( 1, 544) Prob > F R-squared Adj R-squared Root MSE = = = = = = 546 522.06 0.0000 0.4897 0.4888 .75141 -----------------------------------------------------------------------------lny | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------lnY60 | .0003215 .0000141 22.85 0.000 .0002938 .0003491 _cons | 6.902612 .0449259 153.64 0.000 6.814363 6.990862 ------------------------------------------------------------------------------ Controlling for saving and population growth . xtreg lny lnY60 lns lnngd if group1==1 Random-effects GLS regression Group variable (i): kraj Number of obs Number of groups = = 540 91 R-sq: Obs per group: min = avg = max = 5 5.9 6 within = 0.0562 between = 0.6457 overall = 0.6097 Random effects u_i ~ Gaussian corr(u_i, X) = 0 (assumed) Wald chi2(3) Prob > chi2 = = 206.77 0.0000 -----------------------------------------------------------------------------lny | Coef. Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------lnY60 | .0002534 .0000278 9.10 0.000 .0001988 .0003079 lns | .2594275 .0403126 6.44 0.000 .1804161 .3384388 lnngd | -.7506613 .1733739 -4.33 0.000 -1.090468 -.4108547 _cons | 5.574069 .4559533 12.23 0.000 4.680417 6.467721 -------------+---------------------------------------------------------------sigma_u | .54971182 sigma_e | .26339936 rho | .81327701 (fraction of variance due to u_i) ------------------------------------------------------------------------------ Hausman test . xthausman Hausman specification test ---- Coefficients ---| Fixed Random lny | Effects Effects Difference -------------+----------------------------------------lns | .1734279 .2594275 -.0859996 lnngd | -.6418418 -.7506613 .1088195 Test: Ho: difference in coefficients not systematic chi2( 2) = (b-B)'[S^(-1)](b-B), S = (S_fe - S_re) = 31.07 Prob>chi2 = 0.0000 Controlling for saving and population growth . xtreg lny lnY60 lns lnngd if group1==1, fe Fixed-effects (within) regression Group variable (i): kraj Number of obs Number of groups = = 540 91 R-sq: Obs per group: min = avg = max = 5 5.9 6 within = 0.0571 between = 0.6378 overall = 0.5657 corr(u_i, Xb) = 0.6823 F(2,447) Prob > F = = 13.54 0.0000 -----------------------------------------------------------------------------lny | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------lnY60 | (dropped) lns | .1734279 .0432074 4.01 0.000 .088513 .2583427 lnngd | -.6418418 .1774066 -3.62 0.000 -.9904963 -.2931873 _cons | 6.26194 .471011 13.29 0.000 5.336269 7.18761 -------------+---------------------------------------------------------------sigma_u | .88996904 sigma_e | .26339936 rho | .91945985 (fraction of variance due to u_i) -----------------------------------------------------------------------------F test that all u_i=0: F(90, 447) = 27.41 Prob > F = 0.0000 Controlling for saving and population growth . reg lny lnY60 lns lnngd if group1==1 Source | SS df MS -------------+-----------------------------Model | 387.410632 3 129.136877 Residual | 202.178054 536 .377197862 -------------+-----------------------------Total | 589.588686 539 1.09385656 Number of obs F( 3, 536) Prob > F R-squared Adj R-squared Root MSE = = = = = = 540 342.36 0.0000 0.6571 0.6552 .61416 -----------------------------------------------------------------------------lny | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------lnY60 | .0001856 .0000155 11.95 0.000 .0001551 .0002161 lns | .6007277 .0406437 14.78 0.000 .5208872 .6805683 lnngd | -1.204025 .2259398 -5.33 0.000 -1.647861 -.7601891 _cons | 5.195703 .5954142 8.73 0.000 4.026072 6.365335 ------------------------------------------------------------------------------ Controlling for human capital . reg lny lnY60 lns lnngd lnSCHOOL if group1==1 Source | SS df MS -------------+-----------------------------Model | 376.353388 4 94.088347 Residual | 117.526225 466 .252202199 -------------+-----------------------------Total | 493.879613 470 1.05080769 Number of obs F( 4, 466) Prob > F R-squared Adj R-squared Root MSE = = = = = = 471 373.07 0.0000 0.7620 0.7600 .5022 -----------------------------------------------------------------------------lny | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------lnY60 | .0001264 .0000134 9.46 0.000 .0001002 .0001527 lns | .3148332 .0456057 6.90 0.000 .2252149 .4044514 lnngd | -1.093758 .2040205 -5.36 0.000 -1.494672 -.6928435 lnSCHOOL | .3618771 .0245546 14.74 0.000 .3136256 .4101285 _cons | 4.347212 .5360951 8.11 0.000 3.293749 5.400675 ------------------------------------------------------------------------------ Restricted regression . reg lny lnY60 rhs1 rhs2 Source | SS df MS -------------+-----------------------------Model | 375.396246 3 125.132082 Residual | 118.483367 467 .253711707 -------------+-----------------------------Total | 493.879613 470 1.05080769 if group1==1 Number of obs F( 3, 467) Prob > F R-squared Adj R-squared Root MSE = = = = = = 471 493.21 0.0000 0.7601 0.7586 .5037 -----------------------------------------------------------------------------lny | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------lnY60 | .0001386 .0000118 11.72 0.000 .0001154 .0001619 rhs1 | -.7030645 .0375851 -18.71 0.000 -.7769213 -.6292077 rhs2 | .36919 .0243385 15.17 0.000 .3213636 .4170165 _cons | 5.374257 .0975428 55.10 0.000 5.18258 5.565934 ------------------------------------------------------------------------------ Conclusions: We proved that augmented Solow model describes growth well, both in directions of influence and magnitudes. We also showed that convergence do happen in reality once we compare similar countries in terms of population growth and saving rates. Cook book procedure for a research project: Begin as soon as possible: data sets are ‘like a box of chocolates – you never know what you’re gonna get’ Be patient: ‘read me’ files can be tricky Attend STATA classes and learn programming or become a ‘copy/paste’ master Make friends – there is a huge difference between knowing how to do something & doing it. So…& so it is – just like Solow said it should be… GOOD LUCK!!!!!!
