Country-specific convergence speeds

We take the Solow model with a Cobb-Douglas production function, in the notation of the note on simulating the Solow model: \(s\) is the saving rate, \(n\) the growth rate of population, \(x\) that of labour efficiency \(A\), \(\delta\) the depreciation rate and \(\alpha\) the elasticity of output with respect to capital. In the neighbourhood of the balanced growth path, the gap between output per efficiency unit of labour, \(\hat y = y/A\), and its steady-state value shrinks at the rate \(\beta = (1-\alpha)(n+x+\delta)\), the convergence speed: \(\log\hat y(t) - \log\hat y^{\star} \approx e^{-\beta t}\bigl(\log\hat y(0) - \log\hat y^{\star}\bigr)\). Going back to income per head \(y\), with \(A(t) = A(0)e^{xt}\), the average annual growth rate between dates \(0\) and \(t\) is

where \(\log\hat y^{\star} = \frac{\alpha}{1-\alpha}\bigl(\log s - \log(n+x+\delta)\bigr)\). Initial labour efficiency is not observed: for country \(i\), we write \(\log A_i(0) = a + \varepsilon_i\), where \(\varepsilon_i\) is a disturbance with zero mean.

If the convergence speed is the same in every country, the factor \((1-e^{-\beta t})/t\) is a constant and the growth equation is linear:

with \(a_1 = -(1-e^{-\beta t})/t\), \(a_2 = -a_1\alpha/(1-\alpha)\), \(a_3 = -a_2\) and \(u_i = -a_1\varepsilon_i\). This is the conditional convergence regression of Mankiw, Romer and Weil (1992), estimated by ordinary least squares (OLS). The convergence speed follows, \(\hat\beta = -\log(1 + t\hat a_1)/t\). To compute \(\log(n_i + x + \delta)\), the three authors impose \(x + \delta = 0.05\) in every country, and explain why (p. 410):

We assume that g and δ are constant across countries. g reflects primarily the advancement of knowledge, which is not country-specific. And there is neither any strong reason to expect depreciation rates to vary greatly across countries, nor are there any data that would allow us to estimate country-specific depreciation rates.

Yet the convergence speed depends on \(n_i\), which differs across countries, and the Penn World Tables (PWT) now provide a country-specific depreciation rate. Let us start by reproducing the estimation. The following code reads the data sheet of the Excel workbook of version 11.0 of the PWT, which covers 185 countries between 1950 and 2023.

pwt = pd.read_excel('pwt110.xlsx', 'Data')

As in the course, we keep the period 1960-2023 and the countries for which GDP, rgdpo, population, pop, and the investment share of GDP, csh_i, are available every year. For each country, income per head is the ratio of GDP to population, the saving rate is the average of csh_i over the period, the growth rate of population is its average annual rate between 1960 and 2023, and the depreciation rate is the average of delta, the average depreciation rate of the capital stock. The course measures the growth of income per head by its average annual rate, \((y(t)/y(0))^{1/t} - 1\), which we keep in order to reproduce its results, and we also compute the log difference that appears in the growth equation.

FIRST, LAST = 1960, 2023
t = LAST - FIRST
data = (pwt[(pwt['year'] >= FIRST) & (pwt['year'] <= LAST)]
        .set_index(['countrycode', 'year'])[['rgdpo', 'pop', 'csh_i', 'delta']]
        .sort_index())
available = data.notna().groupby('countrycode').sum() == t + 1

def summarise(country):
    y = (country['rgdpo']/country['pop']).to_numpy()
    population = country['pop'].to_numpy()
    return pd.Series({'y0': y[0],
                      's': country['csh_i'].mean(),
                      'n': (population[-1]/population[0])**(1/t) - 1,
                      'delta': country['delta'].mean(),
                      'g_geo': (y[-1]/y[0])**(1/t) - 1,
                      'g': np.log(y[-1]/y[0])/t})

kept = available[['rgdpo', 'pop', 'csh_i']].all(axis=1)
countries = data.loc[kept[kept].index].groupby('countrycode').apply(summarise)
countries = countries[countries['s'] > 0]
countries['with_delta'] = available.loc[countries.index, 'delta']
print(len(countries), list(countries.index[~countries['with_delta']]))

The sample contains 111 countries. The following function computes the OLS estimator and its variance matrix, with the normalisation used in the course (the sum of squared residuals, SSR, is divided by the number of observations, without the degrees-of-freedom correction).

def ols(X, y):
    b = np.linalg.solve(X.T @ X, X.T @ y)
    e = y - X @ b
    V = np.linalg.inv(X.T @ X)*(e @ e)/len(y)
    return b, V, e @ e

def speed(a1):
    return -np.log(1 + t*a1)/t

X = np.column_stack([np.ones(len(countries)), np.log(countries['y0']),
                     np.log(countries['s']), np.log(countries['n'] + 0.05)])
b, V, _ = ols(X, countries['g_geo'].to_numpy())
R = np.array([0, 0, 1, 1])
print(b.round(4), (b/np.sqrt(np.diag(V))).round(2))
print(speed(b[1]), R @ b/np.sqrt(R @ V @ R))

We recover the results of the course: \(\hat a_1 = -0.0097\), with a \(t\) statistic of \(-5.86\), that is a convergence speed of 1.5% per year, and the restriction \(a_2 + a_3 = 0\) is rejected (the \(t\) statistic of \(\hat a_2 + \hat a_3\) is \(-3.91\)).

The depreciation rate is not available every year for Equatorial Guinea: from now on the sample contains 110 countries, and growth is measured by the log difference, which is the exact quantity of the growth equation. With these two changes, the speed estimated by OLS goes from 1.51% to 1.43% per year. We set \(x = 0.02\), the value used by Mankiw, Romer and Weil, and keep for each country its own population growth and depreciation rates.

sample = countries[countries['with_delta']]
g, s, n, delta = (sample[v].to_numpy() for v in ('g', 's', 'n', 'delta'))
ly0 = np.log(sample['y0'].to_numpy())
N, x = len(g), 0.02
c = n + x + delta
codes = sample.index
print(N, np.median(delta), delta.min(), codes[delta.argmin()], delta.max(), codes[delta.argmax()])
print(n.min(), n.max(), np.std(n), np.std(delta), np.corrcoef(n, delta)[0, 1])
print(np.corrcoef(g, n)[0, 1], np.corrcoef(g, delta)[0, 1], np.corrcoef(g, np.log(c))[0, 1])

The median depreciation rate is 4.1%, and ranges from 1.6% in Haiti to 8.9% in Fiji. It varies more than the growth rate of population (a standard deviation of 1.2 points against 0.9), and the two are barely correlated. In these data, differences in convergence speeds across countries therefore come mostly from depreciation, which Mankiw, Romer and Weil assumed to be common.

If we take the model seriously, the convergence speed of country \(i\) is \(\beta_i = (1-\alpha)(n_i+x+\delta_i)\), and the growth equation becomes

with \(u_i = \varepsilon_i(1-e^{-\beta_i t})/t\). The factor multiplying the bracket differs across countries and depends on \(\alpha\): the equation is no longer linear in the parameters, and OLS no longer applies. There are still only two parameters to estimate, \(\alpha\) and \(a\), since the speed of each country follows from \(\alpha\) and from its own \(n_i\) and \(\delta_i\). We do not need to estimate 110 convergence speeds.

logliks = {}
for name, z in (('M0', np.log(n + 0.05)), ("M0'", np.log(c))):
    X = np.column_stack([np.ones(N), ly0, np.log(s), z])
    b, V, ssr = ols(X, g)
    logliks[name] = -N/2*np.log(ssr/N)
    print(name, b.round(4), (b/np.sqrt(np.diag(V))).round(2), speed(b[1]),
          b[2]/(b[2] - b[1]), R @ b/np.sqrt(R @ V @ R), ssr, logliks[name])

STARTS = {'M1': [[th, 8, 2, -2] for th in (0.05, 0.1, 0.3)],
          'M2': [[th, 7, al] for th in (0.05, 0.1, 0.3) for al in (0.4, 0.7)],
          'M3': [[7, al] for al in (0.3, 0.5, 0.7)]}
estimates = {}
for name in ('M1', 'M2', 'M3'):
    start, trials = nls(name, g, ly0, STARTS[name])
    fit = mle(name, g, ly0, start.x)
    estimates[name], logliks[name] = fit.x, -fit.fun
    classical, robust = standard_errors(fit.x, name, g, ly0)
    print(name, start.x.round(4), fit.x.round(4), classical.round(4),
          robust.round(4), logliks[name], sorted({round(2*e.cost, 6) for e in trials}))

NLS, which provides the starting point, reaches the same minimum whatever the starting value, and the maximisation of the likelihood then starts from its solution. The table below summarises the estimates, with the standard errors from the Hessian in brackets; the robust standard errors differ little (0.042 instead of 0.049 for the \(\alpha\) of M2), which suggests that the form of the variance given by the model is not contradicted by the data. For M0 and M0', the speed is common to all countries and \(\alpha\) follows from the coefficients, \(\alpha = a_2/(a_2 - a_1)\) (see exercise 1); for M1, M2 and M3, the reported speed is the median of the \(\beta_i\).

def lr_test(restricted, free):
    LR = 2*(logliks[free] - logliks[restricted])
    return LR, 1 - stats.chi2.cdf(LR, 1)

print(lr_test('M2', 'M1'), lr_test('M3', 'M2'))
theta = {'M1': estimates['M1'][0], 'M2': estimates['M2'][0],
         'M3': 1 - estimates['M3'][1]}
for name, th in theta.items():
    beta = th*c
    print(name, np.median(beta), beta.min(), codes[beta.argmin()],
          beta.max(), codes[beta.argmax()], np.log(2)/np.median(beta))

Three results stand out.

Replacing the common 0.05 by country-specific rates worsens the fit. The log-likelihood goes from 480.2 (M0) to 466.5 (M0'), and the coefficient on \(\log(n_i + x + \delta_i)\) is no longer significant (its \(t\) statistic is \(-0.76\)). The restriction \(a_2 + a_3 = 0\) is no longer rejected (\(1.06\)), but mostly because this regressor no longer explains much. In M0, \(\log(n_i + 0.05)\) varies only with population growth, which is negatively correlated with the growth of income per head (the correlation coefficient is \(-0.38\)). In M0', its variation comes mostly from depreciation, which is on the contrary positively correlated with growth (\(0.22\)), so that \(\log(n_i + x + \delta_i)\) is barely correlated with growth at all (\(-0.06\)). The model nevertheless predicts that faster depreciation lowers long-run income, and hence growth. The PWT depreciation rate is an average of asset-specific rates, weighted by the composition of capital: it reflects the structure of investment as much as the wear and tear of capital.

Country-specific speeds barely fit better than a common speed. With the same regressors, M1 and M0' have the same number of parameters and close log-likelihoods (467.75 against 466.46). The data hardly distinguish between the two structures. The interest of country-specific speeds lies in their consistency with the model, and in the test they allow: in M2 the parameter \(\theta\) that governs the speed is estimated independently of \(\alpha\), and the Solow model imposes \(\theta = 1 - \alpha\). The restriction of M2 on the coefficients in the bracket is not rejected (likelihood ratio of 0.71, p-value of 0.40).

The full Solow model is rejected. The restriction \(\theta = 1-\alpha\) costs 7.1 points of log-likelihood, that is a likelihood ratio of \(14.2\) and a p-value of 0.00016. Without the restriction, the median speed is 0.84% per year, a half-life of 82 years, and \(\alpha\) is estimated at 0.71. With the restriction, the median speed is 2.9% per year (a half-life of 24 years), and ranges from 1.7% in Greece to 4.3% in Fiji. The figure below shows the log-likelihood of M2 as a function of \(\alpha\) and \(\theta\) (the constant \(a\) is optimised for each pair). The maximum is far from the line \(\theta = 1-\alpha\): the value of \(\alpha\) that accounts for income differences across countries implies far too fast a convergence.

This is, in another guise, the finding that leads Mankiw, Romer and Weil to the model augmented with human capital: the convergence speed becomes \(\beta_i = (1-\alpha-\lambda)(n_i+x+\delta_i)\), where \(\lambda\) is the elasticity of output with respect to human capital, and a larger share for capital broadly defined slows convergence down.

All these estimates assume that the disturbance \(\varepsilon_i\) is correlated neither with the saving and population growth rates, nor with initial income. The course states the second assumption explicitly, \(\mathrm{Cov}(\varepsilon_i, \log y_j(0)) = 0\). It is nevertheless hard to defend: since \(\log y_i(0) = a + \varepsilon_i + \log\hat y_i(0)\), a country whose labour efficiency is high has, other things being equal, a high initial income. In the growth equation, initial income is then correlated with the error term, which contains \(\varepsilon_i\), and the estimators no longer converge to the true values of the parameters.

To measure the effect of this correlation, let us simulate data from model M3, keeping the estimated values of \(\alpha\) and \(a\), the observed saving, population growth and depreciation rates, and a standard deviation of \(\varepsilon_i\) taken from the residuals. Incomes in 2023 are computed with the exact solution of the Solow equation, not with its log-linear approximation: writing \(z = \hat k^{1-\alpha}\), \(z(t) = z^{\star} + (z(0) - z^{\star})e^{-\beta_i t}\), where \(z^{\star} = s_i/(n_i+x+\delta_i)\) (see the note on simulating the Solow model). We consider two cases:

• in the first, initial income is the observed one, and it is initial output per efficiency unit of labour, \(\log\hat y_i(0)\), that adjusts to \(\varepsilon_i\): the assumption of the course holds;
• in the second, \(\log\hat y_i(0)\) is fixed, and initial income contains \(\varepsilon_i\).

In each case we simulate 500 samples, estimate M2 and M3 by maximum likelihood, and test \(\theta = 1-\alpha\) at the 5% level.

a3, alpha3 = estimates['M3']
beta3 = (1 - alpha3)*c
epsilon = (g - growth(estimates['M3'], 'M3', ly0))/weights(estimates['M3'], 'M3')
sigma = epsilon.std()
rng = np.random.default_rng(2014)

def simulation(endogenous):
    """Simulated growth rates, from the exact solution, and initial incomes."""
    e = rng.normal(0, sigma, N)
    if endogenous:
        lyh0 = ly0 - a3
        ly0_sim = lyh0 + a3 + e
    else:
        ly0_sim = ly0
        lyh0 = ly0 - a3 - e
    z0 = np.exp(lyh0*(1 - alpha3)/alpha3)
    z = s/c + (z0 - s/c)*np.exp(-beta3*t)
    lyt = a3 + e + x*t + alpha3/(1 - alpha3)*np.log(z)
    return (lyt - ly0_sim)/t, ly0_sim

simulations = {}
for endogenous in (False, True):
    theta2, alpha3_sim, rejections = [], [], 0
    for r in range(500):
        g_sim, ly0_sim = simulation(endogenous)
        m2 = mle('M2', g_sim, ly0_sim, [0.1, 7, 0.6])
        m3 = mle('M3', g_sim, ly0_sim, [7, 0.6])
        LR = 2*(m3.fun - m2.fun)
        rejections += LR > stats.chi2.ppf(0.95, 1)
        theta2.append(m2.x[0])
        alpha3_sim.append(m3.x[1])
    simulations[endogenous] = np.array(theta2)
    print(endogenous, np.mean(alpha3_sim), np.median(theta2), rejections/500)
print(sigma, 1 - alpha3, simulations[False].min(), np.percentile(simulations[False], 5),
      simulations[True].max(), np.percentile(simulations[True], 95))

In the first case, the estimator of \(\alpha\) in M3 is almost unbiased (0.655 on average for a true value of 0.653): the log-linear approximation is not the problem. The test rejects the restriction \(\theta = 1-\alpha\) in 8% of the samples, somewhat more than the 5% announced: with 110 countries, the approximation of the distribution of the likelihood ratio by a \(\chi^2\) remains crude. In the second case, the estimate of \(\alpha\) rises to 0.74, the median of \(\hat\theta\) falls to 0.06 whereas the true value is 0.35, and the test rejects the restriction in every sample. The figure below compares the two distributions of \(\hat\theta\) with the estimate obtained on the data.

The estimate obtained on the data, 0.102, is below every estimate of the first case, the smallest of which is 0.17, and above every estimate of the second, the largest of which is 0.09. It could come from a Solow model in which initial income contains only part of \(\varepsilon_i\): the rejection of the full model would then come from that correlation, and not from a failure of the model. It could also come from a model in which the convergence speed is not the one of the Solow model: in a cross-section, we cannot tell. Nor can we instrument initial income by older incomes, which contain the same permanent component \(\varepsilon_i\). The second source of endogeneity, the correlation of \(\varepsilon_i\) with the saving and population growth rates, which Mankiw, Romer and Weil rule out by assumption, raises the same problem.

The usual answer is to follow countries over several periods. Splitting 1960-2023 into periods of length \(\tau\), the growth equation reads

with \(\rho_i = e^{-\beta_i\tau}\), a country effect \(\mu_i = (1-\rho_i)\log A_i(0)\), which absorbs \(\varepsilon_i\), and a trend term \(\gamma_{i,k}\) known up to \(\alpha\). This fixed effect can be removed by differencing the equation, and incomes of earlier periods then become valid instruments. The literature shows, however, that the answer depends a great deal on the method. With fixed effects, Islam (1995) finds speeds of 4 to 5% per year on the large samples, and above 9% for the OECD countries. With the generalised method of moments (GMM) applied to the equation in differences, Caselli, Esquivel and Lefort (1996) find about 10%. Bond, Hoeffler and Temple (2001) show that the instruments are then weak, and find 2 to 4% with the system GMM estimator. The simulations of Hauk and Wacziarg (2009) indicate that fixed effects and difference GMM greatly overstate the convergence speed, and that the cross-section estimator, the one used in this note, is the least biased in the presence of measurement error. Finally, when each country has its own speed, unrelated to the model, Lee, Pesaran and Smith (1997) find average speeds of the order of 30% per year, very imprecisely estimated. The structure \(\beta_i = (1-\alpha)(n_i+x+\delta_i)\) lies between these two extremes: it allows speeds to differ, but ties them to a single parameter. Estimating it on a panel is the subject of the next note.

Taking the convergence speed of the Solow model seriously only requires a change of estimator: a common capital share is enough to give each country its own speed, and the model also says how countries should be weighted, which maximum likelihood exploits. On the PWT data, these country-specific speeds barely fit better than a common speed, and the full Solow model is rejected, because the capital share that accounts for income differences implies too fast a convergence. This verdict is fragile, however: the same estimation, applied to data simulated from an exact Solow model, leads to the same rejection as soon as initial income contains labour efficiency. Only a panel estimation can separate the two explanations.

The conditional convergence regression is due to Mankiw, N. G., Romer, D. and Weil, D. (1992), "A Contribution to the Empirics of Economic Growth", Quarterly Journal of Economics, 107(2), 407-437. The panel estimates cited above are those of Islam, N. (1995), "Growth Empirics: A Panel Data Approach", Quarterly Journal of Economics, 110(4), 1127-1170, of Caselli, F., Esquivel, G. and Lefort, F. (1996), "Reopening the Convergence Debate: A New Look at Cross-Country Growth Empirics", Journal of Economic Growth, 1(3), 363-389, of Bond, S., Hoeffler, A. and Temple, J. (2001), "GMM Estimation of Empirical Growth Models", CEPR Discussion Paper 3048, and of Lee, K., Pesaran, M. H. and Smith, R. (1997), "Growth and Convergence in a Multi-Country Empirical Stochastic Solow Model", Journal of Applied Econometrics, 12(4), 357-392. The simulations are those of Hauk, W. and Wacziarg, R. (2009), "A Monte Carlo Study of Growth Regressions", Journal of Economic Growth, 14(2), 103-147. For an overview, see Durlauf, S., Johnson, P. and Temple, J. (2005), "Growth Econometrics", in Aghion, P. and Durlauf, S. (eds), Handbook of Economic Growth, vol. 1A, chapter 8, 555-677.

The data come from version 11.0 of the Penn World Tables (Groningen Growth and Development Centre, 2025, doi:10.34894/FABVLR), described in Feenstra, R., Inklaar, R. and Timmer, M. (2015), "The Next Generation of the Penn World Table", American Economic Review, 105(10), 3150-3182. On the Solow model itself, see the notes on the Solow model and on its simulation.