The document investigates whether poorer economies grow faster than wealthier ones, emphasizing the importance of estimating convergence speeds in growth theory through various econometric methods. It reviews the conditional convergence approach and outlines the estimation process using a panel dataset, addressing potential biases from model restrictions and measurement errors. Monte Carlo experiments assess the performance of different estimators, concluding that the difference-GMM estimator is the most consistent and effective in larger samples, while revealing that convergence rates lie between 0.3 percent and 1.6 percent annually.

1. Advanced Microeconometrics project:
Convergence in Growth Theory
Winter semester 2008
Laurent Cyrus & Molnar Gyorgy
1. Introduction
Do poor economies grow faster than rich economies? If so, how much faster, and can they catch
up in the long-run? Estimating the speed of convergence has been one of the main objectives of
the long developing literature concerning growth empirics. According to the conditional
convergence approach: "if countries did not vary in their investment and population growth rates,
there would be a strong tendency for poor countries to grow faster than rich ones1" In a series of
contributions2 that have shaped the research agenda in growth empirics, Robert Barro together
with Xavier Salai-Martin has argued that countries converge to their steady-state level of per-
capita income at a rate of approximately 2 or 3% per year.
The results in the seminal contribution of MRW are derived using Ordinary Least Squares, and
the estimated parameters potentially suffer from omitted variable bias, endogenous right-hand-
side variables and measurement error. A large fraction of the growth literature under the rubric of
"conditional convergence" includes a variety of variables in the regression equation that might
affect the steady state value of output and/or the initial technology level to overcome the omitted
variable bias3. However, no matter how ingeniously the variables are selected, some components
having to do with features unique to the country would remain unexplained and find its way in
the error term. The alternative approach is to treat these components as an unobservable country
specific or fixed effect. Models estimated by static panel methods will no longer be biased by
omitted variables that are time-invariant, but it still exposes the analysis to endogeneity issue.
Meanwhile permanent additive measurement errors are controlled for because it is taken into
account by time the invariant fixed effects.
The strict exogeneity assumption has to be abandoned if the regressors include lagged dependent
variables. One solution is to allow the error term to be correlated with future values of the
1
Mankiw, Romer, Weil, 1992 (MRW)
2
Barro, Sala-Martin, 1995
3
Such as measures of political stability, degree of financial intermediation, or distance from the equator

2. explanatory variables. This allows for feedback effects from past shocks. In fact we may have to
further relax this assumption if there is contemporaneous correlation between the error term and
right hand side variables. Consistent estimation requires the use of instrumental variables Hsiao-
Anderson (1982) proposed an instrumental variable estimator for the dynamic panel data model
that uses the lagged First-Difference of the lagged dependent variable as the instrument.
However Arellano (1989) has shown that this estimator suffers from large variances and he
recommended the use of the lagged level instead because otherwise the estimator does not use all
of the available moment restrictions. Based on this estimator Arellano and Bond (1991)
developed a Generalized Methods of Moments estimation procedure that used levels instead of
differences, and they advocated that all available instruments available at time t should be used
instead of a single instrument per time period. One must take the first differences of the
equations, and then in theory, consistent estimation can be done to the extent that we set up the
correct moment restrictions between the differenced error term and instruments for the
endogenous regressors. More restrictive assumptions will imply the validity of additional
moment restrictions. This will increase efficiency if they are valid, however they can also imply
inconsistency if they are not. Temporary measurement errors can also be controlled for by
further altering the instrument matrix. Fortunately, the validity of the applied instrument matrix
can be tested for by the Sargan test of over-identifying restrictions. Arellano and Bover(1995)
point out that when the time series are persistent, the first differenced GMM estimator can have
poor properties because lagged levels of the series provide only weak instruments for the first-
differences. This weakness may cause large finite sample biases in the estimated parameters.
2. Estimation
We use an international panel based on the Barro-Lee 1994 data-set, which has become a
standard data set for studying the growth of nations. The database includes major
macroeconomic variables measured in a consistent basis for countries of the world. We estimate
the speed of convergence using 97 countries. Because of the shortness of the panel there is a
possibility for finite sample biases and large standard errors. In addition to this, due to data
unavailability for certain countries, the panel dataset is unbalanced. The panel is constructed by
by taking five year averages of the variables over the 1960-1985 time period. A small number of

3. time periods must be chosen to avoid modeling business cycle dynamics. A neoclassical growth
model can be expressed in the following form:
∆ , 1 , , , for 1, … , and 2, … ,
The dependent variable is the growth rate of GDP. The right-hand-side variables include the
logarithm of GDP, the logarithm of the savings rate, and the logarithm of a measure that captures
the population growth rate plus a common factor for the sum of technical change & depreciation
rate. This specification allows for country and period specific effect. We can eliminate the need
for time dummies by expressing all variables as deviations from time means. The regression
equation from a neoclassical growth model can be rewritten as a dynamic panel data model:
, , , , for 1, … , and 2, … ,
By taking first differences one removes the unobservable time-invariant country specific effect.
We then set up a series of moment restriction between the error term and instruments of right
hand side variables, using the level series. We use lags of two periods or more for all explanatory
variables because savings and population growth are possibly contemporaneously correlated with
the error term, there is simultaneity between the determination of the above variables and the
growth rate of a country.
We estimate the speed of convergence in a neoclassical growth model using several different
estimation techniques. The results for the OLS, Within Group, First-difference, Hsiao-Anderson,
and first-differenced GMM for predetermined & endogenous variable estimators for the
neoclassical growth model are listed in Table 1. We apply the orthogonal deviations
transformation when using the first-differenced-GMM estimator to remove any serial correlation
that may be induced by differencing. To detect finite sample biases, we can compare the first-
differenced GMM results to alternative estimators. It has been shown that OLS produces an
upward biased estimate of the autoregressive parameter on the lagged dependent variable, while
the within groups estimates gives downward bias. Unfortunately the first-differenced GMM
estimator lies below the corresponding Within Groups estimate, which suggests that it is strongly
biased. This is consistent with Arellano and Bover's remark concerning finite sample bias with
weak instruments because output and the other right-hand-side variables are highly persistent.
The Hsaio-Anderson, Difference-GMM for predetermined variables, and Difference-GMM for
endogenous variables estimators are determined with a different configuration of the instrument
matrix. The estimated Beta for convergence is extremely sensitive to its specification. Because

4. the Difference-GMM estimates are quiet close to the Within Groups estimate it is also possible
that we are over fitting' the model with too many instruments. Over fitting the model can be a
source of finite sample bias. If this is the case, decreasing the number of instruments can reduce
this source of bias. The Sargan test of overidentifying restrictions does not detect the invalidity
of the instrument matrices for the two GMM estimations.
3. Experimental evidence
To assess the performance of our estimator we run a series of Monte Carlo experiments with different
sample sizes. This requires generating a true model that features the problematic issues that one
encounters when estimating a growth model in practice. An unobserved time-invariant effect is created
for each individual of the true model. ~. . . 0, σ . After assuming a starting value for y, using
the idea of recursive solution of a time series model we generate an AR model to capture the 'dynamic'
feature of the dynamic panel data model:
, , Φ , with , , and , ~ 0,1 , 0
To induce endogeneity in one of the explanatory variables, we make the error term Vi t a function of x3 so
that there is contemporaneous correlation between them. Finally with regard to x2 and x3 we considered
the following generating equation:
, , with , ~ 0, σ
By generating these variables with an autoregressive parameter we aim to create persistence in the right
hand side variables. We run this design for the sample sizes of N=100 and N=750 to capture both the
finite sample and the asymptotic performance. , independent over time: Φ 0, 1, σ 1,
0.8, 1, 0.50, 0.8, σ 0.9 and 5. Tables 1 and 2 summarize the
results obtained from 250 replications.
Table 2 and 3 report the mean bias and the standard deviations of the Monte Carlo estimates for n=100,
500. We report results for OLS, Within-Groups, and Difference-GMM(endogenous regressors)
estimators. In the case of n=100 the GMM estimator significantly outperforms the other estimators when
estimating the coefficient on the lagged dependent. However even so it is not an unbiased estimate, and
the estimator shows results indicating a downward finite-sample bias for all other variables as well. In
case of the other regressors, there are no major differences in the mean bias for the different estimators.
The Difference-GMM estimator results with the highest standard deviations of the Monte Carlo estimates.
This is clearly presented in Graph 1 which plots the kernel densities of the coefficients. The probability

5. density spreads out, it has long tails for Difference-GMM. On the first subplot we can also view the well
known finite sample bias result on lagged dependent variables: the mean of the Difference-GMM
estimates lies between those of the OLS and WG estimates. The result for n=750 are much more
promising for the Diff-GMM estimator. It is the only estimator that is consistent, the mean bias is
practically zero. The other estimators are not capable of decreasing the size of their bias despite the large
increase in sample size. The Difference-GMM estimator still reports the largest standard deviations of the
Monte Carlo estimates, but it is much more efficient than it was for the finite sample. It is also worth
mentioning that we have been working with the one-step Difference-GMM estimator because for finite
samples it converges faster to its asymptotic distribution than the two-steps estimator. Using the two-steps
estimator there could be significant efficiency gains in practice for finite samples.
Conclusion:
The Monte Carlo evidence suggests that the First-Difference estimator of the lagged dependent variable
can become imprecise and subject to finite sample bias, when the series is highly persistent. This
persistence causes even a valid instrument to be only weakly correlated with the endogenous right hand
side variables. The finite sample bias is found to be downwards, in the direction of the within-groups
estimator. It requires a large sample to overcome the unbiasedness. The results for Difference-GMM
estimation indicates that with fixed T the estimators are consistent and asymptotically normal as N goes to
infinity.
Thus in the case of our sample we can only conclude that the rate of convergence is somewhere between
the values estimated by OLS and the Within-Group Estimator. This means that it lies somewhere between
0.3 percent and 1.6 percent per period. There have been attempts to correct for the weakness of
instruments in persistent series. To attain more precise results one could either add additional nonlinear
moment restrictions or it is possible to estimate so called System Generalized Method of Moments
estimators.

6. References:
A Contribution to the Empirics of Economic Growth, N. Gregory Mankiw; David Romer; David
N. Weil, 1992
Economic Growth, Robert Barro and Xavier Sala-i-Martin, 1995
Panel Data Econometrics, Arellano Manuel, 2003
Reopening the Convergence Debate: A New Look at Cross-Country Growth Empirics, Caselli
Francesco, Esquivel Gerardo, Lefort Fernando, 1996
Another look at the Instrumental Variable Estimation of Error-Components Models, Arellano
Manuel, Bover Olympia,1995
Some Tests of Specification for Panel Data: Monte Carlo Evidence and an Application to
Employment Equations, Arellano Manuel, Bond Stephen,1991
GMM Estimation of Empirical Growth Models, Bond Stephen, Hoeffler Anke, Temple Jonathan,
2001

7. Table 1: Estimation
Dependent variable: ,
UNBALANCED PANEL DATA
Variable OLS WG FD
Observations 479 479 382
ln , -0.030 -0.318 -0.858
- 0.013 0.055 0.077
ln , 0.089 0.127 0.157
- 0.015 0.038 0.035
ln -0.111 -0.083 0.065
- 0.055 0.146 0.128
0.006 0.077 0.391
BALANCED PANEL DATA
Variable OLS WG FD HA Diff GMM Predet Diff GMM Endog
1-step 2-step 1-step 2-step
Observations 470 470 376 376 376 376 376 376
ln , -0.030 -0.316 -0.857 -0.563 -0.288 -0.264 -0.411 -0.363
- 0.013 0.056 0.078 0.198 0.067 0.063 0.097 0.096
ln , 0.092 0.129 0.162 0.023 0.138 0.206 0.129 0.150
- 0.017 0.039 0.036 0.085 0.044 0.040 0.050 0.050
ln 0.108 -0.074 0.077 0.249 0.055 -0.092 -0.424 -0.477
-0.056 0.147 0.128 0.307 0.134 0.147 0.200 0.215
Sargan test 0.000 0.000 0.000 0.000 0.000 0.000 20.132 36.084
p-value 0.000 0.000 0.000 0.000 0.000 0.000 0.386 0.113
0.006 0.076 0.389 0.166 0.068 0.061 0.106 0.090
Table 2. Monte Carlo Experiment for n=100
ln , ln , ln
0.8961 0.8911 -0.4514
0.6586 0.9102 -0.4560
0.7700 0.8932 -0.4675
OLS , ,
Mean bias 0.096137 -0.1088 0.04856
MCSD4 0.016326 0.1565 0.18758
WG , ,
Mean bias -0.14136 -0.08979 0.04401
MCSD 0.03530 0.18574 0.19549
GMM , ,
Mean bias -0.02997 -0.10677 0.03254
MCSD 0.06621 0.54248 0.61133
4
MCSD = Monte Carlo Standard Deviation

8. Graph 1. Monte Carlo Experiment for n=100
Table 3. Monte Carlo Experiment for n=750
, ,
0.8965 0.8810 -0.4461
0.6614 0.9036 -0.4505
0.7982 0.9915 -0.4994
OLS , ,
Mean bias 0.096458 -0.11898 0.05394
MCSD 0.005339 0.057718 0. 060671
WG , ,
Mean bias -0.13856 -0.096365 0.049504
MCSD 0.012036 0.072379 0.074001
GMM , ,
Mean bias -0.001844 -0.00851 0.00063
MCSD 0.020408 0.18676 0.23213

9. Graph 2. Monte Carlo Experiment for n=100