Advanced microeconometric project


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This paper is a methodological exercices presenting the results obtained from the estimation of the growth convergence equation using different methodologies.
A dynamic balanced panel data is estimated using: OLS, WithinGroup, HsiaoAnderson, First Difference, GMM with endogenous and GMM with predetermined instruments. An unbalanced panel is also realized for OLS, WG and FD.
Results are discused in light of Monte Carlo studies.

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Advanced microeconometric project

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