System GMM estimation of Empirical Growth Models

Transcription

1 System GMM estimation of Empirical Growth Models ELISABETH DORNETSHUMER June 29, Introduction This study based on the paper "GMM Estimation of Empirical Growth Models" by Stephan Bond, Anke Hoeffler and Jonathan Temple should should give examples how you can apply the system GMM estimation to empirical growth models like the Solow and the augmented Solow growth model and compares the estimation outputs of different estimation methods like the first-differenced GMM method, OLS estimation and the Within estimation with each other. This paper highlights the problem in using the first-differenced GMM panel data estimator to estimate cross-country growth regressions. The first-differenced GMM estimator can be poorly behaved, if the time series are persistent. The reason for this is that the lagged levels of the series provide large instruments for the subsequent first-differences. This problem can get very serious in practice, and so the authors Bond, Hoeffler and Temple suggest to use a more efficient GMM estimator, the system GMM estimator, to exploit stationarity restrictions. This approach should then give more reasonable results than the first differenced GMM estimator in empirical growth models. 2 Estimating growth models by system GMM The growth equation we wish to estimate has the following form: y it = γ t + (α 1)y i,t 1 + x itβ + η i + ν it (1) for i = 1,..., N and t = 2,..., T where y it is the log difference per capita GDP over a five year period, y i,t 1 is the logarithm of per capita GDP at the start of that period, and x it is a vector of characteristics measured during, or at the start of, the period. The unobserved county-specific effects, η i reflect differences in the initial levelof efficiency, 1

2 whilst the period-specific intercepts γ t capture productivity changes that are common to all countries. Country and time effects may also reflect country-specific and period-specific components of measurement errors. The above model can be written equivalently as: for i = 1,..., N and t = 2,..., T y it = γ t + αy i,t 1 + x itβ + η i + ν it (2) In order to allow for the levels of the x it variables (and y it ) to be correlated with the unobserved country-specific effects and to permit suitably lagged firstdifferences of x it (and y it ) to be used as instruments in the levels equations we have to make the following assumptions: We need to have constant means of both the y it and x it series through time for each country. This would be sufficient for the validity of the moment conditions E(η i y it ) = 0 and E(η i x it ) = 0. Blundell and Bond (2000) show that this assumption of the constant means in the x it y it series for the validity of of the additional moment conditions exploited by the system GMM estimator is not necessary, if you consider the equation above in first-differences: for i = 1,..., N and t = 3,..., T y it = γ t γ t 1 + α y i,t 1 + x itβ + ν it (3) Given E(η i x it ) = 0 for all t,than E(η i y it ) = 0 is required. This will hold also if the means of the x it and hence the y it variables are not constant, even after removing common time-specific components. The assumption that E(η i y it ) = 0 does not imply that the country-specific effects play no role in output termination. This effects will be one determinant of the steady-state level of output per efficiency unit of labor, conditional on initial output and other steady-state determinants like investment and population growth. The assumption implies that there is no correlation between output growth and the country-specific effect in the absence of conditioning on other variables. Such a correlation would lead to implausible long-run implications. 3 Estimating the Solow growth model Bond, Hoeffler and Temple estimated the Solow and the augmented Solow growth model and used therefore the same date as CEL (Caselli, Esquivel and Lefort, 1996). The variables are expressed as deviations from time means, which eliminates the need for time dummies. Their results of their different estimation methods for the basic Solow growth model are reported in Table 1. 2

3 This table reports in the first three columns the results of using OLS levels, Within Groups and first-differenced GMM estimators. In this table we can see that the point estimate lies below the corresponding Within Groups estimate, which itself is likely to be seriously biased downwards in a short panel like it is used here. The fourth column of the table reports the results from using a system GMM estimator. From the results we can see that the coefficient on the initial income lies above the corresponding Within Groups estimate and below the corresponding OLS levels estimate. The additional instruments seem to be valid and highly informative. All together, the results show that there is a serious finite sample bias problem caused by weak instruments in the first-differenced GMM results. This problem can be solved by using the system GMM estimator. By treating the population growth rate and the investment rate as endogenous variables, these estimates already allow for the possibility of a serially uncorrelated measurement error in either of this explanatory variables. In the last column of table 1 Bond, Hoeffler and Temple considered the possibility of a serially uncorrelated measurement error in the per capita GDP series. This 3

4 imposes that the level of this series dated at t-2 is invalid as an instrument for the first-differenced equations and the first-difference of this series dated t-1 is invalid as an instrument for the levels equations. The final column reports the result of the system GMM estimator when these instruments are excluded. If we compare the results of the fourth and fifth column we can see that they are very similar, which again shows us that there is no serious problem resulting from the transient measurement error in the per capita GDP series. The results of the system GMM estimation indicate a rate of convergence of around two per cent a year, which is similar to the standard cross section finding. They also indicate that the investment rate has a significant positive effect on the steady state level of per capita GDP, even after controlling for unobserved country-specific effects and allowing for the likely endogeneity of investment. Table 2 shows us the results for the estimations of the augmented Solow growth model, where the logarithm of the secondary-school enrollment rate is included as an additional explanatory variable. Here again the system GMM estimates in the final column are the preferable results. Our system GMM estimates show that the particular human capital measure used here can be omitted from the specification model. This suggests that we may be able to strengthen the instrument set used to estimate the basic Solow growth model 4

5 with first-differences, by including the lags of school enrollment as instruments. Table 3 shows the results of the basic first-differenced and system GMM results, using the slightly smaller sample for which school enrollment is measured. The results are very similar to those shown in Table 1. The final column shows the first-differenced GMM resulsts using an extended instrument set, which also includes the lags of school enrollment. With this extended instrument set, the results are much closer to them of the system GMM estimation. 4 Conclusion Bond, Hoeffler and Temple pointed out in their paper "GMM Estimation of Empirical Growth Models" that the first-differenced GMM estimates of the coefficient on the lagged dependent variable tend to lie below the corresponding Within Groups estimates. This suggests that the first-differenced GMM estimates are seriously biased. One explanation for this could be that the instruments are weak. Bond, Ho- 5

6 effler and Temple considered two possible solutions to this problem, which both use more informative sets of instruments. The first solution is to use the system GMM estimator developed by Arellano and Bond and Blundel and Bond (1998). This estimator uses lagged first-differences of the variables as instruments for equations in levels, in combination with the usual approach. The additional instruments are valid under a restriction on the initial conditions which is potentially consistent with the Solow growth framework. The second solution which is proposed in this paper is to strengthen the instrument set used for the equations in first-differences by using other variables that are not included in the model, e.g. through the use of lags of school enrollment as instruments in estimating the basic Solow model. In both cases, the estimates of the coefficient on the lagged dependent variable then lie above the Within group estimates. This shows us that the system GMM approach is probably preferable in this context. 6