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Applied Econometrics - Panel Data 2Jennifer Smith - University of Warwick
“Some strange properties of panel data estimators” Robertson and Symons (1992)
A precursor of Pesaran and Smith, RS look at the fixed-T case as well as the large-T case.
First, taking the case where T is large and N is small (N=2) (i.e. using T-asymptotics):
True model:
(55)
where is independent across i.
Estimated model:
(56)
- dynamic
- imposes slope homogeneity
Assume
where and (varies over i).
[SURE with an identity covariance matrix is equivalent to OLS on this 2-equation system.]
As , there are biases in the means of the individual coefficients:
bias() > 0
bias() < 0
If =1 (i.e. regressors are random walks)
plim() = 1
plim() = 0
The estimated dynamics are very misleading, as are the estimated effects of the exogenous variables and the estimated long-run coefficients.
Monte Carlo experiments showed that the dynamics were still misleading for quite small T (T=40).
Now, take the case of fixed T, large N:
True model:
where , and
Assume
where and .
Estimated model:
When =1, as
plim() > 0 biased upwards (true = 0)
plim() < biased downwards
As (and ), the same result as before is obtained:
plim() = 1
plim() = 0
But if is white noise (i.e. if =0), there are no biases.
Monte Carlo experiments (for T=5, N=50, 100 and 200) confirm that:
- dynamics are overstated even for relatively small N, when T is fixed (N=50, T=5).
- biases disappear as (i.e. as the regressor approaches white noise)
- biases remain important for =0.5.
Finally, RS look at the Anderson-Hsiao estimator. They show that imposing a false homogeneity restriction renders the AH-recommended instruments invalid unless is white noise or follows a random walk (i.e. if =0 or =1). But if =1, using lagged differences as instruments will not work since these are orthogonal to the instrumented variable (the difference), since the first-differenced variable is white noise. In that case, the AH estimator has infinite asymptotic variance (as discovered by Arellano (1989) (referenced earlier) for high values of ). In addition, if =1, using lagged levels will not work either, since correlation between the integrated instrument (the I(1) level) and the stationary instrumented variable (the I(0) first difference) is asymptotically zero. Monte Carlo experiments confirm that biases are severe for the AH estimator unless is white noise or a random walk (for T=5, N=50). But AH seems reasonably useful if is either 0 or 1.
Application: Real wage determination in OECD countries (Robertson and Symons, 1992)
1958-86 (29 years)- large T
13 OECD countries- small N
RS estimate a ‘rudimentary’ real wage equation:
(57)
where
= real product wage for country i at time t
= capital-labour ratio
= change in the tax and import price wedge
The motivation is that it represents a modified, dynamic, version of a (reduced form) competitive labour market model with CRS and inelastic labour supply; the modification, the addition of the wedge term, means that an increase in taxes can cause real wages to be temporarily higher (and, by implication, employment lower).
RS Table II shows unrestricted SURE estimates for all 13 countries (i.e. equations estimated individually, but cross-equation correlation in the residuals is allowed for, estimating iteratively until convergence is achieved). Estimates certainly vary across countries, but patterns can be discerned in the results.
What happens when the data are pooled? Imposing homogeneity on the capital-labour ratio, a regressor with a high degree of positive serial correlation, should in theory bias the coefficient on the lagged dependent variable upwards, and should bias the coefficient on the capital-labour ratio itself downwards towards zero. In contrast, the variable is almost white noise, so imposing cross-country homogeneity on its coefficient should not induce biases.
The final row of Table II shows the results of imposing homogeneity on all slope parameters. (The cross-equation restrictions are rejected easily.) The coefficient on the lagged dependent variable is indeed biased upwards towards unity, that on the capital-labour ratio is biased downwards, and that on the wedge term is hardly changed. The pooled regression does not give a good picture of the central tendency of the individual country regression estimates.