# Comparison parameters in panel data regression

I have two panel data. The first data is on cities on the period 1990-2000. The second data is observations on the same cities, but from 2001 to 2010. I want to estimate a model like this for each sample. $$y_{it}=b_0+b_1x_{1,it}+b_2x_{2,it}+\epsilon_{it}$$ My aim is to compare parameters of the two models or subperiods, i.e compare if $b_1$ from the first panel (1990-2000) differ to the second one (2001-2010). At the end, I will be able to observe if the parameter in the second data set is stronger than the first data.

Are there any strategy to estimate that model? or any statistical test that allow to answer my question? what command on R or stata can I used to compare these two parameter in panel data framework?

Thanks a lot

It would be great to know what you have considered already. Any introductory econometrics text book is here a good starting point, see e.g. Introductory econometrics: A Modern Approach or Econometric Analysis of Cross Section and Panel Data, both written by Wooldrige.

In terms of your modelling choices, I would start with a simple linear model with fixed effects for cities. The theoretical argument for including city fixed effects would be that there are many likely to be unobserved city-specific time-constant characteristics that affect both dependent and independent variables (for example, governmental institutions, quality of governance, etc). Random effects would in that case be inappropriate. You could also test empirically whether fixed or random effects are more appropriate using a (panel-corrected) Hausman test.

As a next step, you should think about unaccounted time-varying factors. It is unlikely that the standard assumption of independently and identically distributed errors hold. With fixed effects, the idiosyncratic error captures all unobserved factors that vary over time (e.g. political, social factors). Intuitively, as the unobserved time-varying factors tend to depend on past events in a given city and simultaneous events in other cities, error terms will probably exhibit city-wise heteroskedasticity, serial correlation for a given city, and cross-sectional dependence across cities for a given time period. There is also a number of empirical tests to check for different error structures, see e.g. Wooldrigde test for autocorrelation, Breusch-Pagan test for cross-sectional dependence, Pesaran’s test for cross-sectional dependence.

For comparing different coefficients or means from two dependent samples, the dependent (sometimes also called paired) t-test seems to be appropriate here, as the pairs can consist of a before-after comparison, as in your case. So you could test the null hypothesis that there are no differences between both means.

Note however that the t-test is a parametric test, as it assumes that the observed differences are a sample from normal distribution. The approximate validity of the confidence interval and hpypothesis test follows from the Central Limit Theorem. If the sample size is small and/or the true distribution of the differences is far from normal, the test may not be very reliable.

A non-parametric alternative would be (Wilcoxon) signed rank test. This test does not depent on the normality assumption and since differences are replaced by ranks it is insensitive to outliers. It is also almost as powerful as the t-test, even when the normality assumption holds and thus may be a good second check in your case.

• As said, it would be helpful to know what you have considered already, otherwise the question seems to be overly broad, or if my answer gives you the answer you were looking for, please accept it as the answer. Jul 21 '18 at 11:36