# Suggestions for running panel data with small sample size

Okay,say I have a dataset which contains data on unemployment rates, wages, oil prices faced by a country, incidence of civil conflict for about 30 countries opver a period of 10 years. So this is like a panel data with small number of countries over a small number of time periods. I am say interested in running a regression of voter turnout in national elections in each year on all the variables I outlined above. A very simple regression i can run is

$Y_{it} = \alpha_{i} + \delta_{t} + \beta X_{it} + \varepsilon_{it}$

Where $\alpha_{i}$ is country fixed effect. $\delta_{t}$ is year fixed effect, and $X_{it}$ are the relevant independent variables. Now I am wondering whether this regression is valid for such a samll sample of countries especially using these fixed effects since I may lose a lot of precision there. Are there any suggestions on how I could explore the panel nature of this data without losing too much precision.

• You do not have enough detail to sufficiently answer your question. What variables do you have? What is your research question?? Commented Apr 2, 2017 at 22:52
• Hi, thank you for asking. This is a very broad question that I had thoughts about. I dont have such a data, but was thinking what would be the first models that would come to mind to handle a research question with such data. But say you have a variable like voter participation in national elections as the dependent variable. And explanatory variable like GDP, oil prices, unemployment , wages? This is really a thought exercise. Commented Apr 3, 2017 at 8:32

As described, this panel data model regression is valid theoretically. The single biggest advantage of panel data is that it "pools" information, thereby shrinking the error. Of course, with more information the errors would be even smaller. With 30 cross sections and ten years of annual information, it sounds like a balanced matrix. I wouldn't even call that "small." It's just not enough information to initialize the more traditional, univariate approaches to time series modelling such as Box-Jenkins, ARIMA, ARCH, etc.

Are there any other complications? Missing values requiring imputation? Mixed frequencies where the predictors and dependent variable are in differing units of time, e.g., annual vs quarterly? If the latter, then Ghysel's MIDAS (MIxed DAta Sampling) approach might be helpful. Ghysels has many papers about this.

You probably want to explore lead-lag relationships which are agnostic wrt causal flow, i.e., relaxing the typical, directional causal assumptions. Some useful work has been done on this by Sornette in his papers on TOPS (thermal optimal paths). His papers are technically sophisticated but the core ideas are not. You can readily develop "brute force" workarounds for them.

Another consideration would be to explore Pesaran's CD test for weak cross-sectional dependence. His test realistically assumes at least some level of dependence between cross-sections and is, therefore, less stringent than others.

Multivariate tests for unit roots, autocorrelation, and so on, are in an early stage of development wrt panel models. The only references I'm aware of that explicitly address them are from SAS as related to its PROC PANEL module. Here are some links:

http://go.documentation.sas.com/?docsetId=etsug&docsetVersion=14.2&docsetTarget=etsug_panel_details50.htm&locale=en

http://go.documentation.sas.com/?docsetId=etsug&docsetVersion=14.2&docsetTarget=etsug_panel_details68.htm&locale=en

These are probably implementable in other software but it would be incumbent on you to do that.

Beyond that, I don't have too many more suggestions. The balanced nature of your data actually simplifies the challenges a great deal. Consider yourself lucky as it could be a lot worse.

• Hi Thanks for such a detailed reply. What are the complications that could arise with an unbalanced panel data in this set up? Further I am a bit skeptical about using lead and lags since with t = 10, one could lose precision. What are your thoughts on that? Commented Apr 4, 2017 at 10:08
• You would want to limit the range of any leads or lags to only 1 or, at most, 2 periods as a function of the strength of the relationship. Unbalanced panel data is a nontrivial issue that most of the "big" theoretical work doesn't address in any detail. In section 17.7.1 of Wooldridge's book Econometric Analysis of Cross Section...Data he speaks to it. There are a few papers too ... Moses, Gale, Altman Methods for Analysis of Unbalanced Growth Data and Analysis of Unbalanced Simultaneously Clustered and Longitudinal Data Using Quasi-Least Squares in SAS Commented Apr 4, 2017 at 10:22

To me, whether or not the regression is "valid" depends on whether the estimator is consistent, not on whether it is efficient. As long as you don't have any omitted variables, and all the other usual assumptions we usually take for granted, a fixed effects estimator should be consistent. (I'm interpreting $\alpha_i$ as the panel effect here, and $\delta_t$ as a set of indicators.)

If you need more precision, you can then try a random effects model, with $\alpha_i$ drawn from a normal distribution as usual, and uncorrelated with the error term. So long as the assumptions of the random effects model are met, the estimator is consistent and efficient. If the assumptions aren't met, the estimator is inconsistent and therefore the regression is "invalid". You can and should check the validity using a Hausman test that compares the estimates from the fixed-effects model to the estimates from the random-effects model.

I should note a third option, which is a sort of "randomer" effects model. You can treat both $\alpha_i$ and $\delta_t$ as distinct, "crossed" random effects, with all the benefits (increased efficiency) and hazards (inconsistency) that go along with that.