# Pooled time series models and HAC residuals

A canonical text for panel data models is Wooldridge's Econometric Analysis of Cross Section and Panel Data. His book is devoted to econometric issues such as endogeneity, unobserved effects, instrumental variables, and so on. To me, it's very striking that wrt this class of models he never discusses controlling for classic "Box-Jenkins"-type issues such as autocorrelation, trend, stationarity, unit roots, cointegration, etc, nor is he concerned with creating HAC residuals, at least based on the index to the 1st edition. Presumably, these core statistical issues for modeling time series are assumed away as not important enough to discuss.

However and in all fairness, multivariate tests such as a Ljung-Box test for autocorrelation or an "augmented Dickey-Fuller" for panel data don't exist or have yet to be developed, at least to the best of my knowledge.

So, have papers been written, have tests and protocols been developed that are appropriate for panel data and that address controlling for statistical issues such as HAC residuals? Cointegration? And so on?

• If I were you, I would use panel data in place of time series [models] in the title. Apr 4 '16 at 11:21
• @RichardHardy I'm sure you're making an important distinction but it's not apparent to me. If panel data structures underlie pooled time series models, why can't the terms be used interchangeably? Apr 4 '16 at 11:28
• I am no expert in these matters, but I think panel data is more standard and recognizable and hence practical to have in a title (especially if you say the terms are interchangeable). Of course, this is just an opinion. Apr 4 '16 at 11:50
• @RichardHardy By "interchangeable," I don't mean synonymous. I think I understand your point but it isn't compelling enough to make me want to change the verbiage. Actually, I like using both terms as they cover more ground than would be if the topic was more narrowly (precisely?) stipulated. Apr 4 '16 at 11:52
• @DJohnson For me they are not interchangeable, because I can see a clear distinction in statistical methodology used in panel data articles and pooled time series articles. However there is no fixed terminology and although the situation is much better than in fixed/random effects terminology mess, there is some abuse of the terms. Apr 4 '16 at 12:41

1. Practically each estimation method Wooldridge discusses includes a section on robust inference, i.e. how to use covariance matrices which are robust to serial correlation or heteroscedasticity, i.e. HAC type covariance matrices. R package plm implements certain variants of such matrices proposed by Wooldridge.

2. Wooldridge touches on a subject of unit-roots, but only to state that his book is about classical panel data, i.e. where time series are short. All the results in the book are derived using N asymptotics, with T fixed, i.e. when the number of time series is going to infinity and their length stays the same. In that case all the time series issues are treated by different covariance structure of the regression disturbances.

Even now there is no agreed way of how to estimate panel models when the time series are long enough to warrant the the treatment of all the time series issues you have mentioned. If you look for unit-root or cointegration tests for panel data in the literature you will not find one simple test. Usually each author presents several variants, with varying assumptions. The DGPs of different authors usually are incompatible, and so on and so forth. So it is not easy to write a book an such subject, because there is no scientific consensus yet, how to treat such type of problems.

One of the possible reasons for this is that there are no compelling real life examples which would require both time-series and panel data treatment. If you have the data with sufficiently long time series, it is usually much easier just to treat each series separately and it is harder to justify the necessity of pooling the time series. There are several examples where such treatment is feasible, such in gravity models of trade, testing PPP hypotheses, but these are topics of specialized economic fields and the results are not easily generalized to general case.

• Thank you for your closely read and well-informed comments. I agree that Wooldridge discusses robust inference, e.g., on p.152 of the 1st ed., "It is important to know what this robustness allows...in panel data models, an unrestricted (unconditional variance matrix) allows for arbitrary serial correlation and time-varying variances in the disturbances..." To me, this leaves a lot to the imagination. You also state that Wooldridge discusses unit roots but is there a citation for this? Apr 4 '16 at 12:11
• Also, you mention literature addressing "unit-roots and cointegration tests for panel data." Do you have any references? As for "no compelling real life examples," I can't agree with this, unless you are limiting your comment to econometric models. For instance, in marketing science, there are plenty of examples of situations with many long panels of this type, see, e.g., Lee Cooper's book Market Share Analysis available as a free download here... anderson.ucla.edu/faculty/lee.cooper/MCI_Book/BOOKI2010.pdf Apr 4 '16 at 12:13
• @DJohnson, I guess long panels rather than large panels is what is relevant here. Apr 4 '16 at 12:14
• @RichardHardy Edited to reflect your observation. Apr 4 '16 at 12:15
• Wooldridge discusses time series persistence (i.e. unit roots) in page 175, section 7.8.3. The exact quote: ... it may appear that we have not been careful in stating our assumptions. However, we do not need to restrict the dynamic behavior of our data in any way because we are doing fixed-T, large-N asymptotics. There is more in that section together with citations of relevant literature, i.e. works of Pesaran and Smith, and Phillips and Moon. Also Baltagi's book on panel data econometrics has a whole chapter devoted to time series issues. Apr 4 '16 at 13:07