Fixed Effects Regressions with Time and Individual Fixed Effects

[Ley Krthree]

Hello,

The following questions are to essentially check whether I am carrying out the correct econometric regressions free from any errors before computing them on Stata to get some results. Please note that this is my first time doing any econometric analysis on my own accord.

Just to provide some context, I am looking at the effect of macroeconomic uncertainty on trade balance in the Euro Area, using quarterly data between 1999 - 2015 (67 quarters) on 17 Euro Area countries. As most of the variation in my data is time variance rather than cross-sectional variance, I am conducting a log differenced fixed effects regression (with the exception of macroeconomic uncertainty, which I am only taking the log of (not log difference), for theoretical reasons) in order to avoid the time series issue of non-stationarity. I am also including time fixed effects. I will be computing the regression first with import growth as my outcome variable, and then with export growth as my outcome variable, and then with net export growth as my outcome variable (each with their own corresponding specification). Please find a screenshot of these three regressions attached below, where "FE" denotes "Fixed Effect" and Q_t is the quarterly dummy for a specific quarter t.

My questions are:

1. Generally speaking, do my regressions raise any red flags or concerns at all? Does anything need tweaking or need to be expressed differently before including it as below in my study?

2. Does there seem to be any issues of endogeneity in any of these regressions?

3. I have included all control variables from my import growth regression in my export growth regression too, purely based on the fact that imported capital often serves as an input into the export production process, and therefore these import growth control variables may also have an indirect effect on export growth; is repeating these controls in my export growth regression (based on this reasoning alone) valid? Does doing this raise any issues econometrically?

4. I also have two time fixed control variables, real effective exchange rate and trade barriers (both identical cross-sectionally due to all nations being in the Euro Area) and it is my understanding that my quarterly dummies will encompass their effects; does that mean there is no need for me to specify them at all, and more specifically does this mean that the part in brackets under each regression (where I have specified the time fixed effect) is correct, or should I have added these two variables and corresponding coefficients in there too? Also, if I am not specifying these two control variables because they are time fixed, does that mean I do not need to collate data for them and include them in my dataset?

5. I am using investment as a proxy for export capacity due to a lack of data availability of the latter and as the former is a key determinant, is this valid?

6. When I am explaining the reasoning for including each control variable, does it matter if the transmission mechanism of the relationship between that control variable and dependent variable occurs through another control variable in your regression?

7. Now that I have log differenced all my variables (except macroeconomic uncertainty) to remove any chance of non-stationarity, are there any further time series econometric issues that I need to consider or deal with?

8. Is it viable to include both GDP growth and investment growth as control variables I.e. from a perspective of collinearity?

9. The reasoning behind my inclusion of the interaction term between FDI growth and log macroeconomic uncertainty is that the effect of a 1% increase in macroeconomic uncertainty on import growth will be greater at higher levels of FDI growth, due to (theoretically) higher levels of FDI growth leading to greater rates of technological transfers, job creation etc in the host country which allows consumers to react more efficiently to macroeconomic uncertainty and adjust their imports accordingly, thus having a greater effect on import growth; does this reasoning seem valid?

10. I have decided to log difference inflation as well because the theoretical relationship seems to have been between inflation and import levels and so, due to the nature of this relationship, I thought that if I log difference the latter then I'd have to log difference the former; however do you think it would be more intuitive to keep inflation as it is rather than taking the natural log?


I know that these questions do not really relate to Stata per se and rather pertain to econometric methods and general theory, however I would be extremely grateful if you could offer any opinions and thoughts whatsoever as I do not have anybody in particular to turn to for guidance before moving onto the Stata stage, and therefore any constructive criticism and nudges in the right direction based on your own viewpoints of the above would give me great peace of mind when conducting my analysis on Stata. I apologise for the numerous questions too, some of which will probably be silly and/or basic questions, and I appreciate anybody taking the time out to answer or offer their thoughts on any number of these questions.

Many many thanks,
Ley

[Carlo Lazzaro]

Ley:
if you are dealing with gravity model, please see Joao Santos Silva ' posts (who is the guru of this stuff in and out this forum).

[Ley Krthree]

Hi Carlo,

Thank you for your reply and suggestion. Unfortunately, it does not allow me to privately message Joao, and I am under a bit of time pressure as my submission deadline is approaching.

Is there absolutely any chance that you could kindly provide your personal thoughts to any of the above questions to the best of your ability? I know that this is not your area of your expertise, however getting some general thoughts and feedback from an experienced economist like yourself would give me great peace of kind.

Many thanks,
Ley

[Clyde Schechter]

Commenting here only questions 4 and 8 in #1--the rest are really economics/econometrics questions that I lack sufficient knowledge to answer.

4. If the variables are invariant over time, you cannot include them in a fixed effects model: they will be colinear with the fixed effects and Stata will omit them. This is not some idiosyncrasy of Stata: it follows from linear algebra that their effects cannot be estimated in a fixed effects model. As for whether it is worth collating data on them, you might find it helpful to include descriptive statistics for these variables in your presentations of results to provide context to your audience. The value of doing that will depend on what those variables are and whether a typical audience for your work would care about them. Since you were at least considering including them in the model, I guess they are relevant, but strictly on the information provided, I can't say.

8. If the only reason you are including these variables in the model is to adjust for them (I do not use the term "control" when we are talking about observational data where, nothing, in reality, is controlled), then it makes no difference whatsoever whether they are highly colinear with each other. If they are exactly colinear, Stata will omit one. If they are simply highly correlated, their coefficients will be estimated with poor precision, i.e. large standard errors. But if the only reason you are including them is for adjustment purposes, you don't care about the standard errors as you are not testing hypotheses about them, nor trying to actually pin down their effects. Where you might run into difficulty, though this does not often happen in practice, is if these variables are also very highly correlated with the predictor variables whose effects you really are primarily interested in estimating. In that case, their inclusion might inflate the standard error of the coefficient(s) of that (those) other predictors--that would be a problem, but one for which there is, in any case, no good solution that doesn't require a larger or differently selected data set.

[Ley Krthree]

Hi Clyde,

Thank you very much for answering these questions. Regarding Question 4, I suspect you may have misread; I mentioned time fixed variables (varying over time, but invariant across countries) rather than time invariant variables that you mentioned in your answer.

If possible, would you be kind enough to revisit that question again when you have some spare time, please?

Many thanks,
Ley

[Clyde Schechter]

Regarding Question 4, I suspect you may have misread; I mentioned time fixed variables (varying over time, but invariant across countries) rather than time invariant variables that you mentioned in your answer.

Indeed, I did misunderstand that. As your variables are functions of time alone, and do not vary across countries, these values will be colinear with the "quarterly dummies" you refer to, and their effects are entirely encompassed within those quarterly terms. If you include them, Stata will omit them, or it might omit some additional quarterly indicators, beyond the usual one omitted as a reference quarter, instead. Since people find it unaesthetic to have additional time variables omitted, it is usually better not to include these time-fixed variables in the first place. (But,I emphasize that this is purely a matter of taste: the model predictions will be the same regardless whether you have a complete set of quarterly indicators and leave out these time-fixed variables, or whether you include some or all of the time-fixed variables and lose an equal number of quarter indicators.)

[Ley Krthree]

Got it. Thank you very much for clarifying this, Clyde - it is much appreciated!

[Joao Santos Silva]

Dear Ley Krthree,

Clyde already provided very useful feedback and I am afraid there in not much more I can add. These are not really gravity equations and I am not familiar enough with this kind of regression to be able to provide any reasonable advice.

Best wishes,

Joao

[Ley Krthree]

Hi João,

Thanks for your message. That's fine, I appreciate you having a read of my post anyway.

All the best,
Ley