Fixed effect versus clustered standard errors

[Morten Gravesen]

Hi, i am taking a chance asking here, as my teacher seems to be having a nice vacation, not answering my email. I am writing my master thesis, but I have a hard time understanding which regression model to use.

The dataset I am using is of panel structure - 1,000 firms (500 Swedish, 100 Danish, 200 Norwegian and 200 Finish) with years ranging from 2004 to 2017. It is unbalanced and has gabs, because I have removed observations with missing values, book leverage above 1, total assets below 10 million dollars and market-to-book ratios above 10.

The regression I am running is:

book leverage = EFWAMB(t-1) + Market-to-book(t-1) + Tangibility(t-1) + Profitability(t-1) + Size(t-1)

The results from different versions of this model can be seen in the table below.

1. I do not know which model to trust?
2. I am confused to why the OLS estimated coeffecients (column 1) is the same as those from clustering the standard errors on both time and firm (column 9). I thought, that by clustering on these two dimensions, I would be able to remove serial correlation and heteroskedasticity and as such, the coeffecients would be different from those of OLS?
3. I am also confused to why the fixed effects regressions are so different from the OLS .

In general I find the litterature on this matter very unfullfilling as it is a LOT OF IFS and WHYS. There is never a clear answer to get. My teacher says - use fixed - but when I ask why, he can't answer. He is one of those corporate finance dudes, who just by default sticks to fixed effects. However, it does not provide me with the results I am looking for - the paper I am following use OLS with robust and Fama Macbeth and get results similar to those I get from doing this - however, the fixed effects model ruins the variable of importance - EFWAMB - as it turns small and insignificant.

So, if anybody could please take a moment and reflect upon my setting of data - the variables included - and come up with a good recommendation on which model to go with and why, by answering questions 1, 2 and 3 above, I would be more than greatfull.


In case you should ask for it, here are the different statacode used to estimate the models above:

OLS robust:

reg b_lev L1.efwamb L1.mb L1.tang L1.prof L1.size, robust

Fixed effects:

areg b_lev L1.efwamb L1.mb L1.tang L1.prof L1.size, absorb(gvkey)

Fixed effects, cluster year:

xi: areg b_lev L1.efwamb L1.mb L1.tang L1.prof L1.size i.year, absorb(gvkey)

Random effects:

xtreg b_lev L1.efwamb L1.mb L1.tang L1.prof L1.size

Fama Macbeth cross-sectional:

xtfmb b_lev L1.efwamb L1.mb L1.tang L1.prof L1.size

The Fama Macbeth two path regression is estimated manually by first making 1000 time series regressions, which provides me with 5*1000 betas using:

statsby, by(gvkey) saving(betas): reg b_lev L1.efwamb L1.mb L1.tang L1.prof L1.size

merge m:1 gvkey using betas

drop _merge

I then do 14 cross-sectional regressions, one for each time period 2004 - 2017 with the estimated betas from above being the new independant variabes, which provides me with 5*14 new beta values (gamma) using:

statsby, by(year) saving(gamma): reg b_lev b1 b2 b3 b4 b5

I then open the gamme file, and take the average of the 14 betas in each row - this is my beta estimates reported in the model above. To get t-test, I simply divide this coefficient through with the square root of the variance of the betas divided by 14.

4. Why do I not get the same coefficients and t-stats as those calculated using the xtfmb command?


Best regards,
Morten

[Clyde Schechter]

You can use -xtreg, be-. This is a pure between-group effects estimator.

[Morten Gravesen]

Carlo Lazzaro I ran the test and it gave me something very significan with a p-value very very close to 0. The Hausman test also gives me a very significan result. So I am guessing that these models suggest that I use fixed effects? However, as I am not interested in the within firm effect but the between firm effect (to test if firms with a high EFWAMB has lower leverage), I don't find the fixed effects model useful. Is there any other model, that I can use?

[Carlo Lazzaro]

Morten:
let's start from square one:
what does the user-written command -xtoverid- tells you if you adopt -re- specification in your panel data regression (I recommend -xtoverid- because -hausman- does not allow non-default standard errors).
As an aside, please also note the -fe- specification (ie, within estimator) gets rid of any observed and unobserves source of heterogeneity related to time-invariant predictors. It does not shelter you from heterogeneity sources related to time-varying predictors.

[Morten Gravesen]

Thank you all, it is very helpfull advice.

Philip Gigliotti, I understand the difference in OLS and Fixed effects in that perspective now (within versus between). However, the main variable that I am interested in is the "external weighted average market-to-book ratio (EFWAMB)". The others, including firm size is control variables.

The EFWAMB is estimated for each firm in each year by weighting each market-to-book value by the external finance in any given year. Thus for instance, the EFWAMB for firm A in year 2017 uses the whole panel of market-to-book ratios (14 m-to-b ratios from 2004-2017) in estimating the EFWAMB. For year 2009 it only uses 6 m-to-b ratios (2004-2009).

What I want to test is, if the capital structure of firms today is smaller for firms, that issue equity when their market-to-book values are high.

Thus, I am interested in the differences between firms with large and small EFWAMB ratios in any year. By using fixed effects, I am looking at the variance in the EFWAMB from year to year of firm A and the impact on leverage for that firm. I find it hard to figure it out, but there is just something that tells me, that using fixed effects is not the right choice here, at it measures something wrong.

However, using OLS measures in a correct way I think, but gives me ineffecient estimates because of these fixed unobservable factors,

[Clyde Schechter]

For this reason it's usually the only accepted choice of estimator in economics, finance or disciplines dealing with observational data.

Well, epidemiology deals primarily with observational data, but the fixed effects estimator is almost never used. One might argue that it should be, but as a matter of practice it is quite uncommon. (Metanalyses are an exception to this generalization, but even there, random effects estimation is more common.) Consistency of estimates is only one of several qualities one can desire from an estimator, and not always the most important. Fixed effects estimators have many limitations and rigidities that make them unsuitable for many purposes.

But I think this advice is misguided in a more fundamental way. The fixed-effects model provides only estimates of within-panel effects. If the research question specifically addresses between-panel effects (as appears to be the case here) then the fixed effects estimator is giving consistent estimates of the wrong parameter. So the inconsistency of an OLS or random effects estimator just has to be accommodated by including as many covariates as you reasonably can and hope that you are left with errors that are uncorrelated or only weakly correlated with the predictors, and then you live with it.

[Philip Gigliotti]

Regardless of whether you run a fixed effects model or an OLS model, if you havehpanel data you should have cluster robust standard errors. If autocorrelation and heteroscedasticity are a problem, they are a problem regardless of what specification you use. Furthermore, they are standard in finance and economics, theory aside you should never in practice run a regression without them.

OLS measures differences between firms, for instance the coefficient on firm size would measure the difference between large and small firms. This is subject to major endogeneity concerns in observational data as small and large firms differ in many unobservables ways. The coefficient on size in a fixed effect measures the difference between periods in the same firm when it had different sizes. It's much harder to argue that this change of size is correlated with other unobservables changes instead of just the inherent nature of the firm itself. Thus fixed effects is usually the only plausibly consistent estimator. For this reason it's usually the only accepted choice of estimator in economics, finance or disciplines dealing with observational data.

[Carlo Lazzaro]

Morten:
as an an aside to previous excellent advice, please note a(nother) relevant difference between fixed effect (-fe-) and (pooled) OLS:
- -fe- specification allows a limited endogeneity, that is the individual error is correlated with the vector of regressors;
- (pooled) OLS (just like random effect specification) rules totally out correlation with any residual component.
As an aside, the fact that Baker and Wurgler (2002) (as per FAQ, full reference please; I can understand you're under heavy pressure, though) do not explain the reason underlying their model choice (I do not know that paper, so I trust your words) will not shelter you from justifying in your dissertation why you decided to go (say) -fe- and not (pooled) OLS (or the other way round).
I think I would follow Clyde's advice (assuming that you have excluded random effect specifiction once and for all).

[Clyde Schechter]

And the hypothesis again: do firms with hight EFWAMB have lower leverage than firms with low EFWAMB.

For this hypothesis, the fixed effects model would be inappropriate, because the fixed-effects model specifically estimates changes within firms over time. The OLS model's estimates are a mixture of the within-firm effects (which this hypothesis does not address) and between-firm effects (which the hypothesis targets). Better still, I think, would be a between-effects analysis, using -xtreg, be- which provides a pure between-firms effect estimate.

[Morten Gravesen]

The way the EFWAMB is constructed, by weighting each firm by its external finance in any given year, devided by the total of external finance up untill that point in time starting at time 0 in the sample, confuses me even further to how I can use the fixed effects model. If there is any fixed effect from unobservable variables, that influence the market-to-book ratio, this will create the problem of serial correlation in my residuals. And because the EFWAMB is constructed from these market-to-book ratio, would I not remove any effect from this variable when using fixed effects?

Also, as market-to-book ratios, book leverage, size and tangibility do not vary hugely over time, can I even use fixed effects without losing some important information?

market-to-book	profitability	tangibility	size
1.636465	0.1699918	0.2760911	10.80538
1.659328	0.1671353	0.2467667	10.87995
1.169266	0.1664327	0.2586308	11.0566
1.62008	0.099755	0.2731158	10.93715
2.02516	0.1871107	0.2339925	11.0191
1.617319	0.1885505	0.2202311	11.10068
1.730845	0.1469131	0.2153826	11.07558
1.661407	0.1166909	0.1985463	11.06032
1.523745	0.11873	0.1896398	11.17008

[Morten Gravesen]

I must say, that you answer completely confuses me.

What exactly do you mean when you say that the fixed effects model and the pooled OLS are used for different objectives? I have read 10 chapters in different books and plenty of articles too, without finding any explanation what so ever. To me it seems as if you talk about the fixed and random effects model outside the scopes of these models - I thought, the fixed effects model was used to adjust for any unobservable fixed effect that is correlated with the explanatory variables of firm A (constantly over time) but not firm B, whereas the Random effects allows for these unobservables to not be correlated with the independant variables.

The external weighted average market-to-book ratio (EFWAMB) takes high values for firms that have issued equity in periods, when their market-to-book ratio was high. Thus, if firms with high EFWAMB has lower leverage on average than firms with a low EFWAMB, my model should give me a significant negative value. I am interested in point estimates, not predictions of change.


I am ONLY interested in knowing weather past attempts to time the market have a cumulative affect on current leverage. Baker and Wurgler (2002) who developed this ratio, do not AT ALL come up with their reasoning behind their choice of model.


I have ruled out the random effects model, as I am thinking that things like managerial ability and attitude towards risk will influence profitability, size and the market-to-book ratio.

So to be clear - the choise is between a fixed effects model and a pooled OLS with clustered standard errors.

My data is 1,000 firms, 500 Swedish, 100 Danish, 200 Finnish, 200 Norwegian. They are selected from the compustat global database. It is unbalanced and with gaps. This is all I know about the data, now you know the same. And the hypothesis again: do firms with hight EFWAMB have lower leverage than firms with low EFWAMB.

[Jesse Tielens]

At this point it's more about the theory behind the framework, rather than statistical knowledge. But perhaps Clyde Schechter or Carlo Lazzaro can confirm I'm not saying wrong things here

If your hypothesis is that EFWAMB has a negative effect on the dependent variable the leverage ratio, there are several ways to test this. You could say:
-If EFWAMB has a negative effect on leverage, when a firms EFWAMB variable increase we see the opposite effect on leverage.

That is what a fixed model tests. Within the panel (so same firm), we check if years with large EFWAMB variable correlates with low variables of the leverage rate. On the other hand, you could also say the following:
-If EFWAMB has a negative effect on leverage, we should observe that firms with high EFWAMBS have low leverage rates and firms with low EFWAMB have high leverage rates.

The Random Effects model also tests that, in addition to what a fixed effect model already tests. Now, because your testing for a whole different range of things, the results are very different of course. Probably in this case, you could even say that the variance is much greater in the RE-model than in the FE-model. Because firms are unlikely to suddenly obtain much more external financing, but it's a rather slow process I'd guess. So, while there exists a lot of differences between firms and their use of external financing methods. I would suspect there's a lot less change in how the same firm handles its financing over the years. That'd be my best bet as to why there's a very significant coefficient for EFWAMB in the POLS and RE-models and much fewer in the FE-model.

But like I said, the choice for either model should be rooted in your theory and hypotheses.

[Morten Gravesen]

Hi Jesse. Thanks again for your reply. You are correct that the EFWAMB is the weighted average market to book ratio, weighted by external finance in any given year.

What I am looking for is if firms with a high EFWAMB has significantly lower current leverage. I am not interested in seeing an effect from any change. In that sense it is sort of looking back, not used to predict future leverage.

What exactly is your reasoning for saying, that in my case then, I should use pooled OLS rather than fixed effects?

Kind regards, Morten

[Jesse Tielens]

Essentially, a fixed effects model is basically the equivalent of doing a Pooled OLS on a de-meaned model. This way, you're just looking at change between time-periods and ignoring the absolute values. As Clyde already mentioned, a pooled OLS is much more like a Random Effects model in that regard. A pooled OLS is also a mix between a within and a between estimator.

Looking at your results, this quickly becomes clear. The coefficient signs and significance for the Pooled OLS and Random Effect models are not that different.

Assuming 'EFWAMB' is the External Finance Weighted Average Market-to-Book ratio, we can infer some things from your results. The EFWAMB is high for firms that issue lots of equity when their market valuation is very high, sort of like evidence of market timing. You found a very negative coefficient in Pooled OLS and RE-models for the lagged EFWAMB variable, I think this would mean that firms that issued lots of equity in the last year have a lower leverage ratio this year. Whereas the coefficient in the fixed effects model looks at the change EFWAMB variable last year.

But to which of these two questions is more relevant to your research?
-When firms EFWAMB rate changes, what is the effect of this change on the leverage rate?
-Do firms with a higher EFWAMB rate have a higher/lower leverage rate across my time period?

Answering the first question is best done by using a fixed-effects model. The second question requires a random effects or the pooled OLS I'd say.

[Morten Gravesen]

I am very greatful with all your answers. But to be clear the choiseis not between fixed effects or random effects but between fixed effects or OLS with clustered standard errors. I know that the later does correct for serial correlation in the standard errors which is something that I assume to be an issue in my data. However, I am worried that this model does not provide effecient coefficient estimates. The fixed effects on the otherhand gives me very odd results, very different from all other litterature out there (which uses simple OLS with White standard errors).