Solved – Can unit/time dummies be included with PCSE or XTGLS

fixed-effects-modelpanel datarandom-effects-modelstata

I am currently working with a balanced time-series cross-section dataset (or a T dominant panel), consisting in 8 units (countries) and 32 observations (quarters) per unit. Thus, the dataset has 8 x 32 = 256 observations. The dependent variable is continuous and there are 5 regressors. I am estimating my model using Fixed Effects (in Stata, "xtreg, fe"), Random Effects (in Stata, "xtreg, re"), Beck and Katz Panel Corrected Standard Errors ("xtpcse") and GLS ("xtgls"). Since my data is not what Greene would called "well-behaved data" -it shows panel heteroskedasticity, serial correlation and contemporaneous correlation-, I am proecting the models against these issues. Thus, my first question is about how to control for time and unit unobserved heterogeneity with PCSE and GLS models. Or, in different words:

(a) Does it make sense to include time and unit dummies with PCSE and GLS models? Any reference supporting your statement will be appreciated. I have found some other people asking this out there, but no proper answer so far.

(b) Does it make sense to include a time trend and time dummies simultaneously in FE/RE/PCSE/GLS? I have found different answers to this question: ones arguing that it makes sense and some others arguing that it does not. Again, explanations with a minimal reference or link are welcome.

Thank you all!

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(a) PCSE just changes the estimated standard errors and can be used with time/unit dummies and/or time trends; in fact the inclusion of dummies and trends is not really related to the issue of how to get appropriate standard errors. So yes, it makes sense. PCSE were introduced by Beck and Katz 1995 and in fact the use of unit dummies and PCSE together is something like a standard in political science at least. GLS is a different way to estimate both the coefficients and the standard errors, and is also not really related to the model specification (dummies and trends). So you would also include dummies or trends based on substantive considerations, and then you might apply GLS if OLS leads to heteroskedasticity or serial correlation. However, and this is extensively discussed in Beck and Katz 1995, the (F)GLS procedure especially for your kind of data has really bad finite-sample behavior.

(b) Jeff Wooldrdige says: "No, it doesn't make sense to include a full set of time dummies along with any function of time. Putting in time dummies allows for any kind of trend, of which a linear trend is a special case." (You get a linear time trend if all the time dummies happen to haven the same estimate). The inclusion of time dummies is of course "conservative", in a sense, but could be more or less warranted depending on the question you try answer - essentially, do you think that units pick up the treatment because of unobserved stuff that changes over time, that this also influences the dependent variable, and that it somehow is common across units?

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Solved – Heteroskedasticity removed through fixed effect estimation

My question is, though, whether there is any reason to assume that the FE model's error term might not be heteroskedastic.

My interpretation is that what you really want to know is whether heteroscedasticity in the pooled OLS regression implies heteroscedasticity in the FE regression. To that the answer is no. In other words, you cannot test on the pooled OLS regression and conclude that the result also holds for the FE regression.

The model underlying the FE-estimator in its simplest form can be written as $$y_{i,t}=x_{i,t}\beta +\alpha_i + u_{i,t},$$ where we now for simplicity assume $u_{i,t}$ is iid. If you fit a model given by $$y_{i,t}=x_{i,t}\beta +e_{i,t}$$ using pooled OLS and data is generated by the fixed effects model, you have in effect set $e_{i,t} = \alpha_i + u_{i,t}$. Decompose the variance of the error term in the pooled OLS model to get: $$\operatorname{Var}(e_{i,t})=\operatorname{Cov}(\alpha_i + u_{i,t},\alpha_i + u_{i,t})=\operatorname{Var}(\alpha_i)+\operatorname{Var}(u_{i,t})+2\operatorname{Cov}(\alpha_i,u_{i,t}).$$ From this equation it is quite clear that while $u_{i,t}$ is of constant variance (it is even iid), $e_{i,t}$ can very well have non-constant variance. Therefore, evidence of an heteroscedastic error term in the pooled OLS regression is in general not evidence of an heteroscedastic error term in the fixed effects regression.

Solved – Time-invariant variables not being removed in Fixed Effects model. And feasibility of addional time dummies in Fixed Effect/Random modelling

Having an unbalanced panel is not a problem nowadays. In the past, when econometrics had to be done by hand, inverting matrices for unbalanced panels was more difficult but for computers this is not a problem. The only worry connected today with this is the question why the panel is unbalanced: is it due to attrition? If yes, is this attrition random or related to characteristics of the statistical units? For instance, in surveys people with higher education tend to be more responsive and stay in the panel longer for that reason.

Regarding the fixed effects model, have you checked whether the variables that are time-invariant in theory are actual not varying over time? Sometimes coding errors sneak in and then all the sudden a variable varies over time when it shouldn't. One way of checking this is to use the xtsum command which displays overall, between, and within summary statistics. The time-invariant variables should have a zero within standard deviation. If they don't then something went wrong in the coding.

Having a negative Hausman test statistics is a bad thing because the matrices that the test is built on are positive semi-definite and therefore the theoretical values of the test are positive. Negative values point towards model misspecification or a too small sample (related to this is this question).

If you cluster your standard errors you also need a modified version of the Hausman test. This is implemented in the xtoverid command. You can use it like this:

xtreg ln_r_prisperkg_Frst_102202 Dflere_mottak_tur i.landingsfylkekode i.kvartiler_ny markedsk_torsk gjenv_TAC_NØtorsk_år_prct lalder_fartøy i.fangstr r_minst_Frst_torsk gjenv_kvote_NØtorsk_fartøy_prct i.lengde_gruppering mobilitet, fe vce(cluster fartyid)
xtoverid

Rejecting the null rejects the validity of the assumptions underlying the random effects mode.

The xtset command only takes into account the unit id for fixed effects estimation. The time variable does not eliminate time fixed effects. So if you do

xtset id time
xtreg y x, fe

will give you the exact same results as

xtset id
xtreg y x, fe

The time variable is only specified for commands for which the sorting order of the data matters, for instance xtserial which tests for panel autocorrelation requires this. This has been discussed here. So if you want to include time fixed effects, you need to include the day dummies separately via i.day, for example. In this context, the season and year dummies make sense so it's good that you use them.

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Solved – Time-invariant variables not being removed in Fixed Effects model. And feasibility of addional time dummies in Fixed Effect/Random modelling

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