# Is this an example of Pooled OLS on Panel Data?

I am looking at a study that analyzes the effect of an infrastructure index on infant mortality and child mortality rates. The database has (asset) quintile level data for 47 different countries (from surveys that used the same methodology in all countries, althought they took place @ different years), thus there are 47*5=235 observations in total. The authors control for some country level factors (i.e. do not vary among quintiles) such as as income per capita and national literacy rate of females. All other independent variables are country quintile specific.

They then give these results. They say they calculated the regressions using Seemingly Unrelated Regressions (SUR), but they just look like usual OLS regressions to me.

I have the following questions: 1) By using SUR, are they simply running a pooled ols over all of the 235 data points? The authors say they are using "cross country quintile level data", and simply state that they used SUR for regressions.

2) I assume the approach 1) would be incorrect since a commentator for the paper suggested setting up the data as a Panel formed over quintiles and countries; and then using FE instead to account for country level fixed effects. I've never heard of a Panel formed this way, does this make sense? And would the FE method makes sense then?

If anyone's interested the study in question is: http://elibrary.worldbank.org/doi/pdf/10.1596/1813-9450-3163

## 1 Answer

Least-squares estimation on a general SUR model is not the same as Pooled OLS on a Panel Data model (at least given the meaning with which these model labels are usually endowed).

Both models are "System of Equations" models, but the SUR model sprung from the observation that in many cases, the disturbance vector from each regression equation is correlated with the disturbance vectors of the other equations. So the general SUR model does not impose the restriction that the unknown coefficients under estimation are identical across equations, as is the case with the usual Panel Data setup and Pooled OLS. Also, in benchmark Panel Data models, errors are assumed independent across equations.

In symbols, the general SUR model is ($N$ is number of equations/cross sectional samples, $T$ the number of available observations for each cross section)

$$y_{it} = X_{it}\beta_i + u_{it},\;\; i=1,...,N,\;\; t=1,...,T,\;\; E[u_{it}u_{jt}]\neq 0$$

while for a Panel data model and Pooled OLS we usually have

$$y_{it} = X_{it}\beta + u_{it},\;\; i=1,...,N\;\; t=1,...,T,\;\; E[u_{it}u_{jt}]= 0$$

Note the two differences.

A usual example of SUR: you have data on a few big firms from the same national economy related to their financials (e.g. sales, profits, market share, credit lines etc), and their investment spending (dependent variable), and also to certain macroeconomic indicators for this national economy. You can reasonably argue that

a) at least some of the coefficients of an explanatory variable are not the same across equations-since how, say, profitability affects investment spending may depend on the long-term strategy of each company, which is formulated by different decision makers, etc
and
b) that at least partly some shocks/disturbances/other factors that are included in the "error term" in each regression, are common to all equations.

So you are led to the SUR model. Cross-equation restrictions on SUR models may arise -but again, they are not usually of the kind imposed in the Panel Data literature (ideally we should have one unified "Systems of equations" regression estimation theory with all the various models as special cases -but we don't).

The attempted benefit from a SUR specification is gains in estimator efficiency, and a usual estimation method is (Feasible) Generalized Least Squares.

The specific paper you mention suffers from a serious deficiency -the authors do not write down the theoretical equations describing their model, neither do they describe clearly their data set. In the beginning I thought that indeed, they had essentially implemented Pooled OLS -but this is not so, they have run SUR after all, as follows:

Their sample does not have a time-dimension (so the index "$T$" here does not represent time). Their dependent variables are "Infant mortality" , "Child mortality", "Child malnutrition" indices. So they have three equations , $N=3$. For each of the 43 countries they have $5$ such values for each dependent variable (sub sample averages), one for each asset quantile. So the $T$ dimension for each equation should have, as they say, $5 \times 43 = 215$ observations for the dependent variables (in practice they have only $T=175$ due to missing values of the dependent variables).

For each equation $i$, and for each observation $t$, they have as regressors a constant, dummies indicating the asset quantile from which the value of the dependent variable came, etc, without indicating that a specific value of the dependent variable has come from a specific country (but the regressor values of course are related to the country that the value of the dependent variable came from). So no idiosyncratic "country-specific" effect, as such (since in any case there exist regressors that refelct various aspects of each country).

So in, say, Table 3 of the paper, the first three columns (labaled 1,2,3) is a SUR system, while the next three columns (labeled 4,5,6) is another SUR system with some changes in the regressors. So, say, the coefficient on the regressor "GDP per capita" is indeed different for the three equations of the first SUR system.

The data series of the regressors for the three equations in each SUR system is identical except for the dummies indicating "asset quantile". So here SUR estimation is not numerically equivalent to equation-per-equation OLS.

The commentator "Fixed Effects" suggestion proposes a different approach: Specify $45$ cross sections/equations, each cross section a dependent variable from a specific country. For each cross-section, take the other dimension $T=5$ to be the $5$ asset quantiles (so no dummy variables for them here). (or vice versa). Etc. This would appear to suggest that we should specify three distinct FE models for each of the three dependent variables, i.e. not estimate jointly the three dependent variables... which again brings us back to the need to validate the SUR specification, by showing that indeed there appears to be correlation between the disturbance vector across equations -something that the authors appear that they haven't done/presented in the paper.

• Also, what benefit would SUR give the authors (vs. running separate OLS regressions) in this case since: they never really test cross restrictions, there are really no omitted terms (those variable not included in certain regressions is because they are covered under other variables, for ex the "Underweight" independent variable is the same as "Child malnutrition" dependent variable, which is why it's omitted in regressions 3 and 6)? I understand the efficiency gains of SUR, but it seems like they wouldn't apply in this case since the same variables are used in each of the 3 estimations (?) – StatsScared Jul 30 '14 at 2:54
• Don't confuse "using the same regressors" (i.e, the same random variables) with "using the same realizations of the same regressors" -it is in the second case that GLS brings no efficiency gains. – Alecos Papadopoulos Jul 30 '14 at 3:00
• Can you expand, since I have never heard of these concepts being different, especially for the SUR case? – StatsScared Jul 30 '14 at 3:11
• Assume you have two companies. for each company you have investment spending as dependent variable. As explanatory variables you have the profits and the sales . In both equations, the regressors are "the same" in the sense that they have the same nature -in both cases they are profits, and sales. But the time series of the profits for company A includes different values than the profit time series for company B 9and the same for sales. So your regressors are "the same" as variables, but take on different values. CONTD – Alecos Papadopoulos Jul 30 '14 at 3:16
• CONTD Now consider the case where you use as regressors, average profits and average sales. Here again, the regressors are the same, but also, the actual values they take are the same in each equation (the dependent variables differ). It is in this 2nd case that GLS brings no efficiency gains – Alecos Papadopoulos Jul 30 '14 at 3:17