# What is the difference between a pooled OLS regression model and a fixed effect model?

I have a given data set and I am asked to fit a pooled OLS regression model, and then a fixed effect model with specific variables. From the research I've done, I am thinking that a pooled OLS regression is just panel data regression.

I think it should look similar to the code below, but please correct me if I am wrong.

result1 <- plm(scrap ~ hrsemp, effect = "individual", model = "pooling", data = panel_data1)


But then what is a fixed effect model?

There is an important difference. If there is unobserved heterogeneity (i.e. some unobserved factor that affects the dependent variable), and this is correlated with some observed regressor, then POLS is inconsistent, whereas FE is consistent. If there is no unobserved heterogeneity (unlikely), or this is uncorrelated with all regressors, then both POLS and FE are consistent (albeit not efficient).

Assume a simple model:

$$y_{it} = x_{it}\beta + w_i\gamma + (\eta_i + \epsilon_{ij})$$

where $x_{it}$ is a vector of time-variable factors, $w_i$ is a vector of time-invariant factors, $\eta_i$ is the individual effect (or unobserved, time-invariant heterogeneity), and $\epsilon_{ij}$ is an idiosyncratic error (i.e. it is unique to the individual-period).

For simplicity, assume exogeneity of time-variable factors. This is:

$$E[x_{ij} \cdot \epsilon_{ij} ]=0$$

(if not, we need to think about instruments, just as in the non-panel data world).

• Pooled OLS (POLS): if $x_{ij}$ uncorrelated with $\eta_i$, OLS consistent but inefficient (because of serial correlation). Use adjusted POLS. If $x_{ij}$ correlated with $\eta_i$, POLS inconsistent.

• Fixed Effects (FE) (or Within Groups, WG): estimates a de-meaned model. This is, it substracts to each individual the average of the period, for each time-varying variable. As the fixed effect is constant, this method eliminates $\eta_i$!. Therefore, no need to worry about correlation between $x_{ij}$ and $\eta_i$, which we might be concerned it does exist. Notice that FE is basically POLS of de-meaned model, and it is consistent in the case of correlation between the unobserved heterogeneity and a time-varying dependent variable.

Notice: quality of FE estimation depends on extent of variation of regressors over time, as estimation uses de-meaned data, i.e. the time differences. also, all invariant variables are eliminated. For example, FE is less successful in controlling for unobserved ability if we want to estimate the effect of schooling on earnings (assuming they are constant). In consequence, FE imprecise if there is only limited time-series (‘within’) variation.