## Random Effects

My last post was on the fixed effects model. I established that the fixed effects model assumes that variables not included in the regression are correlated with the variables included in the regression, and thus the results of the regression cannot be used to assess the effects of unobserved variables. The random effects model, on the other hand, assumes that unobserved variables are not correlated with observed variables, and allows the regression to be used to investigate the effects of variables not included in the regression.

In my past posts on the panel data model and its specific variations, I explained that the general form of the panel data model is $y_{it} = \alpha_i + \beta'x_{it} + \epsilon_{it}$, and the general form of the fixed effects model is $y_{it} = D \alpha_i + \beta'x_{it} + \epsilon_{it}$. With the random effects model, the general form is $y_{it} = \alpha + \beta'x_{it} + u_i + \epsilon_{it}$. In this model, $\alpha$ is taken to be constant, and $u_i$ is a measurement of random disturbance for each cross-sectional unit.

In choosing whether to use a fixed effects model or a random effects model, one must first test to see if individual effects exist. This is done using a Langrange Multiplier (LM) test. If they do indeed exist, then a Hausman test can be used. The Hausman test uses a hypothesis test to determine whether or not the fixed effects model and the random effects model have the same variance. If their variances are the same, then a random effects model may be used. If not, the more restricting fixed effects model must be used.

For my capstone I intend to look at the mathematics behind regression analysis using panel data, or panel data modeling. A panel of data consists of two components: a cross-section and a time-series. For instance, the data I am using consists of individual observations for each of the 58 counties of California over a span of 8 years (2000-2007). This means that each variable in the regression has 464 ($58 \cdot 8$) observations. An advantage of this approach is the ability to account for variability over time as well as across the cross-section. Also, it allows for analysis of data with a limited number of observations over time (provided there are substantial cross-sectional observations) or a limited number of observations over the cross-section (provided there are sufficient time-series observations).
The general form of a panel data model is $y_{it} = \alpha_i + \beta'x_{it} + \epsilon_{it}$. In the model, $i$ represents the cross-sectional units, $t$ represents the time-series units, $y$ represents the dependent variable, $x$ represents the independent variables, $\alpha$ represents the individual effects coefficients, $\beta'$ represents the set of coefficients for the independent variables, and $\epsilon$ represents the error terms. This is just the general form of the panel data model. The specific variations of the model that I will be looking at will be discussed in a later blog post.