Let us say we have a random intercept model

$$y_{i,t} = x_{i,t}'\beta + \alpha_i + \epsilon_{i,t}$$

where $x_{i,t}'\beta$ is a set of time variant explanatory variables, $\alpha_i$ is the individual specific heterogeneity and $\epsilon_{i,t}$ is an idiosyncratic errot term.

We could inlcude a time constant fixed effect $x_i$, which differs among individuals (for example the gender of $i$).

$$y_{i,t} = x_{i,t}'\beta + x_i + \alpha_i + \epsilon_{i,t}$$

However, if accurate predictions are our only goal, is there anything we gain by doing so? Should we not expect to obtain the same results whether we include $x_i$ or not?

Let us say we have a random intercept model for $n$ individuals

$$y_{i,t} = x_{i,t}'\beta + \alpha_i + \epsilon_{i,t} \hspace{35pt} i = 1,...,n$$

where $x_{i,t}'\beta$ is a set of time variant explanatory variables, $\alpha_i$ is the individual specific heterogeneity and $\epsilon_{i,t}$ is an idiosyncratic errot term.

We could inlcude a time constant fixed effect $x_i$, which differs among individuals (for example the gender of $i$).

$$y_{i,t} = x_{i,t}'\beta + x_i + \alpha_i + \epsilon_{i,t}$$

However, if accurate predictions are our only goal, is there anything we gain by doing so? Should we not expect to obtain the same results whether we include $x_i$ or not?

Let us say we have a random intercept model

$$y_{i,t} = x_{i,t}'\beta + \alpha_i + \epsilon_{i,t}$$

where $x_{i,t}'\beta$ is a set of time variant explanatory variables, $\alpha_i$ is the individual specific heterogeneity and $\epsilon_{i,t}$ is an idiosyncratic errot term.

We could inlcude a time constant fixed effect $x_i$, which differs among individuals (for example the gender of $i$).

$$y_{i,t} = x_{i,t}'\beta + x_i + \alpha_i + \epsilon_{i,t}$$

However, if accurate predictions are our only goal, is there anything we gain by doing so? Should we not expect to obtain the same results whether we include $x_i$ or not?

Thank you very much!

Let us say we have a random intercept model

$$y_{i,t} = x_{i,t}'\beta + \alpha_i + \epsilon_{i,t}$$

where $x_{i,t}'\beta$ is a set of time variant explanatory variables, $\alpha_i$ is the individual specific heterogeneity and $\epsilon_{i,t}$ is an idiosyncratic errot term.

We could inlcude a time constant fixed effect $x_i$, which differs among individuals (for example the gender of $i$).

$$y_{i,t} = x_{i,t}'\beta + x_i + \alpha_i + \epsilon_{i,t}$$

However, if accurate predictions are our only goal, is there anything we gain by doing so? Should we not expect to obtain the same results whether we include $x_i$ or not?