Fixed Effects Interpetation

Question

I have some issues understanding the intuition in Fixed Effects models, and the sources of variation they imply. For a concrete example, consider the following regression specification:

$$r_{ist}=\gamma_{i}+\delta_{st}+\epsilon_{ist}$$

The LHS in the above equation is to be interpreted as the rating that a worker $i$ receives at time period $t$ in sector $s.$ The RHS is composed of three terms. The first term represents a worker fixed effect- think of this as a dummy variable that takes a value of 1 for worker $i$ and 0 for all $j\neq i$. In this sense, the design matrix would consist of dummy variables for all workers (no constant term), and the variable for worker $i$ would take on the value $1$ when considered in this matrix. The second term represents $\delta_{st}$, or a sector-year fixed effect. This is basically a coefficient on a dummy variable for a particular sector-year cell. How is one to interpret this? There are a total of $st$ of such dummy variables. The third is an error term. My first question is, how does on interpret the value of a particular $\hat{\gamma}_{i}?$ Also, how does this relate to removing the variation at the worker level and the sector year level. What variation are we removing? What does it intuitively mean to remove variation? Finally, think of adding a set of covariates as: $$ r_{ist}=x_{ist}'\beta+\gamma_{i}+\delta_{st}+\epsilon_{ist} $$ Now, how is $\beta$ being identified? If we take into account individual and sector fixed effects, how is $x$ varying? Would one not interpret that $\beta$ is identified by holding constant $i$ and holding constant $st$ , we vary $x_{ist}$. But if we are holding both $i$ and $st$ constant, there is no variation in $x.$

I apologize for the multiple questions asked here, but I think the questions are quite inter-related.

Answer 1

Interpretation of FE: $\delta_{st}$ is the mean rating in each sector in each time period.

Contrived Example: Over time, public opinion of workers in the library sector improved and their rating r improved. On the other hand, those in the used-car sector became seen as less reliable and their rating r declined. We capture this with $\delta_{st}$ .

Identification of B (intuition):

• x must vary within individuals i of the same sector s every time period. Otherwise $\delta_{st}$ will be co-linear.
• x must vary within individuals i along time periods t. Otherwise $\gamma_i$ will be co-linear.

Basically, $\gamma_i$ can be identified separately from $\beta$ because individuals change over time. And $\delta_{st}$ can be identified separately from $\beta$ because they individuals are heterogeneous within each time period and sector.