# Fixed Effects Interpetation

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.

• HI: Your questions are interesting and I'd love to see answers myself but my guess is that they'd be too long even if someone can answer them. My suggestion is to obtain some notes or a text on "analysis of variance" and "analyis of covariance". Fixed effects without covariates is equivalent to anova in statistics and fixed effects with covariates is called ancova. Googling those terms will probably kick out some useful documents. I took the material in graduate school but never found a great text on it but that was a long time ago. There may be some now. – mark leeds Jun 9 '18 at 5:15
• @ChinG If possible, could you describe the data set you're working with? Depending on the data, the answers to your questions change (especially the ones asking how something is being identified and where the variation is coming from). – AndrewC Jun 9 '18 at 11:26
• @Ching: Only looked qwuickly but these links look decernt at a glance.are.berkeley.edu/courses/EEP118/current/handouts/… AND jblumenstock.com/files/courses/econ174/FEModels.pdf – mark leeds Jun 9 '18 at 14:28
• @ChinG: you're welcome. I'm gonna read them in the future also. – mark leeds Jun 10 '18 at 1:52

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.

• Thanks a lot! With regards to your first point, x is indexed by ist. How can x vary within each time period and sector, if there is only one observation per ist? I guess in your first point, it shoud read x must vary "across individuals" of the same sector s every time period. – ChinG Jun 11 '18 at 13:13
• I think you're losing the forest for the trees. If (capital) I=10, S=10, T=10, we have a maximum 1000 observations of x, one for each individual in each time period and each sector. (I have a hunch you actually mean x_it, individuals are tied to a sector and don't move.) But here's my point about varying over time. Example: x_{1,2,3} = 13 x_{1,2,4}=14, x increased over time, x varies over time. – RegressForward Jun 11 '18 at 19:59