Below is the definition taken from Wikipedia(with some minor alterations)
Consider the linear unobserved effects model for $N$ observations and $T$ time periods: :$y_{it} = X_{it}\mathbf{\beta}+\alpha_{i}+u_{it}$ for $t=1,\dots,T$ and $i=1,\dots,N$
Where:
- $y_{it}$ is the dependent variable observed for individual $i$ at time $t$.
- $X_{it}$ is the time-variant $1\times k$ (the number of independent variables) regressor vector.
- $\beta$ is the $k\times 1$ matrix of parameters.
- $\alpha_{i}$ is the unobserved time-invariant individual effect. For example, the innate ability for individuals or historical and institutional factors for countries.
- $u_{it}$ is the error term.
The fixed effects (FE) model allows $\displaystyle \alpha _{i}$ to be correlated with the regressor matrix $\displaystyle X_{it}$. Strict exogeneity with respect to the idiosyncratic error term $\displaystyle u_{it}$ is still required.
Reading this definition, I would say that a spacial effect term would be highly correlated to $X_{it}$ and therefore it 'fitted' the definition of a fixed effect. I'm thinking for example in a situation where (innate)ability can be highly correlated to the pollution in a certain region.
However, in the answer to this question, definition 4 of a random effect seems to also 'fit' perfectly to a spatial effect.
Maybe a spatial effect can be both... like a $\alpha_i \sim F(X_{it})$?
