# #StackBounty: #probability #panel-data #identifiability Static panel linear model

### Bounty: 50

This question is about how to show identification of the fixed effects in a static panel linear model.

A1 (model): The model is
$$Y_{it}=alpha_i+X_{it}^top beta+epsilon_{it}$$
for each $$i=1,…,N$$ and $$t=1,…,T$$, where $$i$$ indexes individuals and $$t$$ indexes time periods.

A2 (data): We assume to have an i.i.d. sample of $$N$$ observations $${Y_{i1}, X_{i1},dots, Y_{iT}, X_{iT}}_{i=1}^N$$ with $$N$$ large.

A3 (exogeneity): $$E(epsilon_{it}| X_{i1},…, X_{iT}, alpha_i)=0$$ for each $$t=1,…,T$$ and $$i=1,…,N$$.

Question: In the so called "fixed effect model", $$alpha_1,…, alpha_N$$ are treated as parameters (together with $$beta$$) and possibly estimated. How can we show that $$(alpha_1,…, alpha_N, beta)$$ are identified under A1, A2, A3?

Remark: I think that $$T$$ large is also needed to identify $$alpha_1,…, alpha_N$$. Feel free to add this assumption.

My thoughts and doubts:

I have found several sources discussing how to estimate $$(alpha_1,…, alpha_N, beta)$$ or how to identify/estimate $$beta$$ alone (by differencing out $$alpha_1,…, alpha_N$$), but no papers or books explaining the joint identification of $$(alpha_1,…, alpha_N, beta)$$.

I’m aware of the incidental parameter problem which prevents consistent estimation of $$(alpha_1,…, alpha_N)$$ when $$T$$ is fixed and $$Nrightarrow infty$$. Hence, I suppose that $$(alpha_1,…, alpha_N)$$ are not identified when $$T$$ is fixed and $$Nrightarrow infty$$. The incidental parameter problem disappears if we also let $$Trightarrow infty$$. Does this imply that identification of $$(alpha_1,…, alpha_N)$$ can be established? How?

In what follows, I report my incomplete attempt.

$$Y_iequiv (Y_{i1},…, Y_{iT})$$ and $$X_iequiv (X_{i1},…, X_{iT})$$. $$K$$ is the size of $$beta$$. I assume $$NT> N+K$$.

First, I rewrite the model as
$$Y_{it}=sum_{ell=1}^Nalpha_l 1{i=ell}+X_{it}^top beta+epsilon_{it},$$
where I consider $$alpha_1,…, alpha_N$$ as parameters and the index $$i$$ as a random variable. Second, I rewrite A3
$$E(epsilon_{it}| i, X_{i1},…, X_{iT})=0,$$
for each $$i=1,…,N$$ and $$t=1,…,T$$.

By A3, for each $$i=1,dots, N$$, there exists a realisation of the $$Ttimes K$$ matrix $$X_{i}equiv (X_{i1},…, X_{iT})$$ (which I denote by $$x_{i}equiv (x_{i1},…, x_{iT})$$) such that
$$begin{cases} E(epsilon_{i1}|i=1, X_{i} =x_1)=0 \ vdots\ E(epsilon_{iT}|i=1, X_{i} =x_1 )=0 \ vdots\ E(epsilon_{i1}|i=N, X_{i} =x_N )=0 \ vdots\ E(epsilon_{iT}|i=N, X_{i} =x_N )=0 \ end{cases}$$
In turn, by A1,
$$begin{cases} E(Y_{i1}|i=1, X_{i} =x_1 )=alpha_1+beta x_{11} \ vdots\ E(Y_{iT}|i=1, X_{i} =x_1 )=alpha_1+beta x_{1T} \ vdots\ E(Y_{i1}|i=N, X_{i} =x_N )=alpha_N+beta x_{N1} \ vdots\ E(Y_{iT}|i=N, X_{i} =x_N )=alpha_N+beta x_{NT} \ end{cases}$$
I can more compactly rewrite this system of equations as
$$underbrace{Y}_{NTtimes 1}=overbrace{underbrace{begin{pmatrix} D & X end{pmatrix}}_{NT times (N+K)}}^{equiv Gamma} overbrace{underbrace{begin{pmatrix} alpha \ beta end{pmatrix}}_{(N+K)times 1}}^{equiv phi}.$$
Next,
$$Gamma^top Y= Gamma^top Gamma phi.$$
Thus,
$$phi=(Gamma^top Gamma)^{-1} Gamma^top Y$$
under the assumption that $$Gamma^top Gamma$$ is invertible.

I would be done with the proof if I could claim that $$Y$$ is known for $$N$$ large under assumption A1. I don’t think this is the case, though. I suppose that somehow we also need large $$T$$, but I don’t see clearly how.

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