A Principal Components Approach to Cross-Section Dependence in Panels

Suppose data are available on xi_t and xi_t and we focus on the regressions

yit = θxit + uit         (RI)

yit = bxit + cwt + vit (RII)

which differ in that the latter is augmented by wt = ^P_i^N₌₁ uit/N to proxy the
unobserved random factor z_t. It can be shown that, for our baseline DGP, this
proxy is equal to the first principal component of u up to a scaling factor. To
study the properties of these regressions, we define the sequential probability
limits (plim) for T →∞and any fixed N

S_y^N_y =plimT^-1   _ty_i²_t; S_y^N_x =plimT^-1 _t yitxit           (₈)

T→∞              T→∞

and likewise for the other variables in (6) whose (co)variances are:

E (x_i²_t) = σ²_x = σ_d² + σ_z²

E(yi²t) = β²σ²d +(β + γ)²σz² +σε²

E(xityit) = βσ_d² +(β + γ)σ_z²

E(yitzt)=(β+γ)σ_z²

E(yityjt)=(β + γ)²σ_z²

E(ztdit) = E(ztεit) = E(ditεit) = 0

The auxiliary regression z_t = δx_it +η_it implies that y_it = (β+γδ)x_it+γη_it+^εit ^.
Hence, the OLS estimator θ measures (β + γδ) and the residuals Ui_t estimate
uit = γη_it + εit, with variance σ²_u ≡ E(u_i²_t)=γ² (1 - δ)²σ_z² + γ² δ² σ²_d + σ_ε² and
covariance σij ≡ E(uitujt) = γ² (1 - δ)²σ_z². Then we have wt = ^P_i^N₌₁(γη_it +
ε_it)/N = γ(1 — δ)z_t — γδd_t + ε_t where d_t = N^-1 ɪɪi dit and it follows that

E(wt²) = γ²(1 - δ)²σz² + γ²δ²N^-1σ²d + N^-1σε²
E(yitwt)=(β+γ)(1 — δ)γσ_z² — βγδN^-1σ²_d + N^-1σ_ε²
E(xitwt)  =  γ(1 — δ)σ_z² — γδN^-1σ²_d;  E(ztwt) = γ(1 — δ)σ_z²

For our baseline DGP we have

These results are used in the next sections to analyze the question of how
ʌ ʌ

well does the OLS estimator b (and by comparison θ) measure the true β.

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