Empirical Methods

(c) Standard errors: If εt are independent over time. 1. OLS standard errors α̂i, β̂i. 2. λ̂ : σ(λ̂) = σ(ft) √ T (d) Test α are jointly zero? 1. Answer: look at α̂0cov(α̂, α̂0)−1α̂. Precise forms, α̂0cov(α̂)−1α̂ = T £ 1 + f̄ 0Σ−1 f f̄ ¤−1 α̂Σα̂ ̃χN T −N −K N £ 1 + f̄ 0Σ−1 f f̄ ¤−1 α̂Σ̂α̂ ̃FN,T−N−K Intuition. R t = αi + β 0 ift + ε i t means that α̂i ≈ α + 1 T PT t=1 ε i t (except for beta fitting). Thus cov(α̂) ≈ 1 T Σ. The other terms correct for beta fitting. As usual χ is asymptotic for any iid distribution, F is finite-sample for normal ε.