Omitted variable bias
True model y = β₀ + β₁x₁ + β₂x₂ + u. Drop the relevant x₂ and the short regression hands x₂'s credit to x₁. The bias is β₂·δ, where δ = ρ·(σ₂/σ₁) is the auxiliary slope of x₂ on x₁.
Assumption (held fixed): β₁ = 1.00, σ₁ = σ₂ = 1, so δ = ρ. This is a population (plim) result, no sampling noise.
Short-regression slope on x₁1.40bias +0.40
Auxiliary slope δ = ρ·(σ₂/σ₁) 0.50Bias = β₂·δ +0.40
Omitted effect β₂+0.80
Correlation ρ = corr(x₁, x₂)+0.50
Omitting x₂ pushes the slope on x₁ above the truth: the short estimate 1.40 misses β₁ = 1.00 by +0.40. The sign follows sign(β₂)·sign(ρ); only ρ = 0 or β₂ = 0 closes the gap. More data will not help, OLS is inconsistent here.