Let g_{i, t} = (y_t − βx_t)x_t−i (i = 0,1,…). For any finite set I of nonnegative integers, let

be the optimal generalized method of moments estimator of β based on the moment conditions

By the results of Breusch, Qian, Schmidt, and Wyhowski (1999), the ordinary least squares estimator,

, is at least as efficient as any

if, and only if, for all i ≥ 1, the condition Eg_{i, t} = 0 is redundant given Eg_{0, t} = 0; i.e.,

where D_i = E(∂g_{i, t} /∂β) and

. Let γ_i = Cov(x_t, x_t−i). Then, for i ≥ 0,

so (1) is seen to hold. The last equality in (2) follows by writing

with x_t and u_t independent. That this is possible follows from the bivariate normality of x_t−i and x_t.