4.1 Why exact theory breaks down

In the autoregressive equation the regressor in period \(t + 1\) is \(Y_t\), which contains \(u_t\). So \(u_t\) is correlated with a regressor, not in its own equation but in the next. (A1) requires \(E(u_t \mid Y_0, Y_1, \dots , Y_{T-1}) = 0\), and this fails because \(Y_t\), which is in the conditioning set, depends on \(u_t\). The proof of unbiasedness in Theorem 2.3.2 therefore does not apply, and in fact the least squares estimator of \(\beta \) is biased in every finite sample (Software 4.7.1 measures the bias).

What survives is weaker but still useful: \(u_t\) is uncorrelated with the regressor in its own equation, \(Y_{t-1}\), because \(Y_{t-1}\) depends only on earlier disturbances. That contemporaneous condition is enough for least squares to be consistent, that is, to converge to \(\beta \) as \(T \to \infty \), and for its sampling distribution to be approximately normal in large samples. Making those statements precise is the work of this section.

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