4 Large-Sample Theory and the Autoregressive Equation

The results of Section 2 are exact: unbiasedness and the Gauss–Markov theorem hold for every sample size, and under normality the \(t\) and \(F\) distributions are exact. They rest on assumption (A1), that the disturbances have mean zero conditional on all the regressor values, past and future. In time series that assumption fails as soon as a lagged value of the dependent variable is used as a regressor, and in cross sections normality is rarely credible. Without these assumptions exact results are usually unobtainable, and econometrics instead asks what happens as the sample size grows. This section builds the tools for that question and applies them to the first model that needs them, the autoregressive equation \(Y_t = \alpha + \beta Y_{t-1} + u_t\).

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