3.1 Time as a regressor
The simplest use of a regressor that is not an economic variable is a time trend, \[ Y_t = \alpha + \beta t + u_t, \qquad t = 1, \dots , T, \] which estimates the average change in \(Y\) per period. Because the values of \(t\) are known and evenly spaced, the arithmetic simplifies, especially if time is measured from the middle of the sample.
Theorem 3.1.1. If the time index takes the values \(T_i\) with \(\sum T_i = 0\) (for example \(-2, -1, 0, 1, 2\) for five years), the least squares estimates of the trend line \(Y = \alpha + \beta T + u\) are \[ b = \frac {\sum Y_iT_i}{\sum T_i^2}, \qquad a = \bar Y . \]
Proof. With \(\bar T = 0\), the deviation \(T_i - \bar T\) is \(T_i\) itself, so \[ b = \frac {\sum (Y_i - \bar Y)(T_i - \bar T)}{\sum (T_i - \bar T)^2} = \frac {\sum Y_iT_i - \bar Y\sum T_i}{\sum T_i^2} = \frac {\sum Y_iT_i}{\sum T_i^2} . \] Then \(a = \bar Y - b\bar T = \bar Y\). □
Centring the time index changes the meaning of the intercept, which becomes the fitted value at the middle of the sample rather than at some arbitrary origin, but it changes nothing else. The slope, the fitted values and the residuals are the same however time is numbered, by Theorem 1.3.4 applied to \(T\).
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