Least Squares for Linear Trend Estimation

Procedure

For a linear trend model $Z_t={\beta }_0+{\beta }_{1}t+X_t$:

1. Minimize the sum of squared residuals: $Q({\beta }_0,{\beta }_1)={\sum }_{t=1}^n(Z_t-{\beta }_0-{\beta }_{1}t{)}^2$
2. Set partial derivatives with respect to ${\beta }_0$ and ${\beta }_1$ to zero.
3. Solve for the estimates: $$\begin{array}{rl}{\hat{\beta }}_1& =\frac{{\sum }_{t=1}^n(Z_t-\bar{Z})(t-\bar{t})}{{\sum }_{t=1}^n(t-\bar{t}{)}^2}\\ {\hat{\beta }}_0& =\bar{Z}-{\hat{\beta }}_1\bar{t}\end{array}$$