Least Squares for Quadratic Trend Estimation

Procedure

For a quadratic trend model $Z_t={\beta }_0+{\beta }_{1}t+{\beta }_2{t}^2+X_t$:

1. Minimize the sum of squared residuals: $Q({\beta }_0,{\beta }_1,{\beta }_2)={\sum }_{t=1}^n(Z_t-{\beta }_0-{\beta }_{1}t-{\beta }_2{t}^2{)}^2$
2. Set partial derivatives with respect to ${\beta }_0,{\beta }_1,{\beta }_2$ to zero.
3. Solve the resulting system of linear equations (Normal Equations) to find ${\hat{\beta }}_0,{\hat{\beta }}_1,{\hat{\beta }}_2$.