Conditional Sum of Squares Function

Definition

Conditional Sum of Squares Function measures the discrepancy between observed and predicted values, conditioned on the first observation.

For AR(1) model $Y_t-\mu =\varphi (Y_{t-1}-\mu )+e_t$, the conditional sum of squares is:

$S_c(\varphi ,\mu )={\sum }_{t=2}^n[Y_t-\mu -\varphi (Y_{t-1}-\mu ){]}^2$

Why "Conditional"

The function conditions on the observed value $Y_1$ and sums squared deviations from observations $Y_2,\dots ,Y_n$. It treats AR models as regression models where $Y_{t-1}$ is the predictor.

Properties

• Minimizing $S_c$ yields conditional least squares estimates
• For stationary processes with large $n$, the difference from unconditional methods is negligible