Unconditional Sum-of-Squares Function

Definition

Unconditional Sum-of-Squares Function includes all observations by incorporating the marginal distribution of $Y_1$.

For AR(1) model with NIID$(0,{\sigma }_e^2)$ white noise:

$S(\varphi ,\mu )=(1-{\varphi }^2)(Y_1-\mu {)}^2+{\sum }_{t=2}^n[Y_t-\mu -\varphi (Y_{t-1}-\mu ){]}^2$

Key Difference from Conditional

The term $(1-{\varphi }^2)(Y_1-\mu {)}^2$ accounts for the marginal distribution of $Y_1$, which follows $N\left(\mu ,\frac{{\sigma }_e^2}{1-{\varphi }^2}\right)$.

Derivation from Likelihood

The unconditional sum of squares is derived from the full likelihood:

$L(\varphi ,\mu ,{\sigma }_e^2)=f(Y_1)\cdot f(Y_2,\dots ,Y_n|Y_1)$

Where:

• $f(Y_1)$ is the marginal density
• $f(Y_2,\dots ,Y_n|Y_1)$ is the conditional density of $e_2,\dots ,e_n$

Properties

• Minimizing $S(\varphi ,\mu )$ yields unconditional least squares estimates
• The term $(1-{\varphi }^2)(Y_1-\mu {)}^2$ makes the equations nonlinear in $\varphi$ and $\mu$
• Requires numerical optimization