Unconditional Least Squares

Procedure

Unconditional Least Squares minimizes the unconditional sum of squares function $S(\varphi ,\mu )$ as a compromise between conditional least squares and full maximum likelihood.

Procedure

Step 1: Set Up the Function

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

Step 2: Minimize Numerically

Set derivatives to zero:

$\frac{\partial S}{\partial \varphi }=0$ $\frac{\partial S}{\partial \mu }=0$

No Closed-Form Solution

The term $(1-{\varphi }^2)(Y_1-\mu {)}^2$ makes these equations nonlinear in $\varphi$ and $\mu$. Numerical optimization is required.

Step 3: Use Numerical Methods

Apply iterative algorithms:

• Newton-Raphson
• Gradient descent
• Gauss-Newton

Advantages

• Uses all observations (unlike conditional LS)
• Less computational burden than full MLE
• Better for short series or seasonal models

Related

• Unconditional Sum-of-Squares Function
• Conditional Least Squares
• Least Square Method