Large Sample Properties of Parameter Estimates

Properties

Large Sample Properties of Parameter Estimates describe the asymptotic behavior of estimators from maximum likelihood and least squares methods.

Asymptotic Properties

For large $n$, estimates from maximum likelihood and least squares (conditional or unconditional) are:

• Approximately normal
• Unbiased
• Consistent

Equivalence

For large samples, conditional least squares, unconditional least squares, and maximum likelihood produce identical estimates asymptotically.

Variance-Covariance Structure

The asymptotic variance-covariance matrix of parameter estimates depends on the model structure.

AR(p) Models

Parameters ${\varphi }_1,\dots ,{\varphi }_p$ have variance-covariance approximately $\frac{1}{n}{\sigma }^2{\Gamma }^{-1}$ where $\Gamma$ is the autocovariance matrix.

MA(q) Models

Parameter estimates have higher variance than AR models due to nonlinear estimation.

ARMA(p,q) Models

Combined variance structure from both AR and MA components.

Practical Implications

Sample Size	Method Preference	Reason
Small $(n<50)$	MLE or Unconditional LS	Uses all information
Medium $(50\le n<200)$	Any method	Results converge
Large $(n\ge 200)$	Conditional LS acceptable	Simple, fast

Related

• Maximum Likelihood Method
• Conditional Least Squares
• Unconditional Least Squares