Noise Variance Estimation

Definition

Noise Variance Estimation determines ${\sigma }_e^2$ after model coefficients have been estimated.

After obtaining coefficient estimates, the noise variance is estimated using the relationship between theoretical variance ${\gamma }_0=\text{Var}(Y_t)$ and sample variance:

$s^2=\frac{1}{n-1}{\sum }_{t=1}^n(Y_t-\bar{Y}{)}^2$

AR(p)

${\hat{\sigma }}_e^2=(1-{\hat{\varphi }}_1{r}_1-{\hat{\varphi }}_2{r}_2-\cdots -{\hat{\varphi }}_p{r}_p)s^2$

MA(q)

${\hat{\sigma }}_e^2=\frac{s^2}{1+{\hat{\theta }}_1^2+{\hat{\theta }}_2^2+\cdots +{\hat{\theta }}_q^2}$

ARMA(1,1)

${\hat{\sigma }}_e^2=s^2\cdot \frac{1-{\hat{\varphi }}_1^2}{1-2{\hat{\varphi }}_1{\hat{\theta }}_1+{\hat{\theta }}_1^2}$

Related

• Parameter Estimation by Method of Moments