Properties

Process

Bilinear

Linear ARMA(2, 1)

Structure

$X_t=\beta X_{t-2}{e}_{t-1}+e_t,e_t~N\left(0,{\sigma }^2\right)$ ,

$Y_t=X_t^2~\text{ARMA}\left(2,1\right)$ with ${\varphi }_1=0$

$Y_t=\lambda +{\varphi }_2{Y}_{t-2}+{\theta }_1{a}_{t-1}+a_t$ , $E\left(a_t\right)=0$ , $Var\left(a_t\right)={\sigma }_1^2$

Mean

${\mu }_Y=E\left(Y_t\right)=E\left(X_t^2\right)=\frac{{\sigma }^2}{1-{\sigma }^2{\beta }^2};{\sigma }^2{\beta }^2<1$

${\mu }_Y=E\left(Y_t\right)=\frac{\lambda }{1-{\varphi }_2}$ , $\left[\lambda =\left(1-{\varphi }_2\right){\mu }_X\right]$

Autocovariance

$R_Y\left(k\right)=\{\begin{array}{l}\frac{2{\sigma }^4}{{\left(1-{\sigma }^2{\beta }^2\right)}^2\left(1-3{\sigma }^4{\beta }^4\right)},{\sigma }^2{\beta }^2<\frac{1}{\sqrt{3}},k=0\\ \frac{2{\sigma }^6{\beta }^2}{{\left(1-{\sigma }^2{\beta }^2\right)}^2},{\sigma }^2{\beta }^2<1,k=1\\ {\sigma }^2{\beta }^2{R}_Y\left(k-2\right),k=2,3,\cdots \end{array}$

$R_Y\left(k\right)=\{\begin{array}{l}\frac{{\sigma }_1^2\left(1+{\theta }_1^2\right)}{1-{\varphi }_2^2},\left|{\varphi }_2\right|<1,k=0\\ \frac{{\sigma }_1^2{\theta }_1}{1-{\varphi }_2},{\varphi }_2\ne 1,k=1\\ {\varphi }_2{R}_Y\left(k-2\right),k=2,3,\cdots \end{array}$

Autocorrelation

${\rho }_Y\left(k\right)=\{\begin{array}{l}1,k=0\\ {\sigma }^2{\beta }^2\left(1-3{\sigma }^4{\beta }^4\right),k=1\\ {\sigma }^2{\beta }^2{\rho }_Y\left(k-2\right),k=2,3,\cdots \end{array}$

${\rho }_Y\left(k\right)=\{\begin{array}{l}1,k=0\\ \frac{{\theta }_1\left(1+{\varphi }_2\right)}{1+{\theta }_1^2},k=1\\ {\varphi }_2{\rho }_Y\left(k-2\right),k=2,3,\cdots \end{array}$