Time Series: Computing the Autocorrelation Function for Stationary Process

Question

I'm trying to find the autocorrelation function $\rho_x(h)$ for the stationary process

$X_t + 0.1X_{t-1} = Z_t$

So I have that

$\rho_x(h) = Corr[X_t, X_{t+h}] = \frac{\gamma_x(h)}{\gamma_x(0)}$ where $\gamma_x(h) = \frac{\phi^{|h|}\sigma_z^2}{1-\phi^2}$ and $\gamma_x(0) = \frac{\sigma_z^2}{1-\phi^2}$

This simplfies to

$\rho_x(h) = \phi^{|h|}$

but I'm not sure how to compute this.

I've been trying to use the recursive definition $X_t = \phi^hX_{t-h} = \sum_{j=0}^{h-1}\phi^jZ_{t-j}$, subbing in $0.1$ for $\phi$, but I have no idea how to figure this out.

Answer 1

A few preliminaries

$X_t = \sum_{j = 0}^{+\infty}\phi^jZ_{t-j}$

You can prove this one this one just by writing

\begin{eqnarray} X_t &=& \sum_{j = 0}^{+\infty}\phi^jZ_{t-j} = \phi^0Z_{t-0} + \sum_{j = 1}^{+\infty}\phi^jZ_{t-j} \\ &=& Z_t + \phi \sum_{j = 1}^{+\infty}\phi^{j-1}Z_{t-j} \stackrel{k=j-1}{=} Z_t + \phi \sum_{k = 0}^{+\infty}\phi^{k}Z_{t- (k + 1)}\\ &=& Z_t + \phi\sum_{k=0}^{+\infty}\phi^kZ_{(t-1)-k} \\ &=& Z_t +\phi X_{t-1} \tag{1} \end{eqnarray}

where $Z_t$ is white noise with $\mathbb{E}[X_t] = 0$ and ${\rm Var}[X_t] = \sigma^2$

Roots of the characteristic equation

the characteristic equation for this process is

$$ 1 -\phi z =0 \tag{2} $$

which has only one root $\lambda = 1/\phi$. The process is thus covariance-stationary if $|\lambda| > 1$, or equivalently if $|\phi| < 1$

${\rm Var(X_t) = \sigma^2/(1-\phi^2)}$

If $|\phi|<1$ then $\{X_t\}$ is covariante stationary, and ${\rm Var}[X_t] = \sigma_X^2$ for all time $t$, therefore

\begin{eqnarray} {\rm Var}[X_t] &=& {\rm Var}[\phi X_{t-1}] + {\rm Var}[Z_t] \\ \sigma_X^2 &=& \phi^2 \sigma_X^2 + \sigma^2 \\ \sigma_X^2 &=& \frac{\sigma^2}{1 - \phi^2} \tag{3} \end{eqnarray}

${\rm Cov}(X_t, X_{t-1}) = \phi \sigma_X^2$

\begin{eqnarray} {\rm Cov}(X_t, X_{t- 1}) &=& {\rm Cov}(\phi X_{t- 1} + Z_t, X_{t-1}) = \phi {\rm Cov}(X_{t-1}, X_{t-1}) = \phi {\rm Var}[X_{t-1}] = \phi \sigma_X^2 \tag{4} \end{eqnarray}

We have then \begin{eqnarray} {\rm Cov}(X_t, X_{t- 1}) &=& \phi {\rm Cov}(X_{t-1}, X_{t-1}) = \phi \sigma_X^2\\ {\rm Cov}(X_t, X_{t- 2}) &=& {\rm Cov}(\phi X_{t - 1} + Z_t, X_{t-2}) = \phi{\rm Cov}( X_{t - 1}, X_{t-2}) \stackrel{(4)}{=} \phi^2\sigma_X^2 \\ &\vdots& \\ {\rm Cov}(X_t, X_{t- h}) &=& \phi^h \sigma_X^2 \tag{5} \end{eqnarray}

${\rm Corr}(X_t, X_{t-1}) = \phi^h$

If you call

$$ {\rm Cov}(X_t, X_{t- h}) =\gamma(h) \tag{6} $$

Then

$$ {\rm Corr}(X_t,X_{t-h}) = \frac{\gamma(h)}{\gamma(0)} = \phi^h \tag{7} $$

with

$$ \gamma(0) = {\rm Var}[X_t] = \sigma_X^2 $$