0

Let $X_t = Z_t + \theta Z_{t-1}$ where $ \left\{ Z_t \right\} \approx WN(0, \sigma^2)$. Find variance $ VarX_t$ and covariance function.

Of course we have $EX_t = 0$. Then $VarX_t = EX_t^2 = E(Z_t^2 + 2 \theta Z_t Z_{t-1} + \theta^2Z_{t-1}^2) = \sigma^2 + \theta^2 \sigma^2$

I can't understand the last equal.

I suppose that $E(Z_t^2 + 2 \theta Z_t Z_{t-1} + \theta^2Z_{t-1}^2) = EZ_t^2 + 2 \theta EZ_tZ_{t-1} + \theta^2 EZ_{t-1}^2 = E(Z_t - EZ_t)^2 + 2 \theta EZ_tZ_{t-1} + \theta^2 E(Z_{t-1} - EZ_{t-1})^2 = \sigma^2 + 2 \theta EZ_tZ_{t-1} + \theta^2 \sigma^2 $.

Hence I suppose that $EZ_tZ_{t-1} = 0$ but why? $Z_t$ and $Z_{t-1}$ are independent?

If you explain me that, maybe i will know how find covariance function.

Thanks in advance!

Thomas
  • 2,556

1 Answers1

1

The W in WN$(0,\sigma^2)$ stands for "white" as in "white noise" and means that the process $(Z_t)$ is independent. Since $(Z_t)$ is also centered, yes, $E[Z_tZ_{t-1}]=0$ and yes, $E[Z_t^2]=\sigma^2$.

Did
  • 279,727