Regression model + expected value, variance and autocorrelation of the error term

Consider this regression model $$Y_t=X_t\beta+\epsilon_t, ~~~~~~~~~~\epsilon_t \sim WN(0, \sigma^2_{\epsilon})$$

with 3 different specifications of the error term:

1. $$\epsilon_t=\alpha_1\epsilon_{t-1}+\alpha_2\epsilon_{t-2}+\eta_t,~~~~~~~~~~~~~~~~~~~~~~~~~~\eta_t \sim WN(0, \sigma^2_{\eta})$$
2. $$\epsilon_t=\eta_t+\theta_1\eta_{t-1}+\theta_2\eta_{t-2},~~~~~~~~~~~~~~~~~~~~~~~~~~\eta_t \sim WN(0, \sigma^2_{\eta})$$
3. $$\epsilon_t=\alpha_1\epsilon_{t-1}+\eta_t~~~~~~~~~~$$ with $$|\alpha|=1, ~~~~~~~~~~\eta_t \sim WN(0, \sigma^2_{\eta})$$

a) Compute the expected value, the variance and the autocorrelation coefficients of $$\epsilon_t$$ for each specification

b) Transform the model $$Y_t=X_t\beta+\epsilon_t$$ in first differences and let $$\varepsilon_t$$ be the transformed error. Show that $$\varepsilon_t$$ is not white noise.

My attempt was the following:

POINT A) - (1) FIRST SPECIFICATION

$$E(\epsilon_t) =E(\alpha_1\epsilon_{t-1}+\alpha_2\epsilon_{t-2}+\eta_t)$$

$$~~~~~~~~~~~~~~~~=\alpha_1E(\epsilon_{t-1})+\alpha_2E(\epsilon_{t-2})+E(\eta_t)$$

$$~~~~~~~~~~~~~~~~=0$$

$$\operatorname{Var}(\epsilon_t)=\operatorname{Var}(\alpha_1\epsilon_{t-1}+\alpha_2\epsilon_{t-2}+\eta_t)$$

$$~~~~~~~~~~~~~=\alpha_1^2 \operatorname{Var}(\epsilon_{t-1})+\alpha_2^2\operatorname{Var}(\epsilon_{t-2})+\operatorname{Var}(\eta_t)$$

$$~~~~~~~~~~~~~=\alpha_1^2\cdot\sigma^2_{\epsilon}+\alpha_2^2\cdot\sigma^2_{\epsilon}+\sigma^2_{\eta}$$

$$\rho(h)=\frac{\operatorname{Cov}(\epsilon_t,\epsilon_{t-h})}{\operatorname{Var}(\epsilon_t)}$$ From this point, how can I compute $$\operatorname{Cov}(\epsilon_t,\epsilon_{t-h})$$?

For the other ones

Basically, I have to follow the same reasoning (hoping that the previous calculus are correct)

POINT B) - SHOW THAT $$\varepsilon_t$$ IS NOT WHITE NOISE

$$Y_t-Y_{t-1}=(X_t-X_{t-1})\beta+\varepsilon_t,~~~~~~~~~~\varepsilon_t=\epsilon_t-\epsilon_{t-1}$$

How can I prove that the transformed error term is not white noise?

Any help would be appreciated

• I don't understand your model. How can $\epsilon_t$ be both white noise and follow the specifications? Feb 27, 2019 at 15:28

Your variance is wrong: $$\epsilon_{t-1}$$ and $$\epsilon_{t-2}$$ are not independent!
Hint: $$\mathbf{E}\big[\eta_t \epsilon_t] = \mathbf{E}\big[\eta_t(\alpha_1\epsilon_{t-1} + \alpha_2\epsilon_{t-2} + \eta_t)] = \sigma_\eta^2;$$ because $$\epsilon_k$$ is independent of $$\eta_j$$ if $$j>k$$ (think about why this is the case!); similarly, \begin{aligned} \mathbf{E}\big[\eta_{t-1} \epsilon_t] &= \mathbf{E}\big[\eta_{t-1}(\alpha_1\epsilon_{t-1} + \alpha_2\epsilon_{t-2} + \eta_t)] = \alpha_1\sigma_\eta^2;\\ \mathbf{E}\big[\eta_{t-2} \epsilon_t] &= \mathbf{E}\big[\eta_{t-2}(\alpha_1\epsilon_{t-1} + \alpha_2\epsilon_{t-2} + \eta_t)] = \alpha_2\sigma_\eta^2;\\ \end{aligned} Hence, \begin{aligned} \mathrm{Var}(\epsilon_t) &= \mathbf{E}\big[\epsilon_t \epsilon_t\big]\\ &= \mathbf{E}\big[(\alpha_1 \epsilon_{t-1} + \alpha_2\epsilon_{t-2} + \eta_t) \epsilon_t\big]\\ &= \mathbf{E}\big[(\alpha_1 (\alpha_1 \epsilon_{t-2} + \alpha_2 \epsilon_{t-3} + \eta_{t-1}) + \alpha_2(\alpha_1\epsilon_{t-3} + \alpha_2\epsilon_{t-4} + \eta_{t-2}) +\eta_t\big) \epsilon_t\big] \\\ &=\cdots \end{aligned}