A question about nonstationary test in time series.

Question

Suppose $x_t=\alpha x_{t-1}+\varepsilon_t-\theta\varepsilon_{t-1}$, where $\varepsilon_t\sim N(0,1)$, i.i.d, $|\theta|<1$ and $x_0=0$. Consider a regression of $x_t$ onto $x_{t-1}$, ie. an estimator $$\hat\alpha=\frac{\sum_{t=2}^nx_tx_{t-1}}{\sum_{t=2}^nx_{t-1}^2}$$ a) Assume that $|\alpha|<1$. Is $\hat\alpha$ consistent for $\alpha$? Find a limiting distribution for $\hat\alpha$.

b) Now assume that $\alpha=1$. Is $\hat\alpha$ consistent in this case. Find the limiting distribution of $\hat\alpha$.

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Can anybody give some hints of which theorem can be used to solve this problem?

Answer 1

Maybe it will help\guide you:

1. Using WLLN and the continuous mapping for $g(x)=1/x$ theorem $$ \frac{\frac{1}{n}\sum x_t x_{t-1}}{\frac{1}{n}\sum x_t} \xrightarrow{p}\frac{\mathbb{E}[x_tx_{t-1}]}{\mathbb{E}[x_t^2]}, $$ where $$ \mathbb{E}[x_tx_{t-1}] = \mathbb{E}[\alpha (x_{t-1} + \epsilon_t - \theta \epsilon_{t-1})x_{t-1}] =\alpha\mathbb{E}[x_{t-1}^2], $$ hence as $|\alpha| <1$ implies stationarity, we have that $\mathbb{E}x_t^2 = \mathbb{E}x_{t-1}^2$, so $$ \hat{\alpha}_n \xrightarrow{p}\alpha. $$ I would guess that the limiting distribution is $$ \sqrt{n}(\hat{\alpha} - \alpha)\xrightarrow{D}\mathcal{N}(0, 1/\mathbb{E}x_t^2 ). $$

In the second part I think that you cannot use the stationarity argument to prove constitency.