I have the following problem on my Statistics I problem set:

Suppose that $X_t = \mu + U_t$, where $U_t = V_t + \rho V_{t-1}$ and $V_t$ are iid standard normal variables.

  1. Apply a CLT to find the limiting distribution of $\sqrt{n} (\bar{X}_n -\mu)$
  2. Let $\hat{\theta}_n = (\hat\mu_n, \hat\rho_n)$ be the MLE for $\theta = (\mu, \rho)$. Find the asymptotic distribution of $\sqrt{n}(\hat{\theta}_n - \theta)$.
  3. Compare the asymptotic distributions of $\sqrt{n}(\hat\mu_n - \mu)$ and $\sqrt{n}(\bar{X}_n - \mu)$. Explain your answer.

I cannot prove part 1, since $X_{t}$ variables are not iid, as required by the standard CLT. I understand that we should use Lindeberg-Feller CLT or something stronger to prove this result.

Can anyone do it with standard CLT?

  • $\begingroup$ For part 1: If you sum $X_t$ over $t \in \{1, 2, ..., T\}$ you will get an expression involving a sum of the $V_t$ variables. You will also get a "first term" and "last term" that are slightly different from the "middle" terms (but so what?). There is no need to use any type of CLT since everything in sight is a sum of independent Gaussians. You would only need CLT if the iid $\{V_t\}$ variables were non-Gaussian. $\endgroup$ – Michael Jun 2 '18 at 3:55

Write $$ \sum_{t=1}^nU_t=\sum_{t=1}^nV_t+\rho\sum_{t=1}^nV_{t-1}. $$ In the second sum, do the change of index $j\leftarrow t-1$ to get $$ \sum_{t=1}^nU_t=\sum_{j=1}^nV_j+\rho\sum_{j=0}^{n-1}V_{j}=\rho V_0+V_n+\left(1+\rho\right)\sum_{j=1}^{n-1}V_{j}. $$ It follows that $$ \sqrt n\left(\overline{X_n}-\mu\right)=\frac 1{\sqrt n}\sum_{t=1}^nU_t =\frac{1+\rho}{\sqrt n}\sum_{j=1}^{n-1}V_{j}+\frac{1}{\sqrt n}\left(\rho V_0+V_n\right). $$ The random variable in the right hand side is Gaussian and centered. It remains to compute the variance and itslimit.

  • $\begingroup$ Thanks! How about the two other items? Any ideas? $\endgroup$ – Raul Guarini Jun 30 '18 at 22:01

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