Question 1.1.
\[\begin{eqnarray} \mathrm{Var}\left(\hat{\mu}\left(Y_{1:N}\right)\right)&=&\mathrm{Var}\left(\frac{1}{N}\sum_{n=1}^{N}Y_{n}\right) \\ &=&\frac{1}{N^{2}}\mathrm{Cov}\left(\sum_{m=1}^{N}Y_{m},\sum_{n=1}^{N}Y_{n}\right) \mbox{ using P1 and P3} \\ &=&\frac{1}{N^{2}}\sum_{m=1}^{N}\sum_{n=1}^{N}\mathrm{Cov}\left(Y_{m},Y_{n}\right) \mbox{ using P4} \\ &=&\frac{1}{N^{2}}\left(N\gamma_{0}+2\left(N-1\right)\gamma_{1}+\ldots+2\gamma_{N-1}\right) \mbox{ using P2 to give $\gamma_h=\gamma_{-h}$} \\ &=&\frac{1}{N}\gamma_{0}+\frac{2}{N^{2}}\sum_{h=1}^{N-1}\left(N-h\right)\gamma_{h} \end{eqnarray}\]
Question 1.2. By definition, \[ \hat{\gamma}_{h}\left(y_{1:N}\right)=\frac{1}{N}\sum_{n=1}^{N-h}\left(y_{n}-\hat{\mu}_{n}\right)\left(y_{n+h}-\hat{\mu}_{n+h}\right). \] Here, we consider the null hypothesis where \(Y_{1:N}\) is IID with mean \(0\) and standard deviation \(\sigma\). We therefore use the estimator \(\hat\mu_n=0\) and the autocovariance function estimator becomes \[\begin{eqnarray} \hat{\gamma}_{h}\left(y_{1:N}\right) &=& \frac{1}{N}\sum_{n=1}^{N-h}y_{n}y_{n+h}, \end{eqnarray}\] We let \(\sum_{n=1}^{N-h}Y_{n}Y_{n+h}=U\) and \(\sum_{n=1}^{N}Y_{n}^{2}=V\), and carry out a Taylor first order eof \[\hat\rho_h(Y_{1:N}) = \frac{\hat\gamma_h(y_{1:N})}{\hat\gamma_0(y_{1:N})} = \frac{U}{V}\] about \((\mathbb{E}[U],\mathbb{E}[V])\). This gives \[ \hat{\rho}_{h}(Y_{1:N}) \approx\frac{\mathbb{E}\left(U\right)}{\mathbb{E}\left(V\right)}+\left(U-\mathbb{E}\left(U\right)\right)\left.\frac{\partial}{\partial U}\left(\frac{U}{V}\right)\right|_{\left(\mathbb{E}\left(U\right),\mathbb{E}\left(V\right)\right)}+\left(V-\mathbb{E}\left(V\right)\right)\left.\frac{\partial}{\partial V}\left(\frac{U}{V}\right)\right|_{\left(\mathbb{E}\left(U\right),\mathbb{E}\left(V\right)\right)}. \] We have \[ \mathbb{E}\left(U\right)=\sum_{n=1}^{N-h}\mathbb{E}\left(Y_{n}\, Y_{n+h}\right)=0, \] \[ \mathbb{E}\left(V\right)=\sum_{n=1}^{N}\mathbb{E}\left(Y_{n}^{2}\right)=N\sigma^{2}, \] \[ \frac{\partial}{\partial U}\left(\frac{U}{V}\right)=\frac{1}{V}, \] \[ \frac{\partial}{\partial V}\left(\frac{U}{V}\right)=\frac{-U}{V^{2}}. \] Putting this together, we have \[\begin{eqnarray} \hat{\rho}_{h}(Y_{1:N})&\approx&\frac{\mathbb{E}\left(U\right)}{\mathbb{E}\left(V\right)}+\frac{U}{\mathbb{E}\left(V\right)}-\frac{\left(V-\mathbb{E}\left(V\right)\right)\mathbb{E}(U)}{\mathbb{E}(V)^{2}} \\ &=&\frac{U}{N\sigma^{2}}. \end{eqnarray}\] This gives us an approximation, \[ \mathrm{Var}\left(\hat{\rho}_{h}(Y_{1:N})\right)\approx\frac{\mathrm{Var}\left(U\right)}{N^{2}\sigma^{4}}. \] We can now compute \[\begin{eqnarray} \mathrm{Var}\left(U\right)&=&\mathbb{E}\left[\left(\sum_{n=1}^{N-h}Y_{n}Y_{n+h}\right)^{2}\right] \\ &=&\mathbb{E}\left[\sum_{n=1}^{N-h}Y_{n}^{2}Y_{n+h}^{2}\right]+2\mathbb{E}\left[\sum_{i=1}^{N-h-1}\sum_{j=i+1}^{N-h}Y_{i}Y_{i+h}Y_{j}Y_{j+h}\right] \end{eqnarray}\] Since \(Y_{1:N}\) are i.i.d. and mean zero, each term in the second sum has zero expectation and \[\begin{eqnarray} \mathrm{Var}\left(U\right)&=&\mathbb{E}\left[\sum_{n=1}^{N-h}Y_{n}^{2}Y_{n+h}^{2}\right] \\ &=&\sum_{n=1}^{N-h}\mathbb{E}\left(Y_{n}^{2}\right)\mathbb{E}\left(Y_{n+h}^{2}\right) \\ &=&\left(N-h\right)\sigma^{4} \end{eqnarray}\] Therefore, \[ \mathrm{Var}\left(\hat{\rho}_{h}(Y_{1:N})\right)\approx\frac{\left(N-h\right)}{N^{2}} \] When \(n\rightarrow\infty\), \(\mathrm{Var}\left(\hat{\rho}_h(Y_{1:N})\right)\rightarrow\frac{1}{N}\), justifying a standard deviation under the null hypothesis of \(1/\sqrt{N}\).
B. A 95% confidence interval is a function of the data that constructs a set which (under a spedified model) covers the true parameter with probability 0.95.
Here, the interval \([-1/\sqrt{N},1/\sqrt{N}]\) does not depend on the data. For any given model, it therefore covers \(\rho_h\) either with probability 1 or 0.
The interval \([\hat\rho_h(y_{1:N})-1/\sqrt{N},\hat\rho_h(y_{1:N})+1/\sqrt{N}]\) covers zero if and only if \(\hat\rho^{}_h\) falls between the dashed lines. In this sense, the dashed lines have some meaning relevant to construction of a confidence interval with local coverage of 95% at \(\rho_h=0\) for \(N\) large. However, the lines really represent an acceptance region of a test under a null hypothesis of independence. This is a different thing from a confidence interval.
Question 1.3.
All statements of sources were given full credit as long as they were consistent with the solutions presented.