Question 1.1.
This question uses basic properties of covariance P1–P4 which are given in the homework. Many students will complete it without requiring collaboration or online sources. Simply saying “No sources used” is acceptable if that is the case.
\[\begin{eqnarray} \mathrm{Var}\left(\hat{\mu}\left(Y_{1:N}\right)\right)&=&\mathrm{Var}\left(\frac{1}{N}\sum_{n=1}^{N}Y_{n}\right) \\ &=&\frac{1}{N^{2}}\mathrm{Cov}\left(\sum_{m=1}^{N}Y_{m},\sum_{n=1}^{N}Y_{n}\right) \mbox{ using P1 and P3} \\ &=&\frac{1}{N^{2}}\sum_{m=1}^{N}\sum_{n=1}^{N}\mathrm{Cov}\left(Y_{m},Y_{n}\right) \mbox{ using P4} \\ &=&\frac{1}{N^{2}}\left(N\gamma_{0}+2\left(N-1\right)\gamma_{1}+\ldots+2\gamma_{N-1}\right) \mbox{ using P2 to give $\gamma_h=\gamma_{-h}$} \\ &=&\frac{1}{N}\gamma_{0}+\frac{2}{N^{2}}\sum_{h=1}^{N-1}\left(N-h\right)\gamma_{h} \end{eqnarray}\]
Question 1.2.
This question is not straightforward, and most students are expected to make use of collaboration and/or online sources.
This solution is based on ionides.github.io/531w21/hw01/sol01.html. It is okay to look at the previous solution, if you acknowledge it. In that case, it is okay for your report to discuss the relationship between your work and the previous solution.
On a fairly complex question like this, if your answer looks very much like the solution but you do not acknowledge looking at the solution or discussing the homework with someone who did, then that may raise concerns. These concerns could become serious, even potentially amounting to plagiarism.
If you work things out without consulting any sources, and say so, then this may make writing up your report easier since your solution automatically counts as an original contribution. As pointed out in the syllabus and homework assignment, you must say explicitly that you used no sources, if that is the case. Otherwise your homework will be graded as having failed to list sources.
It is possible to get full points while making heavy use of sources, but this cannot be taken for granted. The burden is on you to demonstrate original contributions of your own that go beyond the sources. Discussing where you got stuck, and what you learned, are legitimate contributions.
You can reference the notes [1] or Shumway and Stoffer (2017) [2] where appropriate. Think of your homework report as a small piece of academic writing. Strong academic writing connects your own work to the wider academic community.
Referencing Wikipedia is acceptable. Indeed, Wikipedia is often a good source. It is so widely used that if you find a problem with the Wikipedia article, or an inconsistency between the Wikipedia version and other sources, that is well worth comment.
References do not need to be internet accessible. For example, it could be a paper book. However, most references nowadays are online.
Now moving on to the solution. 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 independent and identically distributed 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 first order Taylor expansion [3] of \[\hat\rho_h(Y_{1:N}) = \frac{\hat\gamma_h(y_{1:N})}{\hat\gamma_0(y_{1:N})} = \frac{U}{V}\] 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 now look to compute \[ \mathrm{Var}\left(U\right)= \mathrm{Var}\left(\sum_{n=1}^{N-h}Y_{n}Y_{n+h}\right). \] Since \(Y_{1:N}\) are independent and mean zero, we have \(\mathbb{E}[Y_{n}Y_{n+h}] = 0\) for \(h\neq 0\). Therefore, for \(m\neq n\), \[ \mathrm{Cov}\left(Y_{m}Y_{m+h},Y_nY_{n+h}\right) = \mathbb{E}\left[ Y_{m}Y_{m+h}\, Y_nY_{n+h}\right] = 0. \] Thus, the terms in the sum for \(\mathrm{Var}\left(U\right)\) are uncorrelated for \(m\neq n\) and we have \[\begin{eqnarray} \mathrm{Var}\left(U\right) &=& \sum_{n=1}^{N-h} \mathrm{Var}\left(Y_nY_{n+h}\right) \\ &=& (N-h) \, \mathbb{E}\left[Y_n^2Y_{n+h}^2\right] \\ &=& (N-h) \, \sigma^4 \end{eqnarray}\] 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 specified model) covers the true parameter with probability 0.95. [4]
Here, the interval \(\big[-1.96/\sqrt{N},1.96/\sqrt{N}\big]\) does not depend on the data. For any given model, it therefore covers \(\rho_h\) either with probability 1 or 0.
The interval \(\big[\hat\rho_h(y_{1:N})-1.96/\sqrt{N},\hat\rho_h(y_{1:N})+1.96/\sqrt{N}\big]\) 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.
The participation report was graded as part of the participation score, following the participation rubric.
References.
Notes for STATS/DATASCI 531, Modeling and Analysis of Time Series Data
Shumway, R.H., and Stoffer, D.S., 2017. Time series analysis and its applications (4th edition). New York: Springer.
LibreTexts, Taylor Polynomials of Functions of Two Variables, Equation (4)
https://en.wikipedia.org/wiki/Confidence_interval#Definition