4.6 Exercises: Time Series Concepts

Exercise 4.1 Suppose the time series $\left\{ X_{t}\right\} _{t=-\infty}^{\infty}$ is independent white noise. That is $X_{t}\sim i.i.d.\,(0,\sigma^{2})$. Define two new time series $\left\{ Y_{t}\right\} _{t=-\infty}^{\infty}$ and $\left\{ Z_{t}\right\} _{t=-\infty}^{\infty}$, where $Y_{t}=X_{t}^{2}$ and $Z_{t}=|X_{t}|$. Are $\left\{ Y_{t}\right\} _{t=-\infty}^{\infty}$ and $\left\{ Z_{t}\right\} _{t=-\infty}^{\infty}$ also independent white noise processes? Why or why not?

Exercise 4.2 Realizations from four stochastic processes are given in Figure . Which processes appear to be covariance stationary and which processes appear to be non-stationary? Briefly justify your answers.

Exercise 4.3 Consider the MA(1) model $$\begin{eqnarray*} Y_{t} & = & 0.05+\varepsilon_{t}+\theta\varepsilon_{t-1},\,-1<\theta<1\\ \varepsilon_{t} & \sim & iid\,N(0,(0.10)^{2}). \end{eqnarray*}$$ This process has mean $E[Y_{t}]=0.05.$

1. Calculate $\mathrm{var}(Y_{t})$ and $\rho_{1}=\mathrm{cor}(Y_{t},Y_{t-1})$ for $\theta=0.5$ and $\theta=0.9$.
2. Using the R function arima.sim(), simulate and plot 250 observations of the MA(1) process with $\theta=0.5$ and $\theta=0.9$ . Briefly comment on the behavior of the simulated data series. Does it look covariance stationary? Does it show evidence of time dependence?

Exercise 4.4 Consider the AR(1) model $$\begin{eqnarray*} Y_{t}-0.05 & =\phi(Y_{t-1}-0.05)+\varepsilon_{t},\,-1<\phi<1\\ & \varepsilon_{t}\sim iid\,N(0,(0.10)^{2}). \end{eqnarray*}$$ This process has mean $E[Y_{t}]=0.05.$

1. Calculate $\mathrm{var}(Y_{t})$ for $\phi=0.5$ and $\phi=0.9$.
2. Calculate $\rho_{j}=\mathrm{cor}(Y_{t},Y_{t-j})$ for $\phi=0.5$ and $\phi=0.9$ and for $j=1,\ldots,5$.
3. Using the R function arima.sim(), simulate and plot 250 observations of the AR(1) process with $\theta=0.5$ and $\theta=0.9$ . Briefly comment on the behavior of the simulated data series. Does it look covariance stationary? Does it show evidence of time dependence? How is it different from the MA(1) process

Exercise 4.5 Figure shows a realization of a stochastic process representing a monthly time series of overlapping 2-month continuously compounded returns $r_{t}(2)=r_{t}+r_{t-1}$, where the 1-month continuously compounded returns ${r_{t}}$ follow a Gaussian White noise process with variance 1.

1. Based on the sample autocorrelations, which time series process is most appropriate for describing the series: MA(1) or AR(1)? Justify your answer.
2. If you think the process is an AR(1) process, what do you think is the value of the autoregressive parameter? If you think the process is a MA(1) process, what do you think is the value of the moving average parameter?