% This is the take-home midterm for Stanford Stat 207

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\centerline{\bf{STANFORD UNIVERSITY}}
\centerline{\bf{STATISTICS DEPARTMENT}}
\centerline{ }
\centerline{\bf{Statistics 207 Introduction to Times Series Analysis}}
\centerline{\bf{Summer 2001}}
\centerline{ }
\centerline{\bf{Midterm}}
\centerline{ }
\noindent
{\bf{Instructions:}} This is a closed-book and closed-notes exam.  You may use 2 (or 3)
pages of notes on the exam.

\begin{enumerate}
\item Consider a time series $\{x_t\}$ consisting of a quadratic trend and additive independent
white noise $\{w_t\}$ with mean $0$ and variance $\sigma^2$.  That is
\begin{equation}
x_t = \alpha_0 + \alpha_1 t + \alpha_2 t^2 + w_t
\end{equation}
where $\alpha_0$, $\alpha_1$, and $\alpha_2$ are unknown parameters.
\begin{enumerate}
\item Show that $\{x_t\}$ is nonstationary.
\item Suggest a transformation of $\{x_t\}$ that is stationary and prove that it is in fact
stationary.
\end{enumerate}
\item Identify the following ARMA models and determine if they are causal and/or invertible.
\begin{enumerate}
\item $(1-1.05B+0.4B^2)x_t = w_t$
\item $(1-1.05B)x_t=w_t$
\item $(1+0.8B)x_t=(1-0.25B)w_t$
\end{enumerate}
\item Consider the ARIMA(0,1,1) model written as
\begin{equation}
x_t = x_{t-1} + w_t - \theta w_{t-1}
\end{equation}
where $|\theta| < 1$.
\begin{enumerate}
\item Show that the model is invertible and solve for the $\{\pi_j\}$ coefficients.
\item Re-write the invertible representation of the model where $x_t$ is equal to past
values of time series and white noise.
\item Determine the one-step ahead prediction formula by taking the conditional expectation.
Note that it can be written in the form
\begin{equation}
\tilde{x}_{n+1} = (1-\theta)x_n + \theta \tilde{x}_{n-1}
\end{equation}
for , $n>1$.
\item Indicate how the prediction formula could be used in practice.  In particular indicate
where truncation will be need to make the calculations.  (Remark: The prediction formula
derived is a forecasting method known as exponentially weighted moving averages, (EWMA).)
\end{enumerate}
\item Consider the following pairs of ACF's and PACF's.  In each case indicate whether the
pairs of diagrams is associated with an AR or MA process, state the order of the process, and
in each case write an appropriate model that you think would fit the data well.
\end{enumerate}

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