STAT 958:565 Midterm I
Spring 2022

Problem 1) Explicitly write out the form of the theoretical autocorrelation function for an AR(1) process. – 2 points

Problem 2) Suppose it is known a time-series follows an MA(1) process, where the value of θ1 is unknown. However, the
first lag autocorrelation, ρ(1), is known.
a) Find the possible value(s) that θ1 can be as an expression involving the first lag autocorrelationρ(1)? – 2 points
b) Will the value(s) arrived at for θ1 in (a) yield an invertible stochastic process? Explain. – 2 points

Problem 3) Show that for an AR(2) process, the autocorrelation function can never be 0 for two consecutive lags.
– 3 points

Problem 4) Consider an MA(2) process. What are the values of θ1 and θ2 that will yield an invertible stochastic process?
Explain your answer. – 4 points

Problem 5) Consider the ARMA process below.
()� = ()
Where
() = 3 5 − 116 4 + 83 − 553 2 + 30 − 96 �1 √2� �
And
() = (7!)6 + 25 − ()4 + 83 − 17132 + � 16� �

a) Is this process stationary? Explain how you arrive at your conclusion. – 4 points
b) Is this process invertible? Explain how you arrive at your conclusion. – 1 points

Grading note: Both parts can be resolved analytically (i.e. without having to use a computer). Extra credit will be
granted if done so. If this is proving difficult, full credit will be granted for arriving at the answers computationally
provided the code you used to arrive at your answer is included with your submission. However, to get full credit, the
computational algorithm to arrive at the needed results must be originally developed - that is this code cannot rely on
any built-in functions that are already available within the computing software that will simply provide the results
needed via a simple call to this function.
(Hint, one of the roots of the AR operator is rational).

Problem 6) Included with these problems is a dataset containing the daily closing price of Walmart stock from
05FEB2001 to 01FEB2002. – 12 points
In order to inform future decisions regarding taking positions in Walmart stock, you are charged with the task of
developing an ARIMA model to use for forecasting values of the Walmart’s closing price. In narrative form, describe the
process you use develop candidate models and arrive at a final choice. Describe in detail the methods you utilized and
include in the exposition any graphs/outputs that you used to inform the decisions made.
(Note that closing prices are only provided for trading days, there are no records for weekends and other holidays when
the markets are closed. Nonetheless, regardless of whether there are non-trading days between two consecutive records
in the dataset, treat them as consecutive (that is the indexing set is not really the actual calendar days, but rather the
trading days).