Statistics 451
Spring 2013
Final Examination
Name
You must show all of your work
1. Let Zt denote the amount of sales in month t, measured in millions of dollars. Consider the
seasonal time series model for sales defined by
(1 − φ1 B)Wt = θ0 + (1 − Θ1 B12 )at ,
at ∼ nid(0, σa2 )
with the differencing scheme
Wt = (1 − B12 )Zt .
(a) Write down the unscrambled equation for Zt (note that Wt should not be included in
this unscrambled equation for Zt ).
(b) What is the interpretation of θ0 in this model? Be precise by including the units of θ0
in your answer.
2. Consider the ARMA model
(1 − φ1 B)(1 − B)Zt = (1 − θ1 B)at ,
at ∼ nid(0, σa2 ).
(a) Find the roots of the AR and the MA polynomials.
(b) Use the information from part (a) to make a statement about the stationarity of Zt .
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3. A time series Zt can be described by an AR(1) model φ1 (B)Zt = at where at ∼ nid(0, σa2 ).
Assume that a realization Z1 , Z2 , . . . Z250 is available to estimate all of the parameters in this
model and to compute forecasts for Z252 , using Z250 as the forecast origin.
(a) Give an expression for Z252 , based on the AR(1) model.
(b) Give an expression that can be used to compute Zb250 (2), the forecast for Z252 .
(c) Use (a) and (b) to derive an expression for e250 (2), the forecast error in Zb250 (2), as a
function of φ1 and unobserved residuals.
(d) Use (c) to derive an expression for the variance of e250 (2) as a function of φ1 and σa .
(e) Give an expression for a 95% prediction interval for Z252 .
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4. The following seasonal time series model can be used to describe Zt , the monthly average
temperatures in Ames, Iowa.
Wt = (1 − Θ1 B12 )at ,
at ∼ nid(0, σa2 )
with the differencing scheme
Wt = (1 − B12 )Zt
where Zt is measured in degrees F.
(a) Is Zt stationary? Explain why or why not.
(b) Write down the unscrambled equation for Zt .
(c) Write down the unscrambled equation for Wt (note, that Zt should not be included in
this unscrambled equation).
(d) Derive an expression for Var(Wt ).
(e) Derive expressions for ρ12 and ρ13 for the Wt process.
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5. Consider the following intervention model.
Zt = ν(B)It + Nt
= ν(B)It + ψ(B)at =
ω0
1 − θ1 B
It +
at ,
(1 − δ1 B)
(1 − B)
at ∼ nid(0, σa2 ).
(a) Find the unscrambled equation giving Zt as a function of only the parameters and a
finite number of lagged values of Zt , It and at .
(b) Briefly explain (and draw a graph to illustrate) the behavior of the transfer function
ν(B) =
ω0
(1 − δ1 B)
to a step input function if δ1 = 0.10 and ω1 = 2. Compare this with the response to an
impulse input.
(c) Assuming that It can be predicted without error, derive the prediction standard error
for 1 and 2-step ahead forecasts for Zt .
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6. The partial autocorrelation function (PACF) is an important tool for time series model identification.
(a) Briefly explain the practical interpretation and purpose of the PACF.
(b) If you only had a multiple regression computer program available to do computations,
explain how you could use it to get a good approximation for the PACF.
7. Regression methods are commonly used to make predictions, based on time series data. For
many such models, prediction of future values of the response Y will depend on predictions
of future values of explanatory variables X. For example if a model is fit relating total
electricity consumption for a month to average temperature for that month, in order to make
the prediction electricity consumption for next month one requires a prediction of average
temperature for next month. Such a prediction can be obtained by using an appropriate
univariate SARIMA model. If these predictions are used with standard regression analysis
procedures (e.g., lm() and predict() in R or Fit Model in JMP) to compute a prediction
interval for next month’s electricity consumption, the prediction intervals could be misleading.
Explain why.
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