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Assignment 6 for Statistics 451, Spring 2019
1. Show how to express the AR(1) model as an infinite MA model. Explain why knowing that
you can do this might be useful.
2. The roots of a quadratic equation aX 2 + bX + c = 0 can be found by using the familiar
equation
√
−b ± b2 − 4ac
2a
Use this equation (or the corresponding equation in the course notes) to find the roots of the
AR polynomial [i.e. the values of B such that (1 − φ1 B − φ2 B2 ) = 0] for the following values of
φ1 and φ2 . Also determine if the roots are outside of the unit circle of not and, correspondingly,
whether the values of φ1 and φ2 fall inside or outside of the “AR(2) triangle.” Check your work
by graphing the polynomial and finding the roots with the RTseries function arma.roots.
(a) φ1 = 1.6 and φ2 = 0.4
(b) φ1 = 0.0 and φ2 = 0.3
(c) φ1 = 0.4 and φ2 = 0.9
3. Assume that Zt = θ0 + at , at ∼ nid(0, σa2 ), and that Wt = (1 − B)Zt .
(a) Write down a simpler form for Wt by eliminating Zt .
(b) What can you say about the model for Wt ? Is it an ARMA model? Explain.
(c) Is the model for Wt invertible or not? Why or why not?
(d) Derive the mean of Wt .
(e) Derive the variance of Wt .
(f) Derive the covariance between Wt and Wt+1 .
4. Show that for any ARMA model, Cov(Zt , at ) = σa2 .
5. Consider the model (1 − B)(1 − φ1 B)Zt = at . Is the model for Zt stationary? Why or why
not?
6. Briefly explain the relationship between the Yule-Walker equations and Durbin’s formula to
compute the PACF.
7. Consider the following four time series.
• The file CarrotPrices.txt contains monthly data on the retail price of carrots from January
1999 to June 2007 in units of dollars per hundred pounds.
• The file gasrx.txt gives the rate of gas input for a gas furnace (units are cubic feet per
minute) with it’s controller turned off. Readings were taken every nine seconds.
• SavingsRate.txt gives quarterly data on personal savings rates, in percent for the United
States from 1955 to 1979.
• INGA.AmsterdamDiff.txt gives the daily changes in IGA stock price in 2010.
For all four time series, you should
(a) Plot the data. Comment on what you see in the plot (i.e., describe interesting features).
1
(b) Look at the ACF and PACF functions of the data to tentatively identify an ARMA
model or models. Explain which model or models you would identify. Explain how you
made your choice (i.e., which is dying down and which is cutting off, if one is cutting
off).
2
...

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