25573 Time Series Econometrics
Autumn 2020
Tutorial 6

❖ Submit your solutions electronically via the Tutorial 6 link in the Assessment folder on
UTSOnline by 23:59 on Sunday, 3 May 2020.

❖ Your submission should take the form of a single PDF file that may contain scanned
handwritten answers, answers typeset with Word, extracts of R code, and graphs and tables generated with R.

❖ All R code used to do the exercises should be submitted in a separate R file.

❖ Ensure that your name and student ID appear on all submitted documents (you should include them as a comment in your R file).
Exercise 1.
Consider the AR(2) model
zt = φ1zt 1 + φ2zt 2 + εt,
where {εt} is a white noise. The characteristic equation
1 φ1z φ2z2 = 0
has two solutions
z1 = φ1 + »φ21 + 4φ2 2φ2
and z2 = φ1 »φ21 + 4φ2 2φ2 .
Demonstrate that the lag polynomial operator factorises as
φ(L) ≡ 1 φ1L φ2L2 = (1 ∞ ω1L)(1 ∞ ω2L),
where ω1 B 1/z1 and ω2 B 1/z2 are the characteristic roots.

Exercise 2.

Factorise the autoregressive lag polynomial operator φ(L) for the ARMA(2, 1) series {zt} given
by
zt = 0.8zt 1 0.2ztt2 + εt 0.2εt 1,
where {εt} is a white noise, and determine its order of integration d. Derive the ARIMA(p, d, q)
model for the differenced series {∆dzt}.

Exercise 3.

Factorise the autoregressive lag polynomial operator φ(L) for the ARMA(3, 2) series {zt} given
by
zt = 1.75zt 1 0.5zt 2 2 0.25zt 3 + εt 0.5εt 1 + 0.25εt 2,
where {εt} is a white noise, and determine its order of integration d. Derive the ARIMA(p, d, q)
model for the differenced series {∆dzt}.

Exercise 4.

MacKinnon (2010) provides the following method for approximating the critical value for the
Dickey-Fuller test statistic
DF B φˆ1 1
SE”(φ1)
associated with the regression (5.7b) in the slides for Lecture 5:
DFα ≈ β∞(α) + β1(α) T + β2(α) T2 + β3(α) T3 ,
where T is the sample size of the regression and α is the level of significance. The values
of β∞(α), β1(α), β2(α) and β3(α) are presented in Table 1, for the 1%, 5% and 10% levels of
significance. Write an R function DFCritVals that takes the sample size of a Dickey-Fuller
regression as input and returns an array containing the 1%, 5% and 10% critical values as
output.
α β∞(α) β1(α) β2(α) β3(α)
1% %2.56574 42.2358 83.627 0
5% %1.94100 00.2686 63.365 31.223
10% %1.61682 0.2656 62.714 25.364

Table 1: The parameter values obtained by MacKinnon (2010) for computing critical values
for the Dickey-Fuller test statistic.

Exercise 5.

The worksheet Exercise 5 in the Excel workbook Tutorial 6.xlsx contains five series generated
by AR(1) models of the form
xt = φxt 1 + εt,
where |φ| ≤ 1 and {εt} is a white noise process. One of the models is unit root non-stationary,
while the remaining four are stationary. For each series, you should do the following:
❖ Run the regression (5.7b) in the slides for Lecture 5.
2
❖ Calculate the Dickey-Fuller test statistic DF.
❖ Use your function DFCritVals to determine whether or not you can reject null hypothesis
H0 : φ1 = 1 at the 5% level of significance.
❖ Interpret the above result.
Which of the five time series contains a unit root?
References
MacKinnon, J. G. (2010). Critical values for cointegration tests. Working paper.