25573 Time Series Econometrics
Autumn 2020 Tutorial 6
y Submit your solutions electronically via the Tutorial 6 link in the Assessment folder on UTSOnline by 23:59 on Sunday, 3 May 2020.
y 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.
y All R code used to do the exercises should be submitted in a separate R file.
y 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.
has two solutions
−φ1 + »φ21 +4φ2 −φ1 − »φ21 +4φ2 z1=2φ andz2=2φ.
22 Demonstrate that the lag polynomial operator factorises as
φ(L)≡1−φ1L−φ2L2 =(1−ω1L)(1−ω2L), where ω1 1/z1 and ω2 1/z2 are the characteristic roots.
Consider the AR(2) model
where {εt} is a white noise. The characteristic equation
zt = φ1zt−1 + φ2zt−2 + εt, 1−φ1z−φ2z2 =0
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Exercise 2.
Factorise the autoregressive lag polynomial operator φ(L) for the ARMA(2, 1) series {zt} given by
zt = 0.8zt−1 − 0.2zt−2 + 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 − 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
φˆ 1 − 1 DF
SE(φ1 ) associated with the regression (5.7b) in the slides for Lecture 5:
DFα ≈ β∞(α) + β1(α) + β2(α) + β3(α), T T2 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.
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: y Run the regression (5.7b) in the slides for Lecture 5.
”
α
β∞ (α)
β1 (α)
β2 (α)
β3 (α)
1%
5% 10%
−2.56574 −1.94100 −1.61682
−2.2358 −0.2686 0.2656
−3.627 −3.365 −2.714
0 31.223 25.364
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y Calculate the Dickey-Fuller test statistic DF.
y Use your function DFCritVals to determine whether or not you can reject null hypothesis
H0 : φ1 = 1 at the 5% level of significance.
y 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.
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