Semester Two Final Examination, 2020 ECON7350 Applied Econometrics for Macroeconomics and Finance
Exam information
Course code and title
Applied Econometrics for Macroeconomics and Finance
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Exam technology
Semester 2, 2020
Online, non-invigilated, open-book, final examination File upload to Blackboard Assignment
Exam date and time
This examination is scheduled to commence at 12:00pm AEST on 18 November 2020.
The examination duration will be Working time (120 minutes) + additional online allowance of 30 minutes = TOTAL exam duration: 2 hours and 30 minutes.
30 minutes additional time has been incorporated in recognition of the online environment and the different circumstances that students face in their home environments. This includes allowances for network or connection issues.
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Total questions: 4 Total marks: 100
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Semester Two Final Examination, 2020 ECON7350 Applied Econometrics for Macroeconomics and Finance
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Semester Two Final Examination, 2020 ECON7350 Applied Econometrics for Macroeconomics and Finance
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Semester Two Final Examination, 2020 ECON7350 Applied Econometrics for Macroeconomics and Finance
Answer ALL questions.
Show all your working. Explain your answers.
Marks are as indicated. (Total marks: 100) Useful tables are provided at the end of the exam.
Question 1 [25 marks]
The following Figures 1 and Tables 1, 2 and 3 provide information on quarterly log of the consumer price index in Australia {𝑝𝑡 = 𝑙𝑛(𝑐𝑝𝑖𝑡 )} for 𝑡 = 1, … , 𝑇, where the sample period is from September 1948 to June 2020.
5 4 3 2 1 0
Figure 1: A plot of 𝑝𝑡.
Table 1: The sample ACF, PACF and Q-Statistics for 𝑝𝑡. EXAMINATION CONTINUES ON NEXT PAGE
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Dec-52 Mar-57 Jun-61 Sep-65 Dec-69 Mar-74 Jun-78 Sep-82 Dec-86 Mar-91 Jun-95 Sep-99 Dec-03 Mar-08 Jun-12 Sep-16
Semester Two Final Examination, 2020 ECON7350 Applied Econometrics for Macroeconomics and Finance
Table 2: The sample ACF, PACF and Q-Statistics for Δ𝑝𝑡 = 𝑝𝑡 − 𝑝𝑡−1.
p-value for the Ljunge-Box test Q-Statistic at lag 10
ARIMA(1,0,0) ARIMA(1,0,1) ARIMA(1,0,2) ARIMA(1,1,0) ARIMA(1,1,1) ARIMA(1,1,2) ARIMA(2,0,0) ARIMA(2,0,1) ARIMA(2,0,2) ARIMA(2,1,0) ARIMA(2,1,1) ARIMA(2,1,2)
-1506.56 -1625.35 -1704.05 -1818.14 -1875.95 -1883.15 -1819.44 -1865.00 -1869.78 -1872.74 -1881.28 -1879.24
-1495.57 -1610.7 -1685.73 -1807.16 -1861.31 -1864.85 -1808.47 -1846.68 -1847.8 -1858.1 -1862.98 -1857.29
1.000 1.000 1.000 0.000 0.020 0.347 0.000 1.000 1.000 0.005 0.194 0.133
Table 3: Information criteria and the p-value for the Q-Statistics for the 10th lag of the residual autocorrelation for different estimated models of 𝑝𝑡.
EXAMINATION CONTINUES ON NEXT PAGE
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Semester Two Final Examination, 2020 ECON7350 Applied Econometrics for Macroeconomics and Finance
a) Choose the best among the 12 estimated models in Table 1.
b) Justify fully your reasons for your choice of model.
c) Explain the process by which you came to your choice.
d) Consider the following model for 𝑝𝑡 and values of 𝜀𝑇, 𝑝𝑇 and 𝑝𝑇−1 :
𝑝𝑡 = 0.002 + 1.9772𝑝𝑡−1 − 0.9873𝑝𝑡−2 + 𝜀𝑡 − 0.6301𝜀𝑡−1 𝜀𝑇 = −0.02, 𝑝𝑇 = 4.739701 and 𝑝𝑇−1 = 4.758749
(2 marks) (4 marks) (5 marks)
(1.1) (1.2)
i) Compute the unconditional mean for 𝑝𝑡, 𝜇 = 𝐸(𝑝𝑡). Show all working. (5 marks)
ii) Using information up to period 𝑇, forecast 𝑝𝑇+1. Show all working.
e) Assume the log of quarterly real per capita consumption {𝑐𝑡} is characterised by an AR(2) process:
𝑐𝑡 =𝜇+𝜙1𝑐𝑡−1 +𝜙2𝑐𝑡−2 +𝜀𝑡 (1.3)
A unit root exists if a solution 𝑧 to the condition 1 − 𝜙1𝑧 − 𝜙2𝑧2 = 0 lies on the unit
circle; that is if |𝑧| = 1.
i) Rewrite the model (1.3) to permit an Augmented Dickey-Fuller test for a unit
root. (2 marks)
ii) State clearly the null hypothesis for this test in terms of the parameters of the transformed model. (2 marks)
iii) If 𝑇 = 198, the Dickey-Fuller test statistic is -2.7 and we adopt a 5% level of significance, what conclusion would you make from this test regarding 𝑐𝑡 ?
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Semester Two Final Examination, 2020 ECON7350 Applied Econometrics for Macroeconomics and Finance
Question 2 [25 marks]
A researcher is interested in the stability of the real interest rate. Let 𝜋𝑡 denote the inflation rate and 𝑟 denote the nominal interest rate. The real interest rate is 𝑟 − 𝜋 .
She estimates two models. First the following static regression
𝑟 =𝑎+𝑏𝜋 +𝑢 𝑡𝑡𝑡
and then the following ARDL model,
(ARDL) where 𝑎, 𝑏, 𝑐 and 𝑑 are coefficients to be estimated and 𝜀𝑡 is a disturbance.
The error correction form for the ARDL model is
Δ𝑟 =𝑎+𝜃Δ𝜋 +𝛼(𝑟 −𝛽𝜋 )+𝜀 (ECM)
𝑟 =𝑎+𝑏𝜋 +𝑐𝑟 +𝑑𝜋 +𝜀 𝑡 𝑡 𝑡−1 𝑡−1 𝑡
𝑡 𝑡 𝑡−1 𝑡−1 𝑡
The researcher estimates the REGN using 𝑇 = 198 observations and obtains the following results:
𝑟 =4.32+0.008𝜋 +𝑢̂ 𝑡𝑡𝑡
Δ𝑢̂𝑡 = 𝛾̂𝑢̂𝑡−1 + 𝜙1Δ𝑢̂𝑡−1 + 𝜙2Δ𝑢̂𝑡−2 + 𝜈𝑡
where 𝛾̂ = −0.000189 and with standard error 𝑠𝑒(𝛾̂) = 0.0009 such that the 𝑡 – statistic for 𝐻0: 𝛾 = 0 is 𝑡 = −0.21.
a) Transform from the ARDL to the ECM. Showing all working. (8 marks)
b) Write 𝛼 and 𝛽 as functions of 𝑏, 𝑐 and 𝑑. (4 marks)
c) Whatrestrictionsmustbeplacedoncoefficients𝑎,𝑏,𝑐and𝑑tosatisfythenull hypothesis that the real interest rate, 𝑟 − 𝜋 , is a stable equilibrium relationship?
Explain your answer (hint, you may find your answer in b) useful).
d) Consider the estimation results for the REGN. If both 𝑟 and 𝜋 are 𝐼(1), what can you 𝑡𝑡
conclude about the presence of a stable long-run relationship between 𝑟 and 𝜋 ? 𝑡𝑡
Explain how you came to this conclusion using the evidence provided. (8 marks)
EXAMINATION CONTINUES ON NEXT PAGE
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Semester Two Final Examination, 2020 ECON7350 Applied Econometrics for Macroeconomics and Finance
Question 3 [25 marks]
a) Figure 2 plots 5000 observations on daily log returns of the BHP share price. What can we say about the pattern of volatility in the returns and what features would you expect the marginal distribution of the return to display? (4 marks)
Figure 2: Plot of the daily log returns of the BHP share price; 𝑟 . 𝑡
b) Explain carefully how a researcher can formally test for the presence of ARCH effects in the variance of returns. (4 marks)
c) Demonstrate the sense in which the GARCH model is a generalisation of the ARCH framework. As a part of your answer you should (i) show that the GARCH(1,1) model is equivalent to a particular infinite order ARCH, but with restrictions, and (ii) state these restrictions.
d) A researcher has estimated several volatility models for the daily log returns of the
BHP share price; 𝑟. They also compute the ACF and PACF for the squared 𝑡
standardised residuals; nu2.
i) Output from the estimation of three of these models are presented in Tables 4, 5 and 6. Name each of these models and use the output to compute an estimate of the unconditional variance of the error term for Models 1 and 3.
ii) Based on the estimation output presented, which model would you choose to explain the variance of daily log returns? Fully justify your selection.
EXAMINATION CONTINUES ON NEXT PAGE
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Semester Two Final Examination, 2020 ECON7350 Applied Econometrics for Macroeconomics and Finance
Table 4: Estimation output for Model 1.
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Semester Two Final Examination, 2020 ECON7350 Applied Econometrics for Macroeconomics and Finance
Table 5: Estimation output for Model 2.
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Semester Two Final Examination, 2020 ECON7350 Applied Econometrics for Macroeconomics and Finance
Table 6: Estimation output for Model 3.
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Semester Two Final Examination, 2020 ECON7350 Applied Econometrics for Macroeconomics and Finance
Question 4 [25 marks]
Consider two variables – 𝑦𝑡 and 𝑥𝑡 – and the regression 𝑦𝑡 = 𝜇 + 𝛽𝑥𝑡 + 𝜀𝑡
a) Compare and contrast the following three cases by considering the 𝑡-statistic for the hypothesis 𝐻0: 𝛽 = 0, the 𝑅2 and the test for autocorrelation in the residuals:
Case 1: the vector (𝑦𝑡, 𝑥𝑡) is 𝐼(0):
Case 2: the vector (𝑦𝑡,𝑥𝑡) is 𝐼(1) and the variables do not share common trends; Case 3: the vector (𝑦𝑡,𝑥𝑡) is 𝐼(1) and the variables share one common trend.
b) What implications could Case 2 above have for macroeconomic or financial econometric analysis? Discuss.
c) Find a cointegrating vector for the cointegrating relation that exists between (𝑦𝑡,𝑥𝑡) given the following data generating process (DGP) in which the 𝜀𝑖,𝑡 are zero mean, I(0) processes:
DGP: 𝑥𝑡 = 𝑠𝑡 +𝜀𝑥,𝑡 𝑦𝑡 = 2𝑠𝑡 +𝜀𝑦,𝑡 𝑠𝑡 = 𝑠𝑡−1 +𝜀𝑠,𝑡 (8 marks)
Question 5 (no marks)
Specify any assumptions you have made in completing the exam and to which questions those assumptions relate. You may also include queries you may have made with respect to a particular question, should you have been able to ‘raise your hand’ in an examination room.
END OF EXAMINATION Statistical tables on next page.
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Semester Two Final Examination, 2020 ECON7350 Applied Econometrics for Macroeconomics and Finance
Useful Tables
Table 1: 1% and 5% critical values for Dickey-Fuller t-tests. No constant Constant Constant
Sample Size T = 25 T = 50
T = 100 T = 250 T = 500 T = ∞
No trend No trend Trend 1% 5% 1% 5% 1% 5%
-2.66 -1.95 -2.62 -1.95 -2.60 -1.95 -2.58 -1.95 -2.58 -1.95 -2.58 -1.95
-3.75 -3.00 -3.58 -2.93 -3.51 -2.89 -3.46 -2.88 -3.44 -2.87 -3.43 -2.86
-4.38 -3.60 -4.15 -3.50 -4.04 -3.45 -3.99 -3.43 -3.98 -3.42 -3.96 -3.41
Source: W.A. Fuller, Introduction to Statistical Time Series, Wiley, , 1976 ,p. 373
Table 2: 1% and 5% critical values for Dickey- -tests. 𝐻0: 𝛼 = 𝜌 = 0 in 𝐻0: 𝛿 = 𝜌 = 0 in
Sample Size T = 25 T = 50
T = 100 T = 250 T = 500 T =∞
Δ𝑦𝑡 =𝛼+𝜌𝑦𝑡−1 +𝜀𝑡 Δ𝑦𝑡 =𝛼+𝛿𝑡+𝜌𝑦𝑡−1 +𝜀𝑡
1% 5% 1% 5%
7.88 5.18 10.61 7.24 7.06 4.86 9.31 6.73 6.70 4.71 8.73 6.49 6.52 4.63 8.43 6.34 6.47 4.61 8.34 6.30 6.43 4.59 8.27 6.25
Source: D.A. Dickey & W.A. Fuller, “Likelihood ration statistics for autoregressive time series with a unit root,” Econometrica 49 (1981).
Table 3: Dickey-Fuller cointegration test statistics.
Asymptotic critical values for residual unit root tests for cointegration (with a constant term).
Number of variables (incl. 𝑦𝑡)
Source: R. Davidson & J.G.
Econometrics, Oxford University Press (1993).
Significance level 1% 5%
-3.90 -3.34 -4.29 -3.74 -4.69 -4.10 -4.96 -4.42
-3.45 -3.81 -4.13
MacKinnon, Estimation and Inference in
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