EC3060 Econometrics 2: Time Series Summer Examinations 2020/21
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EC3060
Summer Examinations 2020/21
Econometrics 2: Time Series
Section A: Answer ONE question
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1. (a) Find all the fixed points, if any exist, of the following processes and comment on their stability. The initial observation for each process is on the real line.
xt = 1.03xt_1 yt = yt_1 zt = 0.9 + 0.87zt_1 (8 marks) (b) Using the definitions of the variables in part (a), discuss which processes below are stationary. Give your reasoning, xt + ∈t yt + νt zt + ∈t (1) where ∈t ~ N (0, 0.5 + φt 0.2) where %φ% < 1, νt ~ N (0, 1.95). (7 marks) |
Figure 1: ACF: Term Spread
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Figure 2: PACF: term spread
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(c) Consider the sample ACF, Figure 1, and sample PACF, Figure 2, for quarterly U.S. observations on the term-spread of interest rates. Based on this information, how would you begin the process of univariate time-series modelling for this variable? How would you proceed to a preferred model? (10 marks)
(d) Define the autocorrelation function for a time series. Compute autocorrelations up to order 3 and sketch the rest of the autocorrelation function for the following model:
yt = 1.24yt_1 _ 0.79yt_2 + et
where et ~iid N (0, 1). (15 marks)
(e) An AR(1) model was estimated for the data in question (c), and the errors were saved as ‘res 1’. Using the evidence in Table 1, decide if there is any evidence of autocorrelation in the residuals for this regression.
Table 1: Regressions on res 1
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Reg 1 Reg 2 res 1 coeff std error coeff std error |
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L. res 1 .3222367 .0743551 |
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L2.res 1 -.0987785 .076676 |
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L3.res 1 l4.res 1 |
.1951798 .0721825 -.0243178 .0732983 |
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L .Term Spread - .0753591 .0449838 - .000868 .0294703 |
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const .1162728 .0756982 .0047861 .0568089 |
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SS Model SS Residual Nobs |
6.139 61.443 238 |
.00024841 67.583 238 |
(10 marks)
2. (a) Compute the roots of the following processes:
yt = 1.93yt_1 _ 0.93yt_2 + et
xt = 0.32 + 0.75xt_1 + et _ 0.75et_1
Are there any conditions on et such that any of these processes are stationary processes? (10 marks)
(b) Show how to test for stationarity of a variable generated by an AR(4) process. Derive the regression, state the test statistic, null and alternative hypotheses. (5 marks)
(c) Read the ADF test regressions for log U.S. earnings data in Table 2. Decide which is the most appropriate regression and report your decision on stationarity from that regression. (10 marks)
Table 2: ADF regressions
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D.EARN |
Reg 1 coeff std err |
Reg 2 coeff std err |
Reg 3 coeff std err |
Reg 4 coeff std err |
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L. EARN |
-.0506878 .0208045 |
-.0022586 .0028467 |
-.0474477 .0205317 |
-.0021501 |
.0028166 |
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L D EARN |
- 1426351 0653244 |
- 1691756 0649535 |
- 1548501 0639429 |
- 1764324 |
0637278 |
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L2.D. EARN |
.059046 .0648877 |
.0398487 .0649847 |
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trend |
0108581 0046214 |
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0101661 0045651 |
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const |
3.018193 1.11922 |
.4463195 .2354806 |
2.856496 1.105878 |
.4476152 |
.2318925 |
(d) Consider the regressions in Table 3 for the squared residuals 
t(2), from the regression of earnings growth on its lag. What do we learn from these results and how would you model this phenomenon? (10 marks)
Table 3: Squared residual regressions
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Reg 1 Reg 2 Reg 3 Reg 4
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const |
.1993898 .0653396 .1872752 .1038223 .0656378 .1177241 -.0317402 .0664039 -.0119039 .0865047 .0651915 .0000371 8.73e-06 .0000404 |
.0645096 .1806829 .0646876 .2042898 .065191 .1108219 .0646859 .0652234
8.33e-06 .0000416 7.97e-06 .0000465 |
.0634521
7.37e-06 |
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R2 T |
0.0677 237 |
0.0580 238 |
0.0531 239 |
0.0417 240 |
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(e) Why is it important to ensure there is no serial correlation in the residuals in a
regression with lagged dependent variables? What do you do if you find such a problem
in your estimated residuals? (15 marks)
Section B: Answer ONE question
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3. (a) Consider the VAR(2) in n variables: yt = c + A1yt_1 + A2yt_2 + et where et ~iid N(0, Σ). Derive an expression for the k-step ahead forecast error variance, and provide a condition under which this variance tends to a limit. (You do not need to compute the limit). (10 marks) (b) Define Granger Causality. (3 marks) (c) Consider the VAR estimation results, in Table 4, for Outstanding Housing Loans (Loans), House Prices (HPI) and Mortgage Interest Rates (Mort). Summarise the Granger Causality patterns between the variables. RMSE stands for Root Mean Square Error and has been computed using a small-sample degrees of freedom correction. |
2023-06-06