EC3060 Econometrics 2: Time Series Summer Examinations 2019/20
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EC3060
Summer Examinations 2019/20
Econometrics 2: Time Series
Section A: Answer ONE question
1. (a) Characterise the roots of the processes xt and yt
xt = 1.19xt−1 − 0.19xt−2 + t
yt = yt−1 + yt−2 + t
In each case t is i.i.d. white noise. (10 marks)
(b) ‘A random walk is unforecastable’. What does this statement mean? (10 marks)
(c) A researcher is interested in the relationship between the I(1) variables Gross Domestic Product, Real Money Balances, Short and Long-term interest rates. Table 2 shows results of ADF tests for residuals from cointegrating equations estimated with constants and no trends for this system. Table 1 shows an extract from MacKinnon’s Tables of critical values for the ADF test of a regression residual in cases where no constants, τnc, a constant, τc, and a trend and constant, τct, are included in the cointegrating equation. N is the number of variables in the system. Do you see any evidence of cointegration and what do you learn about the potential drawbacks of this test procedure? (10 marks)
Table 1: MacKinnon’s critical values at 5% significance
N 1 2 3 4 5
rnc |
-1.04 |
-1.53 |
-2.68 |
-3.09 |
-3.07 |
rc |
-1 61 |
-2 62 |
-3 13 |
-3 47 |
-3 78 |
rct |
-2.89 |
-3.19 |
-3.50 |
-3.65 |
-3.80 |
Table 2: ADF test results
Coefficient
Depvar on lagged residual Std. Err.
Gross Domestic Product Real Money Balances |
-0.07044 -0 01850 |
0.02883 0 01839 |
Short Rate |
-0.1720 |
0.04529 |
Long Rate |
-0.2224 |
0.05044 |
(d) Consider the data generating process (D.G.P.) yt = o1yt − 1 + o2yt −2 + o3yt −3 + ct where ct is i.i.d. white noise. A researcher observes the series {yt } (but not the D.G.P.) and
suspects there may be unit root. The researcher runs the test regression
Calculate the first order autocorrelation of the researcher’s regression errors. You may
use the notation yk for the kth autocovariance of yt , and are not required further to calculate these quantities. What is the problem with this test regression for the given
D .G . P? How would you test the hypothesis of a unit root? Give details of your test
regression, null and alternative hypotheses, test statistic, and the distribution of the statistic. What are the difficulties in testing for unit roots in economic data? (20 marks)
2. (a) ‘The AR(1) model uses the autocorrelation between yt and yt+k to produce forecasts’. Do you agree? (10 marks)
(b) Do the autocorrelation function and the impulse response function coincide for a general AR(p)? (10 marks)
(c) Consider the GARCH(1,2)
ct |Ψt − 1 ~ N(0, ht )
ht = a0 + a1 ct(2)− 1 + g1 ht − 1 + g2 ht −2
where Ψt − 1 is the information set available after period s - 1. Derive the process for ct(2) .
(10 marks)
(d) Under what conditions will the unconditional variance for this process be well defined? Find this unconditional variance. (10 marks)
(e) What features of economic and financial time series can GARCH processes, and the extensions we have discussed in lectures, capture effectively and what features do they omit? (10 marks)
Section B: Answer ONE question
Please use a separate booklet
3. (a) Derive the k-step ahead forecasts of the n-varible VAR(2) process
yt = c + A1yt−1 + A2yt−2 + t
where t is an n.1 i.i.d. white noise vector. (5 marks)
(b) Show that the forecasts can be written as a weighted average of the mean E(yt) := µ and of the last observations on the process. Use the weights you derive to discuss the asymptotic behaviour of the forecast as k → ∞. (15 marks)
(c) Find an expression for the orthogonalised impulse responses of the system. How many restrictions are imposed by this identification scheme, what do they restrict and how do you interpret them? How could you construct an economic rationale for this sort of identification procedure? (15 marks)
(d) Table 3 extractskey results from the estimation of a VAR(2) in differences of the log of Gross Domestic Product, Inflation and the short-term interest rate. The Table reports results for different specifications of the varables in the VAR. RMSE is the root-mean-square error for the reported equation in each specification; it is calculated using a small sample degrees of freedom correction, so the denominator of the mean is T − k where there are k parameters in each equation in the model. All specifications are estimated on a sample size of 130. Using a small-sample testing approach decide if there is any evidence of Granger causality from the differenced short-term interest rate to the GDP growth rate? Is there any evidence of Granger causality from any variable to the difference of inflation? Critical values for t and F distributions from Carter Hill et. al. Undergraduate Econometrics are presented in Table 4 and Table 5 respectively. Use these to approximate the the inference you would obtain with a more detailed table of critical values. (15 marks)
Table 3: VAR results
Eqn |
Variable |
Model 1 Coef Std Err |
Model 2 Coef Std Err |
Model 3 Coef Std Err |
|||
dlgdp |
l .dlgdp l2 .dlgdp l .dinfl l2.dinfl l.dsrate l2.dsrate const RMSE |
0.4542 0.18484 0.01859 0.04116 0.001163 -0.001823 0.002023 |
0.08703 0.08442 0.05566 0.05507 0.0006356 0.0006361 0.0006654 |
0.4313 0.1794 0.01985 0.01985
0.002204 |
0.08901 0.08733 0.05761 0.05761
0.0006775 |
0.4272 0.1864 |
0.08541 0.08542 |
0.002160 0.0006722 0.00547 |
|||||||
0.005329 |
0.005524 |
||||||
|
.dlgdp l2 .dlgdp l .dinfl l2.dinfl l.dsrate l2.dsrate const RMSE |
0.23774 -0.2517 -0.9269 -0.4054 0.0008845 -0.0002731 -0.0001844 |
0.1283 0.1244 0.08204 0.08117 0.0009368 .0009376 0.0009807 |
0.2422 -0.2485 -0.9235 -0.4036
-0.0002634 |
0.1260 0.1237 0.08157 0.08067
0.0009592 |
0.08128 0.08089
-0.0003212 0.0006864 0.00792 |
|
0.007855 |
0.007822 |
||||||
|
.dlgdp l2 .dlgdp l.infl l2.infl l.dsrate l2.dsrate const RMSE |
-1.166 18.06 8.958 9.846 0.04785 0.1420 -0.1315 |
11.53 11.18 7.372 7.294 0.08418 0.08426 0.09060 |
|
|
||
0.7257 |
Table 4: Critical values for t distributions, a = 0.05
DoF Critical Value
120
与
1.658
1.645
Table 5: Critical values for F distributions, a = 0.05
DoF denominator DoF numerator
|
1 |
2 |
3 |
4 |
5 |
120 |
3.92 |
3.07 |
2.68 |
2.45 |
2.29 |
2023-06-02