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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 zt and yt

zt = 1.19zt − 1 - 0.19zt −2 + et

yt = yt − 1 + yt −2 + et

In each case et 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, rnc , a constant, rc , and a trend and constant, rct , 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)

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 + ct

where ct 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 ) := u and of the last observations on the process. Use the weights you derive to discuss the asymptotic behaviour of the forecast as k → 与. (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 s 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)