Question 6

The following is the GJR extension of a GARCH model for equity returns:

rt = δ0  + ut

σt(2)  = α0  + α1u1 + βσt2−1 + γu1It−1

where

I = 1 if u < 0 and I = 0 otherwise.

ut is white noise and rt is the return on a stock market.

We are given the fitted equation: σˆt2  = 1.2 + 0.11 + 0.5σˆ1 + 0.61It −1

Given that σˆ1  = 0.5 and t −1  = −0.4 , the conditional variance σˆt2     is:

(a) 1.490

(b) 1.200

(c) 3.866

(d) 1.562

(e) 1.466

Question 7

What condition on the parameters will make the conditional variance more responsive to negative ‘news’ shocks than to positive ‘news’ shocks in the EGARCH model below?

log(σt(2)) = ω + α | ut−1 | + βlog(σ1 ) +γ ut−1

σt−1 σt−1

(b) α > 0

(c) α < 0

(d) γ > 0

(e) γ < 0

Question 8

Consider the following bivariate VAR(2):

y1t = α10 + α11y1t −1 + α12y1t −2 + α13y2t −1 + α14y2t −2 + u1t

y2t = α20 + α21y1t −1 + α22y1t −2 + α23y2t −1 + α24y2t −2 + u2t

When y1    does not Granger-cause y2   but y2  does Granger-cause y1 , it is the case that:

(a) α21  and α23   significant; α11  and α13   not significant

(b) α21  and α23 not significant; α11  and α13  significant

(c) α21  and α22   significant; α13  and α14   not significant

(d) α21  and α22   not significant; α13  and α14  significant

(e)       None of the above

Question 9

ARCH and GARCH models are estimated using:

(a)        The Ordinary Least Squares (OLS) method

(b)         The OLS method with an adjustment for heteroscedasticity

(c)         The method of maximum likelihood

(d)         The method of non-linear least squares

(e)         The two stage least squares method, the first stage for the mean equation and the second stage for the variance equation.

Question 10

The ordinary (i.e. usual) standard errors can be used for reliable inference when the residuals from an estimated regression are found to be:

(a)       those from a spurious regression

(b)       uncorrelated and heteroscedastic

(c)       autocorrelated and homoscedastic

(d)       indicative of an incorrect functional form

(e)       uncorrelated and homoscedastic

Question 11

A researcher is interested in forecasting the stock price index in a certain country. The observed stock price index values from 2015 to 2019 are shown in the table along with their forecast values from some forecasting model.

The mean squared forecast error is:

2.97

(b)

(c)

(d)

(e)         None of the above

Question 12

Consider the process yt = 3yt−1 − 2yt−2 + ut , where ut is white noise. Consider the following statements about the stationarity of yt :

(i)        the relevant characteristic polynomial is 1 − 3z + 2z2 with roots equal to 1.0 and 0.5

(ii)        the relevant characteristic polynomial is 1 + 3z − 2z2   with roots equal to 0.5 and 2

(iii)       the relevant characteristic polynomial is z2  − 3z + 2 with roots equal to 1.0 and 2

(iv) yt is a stationary process

(v) yt is a non-stationary process

Which of the above statements are true concerning the stationarity of yt ?

(a)       (i) and (v) only

(b)       (ii) and (v) only

(c)       (iii) and (v) only

(d)       (ii) and (iv) only

(e)       (iii) and (v) only

Question 13

Which of the assumptions below is NOT an assumption for the process yt to be a white noise process?

(a) E(yt ) = µ for all t

(b) E(yt − µ)2  = σ2   for all t

(c) E(yt − µ)(yt −s − µ) = γs for all s ≠ 0

(d) E(yt − µ)(yt −s − µ) = 0 for all s ≠ 0

(e) yt is independently and identically distributed

Question 14

A researcher would like to run an Augmented Dickey-Fuller (ADF) test on the variable yt .    What is the regression that would be estimated and what is the null hypothesis  (H0 ) of the test?

(a)        ∆yt =ψyt−1 + αi ∆yt−i + ut and H0  :ψ = 0,  respectively

=1

(b)        ∆yt =ψ∆yt −1 + ut and H0  :ψ = 1,  respectively

(c)

yt =ψyt −1 + αi yt −i + ut

and H0  :ψ = 0, respectively

(d)        ∆yt =ψyt −1 + ut and H0  :ψ = 0,  respectively

(e)        ∆yt =ψ∆yt−1 + αi yt−i + ut and H0  :ψ = 0,  respectively

=1

Question 15

Consider the GARCH(1,1) model for conditional variance given by

σt2  = ω + αu1 + βσt1

where ut is the residual from a regression equation (i.e. the mean equation) at time t . We have the following parameter estimates ω = 0.06 , α = 0.3 , β= 0.6 , and we are given u0  = 0.4 and σ0  = 0.3 . The conditional variance at time t=1 (i.e. σ1(2) ) is:

(a) 0.600

(b) 0.360

(c) 0.288

(d) 0.234

(e) 0.162

Question 19

Suppose the process for yt is given by

yt = 0.6 + 0.4yt −1 − 0.5ut −1 + ut

where ut is white noise. Compute the expectation of yt +1 , conditional on the information available at time t −1,  namely, that yt −1  = 0.5 and ut −1  = 0.6. The answer is:

(a)       1.0

(b)       0.8

(c)       0.6

(d)       0.5

(e)       0.4

Question 20

Given the following forecasts and actual values of a return series what is the percentage of correct sign predictions?

(a) 10%

(b) 40%

(c) 80%

(d) 60%

(e) 20%

Part B – There are TWO Questions with several parts. Each question is worth 10 marks in total. Answer BOTH questions in the examination booklet provided.

Question 1 (10 marks)

A researcher has data on the daily percentage returns rt on the New York stock exchange index over the last 100 days of trading (i.e. the researcher’s sample of returns is 100).        He/she decides to estimate an AR(1) model for returns i.e. rt = c + φrt −1 + ut −1    and to save  the estimated residuals t .

(a) The researcher decides to obtain the ACF of the squared residuals t(2) . What features of the return data could he/she learn by doing this? (2 marks)

(b) The researcher decides to perform an LM test for seventh-order ARCH effects. Write down the null and alternative hypothesis for this test, a description of any regression you would run, an explanation of how you would construct the test  statistic, a statement on the distribution of the test statistic under the null          hypothesis, the 5% critical value for the test statistic, and what you would         conclude if the null is rejected. (5 marks)

(c) The researcher decided to estimate the AR(1)-GARCH(1,1) model of returns given by

rt = c + φrt −1 + ut

σt(2)  = α0 + α1u1 + βσt2−1

and obtained the following results:

Mean equation

(i)         What is the value of the long-run forecast