ECON334 Financial Econometrics JUNE 2018
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FORMAL EXAMINATION PERIOD: SESSION 1, JUNE 2018
ECON334
Financial Econometrics
Part A – Multiple Choice Questions (20 in total, 1 mark each)
Question 1
The ACF and PACF of a time series are shown in the following graph. They indicate that the
model for the time series is best characterised as:
0.4
0.3
0.2
0.1
0
-0.1
-0.2
-0.3
-0.4
acf
(a) MA(1)
(b) AR( 1)
(c) MA(2)
(d) AR(2)
(e) ARMA(1,1)
Question 2
Suppose that you have estimated the first five autocorrelation coefficients using a series of length 81 observations and found them to be
Lag |
1 |
2 |
3 |
4 |
5 |
Autocorrelation Coefficient |
0.412 |
-0.203 |
-0.332 |
0.005 |
0.543 |
Which autocorrelation coefficients are significantly different from zero at the 5% level?
(a) The first and fifth autocorrelation coefficient
(b) The first, second, third and fifth autocorrelation coefficient
(c) The first, third and fifth autocorrelation coefficient
(d) The second and fourth autocorrelation coefficient
(e) The fourth autocorrelation coefficient
Question 3
Find the roots of the characteristic polynomial of the AR(2) process given by
yt = 3yt−1 − 2yt−2 + ut , where ut is a white noise process. Using the results of your calculation, which of the following statements about the process is true?
(a) the roots are 1 and 0.5, so the process is stationary
(b) the roots are 1 and 0.5, so the process is non-stationary
(c) the roots are 2 and 1 so the process is stationary
(d) the roots are 2 and 1 so the process is non-stationary
(e) None of the above
Question 4
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 is said to Granger-cause y2 and y2 not to Granger-cause y1 , it is the case that:
(a) α13 and α14 significant; α21 and α22 not significant
(b) α21 and α22 significant; α13 and α14 not significant
(c) α21 and α23 significant; α11 and α13 not significant
(d) α11 and α13 significant; α21 and α23 not significant
(e) None of the above
Question 5
What are the steps required to estimate an ARCH/GARCH model?
(a) First specify the correlation and variance equations. The software (EViews in our
case) specifies the Log-Likelihood Function (LLF) and generates estimates of the parameters that maximise the LLF
(b) First specify the correlation and variance equations. The software (EViews in our
case) specifies the Log-Likelihood Function (LLF) and generates estimates of the parameters that minimise the LLF
(c) First specify the mean and variance equations. The software (EViews in our case) specifies the Log-Likelihood Function (LLF) and generates estimates of the parameters that maximise the LLF
(d) First specify the mean and variance equations. The software (EViews in our case) specifies the Log-Likelihood Function (LLF) and generates parameter values that minimise the LLF
(e) None of the above
Question 6
Which of the following can be regarded as an advantage of an EGARCH(1,1) model over a GARCH(1,1) model:
(a) It has fewer parameters for estimation than a GARCH(1,1)
(b) It has one less non-negativity condition on its parameters than does a GARCH(1,1)
(c) It always provides an estimate of conditional volatility which is positive.
(d) It treats positive and negative news shocks equally
(e) None of the above
Question 7
Given the following forecasts and actual values of a return series what is the percentage of correct sign predictions?
Forecast |
Actual |
0.20 0.15 -0.20 -0.06 0.04 |
0.40 -0.20 0.10 -0.10 0.05 |
20%
(b)
(c)
(d) 80%
(e) None of the above
Question 8
Consider the following process:
yt = 0.55 + 0.4ut−2 + 0.2ut−1 + ut
What is the optimal forecast of yt+3 made at time t, given that all information up to and including time t is available; in particular that ut = 0.3, ut−1 = −0.6, ut−2 = 0.4 and ut−3 = 0.8.
(a) 1. 15
(b) 0.73
(c) 0.55
(d) 0.67
(e) 0.80
Question 9
Consider the following process:
yt = 1.0 + 0.5yt−1 + 0.3ut−2 + 0.2ut−1 + ut
What is the optimal forecast of yt+2 , given that all information up to and including time t is available; in particular, it is known that yt = 0 , ut = 0.3, ut−1 = 0.6, and ut −2 = 0.4 ?
0.54
1.24
1.71
2.00
Question 10
The random variable yt follows the ARMA(2,1) process given by:
yt = 5 + 0.3yt−1 + 0.2yt−2 + ut + 0.5ut−1 where ut is a white noise process. The optimal forecast of yt+j as j becomes very large (i.e. as j → ∞) is:
(a) 0.0
(b) 1.0
(c) 5.0
10.0
(e) a very large number (i.e. yt+j → ∞)
Question 11
What is an appropriate approach to testing for ‘ARCH effects’?
(a) Run a regression, save the residuals, regress the squared residuals on their lags and
conduct a hypothesis test to see whether the coefficients on the lagged squared residuals are jointly equal to zero
(b) Run a regression, obtain the fitted values, regress the fitted values on their squared
lags and conduct a hypothesis test to see whether the coefficients on the lagged squared fitted values are jointly equal to zero
(c) Run a regression, save the residuals, regress the residuals on their lags and conduct a hypothesis test to see whether the coefficients on the lagged residuals are jointly equal to zero
(d) Run a regression, obtain the fitted values, regress the fitted values on their lags and conduct a hypothesis test to see whether the coefficients on the lagged fitted values are jointly equal to zero
(e) None of the above
Question 12
Suppose two variables are cointegrated with each other. Which of the following statements is true?
(a) there is a linear combination of the two variables which is stationary
(b) all linear combinations of the two variables are stationary
(c) there is no linear combination of the two variables which is stationary
(d) one of the two variables is stationary, the other is non-stationary
(e) none of the above
Question 13
Consider the process yt = θut−1 + ut where ut is independently and identically distributed with mean zero and variance σ2 for all t. Which of the following statements is true:
(a) yt and ut are both white noise processes
(b) yt is a white noise process but ut is not
(c) ut is a white noise process but yt is not
(d) yt and ut are both serially correlated processes at all leads and lags (e) yt and ut are serially uncorrelated processes at all leads and lags
Question 14
Consider an AR(1) – ARCH(2) model of returns, given by:
yt = c + φyt−1 + ut
σt = α0 + α1ut−1 + α2ut−2
What are the conditions that need to be satisfied to ensure that u has a
variance?
(b) α0 ≥ 0, α1 ≥ 0, α2 ≥ 0, α1 + α2 < 1
(c) α0 ≥ 0, α1 ≥ 0, α2 ≥ 0, α0 + α1 + α2 < 1
(d) α0 > 0, α1 ≥ 0, α2 ≥ 0, α1 + α2 < 1
(e) φ> 0, α1 + α2 < 1
Question 15
A researcher would like to run an augmented Dickey-Fuller 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
(b) ∆yt =ψyt −1 + αi ∆yt −i + ut and H0 :ψ = 1 , respectively
(c) ∆yt =ψ∆yt−1 + αi yt−i + ut and H0 :ψ = 0 , respectively
(d) ∆yt =ψyt −1 + αi yt −i + ut and H0 :ψ = 1 , respectively
(e) yt =ψ∆yt−1 + αi ∆yt−i + ut and H0 :ψ = 0 , respectively
Question 16
A stationary ARMA(1,1) process would have the following characteristics:
(a) the ACF and PACF coefficients are significant at lag 1 and then decay (b) the ACF and PACF coefficients are not significant at any lag
(c) the ACF and PACF coefficients are significant at lag 1 and then the PACF coefficients decay while the ACF coefficients drop to near zero
(d) the ACF and PACF coefficients are significant at lag 1 and then the ACF coefficients decay while the PACF coefficients drop to near zero
(e) the ACF and PACF coefficients are close to one and are significant at all lags
Question 17
Consider the ARCH(2) model for conditional variance given by
σ = ω + α u + α u
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 , α1 = 0.3 , α2 = 0.2 , and we are given u1 = 0.4 and u0 = 0.3 . Calculate the conditional variance at time t=2 (i.e. σ2(2) ). The correct answer is given by:
(a) 0.126
(b) 0.240
(c) 0.168
(d) 0.198
(e) 0.060
Question 18
Which of the following can be regarded as a limitation of the standard GARCH(1,1) model:
(a) It has fewer parameters to be estimated than does a typical ARCH(p) model
(b) It provides an estimate of unconditional volatility but not of conditional volatility
(c) It gives more weight to negative news shocks than to positive news shocks
(d) It gives equal weight to both positive and negative news shocks
(e) None of the above
Question 19
We wish to apply the F-test in order to test H0: F3 + F4 = 2 in the following regression Yt = F1 + F2X2t+ F3X3t+ F4X4t + ut .
If the restricted regression equation utilised by the F-test is given by
Zt = F1 + F2X2t + F3Wt + vt
what are the Zt and t variables?
(a) Zt = 2Yt − X4t , t = X3t − 2X4t
(b) Zt = Yt − 2X4t , t = X3t − X4t
(c) Zt = Yt − X4t , t = 2X3t − X4t
(d) Zt = Yt − 2X2t , t = 2X3t − X4t
(e) None of the above
Question 20
Which of the following can be used to test for non-normality in the residuals from a regression?
(a) LM (Breusch-Godfrey) test
(b) Dickey-Fuller test
(c) Ramsey’s RESET test
(d) Jacque-Bera test
(e) None of the above
Part B – SHORT ANSWER QUESTIONS
Answer four out of five short answer questions in the examination booklet provided.
Question 1
This question is regarding the following equation: Yt = F1 + F2X2t + F3X3t + ut .
– T = 15
– (X′X)− 1 = l 3.5
– (X′ y) = l 2.2 |
3.5
1.0
6.5
6.5 |
Given the information provided above answer the following questions.
(a) What is the value of F̂2 ? (2 marks)
(b) Compute Var(F̂2 ) – the variance of F̂2 . (2 marks)
(c) Test H0: F2 = −4 vs. H1: F2 ≠ −4 using the 5% significance level. (1 mark)
Question 2
The following questions relate to the classical linear regression model (CLRM) assumptions
(a) What does the assumption Var(ut) = a 2 (for all t) imply about Var(u) – the variance
matrix of the residuals vector u . Given an example of Var(u) which satisfied the assumption. (2 marks)
(b) Which of the CLRM assumption is tested using the Breusch-Godfrey test? Specify the
null and alternative hypothesis used in the test, and outline how the test is conducted (including the distribution of the test statistic). (2 marks)
(c) How can we test E(ut) = 0? (1 mark)
Question 3
(a) What is meant by cointegration? Describe in detail the Engle-Granger test for
cointegration between two variables. Be sure to explain each step of the test. (3 marks)
(b) What is meant by a spurious regression? How would you detect that the results from a
regression between two variables are spurious? (2 marks)
Question 4
You have data for the return on a weighted portfolio of all the stocks in the market. The return series is denoted MKT and the sample is daily from 3 January 2010 to 29 December 2017. The total number of daily observations is 2,013. You decide to estimate the mean equation as an MA(1) and the variance equation as in the GJR model. Specifically, you estimate the following model:
MKT = c + u + 9u
σt(2) = α0 + α1u1 + βσt2−1 +γu1It−1
where It −1 = 1 if ut −1 < 0 and = 0 otherwise.
You estimated this model in EViews and obtained the following results.
Dependent Variable: MKT
Method: ML ARCH - Normal distribution (BFGS / Marquardt steps)
Sample: 1/04/2010 12/29/2017
Included observations: 2013
Convergence achieved after 28 iterations
Mean Equation
Coefficient |
Std. Error |
-Statistic |
Prob. |
|
c |
0.049 |
0.015862 |
3.105341 |
0.0019 |
9 |
-0.037 |
0.020680 |
-1.822697 |
0.0683 |
Variance Equation |
||||
α0 |
0.034 |
0.0037 8.969 |
0.0000 |
|
α1 |
-0.035 |
0.0093 -3.625 |
0.0003 |
|
γ |
0.259 |
0.0247 10.497 |
0.0000 |
|
β |
0.859 |
0.0137 62.528 |
0.0000 |
|
R-squared Adjusted R-squared S.E. of regression Sum squared resid Log likelihood Durbin-Watson stat |
0.0016 0.0011 0.9564 1839.6 -2415.2 2.0052 |
2023-06-17