ECON334 Financial Econometrics SESSION 2, NOVEMBER 2019
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FORMAL EXAMINATION PERIOD: SESSION 2, NOVEMBER 2019
ECON334
Financial Econometrics
Part A – Multiple Choice Questions (20 in total, 1 mark each)
Question 1
Consider the following process:
yt = 0.5 + 0.2ut −3 + 0.3ut −2 + 0.4ut −1 + ut
What is the optimal forecast of yt+2 , given that all information up to and including time t is available; in particular that ut = 0.4, ut −1 = 0.6, ut −2 = 0.5 and ut −3 = −0.2.
(a) 0.94
(b) 0.74
(c) 1.3
(d) 0.5
(e) None of the above
Question 2
The random variable yt follows the ARMA(2, 1) process given by:
yt = 0.3 + 0.4yt −1 + 0.5yt −2 − 0.4ut −1 + ut where ut is a white noise process. The unconditional mean of yt is:
(a) 0.33
(b) 0.0
(c) 3.0
(d) 0.8
(e) None of the above
Question 3
A stationary autoregressive process of order two (i.e. an AR(2)) would have the following characteristics:
(a) a decaying ACF and PACF
(b) an ACF and a PACF with two significant spikes
(c) an ACF with one significant spike and an PACF with two significant spikes
(d) a decaying PACF, and an ACF with two significant spikes
(e) a decaying ACF, and a PACF with two significant spikes
Question 4
Which of the following statements are likely to be true in relation to the behaviour of daily returns from shares traded on a major stock exchange?
(i) returns cannot usually be forecast with any accuracy because the time series of stock prices resembles a random walk
(ii) ARCH and GARCH can be useful for modelling common patterns in the variance of returns such as volatility clustering
(iii) share prices usually fluctuate around a deterministic time trend
(iv) there is no pattern in either share returns or in the variance of returns.
(a) (i) only
(b) (iii) only
(c) (iii) and (iv) only
(d) (i) and (ii) only
(e) (i) and (iv) only
Question 5
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 well-defined
conditional and unconditional variance?
(b) α0 ≥ 0, α1 ≥ 0, α2 ≥ 0
(c) α + α < 1
(d) Conditions (a) and (c)
(e) Conditions (b) and (c)
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.
Year |
Observed Value |
Forecast value |
2015 |
100.5 |
102.0 |
2016 |
103 |
102.4 |
2017 |
104 |
107.2 |
2018 |
107 |
106.0 |
2019 |
111 |
112.0 |
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 16
The data generating model for the first difference of yt is given as follows:
∆yt = c + ∆yt −1 + ut
where ut is a white noise process and c ≠ 0.
Which statement is correct?
(a) ∆yt is an I(0) process
(b) ∆yt is an I(1) process with drift
(c) yt is stationary around a deterministic trend
(d) yt is stationary around a constant mean
(e) ∆yt is an I(1) process without drift
Question 17
Consider the following statements about ARCH and GARCH models:
(i) They are used to model the conditional variance of a time series
(ii) They model volatility clustering in the data
(iii) They are applied to stationary time series
(iv) The conditional variance is time varying and the unconditional variance is constant and generally finite
Which of those statements is correct?
(a) (i) only
(b) (i) and (ii) only
(c) (i), (ii) and (iii) only
(d) (i), (ii) and (iv) only
(e) All of the statements are correct
Question 18
A researcher is trying to select an appropriate ARMA model for a data series. Three different models are considered: ARMA(1,2), AR(2) and MA(2). Each model was estimated with an intercept term using 50 observations. The log of the estimated residual variance i.e. ln(Gˆ 2 ) for each model is shown in the table below. Which model would be selected based on the AIC information criteria?
Model |
ln(Gˆ 2 ) |
ARMA(1,2) |
0.91 |
AR(2) |
0.92 |
MA(2) |
0.97 |
(a) ARMA(1,2)
(b) AR(2)
(c) MA(2)
(d) There is not enough information to calculate the AIC value for each model (e) The AIC value is the same for each model so any can be selected
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?
Forecast |
Actual |
-0.10 0.30 0.40 0.05 0.04 |
-0.05 0.15 0.60 -0.02 0.05 |
(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
Coefficient |
Std. Error |
-Statistic |
Prob. |
|
c φ |
0.017 0.183 |
0.009 0.215 |
1.89 0.85 |
0.030 0.198 |
|
||||
Variance Equation |
||||
α0 |
0.645 |
0.077 |
8.38 |
0.000 |
α1 |
0.156 |
0.025 |
6.24 |
0.001 |
β |
0.814 |
0.035 |
23.26 |
0.000 |
(i) What is the value of the long-run forecast
2023-06-24