University of Aberdeen
Final assessment (70%)
BU5565 – Empirical Methods in Finance Research
Session 2020-21
Question 1
RUBRIC: PLEASE READ WITH CARE
There are two sections Answer any two questions All questions carry equal marks Section A
1.1) Light records, a vinyl company, is trying to aid its prediction of sales patterns by looking at the time-series properties of the past 4 years of weekly sales (i.e. 208 weeks). The Table below shows the ACF and the PACF coefficients:
1 2 3 4 5 6 7 8 9
10
ACF
-0.127 0.104 -0.088 0.671 -0.124 0.087 -0.053 0.423 -0.049 0.068
PACF
-0.127 0.089 -0.067 0.663 0.012 -0.013 0.018 -0.052 -0.100 0.013
1.1.2) Using a simple “rule of thumb” determine which, if any, of the ACF and PACF coefficients are significant at the 5% level.
(20%)
1.1.3) Use the Box-Pierce statistic to test the null hypothesis that the first ten autocorrelation coefficients are jointly zero.
(10%)
1.2) The company is also interested in forecasting the demand of its vinyl. The table below presents the demand for each of the last 12 months:
Month
1
2
3
4
5
6
7
8
9
10
11
12
Demand
25
30
32
33
32
31
30
29
28
28
29
31
1
1.2.1) Apply exponential smoothing with smoothing constants of 0.7 and 0.8 to derive forecasts for month 13.
1.2.2) Calculate a four-month moving average for each month.
(20%)
(20%)
1.2.3) Exponential smoothing is used for modelling and generating forecasts of a series. The Box-Jenkins approach suggests the use of ARMA models and a different forecasting technique. Critically discuss the latter approach focusing on the modelling and forecasting processes.
(30%)
Question 2
2.1) Critically discuss the importance of non-stationary in time series and its various implications.
(30%)
2.2) To predict the behaviour of a stock market variable, Michael obtains daily closing data of the main German stock price index (Dax). Data is collected from February 1995 to September 2004, i.e. 2498 observations.
2.2.1) Michael decides to plot the following time series of the Dax index:
Briefly discuss what can be concluded regarding this univariate time series. (15%)
According to Schmitt et al. (2013), the financial markets are examples of highly non-
stationary systems. The authors argue that “sample averaged observables such as
variances and correlation coefficients strongly depend on the time window in which
they are evaluated”.
2
2.2.2) Michael uses the same data to conduct an Augmented Dickey-Fuller (ADF) test. The output is provided below:
Comment on the implications of the estimation results given above. Based on the output results, indicate the null and alternative hypotheses of the test. Is your conclusion in line with the argument of Schmitt et al. (2013)?
(20%)
2.2.3 To draw a conclusion about the most suitable model(s), Michael uses a correlogram as well as the AIC (Akaike Information Criterion) and BIC (Bayesian Information Criteria). The outputs are presented below:
Model
AIC
BIC
AR (1)
1.8856
1.9013
AR (2)
1.8442
1.8677
AR (3)
1.8496
1.8809
ARMA (1,1)
1.8489
1.8723
ARMA (1,2)
1.8482
1.8794
3
Critically discuss the best suitable model(s) based on the above results.
Question 3
Section B
3.1) A researcher is interested in examining the short-term relationship between three cryptocurrencies, Bitcoin returns (y1), Ethereum returns (y2), and Ripple returns (y3). To do this, the researcher estimates a VAR(2) based on their differenced returns. However, the researcher believes that 2 lags is too many and wants to restrict the VAR to 1 lag. Explain how the researcher could test whether this would be valid. The researcher estimates a VAR(2) and a VAR(1) using 121 observations. ln|∑| is -11.992 for the VAR(2), and -11.692 for the VAR(1).
(30%)
3.2) The researcher conducts the Granger causality test on the differenced returns of these three cryptocurrencies (BT-Bitcoin, ETH-Ethereum, RPL-Ripple) during 2016- 2020, and finds the results from Granger causality test as follows:
(35%)
Null Hypothesis: Obs
F-Statistic Prob.
R_ETH does not Granger Cause R_BT 1543
1.04447 0.3521
R_BT does not Granger Cause R_ETH
0.90074 0.4065
R_RPL does not Granger Cause R_BT 1543
5.55800 0.0039
R_BT does not Granger Cause R_RPL
0.73994 0.4773
R_RPL does not Granger Cause R_ETH 1543
1.68963 0.1849
R_ETH does not Granger Cause R_RPL
1.42511 0.2408
Explain the short-term causal relationship between each of two variables (5% significance level).
(30%)
3.3) The researcher also wants to find out if there is a long-term relationship between the three-return series. To examine the long-run relationship between the three series, the analyst uses the Johansen test and gets the following results:
No. of CE(s)
Test Statistics
5% Critical Value
Test Statistics
5% Critical Value
H0: 𝑟 = 0
𝜆𝑚𝑎𝑥 = 29.08
21.14
𝜆𝑡𝑟𝑎𝑐𝑒 = 44.46
29.79
H0: 𝑟 ≤ 1
𝜆𝑚𝑎𝑥 = 12.83
14.26
𝜆𝑡𝑟𝑎𝑐𝑒 = 14.45
15.49
H0: 𝑟 ≤ 2
𝜆𝑚𝑎𝑥 = 2.546
3.84
𝜆𝑡𝑟𝑎𝑐𝑒 = 2.546
3.84
Explain how many cointegrating relationships are there?
4
(30%)
3.4) The researcher also estimates a VECM(1) model on these three series, and finds the results as follows:
Standard errors in ( ) & t-statistics in [ ]
Cointegrating Eq: CointEq1
R_BT(-1) 1.000000
R_ETH(-1) 0.013060
(0.02142)
[ 0.60986]
R_RPL(-1) -0.186089
(0.02033)
[-9.15484]
C -0.001924
Error Correction: D(R_BT) D(R_ETH) D(R_RPL)
CointEq1 -0.998824 -0.607047 -0.056852
(0.03806) (0.07178) (0.07785)
[-26.2460] [-8.45699] [-0.73027]
D(R_BT(-1)) 0.025873 0.296828 -0.021158
(0.02863) (0.05400) (0.05857)
[ 0.90364] [ 5.49633] [-0.36123]
D(R_ETH(-1)) -0.000990 -0.464520 0.050006
(0.01326) (0.02500) (0.02712)
[-0.07465] [-18.5789] [ 1.84408]
D(R_RPL(-1)) -0.123723 -0.100429 -0.574166
(0.01172) (0.02210) (0.02397)
[-10.5591] [-4.54417] [-23.9537]
C -9.29E-06 -1.75E-06 -2.27E-05
(0.00102) (0.00192) (0.00209)
[-0.00910] [-0.00091] [-0.01088]
What is the long-term relationship between these three series (present in equation form)?
(10%)
Question 4
4.1) What stylised facts of financial data cannot be explained using linear time series models?
(20%)
4.2) Which of these features could be modelled using a GARCH (1,1) process? (20%)
4.3) Why, in recent empirical research, have researchers preferred GARCH (1,1) models to pure ARCH(p)?
(10%)
5
4.4) Describe two extensions to the original GARCH model. What additional characteristics of financial data might they be able to capture?
4.5) Consider the following GARCH (1,1) model: 𝑦=𝜇+𝑢, 𝑢⁓𝑁(0,𝜎2)
𝑡𝑡𝑡𝑡
𝜎2=𝛼 +𝛼𝑢2 +𝛽𝜎2
𝑡 0 1𝑡−1 𝑡−1
If 𝑦𝑡 is a daily stock return series, what range of values are likely for the coefficients 𝜇, 𝛼0, 𝛼1, and 𝛽? Based on E-views output below, what is the unconditional variance?
(20%)
GARCH = C(2) + C(3)*RESID(-1)^2 +
C(4)*GARCH(-1)
Variable Coefficient
Std. Error
0.072224
z-Statistic Prob.
C 0.176521
2.444083 0.0145
Variance Equation
C 0.523925
0.286534 0.037335
1.828494 0.0675
RESID(-1)^2 0.125524
0.042487
3.362130 0.0008
GARCH(-1) 0.848657
19.97464 0.0000
R-squared -0.000180
Mean dependent var S.D. dependent var Akaike info criterion Schwarz criterion
0.229536 3.948246 5.364547 5.378394
Adjusted R-squared -0.000180
S.E. of regression 3.948602
Sum squared resid 24042.03
Log likelihood -4134.748
Hannan-Qu
inn criter. 5.369698
Durbin-Watson stat 1.988568
(10%)
4.6) Outline how you would test for the presence of ARCH effects in the residuals 𝑢𝑡 from the following regression:
𝑦𝑡 = 𝛼 + 𝛽′𝑥𝑡 + 𝑢𝑡
Clearly describe the procedure, the variables used, the test statistic and its hypotheses.
END OF EXAMINATION
6
(20%)