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SECTION A - COMPULSORY SECTION Question 1
Provide brief answers to all parts of this question
(a) Explain the workings of the ordinary least squares (OLS) methodology in estimating sample regression functions. What is the difference between OLS estimators and OLS estimates?
(b) Explain the following statement: “Heteroskedasticity in the residuals of the linear regression model could be caused by volatility clustering”.
(c) Describe the concept of structural breaks in time-series regression analysis. How can we test for structural breaks in linear regression models?
(d) Discuss the concept of stationarity. How is stationarity tested in practice?
(e) Describe the concept of seasonality in econometric analysis. How could we account for seasonality in regression models?
SECTION B - ANSWER ONE QUESTION FROM THIS SECTION
Question 2
An analyst estimates a linear regression model where the dependent variable is the spread between the Italian 10-year government bond yield and the German 10-year government bond yield (SPR). The independent variables are the bid-ask spread of the Italian 10-year government bond (BA); the log of the VIX index (LVIX); the difference between the Italian and German expected public debt to GDP ratios (DEBTED); the difference between the Italian and German expected fiscal balance to GDP ratios (BALED); the log of the Italian real effective exchange rate (LRCP); and the difference between the Italian and German growth rates of industrial production (GIND). The regression model also includes a constant. The sample period is January 1999 – August 2011 (monthly frequency). The estimated regression is reported below:
Dependent Variable: SPR
Method: Least Squares
Sample (adjusted): 1999M01 2011M08 Number of observations: 152 after adjustments
Variable Coefficient
Constant 0.105940 BA 0.030022 LVIX 0.497681
Std. Error
3.835675 0.002384 0.071445 0.007366 0.023715 0.796834 0.010706
t-Statistic
0.027620 12.59154 6.965976
-3.780579 -1.411043 -0.078478 -2.923518
p-value
0.9780 0.0000 0.0000 0.0002 0.1604 0.9376 0.0040
0.533026 0.525654 0.276134 0.415391 0.332705 0.806950
DEBTED BALED LRCP GIND
R-squared
Adjusted R-squared
S.E. of regression
Sum squared resid
Log likelihood
F-statistic 70.08795 p-value (F-statistic) 0.000000
-0.027847 -0.033463 -0.062534 -0.031300
0.743602 0.732993 0.271620 10.69773
Mean dependent var S.D. dependent var Akaike info criterion Schwarz criterion Hannan-Quinn criter. Durbin-Watson stat
-13.98617
i. Which of the independent variables are statistically significant in explaining the dependent variable? Why?
ii. The joint significance F-test is reported as 70.08795 with a zero p-value. Briefly explain the hypothesis tested by this F-test and its result in this regression.
iii. The reported R-squared statistic is 0.7436. Explain how the R-squared is calculated and its interpretation. What is the difference between the R- squared and the Adjusted R-squared?
iv. The estimated sample coefficient on LVIX is 0.497681. Using the attached t- distribution table test the null hypothesis that the true population coefficient is 1 against the two-sided alternative that it is different than 1.
Question 3
Let Yt denote excess return on the share of firm A and Xt excess return on the market index in which firm A trades. An analyst estimates the following regression models using T = 360 observations of daily frequency:
Yt =0 +1 Xt+ut (1) Yi =0 +1 Xt+0 Dt +1 (Dt Xt)+ut (2)
where,
The OLS estimates of the two models have as follows (standard errors in brackets):
ˆ
Yt 0.061.15Xt ut (1)
(0.015) (0.07)
Adjusted R-square: 0.11
ˆ
Yt 0.091.02Xt 0.032Dt 0.014(DtXt) (2)
(0.40) (0.25) (0.012) (0.006)
Adjusted R-square: 0.20
a) Explain the financial hypotheses tested by the two models.
b) Provide an interpretation of the reported estimates in relation to the tested financial hypotheses
c) In the light of the estimation results which model would you choose and why?
d) According to each of the models (1) and (2) and assuming a market excess return of 1 per cent, what is the expected value for the excess returns of firm A on Mondays?
e) Using the reported estimates of equation (1) to test the null hypothesis against the alternative that at the 95 and 99 per cent level of statistical significance.
D 1forMondayobservations fort1,2,...,T t 0 o t h e r w i s e
SECTION C - ANSWER ONE QUESTION FROM THIS SECTION Question 4
Consider the following regression output for the excess returns on company General Electric (RGEC) on excess returns on the FTSE100 index (RTFSE). The data sample period is January 1980 to March 1999 (monthly frequency).
Dependent Variable: RGEC
Method: Least Squares
Date: 10/11/07 Time: 19:20
Sample (adjusted): 1980M02 1999M03 Included observations: 230 after adjustments
Variable Coefficient C -0.001349
Std. Error
0.003688 0.072245
t-Statistic
-0.365769 12.17854
Prob.
0.7149 0.0000
RFTSE
R-squared
Adjusted R-squared S.E. of regression Sum squared resid Log likelihood Durbin-Watson stat
0.879839
0.394127 0.391470 0.054761 0.683723 342.7465 1.943777
Mean dependent var
S.D. dependent var
Akaike info criterion
Schwarz criterion
F-statistic 148.3167 Prob(F-statistic) 0.000000
0.007791
0.070199 -2.963013 -2.933117
(a) How is this model called in finance? Write down the model equation and comment on the coefficient values and their meaning.
(b) Comment on the rest of the regression output and the regression statistics (the R-squared, the Durbin-Watson statistic, the F-test).
(c) The following restriction has been imposed on the coefficients. State the null hypothesis and its meaning in financial terms. What do you infer?
Wald Test:
Test Statistic Value F-statistic 1.641867
Null Hypothesis: C(1)=0, C(2)=1 Null Hypothesis Summary:
Normalized Restriction (= 0)
C(1)
-1 + C(2)
df (2, 228)
Value
-0.001349 -0.120161
Probability 0.1959
Std. Err.
0.003688 0.072245
Restrictions are linear in coefficients.
(d) The following tests have been conducted on the residuals. Describe the null and alternative hypotheses in each test. What do you infer by the reported results?
Heteroskedasticity Test: White
F-statistic 0.617161 Obs*R-squared 1.243871 Scaled explained SS 1.700438
Heteroskedasticity Test: ARCH
F-statistic 0.014219 Obs*R-squared 0.028814
Breusch-Godfrey Serial Correlation
F-statistic 0.229514 Obs*R-squared 0.701695
Ramsey RESET Test
Equation: CAPM
Specification: RP C RFTSE Omitted Variables: Powers of fitted
Value F-statistic 0.728238
Prob. F(2,227) Prob. Chi-Square(2) Prob. Chi-Square(2)
Prob. F(2,225) Prob. Chi-Square(2)
LM Test:
Prob. F(3,225) Prob. Chi-Square(3)
values from 2 to 3
0.5404 0.5369 0.4273
0.9859 0.9857
0.8758 0.8728
Likelihood ratio
1.477499
df (2, 226) 2
Probability 0.4839 0.4777
25 20 15 10
5
0
Series: Residuals
Sample 1980M02 1999M03 Observations 230
Mean Median Maximum Minimum Std. Dev. Skewness Kurtosis
Jarque-Bera Probability
1.59e-18 -0.001718 0.152322 -0.217914 0.054641 -0.149332 3.782284
6.719528 0.034743
-0.20 -0.15
-0.10 -0.05
0.00
0.05
0.10
0.15
(e) The following test has been conducted. Describe the null hypothesis. What do you infer?
Chow Breakpoint Test: 1987M10 Equation Sample: 1980M02 1999M03
F-statistic 4.551661 Prob. F(2,226) 0.0115
Question 5
(a) The correlogram of the UK 3-month Treasury Bill (GBY) and its first difference (DGBY), calculated using weekly yield data over the period 1995M01-2010M09 (monthly frequency), are reported below:
GBY Autocorrelation
.|******* .|******* .|******* .|******| .|******| .|******| .|******| .|******| .|***** | .|***** | .|***** | .|***** |
DGBY Autocorrelation
.|** | .|. |
.|* |
.|* |
*|. |
*|. |
.|. |
.|. |
.|. |
.|. |
.|. |
*|. |
Partial Correlation
.|******* *|. | .|. | *|. | .|. | .|. | .|. | .|. | .|. | .|. | .|. | .|. |
Partial Correlation
.|** | *|. | .|* | .|. | *|. | .|. | .|. | .|. | .|. | .|. | *|. | *|. |
AC PAC Q-Stat
1 0.972 0.972 204.25 2 0.941 -0.086 396.38 3 0.913 0.057 578.18 4 0.882 -0.090 748.51 5 0.849 -0.031 907.11 6 0.820 0.060 1056.0 7 0.795 0.038 1196.5 8 0.772 0.024 1329.7 9 0.752 0.034 1456.5
10 0.732 -0.009 1577.5 11 0.713 -0.001 1692.8 12 0.696 0.021 1803.1
AC PAC Q-Stat
1 0.293 0.293 18.519 2 0.008 -0.086 18.531 3 0.132 0.171 22.328 4 0.088 -0.005 24.035 5 -0.113 -0.140 26.834 6 -0.108 -0.045 29.380 7 0.003 0.024 29.382 8 -0.049 -0.044 29.915 9 -0.032 0.041 30.149
10 0.011 -0.006 30.175 11 -0.057 -0.084 30.916 12 -0.156 -0.125 36.402
Prob
0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
Prob
0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.001 0.001 0.000
Comment on what is indicated by the autocorrelation and partial autocorrelation functions and by the Q-statistic in relation to: the series’ stationarity properties; and its Autoregressive (AR), Moving Average (MA) and ARMA components.
(b) The following Augmented Dickey – Fuller tests (including a constant but not a trend) have been conducted:
GBY
Lag Length: 3 (Automatic - based on AIC, maxlag=14)
Augmented Dickey-Fuller test statistic Test critical values: 1% level
5% level
10% level
t-Statistic
-1.225213 -3.461630 -2.875195 -2.574125
Prob.* 0.6636
Prob.* 0.0000
DGBY
Lag Length: 4 (Automatic - based on AIC, maxlag=14)
Augmented Dickey-Fuller test statistic Test critical values: 1% level
5% level
10% level *MacKinnon (1996) one-sided p-values.
t-Statistic
-6.290620 -3.461938 -2.875330 -2.574198
Describe the null hypothesis in each of the tests. What do you infer?
(c) You estimate an ARMA (1,1) model for DGBY and the corresponding output is as follows:
Dependent Variable: DGBY
Method: Least Squares
Sample (adjusted): 1993M03 2010M09 Included observations: 211 after adjustments Convergence achieved after 8 iterations
MA Backcast: 1993M02
Variable Coefficient
C -0.022085 AR(1) -0.142398
Std. Error
0.014386 0.191362 0.169860
t-Statistic
-1.535205 -0.744129 2.820316
Prob.
0.1263 0.4576 0.0053
-0.021991 0.169643 -0.793831 -0.746174 -0.774567 2.000449
MA(1)
R-squared
Adjusted R-squared S.E. of regression Sum squared resid Log likelihood F-statistic Prob(F-statistic)
0.479058
0.101726 0.093089 0.161555 5.428778 86.74917 11.77756 0.000014
Mean dependent var S.D. dependent var Akaike info criterion Schwarz criterion Hannan-Quinn criter. Durbin-Watson stat
The following tests have been conducted on the residuals. Describe the null hypothesis for each test. What do you infer?
Heteroskedasticity Test: White
F-statistic 0.690448 Obs*R-squared 6.327566
Heteroskedasticity Test: ARCH
F-statistic 0.867658 Obs*R-squared 2.620575
Breusch-Godfrey Serial Correlation
F-statistic 2.161187 Obs*R-squared 10.66342
Prob. F(9,201) Prob. Chi-Square(9)
Prob. F(3,204) Prob. Chi-Square(3)
LM Test:
Prob. F(5,203) Prob. Chi-Square(5)
0.7171 0.7067
0.4587 0.4539
0.0598 0.0585
20 16 12
8 4 0
Series: Residuals
Sample 1993M03 2010M09 Observations 211
Mean Median Maximum Minimum Std. Dev. Skewness Kurtosis
Jarque-Bera Probability
0.000301 -0.014058 0.374768 -0.418081 0.160783 0.193324 2.468882
3.794339 0.149993
-0.4 -0.3 -0.2
-0.1
0.0 0.1
0.2
0.3
(d) You now estimate a MA(4) model and the corresponding output is as follows:
Dependent Variable: DGGBY
Method: Least Squares
Sample (adjusted): 1993M02 2010M09 Included observations: 212 after adjustments Convergence achieved after 10 iterations MA Backcast: 1992M10 1993M01
Variable Coefficient C -0.023244
Std. Error
0.018475 0.068625 0.071463 0.071843 0.069086
t-Statistic
-1.258174 4.937538 0.410188 2.187271 2.310039
Prob.
0.2097 0.0000 0.6821 0.0298 0.0219
-0.022877 0.169733 -0.802223
MA(1) MA(2) MA(3) MA(4)
R-squared
Adjusted R-squared S.E. of regression
0.338839 0.029313 0.157140 0.159590
0.126705 0.109830 0.160141
Mean dependent var S.D. dependent var Akaike info criterion
Sum squared resid Log likelihood F-statistic Prob(F-statistic)
5.308538 90.03562 7.508317 0.000011
Schwarz criterion Hannan-Quinn criter. Durbin-Watson stat
-0.723058 -0.770226 2.004757
Compare the empirical performance of this model against the ARMA (1,1) model presented in part (c) above in terms of explaining the movements of DGBY. Discuss the workings of the statistics you use to compare the two models.
0.4 0.25
0.3249 1.0000 0.2887 0.8165 0.2767 0.7649 0.2707 0.7407 0.2672 0.7267 0.2648 0.7176 0.2632 0.7111 0.2619 0.7064 0.2610 0.7027 0.2602 0.6998 0.2596 0.6974 0.2590 0.6955 0.2586 0.6938 0.2582 0.6924 0.2579 0.6912 0.2576 0.6901 0.2573 0.6892 0.2571 0.6884 0.2569 0.6876 0.2567 0.6870 0.2566 0.6864 0.2564 0.6858 0.2563 0.6853 0.2562 0.6848 0.2561 0.6844 0.2560 0.6840 0.2559 0.6837 0.2558 0.6834 0.2557 0.6830 0.2556 0.6828 0.2553 0.6816 0.2550 0.6807 0.2549 0.6800 0.2547 0.6794 0.2545 0.6786 0.2543 0.6780 0.2542 0.6776 0.2541 0.6772 0.2540 0.6770 0.2539 0.6765 0.2538 0.6761 0.2537 0.6757 0.2536 0.6753 0.2533 0.6745
0.15 0.1
1.9626 3.0777 1.3862 1.8856 1.2498 1.6377 1.1896 1.5332 1.1558 1.4759 1.1342 1.4398 1.1192 1.4149 1.1081 1.3968 1.0997 1.3830 1.0931 1.3722 1.0877 1.3634 1.0832 1.3562 1.0795 1.3502 1.0763 1.3450 1.0735 1.3406 1.0711 1.3368 1.0690 1.3334 1.0672 1.3304 1.0655 1.3277 1.0640 1.3253 1.0627 1.3232 1.0614 1.3212 1.0603 1.3195 1.0593 1.3178 1.0584 1.3163 1.0575 1.3150 1.0567 1.3137 1.0560 1.3125 1.0553 1.3114 1.0547 1.3104 1.0520 1.3062 1.0500 1.3031 1.0485 1.3006 1.0473 1.2987 1.0455 1.2958 1.0442 1.2938 1.0432 1.2922 1.0424 1.2910 1.0418 1.2901 1.0409 1.2886 1.0400 1.2872 1.0391 1.2858 1.0382 1.2844 1.0364 1.2816
0.05 0.025 0.01 0.005 0.001 0.0005
6.3138 12.7062 31.8205 63.6567 318.3087 636.6189 2.9200 4.3027 6.9646 9.9248 22.3271 31.5991 2.3534 3.1824 4.5407 5.8409 10.2145 12.9240 2.1318 2.7764 3.7469 4.6041 7.1732 8.6103 2.0150 2.5706 3.3649 4.0321 5.8934 6.8688 1.9432 2.4469 3.1427 3.7074 5.2076 5.9588 1.8946 2.3646 2.9980 3.4995 4.7853 5.4079 1.8595 2.3060 2.8965 3.3554 4.5008 5.0413 1.8331 2.2622 2.8214 3.2498 4.2968 4.7809 1.8125 2.2281 2.7638 3.1693 4.1437 4.5869 1.7959 2.2010 2.7181 3.1058 4.0247 4.4370 1.7823 2.1788 2.6810 3.0545 3.9296 4.3178 1.7709 2.1604 2.6503 3.0123 3.8520 4.2208 1.7613 2.1448 2.6245 2.9768 3.7874 4.1405 1.7531 2.1314 2.6025 2.9467 3.7328 4.0728 1.7459 2.1199 2.5835 2.9208 3.6862 4.0150 1.7396 2.1098 2.5669 2.8982 3.6458 3.9651 1.7341 2.1009 2.5524 2.8784 3.6105 3.9216 1.7291 2.0930 2.5395 2.8609 3.5794 3.8834 1.7247 2.0860 2.5280 2.8453 3.5518 3.8495 1.7207 2.0796 2.5176 2.8314 3.5272 3.8193 1.7171 2.0739 2.5083 2.8188 3.5050 3.7921 1.7139 2.0687 2.4999 2.8073 3.4850 3.7676 1.7109 2.0639 2.4922 2.7969 3.4668 3.7454 1.7081 2.0595 2.4851 2.7874 3.4502 3.7251 1.7056 2.0555 2.4786 2.7787 3.4350 3.7066 1.7033 2.0518 2.4727 2.7707 3.4210 3.6896 1.7011 2.0484 2.4671 2.7633 3.4082 3.6739 1.6991 2.0452 2.4620 2.7564 3.3962 3.6594 1.6973 2.0423 2.4573 2.7500 3.3852 3.6460 1.6896 2.0301 2.4377 2.7238 3.3400 3.5911 1.6839 2.0211 2.4233 2.7045 3.3069 3.5510 1.6794 2.0141 2.4121 2.6896 3.2815 3.5203 1.6759 2.0086 2.4033 2.6778 3.2614 3.4960 1.6706 2.0003 2.3901 2.6603 3.2317 3.4602 1.6669 1.9944 2.3808 2.6479 3.2108 3.4350 1.6641 1.9901 2.3739 2.6387 3.1953 3.4163 1.6620 1.9867 2.3685 2.6316 3.1833 3.4019 1.6602 1.9840 2.3642 2.6259 3.1737 3.3905 1.6577 1.9799 2.3578 2.6174 3.1595 3.3735 1.6551 1.9759 2.3515 2.6090 3.1455 3.3566 1.6525 1.9719 2.3451 2.6006 3.1315 3.3398 1.6499 1.9679 2.3388 2.5923 3.1176 3.3233 1.6449 1.9600 2.3263 2.5758 3.0902 3.2905
1 2 3 4 5 6 7 8 9
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Critical values of t-distribution
Source: Biometrika Tables for Statisticians (1966), Volume 1, 3rd Edition. Reprinted with permission of Oxford University Press.
SECTION A - COMPULSORY SECTION
Question 1
Provide brief answers to all parts of this question
(a) What is heteroscedasticity? What are the potential problems of the latter in estimation and inference and how can these problems be remedied?
(b) What are the main causes of serially correlated errors in financial time series data?
(c) What is a random walk process? Illustrate the latter with an example.
(d) What is an autocorrelation function? How is it used in the context of time series
modelling?
(e) Describe the concept of seasonality, give an example and illustrate how to account for it in a model.
SECTION B - ANSWER ONE QUESTION FROM THIS SECTION Question 2
You estimate a regression of the form given by the equation below in order to evaluate the effect of various firm-specific factors on the firm’s return series. You run a cross- sectional regression with 200 firms
ri =1 +2Si + 3MBi + 4PEi +5BETAi +ui where
ri is the percentage annual return for the stock
Si is the size of firm i measured in terms of sales revenue MBi is the market to book ratio of the firm
PEi is the price-earnings ratio of the firm
BETAi is the stock’s market beta
You obtain the following results (with standard errors in parentheses): ri = 0.080 + 0.801Si + 0.321MBi + 0.164PEi - 0.084BETAi
(0.064) (0.147) (0.136) (0.420) (0.120)
a) What do you conclude about the effect of each variable on the returns of a security? If the stock’s beta increased from 1 to 1.2, what would be the effect on the stock’s return? Is the sign on beta what you would have expected?
b) On the basis of your results what variables would you consider removing from the regression? Explain why it is desirable to do so.
c) Which null hypotheses constitutes the regression F-statistic? What exactly would constitute the alternative hypothesis in this case? Let the residual sum of squares for the restricted and unrestricted regressions are 102.87 and 91.41 respectively. Perform the test. What is your conclusion?
d) Do you expect to find any form of heteroscedasticity in the error terms of this regression model? Explain the reasons behind your answer and name the test that you would use to detect this misspecification.
Question 3
The following estimates (standard errors in parentheses) were obtained using OLS on a cross-section sample of 935 women
Yi 0.014Xi 0.065Ei 0.012Ti 0.188Bi 0.199Mi (0.003) (0.006) (0.003) (0.038) (0.038)
R2 = 25.3%, SER (standard error of regression) = 0.366, RSS (residual sum of squares) = 123.82
where,
Y: log of monthly earnings
X: work experience in years
E: years of education
T: years with current employer
B: dummy variable for race (= 0 if race is white)
M: dummy variable for marital status (=1 if married)
a) What do the coefficient estimates for X, E, and T mean in terms of statistical significance? Report the effect of each variable on earnings
b) Do earnings differ significantly for married and single women? By how much?
c) According to the model above what is the expected monthly salary of a single white woman? What is the expected monthly salary of a single non-white woman with all other attributes being equal? Compare the two answers.
d) Suppose that I add an additional variable in the model and find that the R2 = 30%, Therefore, I conclude that the variable is important in explaining earnings. Comment on the suitability of this approach to check the relevance of a new variable.
SECTION C - ANSWER ONE QUESTION FROM THIS SECTION Question 4
You are estimating a time series model for annual US savings using data on personal savings (savings) and disposable personal income (dpi) in billions of dollars for the period 1970-1995. Accordingly your model and the corresponding EViews output using OLS estimation are as follows:
Dependent Variable: SAVINGS Method: Least Squares Sample: 1970 1995
Included observations: 26
Variable Coefficient
Std. Error
12.76075 0.004237
Mean dependent var S.D. dependent var Akaike info criterion Schwarz criterion F-statistic Prob(F-statistic)
AC PAC
1 0.548 0.548 2 0.197 -0.147 3 0.046 0.004 4 0.059 0.083 5 -0.046 -0.166 6 -0.007 0.134 7 -0.021 -0.083 8 -0.168 -0.230 9 -0.309 -0.115 10 -0.252 -0.015 11 -0.164 -0.036 12 -0.327 -0.369
Savingst 0 1(DPI)t t
t-Statistic
4.891772 8.893776
C DPI
R-squared
Adjusted R-squared S.E. of regression Sum squared resid Log likelihood Durbin-Watson stat
62.42267 0.037679
0.767215 0.757515 31.12361 23248.30
-125.2390 0.859717
Partial Correlation
. |**** | .*|. | .|. |
Prob.
0.0001 0.0000
162.0885 63.20446 9.787614 9.884391 79.09925 0.000000
Q-Stat Prob
8.7544 0.003 9.9374 0.007 10.005 0.019 10.119 0.038 10.193 0.070 10.195 0.117 10.212 0.177 11.348 0.183 15.441 0.080 18.326 0.050 19.626 0.051 25.187 0.014
(i)
Residual Correlogram Sample: 1970 1995 Included observations: 26
Autocorrelation
. |**** | . |**. |
.|. |
.|. |
.|. |
.|. |
.|. | .*|. | .**|. | .**|. | .*|. | ***| . |
.|*. .*|. .|*. .*|. .**| . .*|. .|. | .|. | ***| . |
| | | | | |
(ii)
7
6
5
4
3
2
1
0
Series: Residuals Sample 1970 1995 Observations 26
-60 -40 -20 0 20 40 60
Mean Median Maximum Minimum Std. Dev. Skewness Kurtosis
Jarque-Bera Probability
-7.24e-15 -9.271248 67.39897 -62.23596 30.49479 0.515262 2.862708
1.170900 0.556855
.
(iii)
Ramsey RESET Test:
F-statistic
Log likelihood ratio
7.906451 14.08181
Probability Probability
0.002586 0.000875
(a) Is disposable income a driving factor for savings? Comment on the plausibility of the sign of the slope coefficient.
(b) A relevant question for policy makers is: if a tax cut leads to an increase in disposable income of $1 (billion) by how much are savings expected to increase?
(c) Comment on the test results presented in (i), (ii) and (iii) and describe the problem (or lack thereof) for the model under each test. In your own words describe briefly how each test is carried out.
(d) What remedies do you suggest to improve the specification of the model?
Question 5
You are estimating a time series model for stock market returns of UK insurance companies. The dataset comprises of weekly returns on the insurance index and returns on the FTSE100 over 1986:01 – 2001:12. You assume that insurance returns (Rinsur) can be adequately explained by the current FTSE return, (FTSE100)t and market volatility (VOLATt), respectively.
Dependent Variable: RINSUR
Method: Least Squares
Date: 03/13/06 Time: 18:22
Sample (adjusted): 1/10/1986 12/21/2001 Included observations: 833 after adjustments
Variable
C FTSE100 VOLAT
R-squared
Adjusted R-squared S.E. of regression Sum squared resid Log likelihood Durbin-Watson stat
Coefficient
-0.113777 1.102629 0.082345
0.503237 0.502040 2.568870 5477.248 -1966.380 2.254550
Std. Error
0.089407 0.038110 0.038090
t-Statistic
-1.272571 28.93243 2.161879
Prob.
0.2035 0.0000 0.0309
0.074034 3.640367 4.728404 4.745421 420.4092 0.000000
0.0000
0.0000
0.0551
Breusch-Godfrey Serial Correlation LM Test:
Mean dependent var S.D. dependent var Akaike info criterion Schwarz criterion F-statistic Prob(F-statistic)
(4 lags) Probability
F-statistic
Jarque Bera Test F-statistic
White Heteroskedasticity Test F-statistic
26.8911
86.1963 Probability
9.2531 Probability
Residual Graph
15
10
5
0
-5
-10
-15
1986 1988 1990 1992 1994 1996 1998 2000
RINSUR Residuals
(a) Comment on the estimation output and specifically on the coefficients, the R squared, the regression F-statistic and the DW test.
(b) Comment on the test results presented in the above misspecification tests and describe the problem (or lack thereof) for the model under each test. In your own words describe briefly how each test is carried out.
(c) Explain the behaviour of the model residuals as illustrated by the Residual Graph. Comment on any problem of the model and its possible causes.
(d) What remedies, if any, do you suggest to improve the specification of the model?
Cumulative Normal Distribution
x
x 0.00 0.01
0.0 0.5000 0.5040
0.1 0.5398 0.5438
0.2 0.5793 0.5832
0.3 0.6179 0.6217
0.4 0.6554 0.6591
0.5 0.6915 0.6950
0.6 0.7257 0.7291
0.7 0.7580 0.7611
0.8 0.7881 0.7910
0.9 0.8159 0.8186
1.0 0.8413 0.8438
1.1 0.8643 0.8665
1.2 0.8849 0.8869
1.3 0.9032 0.9049
1.4 0.9192 0.9207
1.5 0.9332 0.9345
1.6 0.9452 0.9463
1.7 0.9554 0.9564
1.8 0.9641 0.9649
1.9 0.9713 0.9719
2.0 0.9772 0.9778
2.1 0.9821 0.9826
2.2 0.9861 0.9864
2.3 0.9893 0.9896
2.4 0.9918 0.9920
2.5 0.9938 0.9940
2.6 0.9953 0.9955
2.7 0.9965 0.9966
2.8 0.9974 0.9975
2.9 0.9981 0.9982
3.0 0.9987 0.9987
3.1 0.9990 0.9991
3.2 0.9993 0.9993
3.3 0.9995 0.9995
3.4 0.9997 0.9997
3.5 0.9998 0.9998
3.6 0.9998 0.9998
The area from -¥ to x
0.02 0.03 0.04 0.05 0.06
0.5080 0.5120 0.5160 0.5199 0.5239 0.5478 0.5517 0.5557 0.5596 0.5636 0.5871 0.5910 0.5948 0.5987 0.6026 0.6255 0.6293 0.6331 0.6368 0.6406 0.6628 0.6664 0.6700 0.6736 0.6772 0.6985 0.7019 0.7054 0.7088 0.7123 0.7324 0.7357 0.7389 0.7422 0.7454 0.7642 0.7673 0.7704 0.7734 0.7764 0.7939 0.7967 0.7995 0.8023 0.8051 0.8212 0.8238 0.8264 0.8289 0.8315 0.8461 0.8485 0.8508 0.8531 0.8554 0.8686 0.8708 0.8729 0.8749 0.8770 0.8888 0.8907 0.8925 0.8944 0.8962 0.9066 0.9082 0.9099 0.9115 0.9131 0.9222 0.9236 0.9251 0.9265 0.9279 0.9357 0.9370 0.9382 0.9394 0.9406 0.9474 0.9484 0.9495 0.9505 0.9515 0.9573 0.9582 0.9591 0.9599 0.9608 0.9656 0.9664 0.9671 0.9678 0.9686 0.9726 0.9732 0.9738 0.9744 0.9750 0.9783 0.9788 0.9793 0.9798 0.9803 0.9830 0.9834 0.9838 0.9842 0.9846 0.9868 0.9871 0.9875 0.9878 0.9881 0.9898 0.9901 0.9904 0.9906 0.9909 0.9922 0.9925 0.9927 0.9929 0.9931 0.9941 0.9943 0.9945 0.9946 0.9948 0.9956 0.9957 0.9959 0.9960 0.9961 0.9967 0.9968 0.9969 0.9970 0.9971 0.9976 0.9977 0.9977 0.9978 0.9979 0.9982 0.9983 0.9984 0.9984 0.9985 0.9987 0.9988 0.9988 0.9989 0.9989 0.9991 0.9991 0.9992 0.9992 0.9992 0.9994 0.9994 0.9994 0.9994 0.9994 0.9995 0.9996 0.9996 0.9996 0.9996 0.9997 0.9997 0.9997 0.9997 0.9997 0.9998 0.9998 0.9998 0.9998 0.9998 0.9999 0.9999 0.9999 0.9999 0.9999
0.07 0.08 0.5279 0.5319 0.5675 0.5714 0.6064 0.6103 0.6443 0.6480 0.6808 0.6844 0.7157 0.7190 0.7486 0.7517 0.7794 0.7823 0.8078 0.8106 0.8340 0.8365 0.8577 0.8599 0.8790 0.8810 0.8980 0.8997 0.9147 0.9162 0.9292 0.9306 0.9418 0.9429 0.9525 0.9535 0.9616 0.9625 0.9693 0.9699 0.9756 0.9761 0.9808 0.9812 0.9850 0.9854 0.9884 0.9887 0.9911 0.9913 0.9932 0.9934 0.9949 0.9951 0.9962 0.9963 0.9972 0.9973 0.9979 0.9980 0.9985 0.9986 0.9989 0.9990 0.9992 0.9993 0.9995 0.9995 0.9996 0.9996 0.9997 0.9997 0.9998 0.9998 0.9999 0.9999
0.09 0.5359 0.5753 0.6141 0.6517 0.6879 0.7224 0.7549 0.7852 0.8133 0.8389 0.8621 0.8830 0.9015 0.9177 0.9319 0.9441 0.9545 0.9633 0.9706 0.9767 0.9817 0.9857 0.9890 0.9916 0.9936 0.9952 0.9964 0.9974 0.9981 0.9986 0.9990 0.9993 0.9995 0.9997 0.9998 0.9998 0.9999
t-distribution
t 0.9
1 0.1584
2 0.1421
3 0.1366
4 0.1338
5 0.1322
6 0.1311
7 0.1303
8 0.1297
9 0.1293
10 0.1289
11 0.1286
12 0.1283
13 0.1281
14 0.1280
15 0.1278
16 0.1277
17 0.1276
18 0.1274
19 0.1274
20 0.1273
21 0.1272
22 0.1271
23 0.1271
24 0.1270
25 0.1269
26 0.1269
27 0.1268
28 0.1268
29 0.1268
30 0.1267
infinite 0.1257
0.5 0.2 1.0000 3.0777 0.8165 1.8856 0.7649 1.6377 0.7407 1.5332 0.7267 1.4759 0.7176 1.4398 0.7111 1.4149 0.7064 1.3968 0.7027 1.3830 0.6998 1.3722 0.6974 1.3634 0.6955 1.3562 0.6938 1.3502 0.6924 1.3450 0.6912 1.3406 0.6901 1.3368 0.6892 1.3334 0.6884 1.3304 0.6876 1.3277 0.6870 1.3253 0.6864 1.3232 0.6858 1.3212 0.6853 1.3195 0.6848 1.3178 0.6844 1.3163 0.6840 1.3150 0.6837 1.3137 0.6834 1.3125 0.6830 1.3114 0.6828 1.3104 0.6747 1.2824
0.1 0.05 6.3137 12.7062 2.9200 4.3027 2.3534 3.1824 2.1318 2.7765 2.0150 2.5706 1.9432 2.4469 1.8946 2.3646 1.8595 2.3060 1.8331 2.2622 1.8125 2.2281 1.7959 2.2010 1.7823 2.1788 1.7709 2.1604 1.7613 2.1448 1.7531 2.1315 1.7459 2.1199 1.7396 2.1098 1.7341 2.1009 1.7291 2.0930 1.7247 2.0860 1.7207 2.0796 1.7171 2.0739 1.7139 2.0687 1.7109 2.0639 1.7081 2.0595 1.7056 2.0555 1.7033 2.0518 1.7011 2.0484 1.6991 2.0452 1.6973 2.0423 1.6464 1.9623
0.01 63.6559 9.9250 5.8408 4.6041 4.0321 3.7074 3.4995 3.3554 3.2498 3.1693 3.1058 3.0545 3.0123 2.9768 2.9467 2.9208 2.8982 2.8784 2.8609 2.8453 2.8314 2.8188 2.8073 2.7970 2.7874 2.7787 2.7707 2.7633 2.7564 2.7500 2.5807
SECTION A - COMPULSORY SECTION Question 1
Answer ALL parts of this question
Provide brief answers to all parts of this question
a) List and interpret the 5 assumptions that are required for the Gauss-Markov theorem to hold.
b) WhataretheimplicationsofGauss-Markovtheoremforthepropertiesofleast squares estimation?
c) Whatisheteroscedasticity?Whataretheconsequencesof heteroscedasticity?
d) Write down and explain the formal definition of stationarity for a financial time series process.
e) Give a description of how the features of stationary and non-stationary processes differ and explain why it is important to know whether a process is stationary if one intends to use regression analysis.
SECTION B - ANSWER ONE QUESTION FROM THIS SECTION Question 2
Answer ALL parts of this question
In a study of the effect of changing the class size on the performance of students in tests carried out on a sample of 420 California elementary school districts, the following regression is obtained (standard errors in parentheses):
yˆi 700.4 1.01xi RSS 105.4 (5.5) (0.27)
where yˆi is estimated average score in the district, xi is the average student-teacher ratio,forthedatainthesample,14xi 25.8.RSSistheresidualssumofsquares.
a) Test for the significance of each parameter and interpret its effect on yˆi .
b) Eachschooldistrictisclassifiedasruralorurban.Twobinaryvariablesare
defined, x2 (equal to 1 if the district is rural, 0 if it is urban) and x3 (equal to 1 if the district is urban, 0 if it is rural). Would you recommend adding both
variables x2 and x3 in the model?
c) Does adding variable x12 in the model cause perfect multicollinearity in the regression? Based on your previous answer: would you recommend adding variable x12 to the model?
d) Decide which variables ( x2 , x3 , x12 ) you can keep in the regression and justify your answer. Test the hypothesis that all added variables ( x2 and/or x3
and/or x12 )inyourmodelarejointlyinsignificant.Theunrestrictedresiduals
sum of squares of the model is 91.41 and critical value for 5% significance level in F-distribution is 3.026.
Question 3
Answer ALL parts of this question
A researcher is analysing the relationship between returns on Vodafone stock and
returns on the FTSE100 using daily data and the regression model below;
yˆt 0.76 1.1xt RSS50.1 (2.3) (0.05)
where, yˆt is estimated excess returns on Vodafone stock and xt is the excess returns
on the market.
a) Interpret the coefficients of the regression and comment on their statistical
significance.
b) Comment on the goodness of fit of the overall regression if total sum of squares is 125.25.
c) TheresearchersuspectsthatVodafonereturnsonMondaysmaybegreater than on other days of the week. Discuss how the researcher can modify the regression model to test this. How could the researcher change the model to test whether the slope coefficient is different on Mondays than on other days of the week.
d) Consider a standard multiple regression model;
yi 1 2x1i 3x2i 4x3i ei
Describe how a researcher could construct an F-test of the null hypothesis
that2 3.
SECTION C - ANSWER ONE QUESTION FROM THIS SECTION Question 4
Answer ALL parts of this question
A researcher has collected 100 consecutive daily returns on a particular UK stock, SGL Industries, and 100 daily returns on the FTSE100 covering the same time period. Halfway through this 100 day sample, SGL released a very positive set of earnings figures. The researcher is interested in examining the impact of these earnings figures on SGL’s stock price. Thus she runs the following regression:
rrM D D e t 1t21,t32,tt
where r is the daily return for SGL, r M is the daily FTSE100 return, D is a dummy t t 1,t
variable taking the value one on the date of the earnings release and zero otherwise and D2,t is a dummy variable that takes the value one only on the day immediately
after the earnings release. The variable et is an error term.
The researcher estimates the model using OLS and obtains the following results:
In the preceding and the following tables, R represents the return on SGL, RM is the return on the FTSE-100, D1 is the first dummy variable mentioned above and D2 is the second dummy variable mentioned above.
(Question 4 continued on page 6)
a) Interpret the coefficients that the researcher has obtained and run the hypothesis tests that you feel are appropriate for understanding the determination of SGL’s returns. Comment on the regression R2 .
b) FormallytestthenullhypothesisthatSGLhasasensitivityof1.0tothe FTSE100.
c) TheresearcherwantstorunaWhitetestinordertoassesswhetherthe regression suffers from heteroscedasticity. Thus she has run the regression below, where the variable RESID contains the residuals from the least squares regression above.
Use these results to test for heteroscedasticity, stating very clearly the null and alternative hypotheses for your test. Use 9.488 for chi-square critical values.
Question 5
Answer ALL parts of this question
You are estimating a time series model for the change in the UK 3-month Treasury Bill using weekly yield data (D3M) for the period 1999-2006. The correlogram for D3M is:
a) Commentonwhatisindicatedbytheautocorrelationandpartial autocorrelation functions and by the Q-statistic. Suggest a possible ARMA specification and justify your answer.
You estimate an ARMA (1,1) and the corresponding EViews output is as follows.
ARMA (1,1):
(Question 5 continued on page 8)
The following misspecification tests have been conducted:
Residuals graph:
b) Comment on the test results presented in the above misspecification tests and describe the problem (or lack of) for the model under each test. State the null hypothesis and describe briefly how each test is carried out. What remedies do you suggest to improve the specification of the model?
c) ExplainthebehaviourofthemodelresidualsasillustratedbytheResidual Graph. Comment on any problem of the model and its possible causes.
Durbin-Watson statistical table
Chi-square distribution table

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SECTION A

QUESTION 1

a) OLS stands for Ordinary Least Squares, the standard linear regression procedure. One estimates a parameter from data and applying the linear model. The method of OLS entails taking each vertical distance from the point to the line, squaring it and then minimizing the total sum of the areas of squares (hence ‘least squares’). This can be viewed as equivalent to minimising the sum of the areas of the squares drawn from the points to the line. Estimators are the formulae used to calculate the coefficients while the estimates, on the other hand, are the actual numerical values for the coefficients that are obtained from the sample.

b) One of the most important feature of many series of financial asset returns that provides a motivation for the ARCH class of models, is known as ‘volatility clustering’ or ‘volatility pooling’. Volatility clustering describes the tendency of large changes in asset prices (of either sign) to follow large changes and small changes (of either sign) to follow small changes. In other words, the current level of volatility tends to be positively correlated with its level during the immediately preceding periods.

c) Many financial and economic time series seem to undergo episodes in which the behaviour of the series changes quite dramatically compared to that exhibited previously. The behaviour of a series could change over time in terms of its mean value, its volatility, or to what extent its current value is related to its previous value. The behaviour may change once and for all, usually known as a ‘structural break’ in a series. Or it may change for a period of time before reverting back to its original behaviour or switching to yet another style of behaviour, and the latter is typically termed a ‘regime shift’ or ‘regime switch’. In econometrics, a structural break is an unexpected shift in a (macroeconomic) time series. This can lead to huge forecasting errors and unreliability of the model in general....