Final Exam (0900-1200hr, January 29, 2003)
2940603 Advanced Econometrics (Assoc. Prof. Pongsa Pornchaiwiseskul)
Instructions:
a) Textbooks, lecture notes and calculators are allowed.
b) Each must work alone. Cheating will not be tolerated.
c) There are four tests. Attempt all the tests. Use only the provided test-books.
d) All the hypothesis testing will use 0.05 as the level of significance.
TEST#1 (20 points)
During week t, weekly stock return (RBt) of banking sector is assumed to follow the following process:
RBt = b0 +b1Dt-1+ b2RMt + ut
[st]2 = a0 + a1[ut-1]2 + a2Dt-1
where
RMt = market return in period t
Dt = 1 if sectoral return is less than the market return
in period t
= 0, otherwise.
ut = independently distributed error term
[st]2 = Var(ut)
Use printouts 1.1-1.3 to answer the following questions.
1.1) Give a valid estimate for the above model. Explain in details.
1.2) It has been claimed that the expected return and the return risk of the banking sector do not depend on whether the observed sectoral return is above or below the market return. Test this claim.
TEST#2 (20 points)
Unemployment rate (UE) and domestic inflation rate (DI) form a VAR(1) as follows:
UEt = b10 + b11 UEt-1 + b12 DIt-1 + e1t
DIt = b20 + b21 UEt-1 + b22 DIt-1 + e2t
where (e1t , e2t) ~ iid bi-variate of shocks
Use printouts 2.1-2.2 to answer the following questions.
2.1) Estimate the model. Check for the validity. Explain in details
2.2) Test whether the long-run correlation between the unemployment rate and the inflation rate is negative. Explain in details.
2.3) How much will the future shocks contribute to the variance of the unemployment rate in 2 periods from now? This is a variance decomposition question with simultaneous effects.
TEST#3 (20 points)
Let Pt be the international price in term of local currency in period t. It is assumed that P follows the following model:
lnPt = b1+ b2lnEt + ut
where
Et = the foreign exchange rate in period t
ut = stationary ARMA(1,2) error term in period t
ut = rut-1 + et + q1e t-1+ q2e t-2
e t = white noise with variance of s2
Based on Printouts 3.1 and 3.2, answer the following questions:
3.1) Estimate all the parameters and the variance of the estimates. Check for validity.
3.2) Given the following information,
T / lnPt / lnEt26 / 2.0 / 2
27 / 2.3 / 2.5
28 / ? / 2.3
calculate the best prediction for lnP28 and its prediction interval.
TEST#4 (20 points)
We are suspecting that endogenous time series (Y1t,Y2t) follow VAR(1) with exogenous variables (X1t,X2t,X1t-1,X2t-1). Explain how you will test it.
PRINTOUT 1.1
======
Dependent Variable: RB
Method: ML - ARCH
Date: 01/28/03 Time: 13:14
Sample(adjusted): 2 200
Included observations: 199 after adjusting endpoints
Convergence achieved after 22 iterations
======
CoefficientStd. Errorz-Statistic Prob.
======
C -0.002699 0.145566 -0.018543 0.9852
RB(-1)<RM(-1) -0.152600 0.137169 -1.112497 0.2659
RM 24.08926 23.91289 1.007375 0.3138
======
Variance Equation
======
C 1.009517 0.074876 13.48244 0.0000
ARCH(1) -0.098778 0.042461 -2.326308 0.0200
RB(-1)<RM(-1) 0.046743 0.180000 0.259685 0.7951
======
R-squared 0.019704 Mean dependent var 0.053044
Adjusted R-squared -0.005692 S.D. dependent var 1.004412
S.E. of regression 1.007267 Akaike info criteri2.837820
Sum squared resid 195.8153 Schwarz criterion 2.937116
Log likelihood -276.3631 F-statistic 0.775859
Durbin-Watson stat 1.977487 Prob(F-statistic) 0.568203
======
PRINTOUT 1.2
======
Dependent Variable: RB
Method: ML - ARCH
Date: 01/28/03 Time: 13:16
Sample: 1 200
Included observations: 200
Convergence achieved after 32 iterations
======
CoefficientStd. Errorz-Statistic Prob.
======
SQR(GARCH) -0.682372 0.562348 -1.213432 0.2250
C 0.545380 0.554730 0.983144 0.3255
RM 33.48573 23.21488 1.442426 0.1492
======
Variance Equation
======
C 1.036749 0.101894 10.17474 0.0000
ARCH(1) -0.085844 0.030997 -2.769479 0.0056
(RESID<0)*ARCH(1) -0.045022 0.047190 -0.954060 0.3401
======
R-squared 0.015897 Mean dependent var 0.052978
Adjusted R-squared -0.009466 S.D. dependent var 1.001886
S.E. of regression 1.006617 Akaike info criteri2.844992
Sum squared resid 196.5758 Schwarz criterion 2.943942
Log likelihood -278.4992 F-statistic 0.626776
Durbin-Watson stat 1.820597 Prob(F-statistic) 0.679525
======
PRINTOUT 1.3
======
Dependent Variable: RB
Method: ML - ARCH
Date: 01/29/03 Time: 13:16
Sample: 1 200
Included observations: 200
Convergence achieved after 26 iterations
======
CoefficientStd. Errorz-Statistic Prob.
======
C -0.033348 0.138380 -0.240988 0.8096
RM 22.53198 23.51115 0.958353 0.3379
======
Variance Equation
======
C 1.037565 0.102964 10.07699 0.0000
ARCH(1) -0.098655 0.010456 -9.435508 0.0000
======
R-squared 0.007892 Mean dependent var 0.052978
Adjusted R-squared -0.007293 S.D. dependent var 1.001886
S.E. of regression 1.005533 Akaike info criteri2.839077
Sum squared resid 198.1748 Schwarz criterion 2.905044
Log likelihood -279.9077 F-statistic 0.519739
Durbin-Watson stat 1.861711 Prob(F-statistic) 0.669179
======
PRINTOUT 2.1
Vector Autoregression Estimates
======
Date: 01/28/03 Time: 13:22
Sample(adjusted): 2 60
Included observations: 59 after adjusting
endpoints
Standard errors & t-statistics in parentheses
======
UE DI
======
UE(-1) 0.484077 -0.186778
(0.10345) (0.08903)
(4.67931) (-2.09803)
DI(-1) -0.312070 0.335431
(0.10708) (0.09215)
(-2.91425) (3.63996)
C 0.542541 0.227590
(0.18083) (0.15561)
(3.00031) (1.46253)
======
R-squared 0.382675 0.263981
Adj. R-squared 0.360627 0.237694
Sum sq. resids 67.11876 49.70590
S.E. equation 1.094783 0.942128
F-statistic 17.35697 10.04248
Log likelihood -87.52069 -78.66066
Akaike AIC 3.068498 2.768158
Schwarz SC 3.174136 2.873796
Mean dependent 0.973097 0.106559
S.D. dependent 1.369150 1.079059
======
Determinant Residual Covaria 0.950494
Log Likelihood -165.9369
Akaike Information Criteria 5.828370
Schwarz Criteria 6.039645
======
PRINTOUT 2.2
Johansen Cointegration Test
======
Date: 01/28/03 Time: 13:28
Sample: 1 60
Included observations: 58
Test assumption: Linear deterministic trend in the data
Series: UE DI
Lags interval: 1 to 1
======
Likelihood 5 Percent 1 Percent Hypothesized
Eigenvalue Ratio Critical ValuCritical ValuNo. of CE(s)
======
0.389833 34.99342 15.41 20.04 None
0.103549 6.340073 3.76 6.65 At most 1
======
*(**) denotes rejection of the hypothesis at 5%(1%) significance level
Unnormalized Cointegrating Coefficients:
======
UE DI
0.075458 0.138147
0.090885 -0.039561
======
Normalized Cointegrating Coefficients: 1 Cointegrating Equation(s)
======
UE DI C
1.000000 1.830790 -1.172253
(0.44837)
Log likelihood-165.0593
======
PRINTOUT 3.1
======
Dependent Variable: LOG(P)
Method: Least Squares
Date: 01/28/03 Time: 13:39
Sample(adjusted): 2 27
Included observations: 26 after adjusting endpoints
Convergence achieved after 19 iterations
Backcast: 0 1
======
Variable CoefficientStd. Errort-Statistic Prob.
======
C 0.556897 0.167547 3.323828 0.0032
LOG(E) -0.404976 0.073326 -5.522935 0.0000
AR(1) 0.578664 0.301539 1.919035 0.0687
MA(1) -0.977222 0.497775 -1.963178 0.0630
MA(2) -0.011509 0.362611 -0.031740 0.9750
======
R-squared 0.637016 Mean dependent var 0.423882
Adjusted R-squared 0.567876 S.D. dependent var 0.283021
S.E. of regression 0.186047 Akaike info criter-0.354592
Sum squared resid 0.726884 Schwarz criterion -0.112650
Log likelihood 9.609693 F-statistic 9.213453
Durbin-Watson stat 2.015074 Prob(F-statistic) 0.000184
======
Inverted AR Roots .58
Inverted MA Roots .99 -.01
======
PRINTOUT 3.2
Coefficient Covariance Matrix
======
C LOG(E) AR(1) MA(1) MA(2)
======
C 0.028072 -3.90E-05 0.030094 -0.064432 0.033179
LOG(E) -3.90E-05 0.005377 -0.000615 -0.005995 0.003812
AR(1) 0.030094 -0.000615 0.090926 -0.134770 0.099009
MA(1) -0.064432 -0.005995 -0.134770 0.247780 -0.171672
MA(2) 0.033179 0.003812 0.099009 -0.171672 0.131487
======
End of Exam
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