Basic Econometrics Final Exam (Trimester 1/2001) BEcon Program, Faculty of Economics Chulalongkorn University

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Basic Econometrics

Final Exam (Trimester 1/2001) BEcon Program, Faculty of Economics

Chulalongkorn University

Instructions:

a) Textbooks, lecture notes and calculators are allowed.

b) Each must work alone. Cheating will not be tolerated.

c) There are four(4) tests. Attempt all the tests.

d) Use only the provided test-books.

e) All the hypothesis testing will use 0.05 as the level of significance.

TEST#1 (20 points)

Return on stock market (R) is believed to closely follow the prime interest rate (I) and its variance over time is believed to be heteroscedastic as follows:

Rt = 1 + 2It + 3[t]² + ut [t]² = 0 + 1[ut-1]² + 2[t-1]² + It

where Rt = return on stock market in quarter t It = prime interest rate in quarter t

ut = unexpected component or error term (independently distributed but not necessarily identical) with var(ut) = [t]²

Use printouts 1.1-1.5 to answer the following questions.

1.1) Give a valid estimate for the above model. Explain in details.

1.2) Test whether the unexpected component of the stock market return in the previous quarter has a significant effect on the current expected return of the stock market.

1.3) Does evidence suggest any significant effect of the prime interest rate on the expected return?

Unemployment rate (UE) and domestic inflation rate (DI) form a VAR(1) as follows:

UEt = 10 + 11 UEt-1 + 12 DIt-1 + 13 DXt-1 + 1t DIt = 20 + 21 UEt-1 + 22 DIt-1 + 23 DXt-1 + 2t where DXt = rate of change in foreign exchange rate in period t

(1t , 2t) – iid bi-variate of shocks

Note that DX is exogenous with respect to UE and DI. Use printouts 2.1-2.3 to answer the following questions.

2.1) Estimate the model. Explain in details.

2.2) If DX is higher than its expected value by 1 unit in the current period, how much the unemployment rate and the inflation rate will change in 10 periods from now. Hint:

Change in DX is equivalent to a combination of shocks from both UE and DI.

2.3) According to Phillips curve, UE and DI are negatively correlated with other things being equal. Does the provided evidence suggest that the unemployment rate and the inflation rate have a negative linear relationship over the long run? Explain in details.

Let Pt be the international price in term of local currency in period t. It is assumed that P follows an ARMA(1,q) with a constant term and a time trend as follows

Pt = 0 + 1Pt-1 + 2t + ut where ut = stationary error term in period t

ut ~ MA(q) Questions

3.1) Suppose that we can also represent Pt as a linear function of time t (a kind of long- term representation) as follows:

Pt = 0 + 2t + vt

where vt = error term in period t Verify that

0 = 0(1 - 1) + 12

2 = 2(1 - 1)

vt ~ ARMA(1,q) without a constant term nor time trend

3.2) If the long-term representation of P exists, show that the short-term model above is equivalent to an Error Correction Model as follows:

Pt = 0 + 1vt-1 + ut

3.3) Based on printouts 3.1-3.4, choose the appropriate order q and estimate all the parameters (,,) and their standard errors. Make sure that you select the printout with valid estimation.

We are suspecting that a time series Y has been generated by an ARIMA(p,d,q) process.

Explain in steps how you will determine the value of d.

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PRINTOUT 1.1

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Dependent Variable: R Method: ML - ARCH Date: 09/21/01 Time: 11:03 Sample: 1 200 Included observations: 200 Convergence achieved after 32 iterations

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CoefficientStd. Errorz-Statistic Prob.

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GARCH -0.068649 0.092608 -0.741284 0.4585 C 1.358171 2.628892 0.516633 0.6054 I 1.408367 0.560580 2.512337 0.0120

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Variance Equation

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C 16.35778 8.006746 2.043000 0.0411 ARCH(1) -0.104136 0.068835 -1.512837 0.1303 GARCH(1) 0.671740 0.204708 3.281458 0.0010 I -3.451832 1.851756 -1.864086 0.0623

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R-squared 0.058100 Mean dependent var 4.669433 Adjusted R-squared 0.028818 S.D. dependent var 3.807874 S.E. of regression 3.752606 Akaike info criteri5.492636 Sum squared resid 2717.836 Schwarz criterion 5.608077 Log likelihood -542.2636 F-statistic 1.984146 Durbin-Watson stat 1.996853 Prob(F-statistic) 0.069728

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PRINTOUT 1.2

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Wald Test:

Equation: TEST1

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Null HypothesisC(1)*C(7)=0 C(3)=0

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F-statistic 6.382780 Probability 0.002069 Chi-square 12.76556 Probability 0.001690

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PRINTOUT 1.3

Coefficient Covariance Matrix

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GARCH C I C ARCH(1) GARCH(1) I

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GARCH 0.008576 -0.202745 0.030444 0.018991 -0.000957 0.000521 - 0.001366

C -0.202745 6.911071 -1.369686 0.391061 0.020978 -0.017530 - 0.175068

I 0.030444 -1.369686 0.314250 -0.136792 -0.003759 0.004567 0.042718

C 0.018991 0.391061 -0.136792 64.10799 -0.251232 -1.223133 - 13.99111

ARCH(1) -0.000957 0.020978 -0.003759 -0.251232 0.004738 0.003538 0.042708

GARCH(1) 0.000521 -0.017530 0.004567 -1.223133 0.003538 0.041905 0.195407

I -0.001366 -0.175068 0.042718 -13.99111 0.042708 0.195407 3.429000

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PRINTOUT 1.4

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C -0.517465 1.458588 -0.354772 0.7228 I 1.720328 0.462550 3.719228 0.0002

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Variance Equation

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C 14.95279 8.008804 1.867044 0.0619 ARCH(1) -0.093416 0.063441 -1.472492 0.1409 GARCH(1) 0.688335 0.224024 3.072593 0.0021 I -3.111322 1.808913 -1.719996 0.0854

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PRINTOUT 1.5

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C 4.666981 0.279605 16.69134 0.0000

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Variance Equation

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C 5.933606 17.26692 0.343640 0.7311 ARCH(1) -0.035832 0.082605 -0.433773 0.6645 GARCH(1) 0.624949 1.149383 0.543726 0.5866

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R-squared 0.000000 Mean dependent var 4.669433 Adjusted R-squared -0.015307 S.D. dependent var 3.807874 S.E. of regression 3.836906 Akaike info criteri5.545001 Sum squared resid 2885.482 Schwarz criterion 5.610967 Log likelihood -550.5001 Durbin-Watson stat 1.968510

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PRINTOUT 2.1

Vector Autoregression Estimates

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Date: 09/21/01 Time: 12:39 Sample(adjusted): 2 60

Included observations: 59 after adjusting endpoints Standard errors & t-statistics in parentheses

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UE DI

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UE(-1) 0.482761 -0.186403 (0.10431) (0.08987) (4.62803) (-2.07415) DI(-1) -0.305201 0.333474 (0.10946) (0.09430) (-2.78830) (3.53621) C 0.355390 0.280931 (0.53116) (0.45762)

(0.66908) (0.61390) DX(-1) 0.030838 -0.008789 (0.08221) (0.07083) (0.37511) (-0.12409)

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R-squared 0.384250 0.264187 Adj. R-squared 0.350664 0.224051 Sum sq. resids 66.94749 49.69198 S.E. equation 1.103280 0.950521 F-statistic 11.44066 6.582405 Log likelihood -87.44532 -78.65240 Akaike AIC 3.099841 2.801776 Schwarz SC 3.240691 2.942626 Mean dependent 0.973097 0.106559 S.D. dependent 1.369150 1.079059

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Determinant Residual Covaria 0.947927 Log Likelihood -165.8571 Akaike Information Criteria 5.893463 Schwarz Criteria 6.175163

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PRINTOUT 2.2

Johansen Cointegration Test

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Date: 09/21/01 Time: 12:41 Sample: 1 60 Included observations: 58 Test assumption: Linear deterministic trend in the data Series: UE DI Exogenous series: DX(-1) Warning: Critical values were derived assuming no exogenous

series

Lags interval: 1 to 1

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Likelihood 5 Percent 1 Percent Hypothesized

Eigenvalue Ratio Critical ValuCritical ValuNo. of CE(s)

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0.389469 35.10033 15.41 20.04 None

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0.058261 3.481634 3.76 6.65 At most 1

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*(**) denotes rejection of the hypothesis at 5%(1%) significance level

L.R. test indicates ? cointegrating equation(s) at 5%

significance level

Unnormalized Cointegrating Coefficients:

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UE DI 0.075620 0.138147 0.090936 -0.040264

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Normalized Cointegrating Coefficients: 1 Cointegrating

Equation(s)

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UE DI C 1.000000 1.826867 -1.171849 (0.44871) Log likelihood-165.0398

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PRINTOUT 2.3

Impulse Response to One S.D. Innovations

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Response of UE:

Period UE DI

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1 1.065224 0.000000 (0.09806) (0.00000) 2 0.539492 -0.278954 (0.12301) (0.09995) 3 0.329465 -0.227692 (0.12001) (0.08871) 4 0.212761 -0.156812

(0.10630) (0.07576) 5 0.139366 -0.104293 (0.08882) (0.06255) 6 0.091608 -0.068804 (0.07120) (0.05007) 7 0.060266 -0.045303 (0.05538) (0.03905) 8 0.039655 -0.029816 (0.04211) (0.02981) 9 0.026094 -0.019621 (0.03147) (0.02236) 10 0.017171 -0.012911 (0.02319) (0.01654)

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Response of DI:

Period UE DI

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1 -0.082709 0.914000 (0.11924) (0.08414) 2 -0.226142 0.304795 (0.10213) (0.08782) 3 -0.175975 0.153639 (0.08621) (0.06339) 4 -0.120096 0.093677 (0.07096) (0.05046) 5 -0.079708 0.060469 (0.05697) (0.04025) 6 -0.052559 0.039605 (0.04457) (0.03148) 7 -0.034603 0.026033 (0.03411) (0.02415) 8 -0.022773 0.017126 (0.02565) (0.01821) 9 -0.014986 0.011269 (0.01900) (0.01354) 10 -0.009861 0.007415 (0.01391) (0.00995)

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Ordering: UE DI

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PRINTOUT 3.1

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Dependent Variable: P

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Method: Least Squares Date: 09/21/01 Time: 12:22 Sample(adjusted): 2 60 Included observations: 59 after adjusting endpoints Convergence achieved after 3 iterations

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Variable CoefficientStd. Errort-Statistic Prob.

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C 0.544748 0.395830 1.376218 0.1742 T 0.028078 0.010966 2.560365 0.0132 AR(1) 0.379810 0.111828 3.396378 0.0013

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Inverted AR Roots .38

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PRINTOUT 3.2

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Dependent Variable: P Method: Least Squares Date: 09/21/01 Time: 12:25 Sample(adjusted): 2 60 Included observations: 59 after adjusting endpoints Convergence achieved after 20 iterations Backcast: 1

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C 0.623544 0.391925 1.590977 0.1173 T 0.026095 0.010901 2.393943 0.0201 AR(1) 0.223999 0.207870 1.077593 0.2859 MA(1) 0.246752 0.230930 1.068515 0.2900

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Inverted AR Roots .22 Inverted MA Roots -.25

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PRINTOUT 3.3

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Dependent Variable: P Method: Least Squares Date: 09/21/01 Time: 12:25 Sample(adjusted): 2 60 Included observations: 59 after adjusting endpoints Convergence achieved after 10 iterations Backcast: 0 1

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C 0.605384 0.405179 1.494113 0.1410 T 0.026611 0.011303 2.354388 0.0222 AR(1) -0.332417 0.202294 -1.643238 0.1061 MA(1) 0.864226 0.138748 6.228750 0.0000 MA(2) 0.414083 0.095717 4.326110 0.0001

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Inverted AR Roots -.33 Inverted MA Roots -.43 -.4 -.43+.48i

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PRINTOUT 3.4

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Dependent Variable: P Method: Least Squares Date: 09/21/01 Time: 12:25 Sample(adjusted): 2 60 Included observations: 59 after adjusting endpoints Convergence achieved after 40 iterations Backcast: OFF (Roots of MA process too large for backcast)

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C 0.079722 0.373835 0.213253 0.8319 T 0.037741 0.010062 3.750663 0.0004 AR(1) 0.657275 0.070069 9.380439 0.0000 MA(1) -0.409069 0.143934 -2.842064 0.0063 MA(2) -0.414119 0.150951 -2.743395 0.0083 MA(3) -0.548431 0.160501 -3.417002 0.0012

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Inverted AR Roots .66 Inverted MA Roots 1.1 -.38 -.57 -.38+.57i Estimated MA process is noninvertible

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End of Exam