Solution a 1 st parametrization y t 1037 d t1 375 d

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Solution: a) 1 st Parametrization: Y t = 103.7 D t1 +3.75 D t1 X t1 +10.51 D t1 X t2 +95.6 D t2 +2.90 D t2 X t1 +13.30 D t2 X t2 B 01 B 11 B 21 B 02 B 12 B 22 2 nd Parametrization: (p-1) Dummy variable (p number of subsamples) D t1 +D t2 = 1 D t1 = 1 - D t2 Y t = 103.7(1 - D t2 )+3.75(1 -D t2 )X t1 +10.51(1-D t2 )X t2 +95.6D t2 +2.90D t2 X t1 +13.30D t2 X t2 Y t = 103.7 - 103.7 D t2 +3.75 X t1 - 3.75 D t2 X t1 +10.51 X t2 - 10.51 D t2 X t2 +95.6 D t2 +2.90D t2 X t1 +13.30D t2 X t2
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ECON 301 - Introduction to Econometrics I May 2013 METU - Department of Economics Instructor: Dr. Ozan ERUYGUR e-mail: [email protected] Lecture Notes 7 Y t = 103.7+3.75 X t1 +10.51 X t2 - 8.1 D t2 - 0.85 D t2 X t1 +2.79 D t2 X t2 2 nd Param. If we illustrate the second parametrization, we can write as follows: Y t = B 01 +B 11 X t1+ B 21 X t2 +(B 02 - B 01 ) D t2 +(B 12 - B 11 )D t2 X t1 +(B 22 - B 21 )D t2 X t2 +u t 1.subsample is base 02 12 22 (D t1 = 1 - D t2 ) Intercept Slope Slope dummy dummy dummy b) First parametrization Y t = B 01 +B 11 X t1 +B 21 X t2 +u t 1st Subsample Y t = B 02 +B 12 X t1 +B 22 X t2 +u t 2nd Subsample Y t = B 01 D t1 +B 11 D t1 X t1 +B 21 D t1 X t2 +B 02 D t2 +B 12 D t2 X t1 +B 22 D t2 X t2 +u t According to second parametrization: SU=S1+S2=7262+1131=8393 Y t = B 01 +B 11 X t1+ B 21 X t2 +(B 02 - B 01 ) D t2 +(B 12 - B 11 )D t2 X t1 +(B 22 - B 21 )D t2 X t2 +u t Y t = B 01 +B 11 X t1+ B 21 X t2 + 02 D t2 + 12 D t2 X t1 + 22 D t2 X t2 +u t (2nd Param.) Joint Test H 0 : 02 = 12 = 22 = 0 H 0 : 02 0 , 12 0 , 22 0 Q=((SR-SU)/SU) * ([T-2(k+1)]/p) = ((8451-8393)/8393) * (170-2[3]/3)= 0.378 F 0.05 (3,164)= 2.66 Q < F Do not reject H 0 at = 0.05 level of significance. So there is no structural change.
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ECON 301 - Introduction to Econometrics I May 2013 METU - Department of Economics Instructor: Dr. Ozan ERUYGUR e-mail: [email protected] Lecture Notes 8 B. Predictive Test We have already discussed Chow’s test of structural stability in the previous lecture. In some cases the sub-period may be too small ( T 2 <k +1) and so Chow suggested the predictive or forecast test . Observation-Specific Dummy An observation-specific dummy is a dummy variable that takes on the value one for a specific observation and zero for all other observations. It is also called period-specific dummy (Kennedy, 2001, pp.226-227). (1) The coefficient estimate for the observation-specific dummy is the forecast error (prediction error) for that observation, and the estimated variance of this coefficient estimate is the estimate of the variance of the forecast error. (2) If the value of the dependent variable for the period (observation) in question is coded as zero instead of its actual value (which may not be known, if we are trying to forecast it) then the estimated coefficient of the period-specific dummy is the forecast of that period’s dependent variable. (3) By testing the estimated coefficient of the observation-specific dummy against zero, using t test, we can test whether or not that observation is “ consistent ” with the estimated relationship. An F
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ECON 301 - Introduction to Econometrics I May 2013 METU - Department of Economics Instructor: Dr. Ozan ERUYGUR e-mail: [email protected] Lecture Notes 9 test (joint test) would be used to test if several observations could be considered consistent with the estimated equation. In
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