ECON301_Handout_12_1213_02

# I individual test consider the following model y t 01

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I. Individual Test Consider the following model; Y t = 01 + 11 X t + 0 D t2 + 1 D t2 X t +u t (2.parametrization) intercept dummy slope dummy a) H 0 : 0 = 0 t 0 =( 0 /se( 0 )) H A : 0 0 t 0 t (T-k-1, /2) if t 0 >t so RH 0 at level of significance of . Thus, 0 is individually different from zero. There is 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 4 b) H 0 : 1 = 0 t 1 =( 1 /se( 1 )) H A : 1 0 t 1 t (T-k-1, /2) if t 1 >t so RH 0 at level of significance of . Thus, 1 is individually different from zero. There is structural change. II. Joint Test Chow suggested the following test 1 : Step 1. Estimate the equation over the complete sample period and retrieve the sum of squared residuals, SR. Step 2. Estimate the equation over both sample sub-periods and retrieve the sum of squared residuals S1 and S2. Step 3. Calculate the following test statistic:     12 / / 2( 1) R S S S p Q S S T k     , or     . 2( R S S S Tk Q p SS 1 Note that: we carry out this Chow test for T>k+1.
ECON 301 - Introduction to Econometrics I May 2013 METU - Department of Economics Instructor: Dr. Ozan ERUYGUR e-mail: [email protected] Lecture Notes 5 which is distributed as F with [p,T-2(k+1)] degrees of freedom. Here, p is the number of restrictions in null hypothesis (hence, p = k +1) where k is the number of slope parameters in the model which is used in Step 1. For example for a model 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 Hence, the Chow test is: H 0 : 02 = 12 = 22 =0 H A : 0 0 , 1 0 , 22 0 If Q>F, then RH 0 at level of significance of . Thus, 02 , 12 , and 22 are jointly different from zero. There is structural change. Example The following regression results are obtained for the subsamples (T 1 and T 2 , T 1 +T 2 is equal to T) of a set of observation, and for the sample as a whole: (1) t Y ˆ = 103.7 +3.75 X t1 +10.51 X t2 SSR= 7262, T 1 = 105 (1.30) (3.85) (2) Y t =95.6+ 2.90 X t1 +13.30 X t2 SSR= 1131, T 2 = 65 (1.97) (6.80)

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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 6 (3) Y t =98.3+ 3.60 X t1 +12.78 X t2 SSR= 8451, T=170 (1.13) (14.20) Standard errors are given in parenthesis and SSR stands for sum of squared residuals. a. Write the first two equations as a single equation using dummy variables. In doing so, also obtain the standard errors of the estimated coefficients for the new equation and the associated SSR. b. Test if the coefficients from the two subsamples are significantly different as a whole. In other words, test for the structural change. 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
ECON 301 - Introduction to Econometrics I May 2013

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I Individual Test Consider the following model Y t 01 11 X...

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