In this case each observation would have its own

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could be considered consistent with the estimated equation. In this case each observation would have its own period-specific dummy. Such tests are sometimes called post-sample predictive tests. Consider the following model ( t =1 991, 1992, …,2000) : 0 1 1 2 2 2001 2001 2002 2002 2003 2003 t t t t Y X X e I e I e I u where the observation-specific dummies of 2001 I , 2002 I and 2003 I are defined as: 2001 1 for 2001 0 otherwise I , 2002 1 for 2002 0 otherwise I , 2003 1 for 2003 0 otherwise I Here, the coefficients of observation-specific dummies denoted by 2001 e , 2002 e and 2003 e are the expected value of forecast errors (prediction errors) of the corresponding years (The proof is beyond the scope of this lecture). In order to test for structural change in 2001, 2002 and 2003; we can test 2001 e , 2002 e and 2003 e individually (t test) and jointly (F test) different from zero.
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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 10 I. Individual Test For example, we can individually test the statistical significance of 2001 e as follows: 0 2001 : 0 H e 0 2001 : 0 H e Generalizing the individual test of the statistical significance of t e is 0 : 0 t H e : 0 t A H e Note that if this null hypothesis is not rejected this does not necessarily imply that the coefficients pertaining to the two subsamples will be equal. If, however, the null hypothesis is rejected this will imply that at least one of the coefficients is different for two subsamples. o In other words, although the rejection of null hypothesis would imply a structural change, the non-rejection of null hypothesis does not necessarily imply parameter stability for two subsamples. II. Joint Test
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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 11 Below, note the summary of the basic steps of the Predictive test to test for the following hypothesis: 0 1 1 3 : ... 0 t H e e e e 0 : at least one of them is non zero H where t e is the expected value of prediction error. Note that For our example, the restricted model is: (1) 0 1 1 2 2 t t t t Y X X u t =1991, 1992, …,200 3 The unrestricted model is (2) 0 1 1 2 2 1 2001 2 2002 3 2003 t t t t Y X X e I e I e I u 1991,1992,...,2003 t Step 1. Estimate the equation (1) and retrieve the sum of squared residuals, R S . Note that for our example t =1991, 1992, …,2003 and hence T=13. Step 2. Estimate the equation (2) and retrieve the sum of squared residuals, U S . Step 3. Calculate the following test statistic: / / 1 R U U S S p Q S T k
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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 12 where 1 T k is the degrees of freedom in unrestricted model 2 and p is the number of restrictions in null hypothesis 3 . The calculated Q statistic is distributed as F with [ p , 1 T k ] degrees of freedom: , 1 p T k Q F   .Reject the null hypothesis of parameter constancy if , 1 p T k Q F   .
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