Note that due to mathematical reasons the ssr of the

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Note that, due to mathematical reasons , the SSR of the following models will be the same: (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 ; SSR= U S (1*) 0 1 1 2 2 t t t t Y X X u 1991,1992,...,2000 t ; SSR= * U S Hence instead of equation (2), we can estimate equation (1*) and use its SSR as the unrestricted model’ s SSR: * U U S S Let us denote the number of observations in equation (1*) as T 1 . In our example, T 1 =10 and T =13. 2 In our example T-k-1=13-5-1=7. 3 In our example p=3.
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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 13 Hence, the steps of predictive test can also be written as follows: Step 1. Estimate the equation over the complete sample period [i.e., estimate equation (1)] and retrieve the sum of squared residuals, R S . Step 2. Estimate the equation over the larger sub-sample and retrieve the sum of squared residuals, * U S . Step 3. Calculate the following test statistic: * * 1 / / 1 R U U S S p Q S T k where 1 1 1 T k is the degrees of freedom 4 in unrestricted model (1*). The calculated Q statistic is distributed as F with [ T 2 , 1 1 1 T k ] degrees of freedom: 1 1 , 1 p T k Q F .Reject the null hypothesis of parameter constancy if 1 1 , 1 p T k Q F . 2. Unknown Breaks: Testing for Break Points A break is another name of a structural change point in data. Breaks can arise either from a change in the population regression coefficients at a specific date or from a gradual evolution of the 4 In our example T 1 -k-1=10-2-1=7.
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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 14 coefficients over a longer period of time (Stock and Watson, 2012, pp. 598-610). One way to detect breaks is to test for discrete changes, or breaks, in the regression coefficients. How this is done depends on whether the date of the suspected break (the break date) is known. In some applications you might suspect that there is a break at a known date. We have seen that (last two lectures), if the date of the hypothesized break is known, then the null hypothesis of no break (no structural change) can be tested using ACOV Chow test and Predictive Chow test. Chow suggested that the predictive test for the case where second subsample size ( T 2 ) is less than k +1. In this case the regression equation cannot be estimated with the second sample and thus the analysis of covariance test (ACOV test) cannot be applied. In this case only the predictive test can be used. He also suggested that the predictive test can be used even when second subsample size ( T 2 ) is larger than k +1 [i.e., T 2 > k +1] but that in this case the ACOV Chow test should be preferred because it is more powerful (Maddala, 1992, p.176) On the other hand, often the date of a possible break (structural change point) is unknown or known only within a range. In this case three tests can be used to determine the break date:
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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 15 (1) Quandt likelihood ratio (QLR) statistic (Quandt, 1960) 5 (2) CUSUM and CUSUMSQ tests (Recursive Residuals) In this semester of ECON301 we will only see CUSUM and CUSUMQ tests.
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