ECON301_Handout_13_1213_02

# Now if model c is correct 3 4 whereas model d is

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Now if Model C is correct, 3 4 0 , whereas Model D is correct if 1 2 0 . This testing can be done by the usual F test , hence the name encompassing F test . B. Davidson MacKinnon J Test To illustrate this test, suppose we want to compare hypothesis or Model C with hypothesis or Model D. The J test proceeds as follows:

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ECON 301 - Introduction to Econometrics I May 2013 METU - Department of Economics Instructor: Dr. Ozan ERUYGUR e-mail: Lecture Notes 4 (1) We estimate Model D ( 0 1 1 2 2 t t t t Y Z Z v ), and from it we obtain the estimated Y values, ˆ D t Y (2) We add the estimated Y value in Step 1 as an additional regressor to Model C ( 0 1 1 2 2 t t t t Y X X u ) and estimate the following model: 0 1 1 2 2 3 ˆ D t t t t t Y X X Y u (6) where the ˆ D t Y values are obtained from Step 1. (3) Using the t test , test the hypothesis that 3 = 0. (4) If the hypothesis that 3 = 0 is not rejected, we can accept (i.e., not reject) Model C as the true model because ˆ D t Y included in (6), which represent the influence of variables not included in Model C, have no additional explanatory power beyond that contributed by Model C. In other words, Model C encompasses Model D in the sense that the latter model does not contain any additional information that will improve the performance of Model C. By the same token, if the null hypothesis is rejected, Model C cannot be the true model (5) Now we reverse the roles of hypotheses, or Models C and D. We now estimate Model C first, use the estimated Y values from this model as regressor in (6), repeat Step 4, and decide whether to accept
ECON 301 - Introduction to Econometrics I May 2013 METU - Department of Economics Instructor: Dr. Ozan ERUYGUR e-mail: Lecture Notes 5 Model D over Model C. More specifically, we estimate the following model: 0 1 1 2 2 3 ˆ C t t t t t Y Z Z Y u (7) where ˆ C t Y are the estimated Y values from Model C. We now test the hypothesis that 3 = 0. If this hypothesis is not rejected, we choose Model D over C. If the hypothesis that 3 = 0 is rejected, choose C over D, as the latter does not improve over the performance of C. Although it is intuitively appealing, the J test has some problems. Since the tests given in (6) and (7) are performed independently, we have the following likely outcomes: Hypothesis: 3 = 0 Hypothesis: 3 = 0 Do not Reject Reject Do not Reject Accept both C and D Accept D, reject C Reject Accept C, reject D Reject both C and D As this table shows, we will not be able to get a clear answer if the J testing procedure leads to the acceptance or rejection of both models. In case both models are rejected, neither model helps us to explain the behavior of Y.

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ECON 301 - Introduction to Econometrics I May 2013 METU - Department of Economics Instructor: Dr. Ozan ERUYGUR e-mail: Lecture Notes 6 Another problem with the J test is that when we use the t statistic to test the significance of the estimated Y variable in models (6) and (7), the t statistic has the standard normal distribution only asymptotically, that is, in large samples. Therefore, the J test may not be very
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