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week06-2 - TA session 6 Econ 103 winter 2010 Wed Feb 10...

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TA session 6 Econ. 103, winter 2010 Wed., Feb. 10, 2010, 10:00 a.m. and 1:00 p.m. in PP2400E. 2 Verbose discussion of certain true/false 2.1 In a single-variable OLS regression, if ˆ β = 0 then R 2 = 0 This is true actually, and it’s easy to see why intuitively. If the coefficient ˆ β equals zero then ˆ β · x equals zero, and the data explain nothing about the dependent variable. Let’s show it mathematically. If β = 0... ˆ y = ˆ c + 0 The constant c is the only parameter left that OLS may use to minimize the SSE. The only c that minimizes the SSE is c = Y . (Think about it.) So ˆ y = Y RSS = X ( y - ˆ y ) 2 = X ( y - y ) 2 that means RSS = TSS = X ( y - y ) 2 also Since R 2 = 1 - RSS T SS and RSS = TSS , then R 2 = 1 - 1 = 0. Let’s have a little discussion about the meaning of R 2 . The purpose of a statistical model is to explain the variation in a dependent variable ( y i ) using several explanatory variables (the x ’s). The variation in the sample, before using any model to explain it, is TSS = X ( y - y ) 2 Then TSS is just the sum of the errors of the sample average. One way to judge an estimator is by the amount of variation in the sample it leaves. So, try to think of y as the “baseline” or “default” statistical model. If none of the independent variables x explain anything about the data, the worst you can do is regress y on a constant c . But if a statistical model succeeds in explaining something about the data, it should be able to form better predictions than the sample average. That
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