Lecture 11 Goodness of fit, estimating the variance of errors

Lecture 11 Goodness of fit, estimating the variance of errors

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Unformatted text preview: Economics 326 Methods of Empirical Research in Economics Lecture 11: Goodness of &t, estimation of 2 Vadim Marmer University of British Columbia May 5, 2010 Fitted values I Consider the multiple regression model with k regressors: Y i = + 1 X 1 , i + 2 X 2 , i + . . . + k X k , i + U i . I Let , 1 , . . . , k be the OLS estimators. I The &tted (or predicted) by the model value of Y is: Y i = + 1 X 1 , i + 2 X 2 , i + . . . + k X k , i . I The residual is: U i = Y i & Y i . I Consider the average of Y : Y = 1 n n i = 1 Y i = 1 n n i = 1 & Y i & U i = Y & 1 n n i = 1 U i = Y , because when there is an intercept, n i = 1 U i = 0. 1/15 Sum-of-Squares I The total variation of Y in the sample is: SST = n i = 1 ( Y i & Y ) 2 (Total Sum-of-Squares). I The explained variation of Y in the sample is: SSE = n i = 1 & Y i & Y 2 (Explained or Model Sum-of-Squares). I The residual (unexplained or error) variation of Y in the sample is: SSR = n i = 1 U 2 i (Residual Sum-of-Squares). I If the regression contains an intercept: SST = SSE + SSR . 2/15 Proof of SST=SSE+SSR I First, SST = n i = 1 ( Y i & Y ) 2 = n i = 1 & Y i + U i & Y 2 = n i = 1 && Y i & Y + U i 2 = n i = 1 & Y i & Y 2 + n i = 1 U 2 i + 2 n i = 1 & Y i & Y U i = SSE + SSR + 2 n i = 1 & Y i & Y U i . I Next, we will show that n i = 1 & Y i & Y U i = . 3/15 Proof of SST=SSE+SSR I Since Y i = + 1 X 1 , i + . . . + k X k , i , n i = 1 & Y i & Y U i = n i = 1 && + 1 X 1 , i + . . . + k X k , i & Y U i = n i = 1 U i + 1 n i = 1 X 1 , i U i + . . . + k n i = 1 X k , i U i & Y n i = 1 U i ....
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Lecture 11 Goodness of fit, estimating the variance of errors

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