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 σ 2 Vadim Marmer University of British Columbia May 5, 2010 Fitted values I Consider the multiple regression model with k regressors: Y i = β + β 1 X 1 , i + β 2 X 2 , i + . . . + β k X k , i + U i . I Let ˆ β , ˆ β 1 , . . . , ˆ β k be the OLS estimators. I The &tted (or predicted) by the model value of Y is: ˆ Y i = ˆ β + ˆ β 1 X 1 , i + ˆ β 2 X 2 , i + . . . + ˆ β k X k , i . I The residual is: ˆ U i = Y i & ˆ Y i . I Consider the average of ˆ Y : ˆ Y = 1 n n ∑ i = 1 ˆ Y i = 1 n n ∑ i = 1 & Y i & ˆ U i ¡ = ¯ Y & 1 n n ∑ i = 1 ˆ U i = ¯ Y , because when there is an intercept, ∑ n i = 1 ˆ 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 & ˆ 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 ˆ 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 & ˆ Y i + ˆ U i & ¯ Y ¡ 2 = n ∑ i = 1 && ˆ Y i & ¯ Y ¡ + ˆ U i ¡ 2 = n ∑ i = 1 & ˆ Y i & ¯ Y ¡ 2 + n ∑ i = 1 ˆ U 2 i + 2 n ∑ i = 1 & ˆ Y i & ¯ Y ¡ ˆ U i = SSE + SSR + 2 n ∑ i = 1 & ˆ Y i & ¯ Y ¡ ˆ U i . I Next, we will show that ∑ n i = 1 & ˆ Y i & ¯ Y ¡ ˆ U i = . 3/15 Proof of SST=SSE+SSR I Since ˆ Y i = ˆ β + ˆ β 1 X 1 , i + . . . + ˆ β k X k , i , n ∑ i = 1 & ˆ Y i & ¯ Y ¡ ˆ U i = n ∑ i = 1 && ˆ β + ˆ β 1 X 1 , i + . . . + ˆ β k X k , i ¡ & ¯ Y ¡ ˆ U i = ˆ β n ∑ i = 1 ˆ U i + ˆ β 1 n ∑ i = 1 X 1 , i ˆ U i + . . . + ˆ β k n ∑ i = 1 X k , i ˆ U i & ¯ Y n ∑ i = 1 ˆ U i ....
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Lecture 11 Goodness of fit, estimating the variance of errors

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