# sec3_1_ - Goodness of t Yi = Yi Yi = Yi Yi Yi Yi Y = Yi Y...

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Goodness of fit Y i = Y i Y i = Y i + b Y i - b Y i Y i - Y = b Y i - Y + Y i - b Y i Square both sides, ( Y i - Y ) 2 = ( b Y i - Y ) 2 + ( Y i - b Y i ) 2 + 2( b Y i - Y )( Y i - b Y i ) And then sum over all of i , n X i =1 ( Y i - Y ) 2 = n X i =1 ( b Y i - Y ) 2 + n X i =1 ( Y i - b Y i ) 2 + 2 n X i =1 ( b Y i - Y )( Y i - b Y i ) It can be shown that 2 n i =1 ( b Y i - Y )( Y i - b Y i ) = 0 (using that u 0 = i = 0 and b Y i u i = 0). Therefore, n X i =1 ( Y i - Y ) 2 = n X i =1 ( b Y i - Y ) 2 + n X i =1 ( Y i - b Y i ) 2 Labeling these, TSS = ESS + RSS or SST = SSE + SSR Total, explained, and residual. In other words, the total variation can be decomposed into the variation that is explained by the model and the variation that is unexplained (related to the residuals) From this, we can create a statistic that tells us how good our model is at explaining the data; simply put R 2 = SSE/SST Look at the stata data http://www.cid.harvard.edu/ciddata/ciddata.html Read it in graph drop _all

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