Simple vs multiple reg estimate Compare the simple regression y=Bo+B,x, with the multiple regression y=Bo+B,,+B2x2 Genera,B1≠ B, unless: B,=0(ie. no partial effect of x,OR x, and x, are uncorrelat ed in the sample Economics 20- Prof anderson 6
Economics 20 - Prof. Anderson 6 Simple vs Multiple Reg Estimate and are uncorrelat ed in the sample ˆ 0 (i.e. no partial effect of ) O R unless : ˆ ~ Generally, ˆ ˆ ˆ with the multiple regression ˆ ~ ~ ~ Compare the simple regression 1 2 2 2 1 1 0 1 1 2 2 0 1 1 x x x y x x y x = = + + = + b b b b b b b b
Goodness-of-Fit We can think of each observatio n as being made up of an explained part, and an unexplained d part, y,=y,+u We then define the following >O-y is the total sum of squares(SST) >O-v is the explained sum of squares(SSE) >u? is the residual sum of squares(SSR) Then sst=sse+ ssr Economics 20- Prof anderson 7
Economics 20 - Prof. Anderson 7 Goodness-of-Fit ( ) ( ) Then SST SSE SSR ˆ is the residual sum of squares (SSR) ˆ is the explained sum of squares (SSE) is the total sum of squares (SST) ˆ ˆ We then define the following : up of an explained part, and an unexplaine d part, We can think of each observatio n as being made 2 2 2 = + − − = + i i i i i i u y y y y y y u
Goodness-of-Fit(continued) How do we think about how well our sample regression line fits our sample data? Can compute the fraction of the total sum of squares (sst)that is explained by the model, call this the R-squared of regression D R2= SSE/SST=1- SSR/SST Economics 20- Prof anderson 8
Economics 20 - Prof. Anderson 8 Goodness-of-Fit (continued) How do we think about how well our sample regression line fits our sample data? Can compute the fraction of the total sum of squares (SST) that is explained by the model, call this the R-squared of regression R2 = SSE/SST = 1 – SSR/SST
Goodness-of-Fit(continued) We can also think of R as being equal to the squared correlatio n coefficien t between the actual y, and the values y C(-y)6-5 Economics 20- Prof anderson 9
Economics 20 - Prof. Anderson 9 Goodness-of-Fit (continued) ( ( )( )) (( ) )(( ) ) − − − − = 2 2 2 2 2 ˆ ˆ ˆ ˆ the actual and the values ˆ the squared correlatio n coefficien t between We can also think of as being equal to y y y y y y y y R y y R i i i i i i
More about R-squared o R2 can never decrease when another independent variable is added to a regression, and usually will increase o Because R2 will usually increase with the number of independent variables, it is not a good way to compare models Economics 20- Prof anderson 10
Economics 20 - Prof. Anderson 10 More about R-squared R2 can never decrease when another independent variable is added to a regression, and usually will increase Because R2 will usually increase with the number of independent variables, it is not a good way to compare models