Least Squares estimators We want to choose values of bo and b, that minimize the sum squared error Ssb1)-∑|y(b0+b1x) Take the derivatives, set them equal to zero and you get ∑ Xi- mean(x))(y: mean(y) 0:= mean(y)-b 1 mean(x) b mean x
Least Squares Estimators • We want to choose values of bo and b1 that minimize the sum squared error SSE b , 0 b 1 i 2 yi b 0 b 1 xi . • Take the derivatives, set them equal to zero and you get b 1 i xi mean () x yi . mean ( ) y i xi mean () x 2 b 0 mean () y b ( ) 1 .mean x MIT
Distribution of error Homoscedasticity Heteroscedasticity xxX Population data points Error terms mit 人 16881
Distribution of Error • Homoscedasticity • Heteroscedasticity 4 2 0 2 4 6 0 0.5 1 1.5 2 Population regression line Population data points Error terms 16.881 MIT
Cautions re: Regression What will result 2·10 if you run a linear y expo. IIod regression on these data sets? 2 0.012684 9.885085 521501 30 l3690990 a D o Scatterplot of data Estimated regression line 0012684 9.885085 mit 人 16881
Cautions Re: Regression 2 104 . • What will result 2 104 if you run a linear y expok 1 104 regression on 0 0 0 2 4 6 8 10 these data sets? 0.012684 xk 9.885085 28.521501 30 20 y quadk 1.369099 10 0 16.881 1 2 3 0 50 Scatterplot of data Estimated regression line 0 2 4 6 8 10 0.012684 xk 9.885085 MIT
Linear regression A ssumptions 1. The average value of the dependent variable y is a linear function ofX. 2. The only random component of the linear model is the error term 8. The values of X are assumed to be fixed 3. The errors between observations are uncorrelated. In addition, for any given value ofx, the errors are are normally distributed with a mean of zero and a constant variance mit 人 16881
Linear Regression Assumptions 1. The average value of the dependent variable Y is a linear function ofX. 2. The only random component of the linear model is the error term ε. The values of X are assumed to be fixed. 3. The errors between observations are uncorrelated. In addition, for any given value ofX, the errors are are normally distributed with a mean of zero and a constant variance. 16.881 MIT