4-1 Chapter 4 Further issues with the classical linear regression model
4-1 Chapter 4 Further issues with the classical linear regression model
4-2 本章目标 继续讨论古典线性回归模型 ·了解确定模型优劣的各种方法 ·普通最小二乘法OLS可能遇到的各种问题及其处理
4-2 本章目标 继续讨论古典线性回归模型 • 了解确定模型优劣的各种方法 • 普通最小二乘法OLS可能遇到的各种问题及其处理
4-3 1 Goodness of fit statistics We would like some measure of how well our regression model actually fits the data. We have goodness of fit statistics to test this: i.e. how well the sample regression function(srf) fits the data. The most common goodness of fit statistic is known as R2.One way to define rl is to say that it is the square of the correlation coefficient between y and y For another explanation, recall that what we are interested in doing is explaining the variability of y about its mean value, y i,, the total sum of squares,TSS总变差: 7SS=∑(01-y) We can split the Tss into two parts, the part which we have explained (known as the explained sum of squares, ESS)and the part which we did not explain using the model (the rss)
4-3 1 Goodness of Fit Statistics • We would like some measure of how well our regression model actually fits the data. * • We have goodness of fit statistics to test this: i.e. how well the sample regression function (srf) fitsthe data. • The most common goodness of fit statistic is known as R2 . One way to define R2 is to say that it is the square of the correlation coefficient between y and . • For another explanation, recall that what we are interested in doing is explaining the variability of y about its mean value, , i.e. the total sum of squares, TSS总变差: • We can split the TSS into two parts, the part which we have explained (known as the explained sum of squares, ESS) and the part which we did not explain using the model (the RSS)*. y $ = ( − ) t t TSS y y 2 y
4-4 Defining R2 That is. SS Ess RSS ∑(-y)=∑(1-y)+∑4 Goodness of fit statistic is R2- ESS TSS ESS TSS- RSS Rss R TSS TSS TSS R must always lie between zero and one. To understand this consider two extremes RSS= TSSie. ESS=0 S0 R= ESS/TSS=0 ESS= SS ie. RSS=0 S0 R2= ESS/TSS= 1
4-4 Defining R2 • That is, TSS = ESS + RSS • Goodness of fit statistic is • R2 must always lie between zero and one. To understand this, consider two extremes RSS = TSS i.e. ESS = 0 so R2 = ESS / TSS = 0 ESS = TSS i.e. RSS = 0 so R2 = ESS / TSS = 1 R ESS TSS 2 = R ESS TSS TSS RSS TSS RSS TSS 2 = = 1 − = − ( − ) = ( − ) + t t t t yt y yt y u 2 2 2 ˆ ˆ
4-5 The Limit Cases: 2=0 and R2=1 y yI
4-5 The Limit Cases: R2 = 0 and R2 = 1 t y y t x t y t x