伍德里奇計量經(jīng)濟學講義8PPT課件

1、1Parallels with Simple Regression b0 is still the intercept b1 to bk all called slope parameters u is still the error term (or disturbance) Still need to make a zero conditional mean assumption, so now assume that E(u|x1,x2, ,xk) = 0 Still minimizing the sum of squared residuals, so have k+1 first o

2、rder conditions第1頁/共28頁2Interpreting Multiple Regressiontioninterpreta a has each is that , thatimplies fixed ,., holding so,.so ,.112221122110ribus ceteris paxyxxxxxyxxxykkkkkbbbbbbbbb第2頁/共28頁3A “Partialling Out” Interpretation22011211122110 regression estimated thefrom residuals theare where, then

3、 ,i.e. , 2 wherecase heConsider txxrryrxxykiiiibbbb第3頁/共28頁4“Partialling Out” continued Previous equation implies that regressing y on x1 and x2 gives same effect of x1 as regressing y on residuals from a regression of x1 on x2 This means only the part of xi1 that is uncorrelated with xi2 are being

4、related to yi so were estimating the effect of x1 on y after x2 has been “partialled out”第4頁/共28頁5Simple vs Multiple Reg Estimatesample in the eduncorrelat are and OR ) ofeffect partial no (i.e. 0:unless Generally, regression multiple with the regression simple theCompare21221122110110 xxxxxyxybbbbb

5、bbb第5頁/共28頁6Goodness-of-FitSSR SSE SSTThen (SSR) squares of sum residual theis (SSE) squares of sum explained theis (SST) squares of sum total theis :following thedefine then Wepart, dunexplainean and part, explainedan of upmade being asn observatioeach ofcan think We222iiiiiiuyyyyuyy第6頁/共28頁7Goodne

6、ss-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第7頁/共28頁8Goodness-of-Fit (continued)22222 val

7、ues theand actual thebetweent coefficienn correlatio squared the toequal being as of think alsocan WeyyyyyyyyRyyRiiiiii第8頁/共28頁9More 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 n

8、umber of independent variables, it is not a good way to compare models第9頁/共28頁10Assumptions for Unbiasedness Population model is linear in parameters: y = b0 + b1x1 + b2x2 + bkxk + u We can use a random sample of size n, (xi1, xi2, xik, yi): i=1, 2, , n, from the population model, so that the sample

9、 model is yi = b0 + b1xi1 + b2xi2 + bkxik + ui E(u|x1, x2, xk) = 0, implying that all of the explanatory variables are exogenous None of the xs is constant, and there are no exact linear relationships among them第10頁/共28頁11Too Many or Too Few Variables What happens if we include variables in our spec

10、ification that dont belong? There is no effect on our parameter estimate, and OLS remains unbiased What if we exclude a variable from our specification that does belong? OLS will usually be biased 第11頁/共28頁12Omitted Variable Bias21111111022110 then, estimatebut we ,asgiven is model true theSupposexx

11、yxxuxyuxxyiiibbbbbb第12頁/共28頁13Omitted Variable Bias (cont)iiiiiiiiiiiiiuxxxxxxxuxxxxuxxy1121122111221101122110becomesnumerator theso , thatso model, true theRecallbbbbbbbb第13頁/共28頁14Omitted Variable Bias (cont) 2112112112111121121121have wensexpectatio taking0,)E( sincexxxxxEuxxuxxxxxxxiiiiiiiiiibbb

12、bbb第14頁/共28頁15Omitted Variable Bias (cont) 12112112111110212 so then on of regression heConsider tbbbExxxxxxxxxiii第15頁/共28頁16Summary of Direction of BiasCorr(x1, x2) 0 Corr(x1, x2) 0Positive biasNegative biasb2 0Negative biasPositive bias第16頁/共28頁17Omitted Variable Bias Summary Two cases where bias

13、is equal to zerob2 = 0, that is x2 doesnt really belong in modelx1 and x2 are uncorrelated in the sample If correlation between x2 , x1 and x2 , y is the same direction, bias will be positive If correlation between x2 , x1 and x2 , y is the opposite direction, bias will be negative第17頁/共28頁18The Mor

14、e General Case Technically, can only sign the bias for the more general case if all of the included xs are uncorrelated Typically, then, we work through the bias assuming the xs are uncorrelated, as a useful guide even if this assumption is not strictly true第18頁/共28頁19Variance of the OLS Estimators

15、Now we know that the sampling distribution of our estimate is centered around the true parameter Want to think about how spread out this distribution is Much easier to think about this variance under an additional assumption, soAssume Var(u|x1, x2, xk) = s2 (Homoskedasticity)第19頁/共28頁20Variance of O

16、LS (cont) Let x stand for (x1, x2,xk) Assuming that Var(u|x) = s2 also implies that Var(y| x) = s2 The 4 assumptions for unbiasedness, plus this homoskedasticity assumption are known as the Gauss-Markov assumptions第20頁/共28頁21Variance of OLS (cont) s other allon regressing from theis and where,1sAssu

17、mption Markov-Gauss Given the22222xxRRxxSSTRSSTVarjjjijjjjjsb第21頁/共28頁22Components of OLS Variances The error variance: a larger s2 implies a larger variance for the OLS estimators The total sample variation: a larger SSTj implies a smaller variance for the estimators Linear relationships among the

18、independent variables: a larger Rj2 implies a larger variance for the estimators第22頁/共28頁23Misspecified Models same there then theyed,uncorrelat are and unless Thus, that so ,model edmisspecifi again theConsider 2111121110 xxVarVarSSTVarxybbsbbb第23頁/共28頁24Misspecified Models (cont) While the variance of the estimator is smaller for the misspecified model, unless b2 = 0 the misspecified model is biased As the sample size grows, the variance of each estimator shrinks to zero, making the variance difference less important第24頁/共28