截面與面板數據法分析-CH3.ppt

Chap 3. Multiple Regression Analysis：Estimation,Advantages of multiple regression analysis build better models for predicting the dependent variable. E.g. generalize functional form. Marginal propensity to consume Be more amenable to ceteris paribus analysis Key assumption: Implication: other factors affecting wage are not related on average to educ and exper. Multiple linear regression model:,OLS Estimator,OLS: Minimize ceteris paribus interpretations: Holding fixed, then Thus, we have controlled for the variables when estimating the effect of x1 on y.,Holding Other Factors Fixed,The power of multiple regression analysis is that it provides this ceteris paribus interpretation even though the data have not been collected in a ceteris paribus fashion. it allows us to do in non-experimental environments what natural scientists are able to do in a controlled laboratory setting: keep other factors fixed.,OLS and Ceteris Paribus Effects,measures the effect of x1 on y after x2, xk have been partialled or netted out. Two special cases in which the simple regression of y on x1 will produce the same OLS estimate on x1 as the regression of y on x1 and x2. -The partial effect of x2 on y is zero in the sample. That is, - x1 and x2 are uncorrelated in the sample. -Example,data1: 1832 rural household reg consum laborage reg consum laborage financialK corr laborage financialK reg consum laborage reg consum laborage laboredu corr laborage laboredu,Goodness-of-fit,R-sq also equal the squared correlation coef. between the actual and the fitted values of y. R-sq never decreases, and it usually increases when another independent variable is added to a regression. The factor that should determine whether an explanatory variable belongs in a model is whether the explanatory variable has a nonzero partial effect on y in the population.,The Expectation of OLS Estimator,Assumption 1-4 Linear in parameters Random sampling Zero conditional mean No perfect co-linearity none of the independent variables is constant; and there are no exact linear relationships among the independent variables Theorem (Unbiasedness) Under the four assumptions above, we have:,Notice 1: Zero conditional mean,Exogenous Endogenous Misspecification of function form (Chap 9) Omitting the quadratic term The level or log of variable Omitting important factors that correlated with any independent v. Measurement Error (Chap 15, IV) Simultaneously determining one or more x-s with y (Chap 16) Try to use exogenous variable! (Geography, History),Omitted Variable Bias: The Simple Case,Omitted Variable Bias The true population model: The underspecified OLS line: The expectation of : (46),前面3.2節(jié)中是x1對x2回歸,The expectation of , where the slope coefficient from the regression of x2 on x1, so then, Only two cases where is unbiased, , x2 does not appear in the true model; , x2 and x1 are uncorrelated in the sample;,前面3.2節(jié)中是x1對x2回歸,Omitted variable bias:,Notice 2: No Perfect Collinearity,An assumption only about x-s, nothing about the relationship between u and x-s Assumption MLR.4 does allow the independent variables to be correlated; they just cannot be perfectly correlated If we did not allow for any correlation among the independent variables, then multiple regression would not be very useful for econometric analysis How to deal with collinearity problem? Drop correlated variable, respectively. (corr=0.7),Notice 3: Over-Specification,Inclusion of an irrelevant variable： does not affect the unbiasedness of the OLS estimators. including irrelevant variables can have undesirable effects on the variances of the OLS estimators.,Variance of The OLS Estimators,Assumption 5 Homoskedasticity: Gauss-Markov Assumptions (for cross-sectional regression): Assumption 1-5 Linear in parameters Random sampling Zero conditional mean No perfect co-linearity Homoskedasticity,Theorem (Sampling variance of OLS estimators) Under the five assumptions above:,More about,The statistical properties of y on x=(x1, x2, , xk) Error variance only one way to reduce the error variance: to add more explanatory variables not always possible and desirable (multi-collinearity) The total sample variations in xj: SSTj Increase the sample size,Multi-collinearity,The linear relationships among the independent v-s. 其他解釋變量對xj的擬合優(yōu)度（含截距項） If k=2： ：the proportion of the total variation in xj that can be explained by the other independent variables High (but not perfect) correlation between two or more of the in dependent variables is called multicollinearity.,Small sample size,Small sample size Low SSTj one thing is clear: everything else being equal, for estimating , it is better to have less correlation between xj and the other V-s.,Notice: The influence of multi-collinearity,A high degree of correlation between certain independent variables can be irrelevant as to how well we can estimate other parameters in the model. x2和x3之間的高相關性并不直接影響x1的回歸系數的方差，極端的情形就是X1和x2、x3都不相關。同時前面我們知道，增加一個變量并不會改變無偏性。在多重共線性的情形下，估計仍然無偏，我們關心的變量系數的方差也與其他變量之間的共線性沒有直接關系，盡管方差會變化，只要t值仍然顯著，共線性不是大問題。 How to “solve” the multi-collinearity? Dropping some v.? 如果刪除了總體模型中的一個變量，則可能會導致內生性。,參見注釋,Estimating : Standard Errors of the OLS Estimators,參見注釋,df=number of observations-number of estimated parameters Theorem 3.3 Unbiased estimation of Under the Gauss-Markov Assumption, MLR 1-5,While the presence of heteroskydasticity does not cause bias in the , it does lead to bias in the usual formula for , which when then invalidates the standard errors. This is important because any regression package compute 3.58 as the default standard error for each coefficient.,Gauss-Markov Assumptions (for cross-sectional regression): 1. Linear i