计量经济学Chapter 2简单回归模型.ppt

* * * * * * * * * * Theorem 2.1 (Unbiasedness of OLS) Interpretation of unbiasedness The estimated coefficients may be smaller or larger, depending on the sample that is the result of a random draw However, on average, they will be equal to the values that charac-terize the true relationship between y and x in the population “On average” means if sampling was repeated, i.e. if drawing the random sample and doing the estimation was repeated many times In a given sample, estimates may differ considerably from true values The Simple Regression Model Variances of the OLS estimators Depending on the sample, the estimates will be nearer or farther away from the true population values How far can we expect our estimates to be away from the true population values on average (= sampling variability)? Sampling variability is measured by the estimator‘s variances Assumption SLR.5 (Homoskedasticity) The value of the explanatory variable must contain no information about the variability of the unobserved factors The Simple Regression Model Graphical illustration of homoskedasticity The variability of the unobserved influences does not depend on the value of the explanatory variable The Simple Regression Model An example for heteroskedasticity: Wage and education The variance of the unobserved determinants of wages increases with the level of education The Simple Regression Model Theorem 2.2 (Variances of the OLS estimators) Conclusion: The sampling variability of the estimated regression coefficients will be the higher, the larger the variability of the unobserved factors, and the lower, the higher the variation in the explanatory variable Under assumptions SLR.1 – SLR.5: The Simple Regression Model Estimating the error variance The variance of u does not depend on x, i.e. equal to the unconditional variance One could estimate the variance of the errors by calculating the variance of the residuals in the sample; unfortunately this estimate would b