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1、第一篇 古典线性回归模型,CHAPTER ONE CLASSICAL TWO-VARIABLE LINEAR REGRESSION MODEL,Two variable: one dependent variable only one explanatory varialbe Linear:Y is linear function of parameters.,IMPORTANT differentiation 1、correlation ：determinate stochastic（no-determinate） 相关关系：确定性 非确定性 2、Correlation and causal
2、ity 相关关系与因果关系 3、Linearity: linearity in the variable linearity in the parameters 线性：变量线性 参数线性,1.1 Two-variable linear regression model,Population: a set of all possible outcomes of a random variable Subpopulation： 1.1.1 population regression function and population regression line PRF： Yt= 0+1Xt+ut
3、stochastic E(YtXt)= 0+1Xt determinate Yt= E(YtXt) +ut SEE T.F 2.1,X: explanatory ,independent variable, fixed-value variable Y: dependent variable ,random variable ，：regression coefficients U: disturbance or error term, is a random variable The task of regression is to estimate the PDF，that is，to es
4、timate the value of unknown ，on the basis of observations on Y and X。,error term u stands for the aggregate effect of all factors which are excluded from model but indeed affect Y: 1、variables excluded from model with effect on Y vagueness of theory, negligible effect, noavailablity of data 2、intrin
5、sic random of human beings behavoir 3、measurement error 4、error of model forms,1.1.2 sample regression function and sample regression line(f 2.4 ,t 2.4,t2.5) SRF See f 2.5,The task of regression is to estimate the PDF on the sample information of Y and X,Primary objective in regression analysis is t
6、o estimate the PRF on the basis of the SRF，or on the sample information of Y and X ,but the estimation of the PRF based on the SRF is at best an approximation。 The next question is how the SRF should be constituted as close as possible to the PRF even though we never know what is the true PRF。 Metho
7、d of estimation： Ordinary least square：最小二乘法 Maximum likelyhood：最大似然法,1.1.2 the meaning of the term linear,Linearity in variables Linearity in parameters conditional means of Y is linear function of parameters.,1.2 estimation methodordinary least square（OLS）,Now, the and PRF is unknown,our task in r
8、egression analysis is to estimate the PRF on the basis of the SRF，or on the sample information of Y and X ,but the estimation of the PRF based on the SRF is at best an approximation。 We assume that the closer the SRF is to a set of sample data on Y and X, the better the SRF fits the PRF.,Scatter-gra
9、ph,Y 0 X,PRF:E(Yi)= 0+1Xi,Enlarged local area,actual,estimate,residual,fundamental thought of OLS： to make SRF as close as possible to PRF, the residual of every point should be as small as possible How to chose to make the sum of squared residuals as small as possible,?,Criterion of minimizing the
10、sum of squared residuals (The sum of squared residuals is a function of estimator of parameters.),1.3 CLRMs Assumption,CLRM :classical linear regression model 1、Model is linear in parameters，X is fixed-value 2、Zero mean value of disturbance ui E（ui|Xi）=0 It means that these factor excluded from the
11、model dont systematically affect the mean value of Y.,3、Homoscedasticity or equal variance of ui (the variance of ui is the same for all observation.) Var（ui|Xi）=Eui-E(ui|Xi)2 =E（ui）2 =2 Heteroscedasticity:different variance of ui , variance of ui varies with X.,Heteroscedasticity:different variance
12、 of ui , variance of ui varies with X. Var（ui|Xi）=Eui-E(ui|Xi)2 =E（ui）2 =i2,Conditional distribution of Disturbance ui,4、No autocorrelation between the disturbances，namely，no serial correlation。 Cov(ui, uj|Xi,Xj) =Eui-E(ui)|Xiuj-E(uj)|Xj =E(ui|Xi)(uj|Xj)=0 ij Y only depends on X and current u withou
13、t relation to other us.,Positive serial correlation,+ui,+ui,-ui,-ui,Negative serial correlation,+ui,+ui,-ui,-ui,Zero correlation,+ui,+ui,-ui,-ui,5、Zero covariance between ui and Xi Because X is fixed value and u is random variable, this assumption is satisfied automatically. This assumption guarante
14、e that the effect of X and u on Y can be separated easily.,6、the number of observations n is greater than the number of parameters to be estimated。,1.4 statistical properties of OLS estimators,Given the assumptions of the classical linear regression model, the estimators of OLS possess ideal or opti
15、mum properties: Best, Linear, unbiased estimators(blue) Guass-Markov theorem,linearity,are linear functions of Yi or ui . Because the later are normal random variable , the estimator of parameters are normal random variables,UNBIASED the expected value of estimator of parameters are equal to its tru
16、e value。,Minimum variance or best,the estimator of OLS has minimum variance in the class of all such linear unbiased estimator , namely, efficient estimator In statistic, the precision or reliability of estimate is measured by its variance or standard error. The smaller the SE, the better the estimate,最小二乘估计量 和 的 方差,Because follow normal distribution, then,is variance of error te
