古典一元线性回归2011川大计量经济学.pptx

第一篇第一篇古典线性回归模型古典线性回归模型 CHAPTER ONECLASSICAL TWO-VARIABLE LINEAR REGRESSION MODEL Two variable:one dependent variableTwo variable:one dependent variable only one explanatory varialbe only one explanatory varialbeLinear:Y is linear function of Linear:Y is linear function of parameters.parameters.IMPORTANT differentiation1、correlation：determinate stochastic（no-determinate）相关关系相关关系：确定性 非确定性非确定性2、Correlation and causality相关关系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 参数线性参数线性1.1 Two-variable linear regression modelPopulation:a set of all possible outcomes of a random variableSubpopulation：1.1.1 population regression function and population regression linePRF：Y Yt t=0 0+1 1X Xt t+u+ut t stochastic E(E(Y Yt tX Xt t)=0 0+1 1X Xt determinatet determinate Y Yt t=E(E(Y Yt tX Xt t)+u+ut tSEE T.F 2.1SEE T.F 2.1X:explanatory ,independent variable,fixed-value variableY:dependent variable,random variable，：regression coefficientsU:disturbance or error term,is a random variableThe task of regression is to estimate the PDF，that is，to estimate 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、intrinsic random of human beings behavoir 3、measurement error 4、error of model forms1.1.2 sample regression function and sample regression line(f 2.4,t 2.4,t2.5)SRF See f 2.5The task of regression is to estimate the PDF on the sample information of Y and XPrimary objective in regression analysis is to estimate the PRF on the basis of the SRF，or on the sample information of Y and Xon 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。Method of estimation：Ordinary least square：最小二乘法：最小二乘法Maximum likelyhood：最大似然法：最大似然法1.1.2 the meaning of the term linearLinearity in variablesLinearity 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 regression 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-graph Y 0 XPRF:E(PRF:E(Y Yi i)=0 0+1 1X Xi iEnlarged local areaactualestimateresidualfundamental thought of OLS：to make SRF as close as possible to PRF,the residual of every point should be as small as possibleHow to chose to make the sum of squared residuals as small as possible Criterion of minimizing the sum of squared residuals(The sum of squared residuals is a function of estimator of parameters.)1.3 CLRMs AssumptionCLRM:classical linear regression model1、Model is linear in parameters，X is fixed-value2、Zero mean value of disturbance u ui iE E（u ui i|X|Xi i）=0=0It means that these factor excluded from the model dont systematically affect the mean value of Y.3 3、Homoscedasticity or equal variance Homoscedasticity or equal variance of of u ui i(the variance of (the variance of u ui i is the same is the same for all observation.)for all observation.)Var Var（u ui i|X|Xi i）=E=Eu ui i-E(-E(u ui i|X|Xi i)2 2 =E =E（u ui i）2 2 =2 2Heteroscedasticity:different variance Heteroscedasticity:different variance of of u ui,i,variance of variance of u ui i varies with X.varies with X.,Heteroscedasticity:different Heteroscedasticity:different variance of variance of u ui,i,variance of variance of u ui i varies with X.varies with X.VarVar（u ui i|X|Xi i）=E=Eu ui i-E(-E(u ui i|X|Xi i)2 2 =E =E（u ui i）2 2 =i i2 2Conditional distribution of Disturbance u ui i4、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 ijY only depends on X and current u without relation to other us.Positive serial correlation +u ui i+u ui i-u ui i-u ui iNegative serial correlation+u ui i+u ui i-u ui i-u ui iZero correlation+u ui i+u ui i-u ui i-u ui i5、Zero covariance between ui and Xi Because X is fixed value and u is random variable,this assumption is satisfied automatically.This assumption guarantee 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 estimatorsGiven the assumptions of the classical linear regression model,the estimators of OLS possess ideal or optimum properties:Best,Linear,unbiased estimators(blue)Guass-Markov theoremlinearity 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 true 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,thenis variance of error term附：OLS估计量的其他性质1.5 THE COEFFICIENT OF DETERMINATION The coefficient of determination is a measure of“goodness of fit”,indicates how “well”the sample regression line fits the data.1.5.1 variation analysis YX0 xi is total variation about observation from the sample means，is the part that can be explained by explanatory variable,is the residual part that cant be explained by model.Total sum of squareexplained sum of squareresidual sum of squareESS stands for the part of TSS that can be explained by Regression,so:The greater ESS,the smaller RSS,the better SRF fits the data1.52 goodness of fit(coefficient of determination）We define the goodness of fit R2 as following:R2 is called goodness of fit or coefficient of determination。将RSSTSSESS代入上式可得：We have the conclusion:The Greater R2，the closer R2 to 1,the better SRF fit sample data1.5.3 correlation coefficient(相关系数）Correlation coefficient quantitively measures the linear correlation between two variables,denoted as rProperty:（1）r and has same sign（2）r:-1r1r=1，variable X and Y are perfect negative correlationr=0，variable X and Y are absolutely independentr=1，perfect positive correlation the closer r to 1，the greater the extent of correaltion of two variable.The positive r means that Y is proportional to X,The negative r means that Y is inversely proportional to X。Pay attention：linearity