1、Chapter 3 一元线性回归模型第一节 回归分析与回归方程第1页，共21页。回归分析:1.根据经济理论或考察样本数据去设定回归方程 Y:dependent variable;:independent :random error or disturbance term(,)Yf XZNoImageNoImageNoImageNoImage,XZ第2页，共21页。A special and simple case(univariate linear regression model):这是本章研究的重点。2.参数估计(Estimation of parameter)3.Testing4.Pre
2、dicting设有样本为 ，则01Ybb X,iiY X01(1,)iiiYbbXin 第3页，共21页。模型的假设：1.2.(同方差)3.4.满足这四条件的LRM称为 经典线性回归模型(CLRM)。()0iE()0ijEij2var()i()0iiE X第4页，共21页。n由假设得 Population regression equation(function)The pity is the parameters are unknown.我们要利用样本来估计参数.如得参数估计值 ,则 称为sample regression equation(function).How to estimate
3、 them?The OLS method.01()E Ybb X01bb和01YbbX第5页，共21页。普通最小二乘法(Ordinary least squares procedure):求 使残差平方和最小:Let Then (OLSE)01bb和22201()()iiiiieY YYbbXandiiiixXXyY Y 2101iiibx yxbYb X第6页，共21页。The properties of the OLSE:1.无偏性(unbiased):2.1100(),()E bbE bb2221022var(),var()iiiXbbxnx2012cov(,)iXb bx 第7页，共2
4、1页。3.关于样本 的线性性:4.Gauss-Markov theorem:如果 是经典线性回归模型(CLRM),则其参数的OLSE 为BLUE。即,在所有线性无偏估计中，OLSE的 方差最小。iY1021,()()ii iiiijxbkYbXk Yknx01(1,)iiiYbbXin 第8页，共21页。Estimation of the variance of the random disturbance term,:We know and it is unknown.Thus,and so on are also unknown.To estimate them,we have to fi
5、rst evaluate .It is not difficult to show that is an unbiased estimator for ,22var()i2221022var(),var()iiiXbbxnx222(2)ien2第9页，共21页。Whereare the residuals.Example3.1(P39)(how to use Eviews)(iieY01()iiieYbb X第10页，共21页。模型的假设：模型的假设：5.Normality assumption:The properties of the OLSE:5.2(0,)iN222110022(,),
6、(,)iiiXbN bbN bxx第11页，共21页。nModel testing(模型的检验模型的检验):总离差分解公式:即,TSS =ESS +RSS TSS:Total sum of squares ESS:Error(residual)sum of squares RSS:Regression(explained)sum of squares222()()()iiiiY YY YY Y222iiiyey第12页，共21页。1.Goodness-of-fit testing(R2检验检验):Coefficient of determination(判定系数):In general,the
7、 larger R2,the better.2.Sample coefficient of correlation:222ESSRSS1TSSTSSiiyRy 2rR 第13页，共21页。3.Hypothesis testing We have known Let (standard error)222110022(,),(,)iiiXbN bbN bxx22(2)ien21()iS bx第14页，共21页。ThenAnd we can test the following hypothesis:Moreover,interval estimator for is 111(2)()bbtt n
8、S b0111:versus:H bcH bc(1)1b112()bt S b第15页，共21页。Forecasting(预测预测)1.Point forecastingSince we know and the sample regression equationthen given ,what about and?As(an unbiased estimator for )01Ybb X01Ybb X1nX1nY1()nEY101110111()()()nnnnnEYEb bXb bXEY 1()nEY第16页，共21页。nand (误差均匀)nNaturally,we use as a
9、point predictor for bothn and .n2.Interval forecastingn(1)Forecast interval for n Forecast error:n 1nY11()0nnE YY1nY1()nE Y1nY111nnneYY第17页，共21页。The variance of the forecast error:Therefore21121011101var()()var()var()var()2cov(,)nnnnneE ebXbXb b2212()11niXXnx22112()1(0,1)nniXXeNnx第18页，共21页。It is a pity that is unknown.Fortunately,we have Thus,Hence,a Forecast interval for is 22(2)ien211122221122(2)()()1111nnnnniieYYt nXXXXnxnx 1nY(1)第19页，共21页。n(2)Forecast interval forn Similarly,we can obtain a forecast interval for :21122()1 1nniXXYtnx1()nE Y(1)1()nE Y第20页，共21页。21122()1nniXXYtnx第21页，共21页。
