Chapter 3 元线性回归模型 节回归分析与回归方程
Chapter 3 一元线性回归模型 第一节 回归分析与回归方程
回归分析 无法显示该图片 1根据经济理论或考察样本数据去设定回归 方程 Y=f(X…,Z)+6 Y: dependent variable Z: independent 8: random error or distur bance term
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 Y f X Z = + ( , , ) X Z
A special and simple case(univariate linear regression model) Y=b+bⅩ+E 这是本章研究的重点 2.参数估计( Estimation of parameter) 3. Testing 4. Predicting 设有样本为{Y2X},则 y=b+bX1+(i=1,,n)
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. Testing 4. Predicting 设有样本为 ，则 Y b b X 0 1 = + + { , } Y Xi i 0 1 ( 1, , ) i i i Y b b X i n = + + =
模型的假设: 1.E(E;)=0 2.var(E)=a2(同方差 3.E(:E,)=01≠j 4.E(X1E;)=0 满足这四条件的LRM称为 经典线性回归模型(CLRM)
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)。 ( ) 0 E i = ( ) 0 E i j i j = 2 var( )i = ( ) 0 E Xi i =
由假设得 E(Y=6+6,x 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 五和1,则=b+bXx称为 sample regression equation(function) How to estimate them? the ols method
◼ 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 them? The OLS method. 0 1 E Y b b X ( ) = + 0 1 ˆ ˆ b b 和 0 1 ˆ ˆ ˆ Y b b X = +
普通最小二乘法 Ordinary least squares procedure) 求b和b使残差平方和最小 ∑c=∑(x-1)=∑y-(+bx Let x,=X-X and y=YY Then b=∑x/∑ Xi (OLSE) -Y-bX
普通最小二乘法(Ordinary least squares procedure): 求 使残差平方和最小: Let Then (OLSE) 0 1 b b 和 2 2 2 0 1 ˆ ˆ ˆ ( ) [ ( )] i i i i i e Y Y Y b b X = − = − + and i i i i x X X y Y Y = − = − 2 1 0 1 ˆ ˆ ˆ i i i b x y x b Y b X = = −
The properties of the OlsE 1.无偏性 unbiased) E(b)=b,, E(bo=bo O ∑X 2.Ⅴar vari 2 ∑ cov(6o, 6= 2
The properties of the OLSE: 1. 无偏性(unbiased): 2. 1 1 0 0 ˆ ˆ E b b E b b ( ) , ( ) = = 2 2 2 1 0 2 2 ˆ ˆ var( ) , var( ) i i i X b b x n x = = 2 0 1 2 ˆ ˆ cov( , ) i X b b x = −
3关于样本{H}的线性性 h=∑k,b=∑(-欢 ∑ 4. Gauss- Markov theorem:如果 y=bo+bx1+E(i=1,…,n) 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
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。 { } Yi 1 0 2 1 ˆ ˆ , ( ) ( ) i i i i i i j x b k Y b Xk Y k n x = = − = 0 1 ( 1, , ) Y b b X i n i i i = + + =
Estimation of the variance of the random disturbance term, o We know o=var(e) and it is unknown Thus, var(b)=o3 i 2 var X and so on are also unknown. to estimate them we have to first evaluate o. It is not difficult to show that a2_o2 n-2) is an unbiased estimator for o
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 first evaluate . It is not difficult to show that is an unbiased estimator for , 2 2 var( ) = i 2 2 2 1 0 2 2 ˆ ˆ var( ) , var( ) i i i X b b x n x = = 2 2 2 ˆ ( 2) i = − e n2
Where e=Y-(6+bXi) are the residuals EXample3. 1(P39) (how to use Eviews)
Where are the residuals. Example3.1(P39)(how to use Eviews) ( i i e Y = − 0 1 ˆ ˆ ( ) i i i e Y b b X = − +