计量经济学PPT-3.1

1、 The Simple Regression Model Deriving OLS Estimates The estimator that we are going to introduce is known as Ordinary Least Squares estimator because it is tting a line through the sample points such that the mean of squared residuals is as small as possible. Consider the function S= 1 n (y n i=1 i

2、b0 b1 xi 2 . Denote the minimizers of this function as 0 and 1 . These minimizers have to satisfy: ( S 2 n 0 1 xi = 0 n i=1 (yi b0 = . S 2 n 0 1 xi = 0 n i=1 xi (yi b = 1 21 / 24 The Simple Regression Model Deriving OLS Estimators The solution of this system of equations is given by 0 1 ¯ = y &

3、#175; 1 x, ¯ y , ¯ x n 1 xi (yi i= n 1 xi (xi i= = Equivalent formulas for 1 : 1 1 = = n 1 (xi i= ¯ x (yi 2 ¯ y ¯ n 1 ( xi x i= n y¯ i=1 xi yi n¯ x . n 2 x i=1 xi n¯ 2 , 1 ¯ ¯ where y = n n 1 yi , x = i= 2 n ¯ i=1 (xi x > 0. 1 n n 1 xi , and

4、it is assumed that i= 22 / 24 The Simple Regression Model Example 2: In order to use the above formula it is useful to construct the following table. ¯ ¯ Note that x = 14 and y = 5.5. i 1 2 3 4 5 xi 10 12 14 16 18 yi 4.5 5 5.5 5.8 6.7 (xi 16 4 0 4 16 40 ¯ x2 (xi ¯ x(yi 4 1 0 0.6

5、4.8 10.4 ¯ y Compute the estimates for 0 and 1 in this particular sample 23 / 24 The Simple Regression Model We can write the Sample Regression Function. yi = 1.86 + 0.26xi Interpretation: The estimated slope parameter ( 2 = 0.26 indicates that as x increases by 1, mean of y increases by 0.26. What is