Chapter 3. Two-Variable Regression Model: The Problem of Estimation

Transcription

1 Chapter 3. Two-Varable Regresson Model: The Problem of Estmaton

2 Ordnary Least Squares Method (OLS) Recall that, PRF: Y = β 1 + β X + u Thus, snce PRF s not drectly observable, t s estmated by SRF; that s, Y = ˆ β ˆ β 1 + X + uˆ And, Y = Yˆ + uˆ

3 On Error Term More If Y = Yˆ + uˆ Then, uˆ = Y Yˆ And, ˆ ˆ ˆ u = Y β1 β X

4 On error term more We need to choose SRF n such a way that, error terms should be as small as possble, That s, The sum of resduals whch s represented by uˆ = ( ) Y Yˆ Should be as SMALL as possble

5 On Error Terms more Therefore, the essental soluton s to fnd a crteron n order to mnmze error dsturbances n SRF. All of the errors are to be as closer as possble to the central lne of SRF

6 Then, Least Squares Crteron Comes as a Soluton Least Squares Crteron s based on: uˆ ( ˆ ) Y Y = = ( ) Y ˆ β ˆ β 1 X Thus, ( ˆ ) β, β uˆ = f 1 ˆ

7 Example to Least Squares Crteron Sum of squares of Error dsturbances of the second model s lower The frst Model s Better? Why?

8 Regresson Equaton u X Y ˆ ˆ ˆ = β β ( ) ( )( ) ( ) = = = x ˆ x y X X Y Y X X X X n Y X Y X n β ( ) X X X n X Y X Y X 1 ˆ Y- ˆ β β = = Sample mean of Y Sample mean of X

9 The Classcal Lnear Regresson Model (CLRM): The Assumptons Underlyng The Method of Least Squares The nferences about the true β 1 and β are mportant because the estmated values of them are needed to be closer and closer to populaton values. Therefore CLRM, whch s the cornerstone of most econometrc theory, makes 10 assumptons.

10 Assumptons of CLRM: Assumpton 1. Lnear Regresson Model The regresson model s lnear n the parameters, that s: Y = β 1 + β X + u Assumpton. X values are fxed n repeated samplng. More techncally, X s assumed to be non-stochastc X: 80$ ncome level Y: 60$ weekly consumpton of a famly X: 80$ ncome level Y: 75$ weekly consumpton of another famly Assumpton s known as: Condtonal Regresson Analyss, that s, condtonal on the gven values of the regressor(s) X.

11 Assumpton 3. Zero Mean value of dsturbance u E( u / ) X = 0

12 Assumpton 4. Homoscedastcty or Equal Varance of u var ( u / X ) = E[ u E( u )/ X ] = =σ ( / X ) E u because of where var stands for varance Assumpton 3

13 Homoscedastcty vs Heteroscedastcty var ( / ) = σ u X

14 Assumpton 5. No Autocorrelaton between the dsturbances

15 Autocorrelaton If : PRF: Y t = β 1 + β X t + u t And f u t and u t-1 are correlated, then Y t depends not only X t, but also on u t-1.

16 utocorrelaton Graphs

17 Assumpton 6. Zero Covarance between u and X.

18 Assumpton 7.

19 Assumpton 8.

20 Assumpton 9.

21 Assumpton 10. There s No Perfect Multcollnearty That s, there s no perfect lnear relatonshp among the explanatory varables. Y = β + β X + β X +... β X + u t n n t Hgh correlaton among ndependent varables causes multcollnearty whch also causes standard errors to be hgh, hypotheses to be neffcent (low t values), etc...

22 Propertes of the Least-Squares Estmators: The Gauss-Markov Theorem Gauss-Markov Theorem s the least squares approach of Gauss (181) wth the mnmum varance approach of Markov (1900). Standard error of estmate s smply the standard devaton of the Y values about the estmated regresson lne and s often used as a summary measure of the goodness of ft of the estmated regresson lne.

23 BLUE (Best Lnear Unbased Estmator) 1. An estmator s lnear, that s, a lnear functon of a random varable, such as the dependent varable Y n the regresson model.. An estmator s unbased, that s, ts average or expected value, E(β ), s equal to the true value, β. 3. An estmator has mnmum varance n the class of all such lnear unbased estmators; an unbased estmator wth the least varance s known as an effcent estmator. Therefore, n the regresson context t can be proved that the OLS estmators are BLUE whch also sets the base of Gauss- Markov Theorem.

24 The Coeffcent of Determnaton, r : A Measure of Goodness of Ft The coeffcent of determnaton, r (two-varable case) or R (multple regresson) s a summary measure that tells how well the sample regresson lne fts the data.

25 The Ballentne Vew of R See Peter Kennedy, Ballentne: A Graphcal Ad for Econometrcs, Australan Economcs Papers, Vol 0, 1981, The name Ballentne s derved from the emblem of the well-known Ballantne beer wth ts crcles.

26 Coeffcent of Determnaton, r TSS = ESS + RSS where; TSS = total sum of squares ESS = explaned sum of squares RSS = resdual sum of squares If TSS = ESS + RSS, then: 1 = = ESS TSS + RSS TSS ( ) Ŷ Y uˆ + ( ) Y Y ( Y Y )

27 On r more: R ndcates the explaned part of the regresson model, therefore, r = And, ESS TSS r ( Yˆ Y ) = ESS ( ) Y Y TSS =

28 Alternatvely, r = 1 uˆ ( Y Y ) r = 1 RSS TSS

29 Coeffcent of Determnaton

30 Coeffcent Of Determnaton

31 HW # 1: Problem 3.0 (Chapter 3) Consumer Prces and Money Supply n Japan 198 to 001