Economic Interpretation of Regression. Theory and Applications

Transcription

1 Economc Interpretaton of Regresson Theor and Applcatons

2 Classcal and Baesan Econometrc Methods Applcaton of mathematcal statstcs to economc data for emprcal support Economc theor postulates a qualtatve relaton. Mathematcal economcs turns economc theor n equatons. Economc statstcs concerns wth collectng, processng and presentng economc data. Econometrcans estmate precse numercal estmates of these relatons. Statement of hpothess Specfcaton of the mathematcal model Specfcaton of econometrc model Data collecton Estmaton of parameters of the econometrc model Hpothess testng Forecastng or predcton Usng model for control or polc analss Mehtodolog of Baesan Econometrcs Baesan pror Sample nformaton Posteror nformaton Econometrcs Theoretcal Appled Classcal Baesan Classcal Baesan

3 Assumptons of a Regresson Model e [ e ] 0 var σ E [ ] e cov( e e j ) 0 for all j E [ e ] 0 s eogenous, not random

4 Test of Normalt and Level of Sgnfcance: Two Tal Test P (.96 z.96) ( α ) f ( ) H 0 : Normalt Accept t f Z < Z Crt Acceptance regons Lower Crtcal or Rejecton regon.5% 95% Upper Crtcal or Rejecton regon.5% α z E ( ) μ σ μ α 4

5 How Regresson Can do Better than Means n Predcton? 5

6 The least square lne s the lne that best fts the data set. The least square lne passes through the average values of varables and. Dfferences between each observaton and the lne s represented b error terms. As some of them are above the lne and others below the lne, postve errors cancel out wth the negatve errors. Each dot n the above graph represents an observaton. Some observatons le above the least square lne and others le below t. These errors represent mssng elements from ths relatonshp. Omtted varables Measurement errors Msspecfcaton 6

7 7 e ( ) e S ( ) 0 > e ( )( ) 0 S ( )( ) 0 S that Mnmsed the Sum of Error Square Or best fts of the the data Choose

8 8 N N N Normal Equatons and Estmators

9 9 Food ependture and ncome: data and predcton E smsqprede Smsqpred 6.48 sumsq sumsq Sum Sum Sum sqprede prede Sqpred pred square square

10 N N 8(55) (9) 8(84) () (.875) The parameters are random varables; the var b samples. 0

11 ' 84 8 ' ' 55 9 ' Data n the Matr Form

12 N ( ) ' ' ( ) ( ) ' ' ' Adj (8)55 (9) (55) 84(9) Solvng the OLS Model usng a Matr

13 Elastct around the mean of and η

14 4 [Total varaton] [Eplaned varaton] [Resdual varaton] df T- K- T-K- ( ) [ ] ( ) [ ] ( ) ( ) e e e Var ( ) ( ) e Var ( ) e 0 Q

15 5 Coeffcent of determnaton ( ) ( ) ( ) ( ) ( ) ( ) R R e ( ) ( ) R 0 R Le Parameters R-Square s also a random varable.

16 6 Lnear, Unbasedness and Mnmum Varance Propertes of an Estmator (BLUE Propert) ( ) E ( ) f Bas Bas w E var σ Lneart: Unbasedness Mnmum Varance ( ) f

17 Interval Estmaton and the level of sgnfcance P t c tc α var( ) P P ( t SE( ) t SE( ) α c [ t SE( ), t SE( )] c c c P ( t t ) α c 7

18 Hpothess Testng About the Mean One-taled hpothess test H 0 : 0 H 0 : 0 Two-taled hpothess test H : A 0 Get the value of Get the standard error of Compute t-rato t t ( T K ) ( ) ~ SE Compare t wth the crtcal value of t from the t- dstrbuton table. reject H 0 f t t c H : A 0 Get the value of Get the standard error of Compute t-rato t t ( T K ) ( ) ~ SE reject H 0 f the computed t-value s greater than or equal to t c, or less than or equal to 8 t c. Compare t wth the crtcal value of t from the t- dstrbuton table.

19 Testng Restrctons n a Multple Regresson Model: F-test 0 Model t t Test Null Hpothess: H 0 : 0 Alternatve hpothess H : or or or an two of them or all are nonzero. At least one of them s sgnfcant. F-test for overall sgnfcance of the model F V V m m ~ F m m ( ) V sum of varaton due to eplanator varables and V sum of varaton not eplaned (squared resduals) m degrees of freedom of eplanator varables (K-) m degrees of freedom of for resdual (N-K) e 9

20 0 ( )( ) 0, S and ( )( ) 0,, S ( )( ) 0,, S Thus normal equatons are N, (),,, (),,,, (4) Dervaton of Normal Equatons n a Multple Regresson Analss

21 Algebrac Method It s easer to solve ths sstem n a devaton form defnng devaton from ( ) the mean, 0 ( ) ;,, 0 ; ( ) 0, ( ),,,, (4 ) In order get value of elmnate b multplng the ( ) b,, and (4 ) b,,,,

22 Algebrac Dervaton of Parameters Use value of the n equaton ( ) to get value of, ( ),,,,,,,,,,, The values of and can be used to fnd the value of.

23 N,,, ( ) ' ' Matr Approach

24 4,,,, (8),,,,,,,,,,, (9),,,,,,,, (0) Usng Cramer s Rule for Dervng the Slope Parameters

25 5,,,, (7.) consequences: Here s constant, and are eplanator varables. Further assume that and are perfectl correlated:,, λ. Then (7.0) and (7.) become as followng: 0 0 λ λ λ λ (7.) 0 0 λ λ λ λ (7.) Ths s the proof of the fact that when two varables are eactl correlated to each other the least square procedure completel breas down. Mutlcollneart

26 LS assumpton: varance of ever th observaton, possble that What s hetroscedastct? var e s constant var σ e for σ but t s σ σ Causes: Learnng, growth, mproved data collecton, outlers, omtted varables; 6

27 What s autocorrelaton Assumpton behnd the OLS cov( e e ) 0 j for all j Autocorrelaton ests when cov( e e j ) 0 for all j e ρe v v 0, ~ N σ ρ correlaton coeffcent between and 7