The Probit Model. Alexander Spermann. SoSe 2009

Transcription

1 The Probt Model Aleander Spermann Unversty of Freburg SoSe 009

2 Course outlne. Notaton and statstcal foundatons. Introducton to the Probt model 3. Applcaton 4. Coeffcents and margnal effects 5. Goodness-of-ft 6. Hypothess tests

3 Notaton and statstcal foundatons. y β + β + K + β + ε G ujarat k k y β + β + K + β + u W ooldrdge t 0 t k tk t. M atr Y Y X β + ε β + ^ ε Y X β + u y ^ β + ε β β β ( β ( 0 3 β β β β 3 3

4 Notaton and statstcal foundatons Vectors Column vector: Transposed (row vector: a [ a a K a ] a n n a a M a n n Inner product: b b [ K ] M b a b a a an ab n 4

5 Notaton and statstcal foundatons densty functon PDF: probablty densty functon f( Eample: Normal dstrbuton: φ ( e σ π ( µ σ Eample: Standard normal dstrbuton: N(0,, µ 0, σ φ ( e π µ 0 5

6 Standard logstc dstrbuton: Eponental dstrbuton: ( 3 0,, ( π σ µ + e e f Notaton and statstcal foundatons dstbutons 6 Eponental dstrbuton: Posson dstrbuton: 0, 0, 0, 0,, ( θ σ θ µ θ θ θ > e f θ σ θ µ θ θ,,! ( e f

7 Notaton and statstcal foundatons CDF CDF: cumulatve dstrbuton functon F( Eample: Standard normal dstrbuton: z e d π ( z Φ The cdf s the ntegral of the pdf. 7

8 Notaton and statstcal foundatons logarthms Rule I: y z log y log + log z Rule II: n y log y nlog Rule III: y a b log y log a + blog 8

9 Introducton to the Probt model bnary varables Why not use OLS nstead? y 0 OLS (lnear 0 Nonlnear estmaton, for eample by mamum lkelhood. 9

10 Introducton to the Probt model latent varables Latent varable: Unobservable varable y* whch can take all values n (-, +. Eample: y* Utlty(Labour ncome - Utlty(Non labour ncome Underlyng latent model: y y *, y* > 0 0, y* 0 β + ε 0

11 Probt s based on a latent model: Introducton to the Probt model latent varables ( ε φ ( 0 ( 0 ( ( * β ε ε β P P y P y P > > + > Assumpton: Error terms are ndependent and normally dstrbuted: β ( β F β β (, ( ( β σ σ β y P Φ Φ because of symmetry

12 Introducton to the Probt model CDF Eample: CDF Φ ( z 0,8 0,5 0, z 0 z z β -0,0,8

13 Introducton to the Probt model CDF Probt vs. Logt F(z les between zero and one CDF of Probt: CDF of Logt: z β β z 3

14 Introducton to the Probt model PDF Probt vs. Logt PDF of Probt: PDF of Logt: 4

15 Jont densty: Introducton to the Probt model The ML prncple [ ] y y y y F F F F y f ( ( ( (, ( β β β 5 Log lkelhood functon: ln( ( ln ln F y F y L +

16 The prncple of ML: Whch value of β mamzes the probablty of observng the gven sample? Introducton to the Probt model The ML prncple ( ( ln + F f y F f y L β 6 0 ( f F F F y F F β

17 Introducton to the Probt model Eample Eample taken from Greene, Econometrc Analyss, 5. ed. 003, ch observatons of a dscrete dstrbuton Random sample: 5, 0,,, 0, 3,, 3, 4, PDF: Jont densty : f f (, θ e θ! θ (,, K, θ f (, θ 0 0 0θ Whch value of θ makes occurance of the observed sample most probable? e 0 θ! 0θ e θ 07,36 0 7

18 Introducton to the Probt model Eample ( ( θ ln L θ 0θ + 0 lnθ, 4 d ln L dθ θ ( θ L ( θ L ( θ lnl θ d ln L( θ 0 dθ θ Mamum 8

19 Applcaton Analyss of the effect of a new teachng method n economc scences Data: Beobachtung GPA TUCE PSI Grade Beobachtung GPA TUCE PSI Grade, , , , , , 3 0 4, , ,06 0 6, ,6 8 7, , , , , , , ,83 7, ,39 7 3, , , ,65 4 3, , , 0 6, ,39 9 Source: Spector, L. and M. Mazzeo, Probt Analyss and Economc Educaton. In: Journal of Economc Educaton,, 980, pp

20 Applcaton Varables Grade Dependent varable. Indcates whether a student mproved hs grades after the new teachng method PSI had been ntroduced (0 no, yes. PSI Indcates f a student attended courses that used the new method (0 no, yes. GPA Average grade of the student TUCE Score of an ntermedate test whch shows prevous knowledge of a topc. 0

21 Applcaton Estmaton Estmaton results of the model (output from Stata:

22 Applcaton Dscusson ML estmator: Parameters were obtaned by mamzaton of the log lkelhood functon. Here: 5 teratons were necessary to fnd the mamum of the log lkelhood functon ( Interpretaton of the estmated coeffcents: Estmated coeffcents do not quantfy the nfluence of the rhs varables on the probablty that the lhs varable takes on the value one. Estmated coeffcents are parameters of the latent model.

23 Coeffcents and margnal effects The margnal effect of a rhs varable s the effect of an unt change of ths varable on the probablty P(Y X, gven that all other rhs varables are constant: P( y E( y ϕ ( β β Recap: The slope parameter of the lnear regresson model measures drectly the margnal effect of the rhs varable on the lhs varable. 3

24 Coeffcents and margnal effects The margnal effect depends on the value of the rhs varable. Therefore, there ests an ndvdual margnal effect for each person of the sample: 4

25 Coeffcents and margnal effects Computaton Two dfferent types of margnal effects can be calculated: Average margnal effect Stata command: margn Margnal effect at the mean: Stata command: mf compute 5

26 Coeffcents and margnal effects Computaton Prncple of the computaton of the average margnal effects: Average of ndvdual margnal effects 6

27 Coeffcents and margnal effects Computaton Computaton of average margnal effects depends on type of rhs varable: Contnuous varables lke TUCE and GPA: n AME ϕ( β β n n Dummy varable lke PSI: n k AME Φ( β Φ( β n [ ] k 0 7

28 Coeffcents and margnal effects Interpretaton Interpretaton of average margnal effects: Contnuous varables lke TUCE and GPA: An nfntesmal change of TUCE or GPA changes the probablty that the lhs varable takes the value one by X%. Dummy varable lke PSI: A change of PSI from zero to one changes the probablty that the lhs varable takes the value one by X percentage ponts. 8

29 Coeffcents and margnal effects Interpretaton Varable Estmated margnal effect Interpretaton GPA If the average grade of a student goes up by an nfntesmal amount, the probablty for the varable grade takng the value one rses by 36.4 %. TUCE 0.0 Analog to GPA,wth an ncrease of.%. PSI If the dummy varable changes from zero to one, the probablty for the varable grade takng the value one rses by 37.4 ppts. 9

30 Coeffcents and margnal effects Sgnfcance Sgnfcance of a coeffcent: test of the hypothess whether a parameter s sgnfcantly dfferent from zero. The decson problem s smlar to the t-test, wheras the probt test statstc follows a standard normal dstrbuton. The z-value s equal to the estmated parameter dvded by ts standard error. Stata computes a p-value whch shows drectly the sgnfcance of a parameter: z-value p-value Interpretaton GPA : sgnfcant TUCE: 0,6 0,533 nsgnfcant PSI:,67 0,008 sgnfcant 30

31 Coeffcents and margnal effects Only the average of the margnal effects s dsplayed. The ndvdual margnal effects show large varaton: Stata command: margn, table 3

32 Coeffcents and margnal effects Varaton of margnal effects may be quantfed by the confdence ntervals of the margnal effects. In whch range one can epect a coeffcent of the populaton? In our eample: Estmated coeffcent Confdence nterval (95% GPA: 0,364-0,055-0,78 TUCE: 0,0-0,00-0,05 PSI: 0,374 0, - 0,66 3

33 Coeffcents and margnal effects What s calculated by mf? Estmaton of the margnal effect at the sample mean. Sample mean 33

34 Goodness of ft Goodness of ft may be judged by McFaddens Pseudo R². Measure for promty of the model to the observed data. Comparson of the estmated model wth a model whch only contans a constant as rhs varable. ln Lˆ( M ˆ( Full : Lkelhood of model of nterest. ln L M Intercept : Lkelhood wth all coeffcents ecept that of the ntercept restrcted to zero. It always holds that ln Lˆ( M ln Lˆ( M Full Intercept 34

35 Goodness of ft The Pseudo R² s defned as: PseudoR R McF ln Lˆ( M ln Lˆ( M Full Intercept Smlar to the R² of the lnear regresson model, t holds that 0 R McF An ncreasng Pseudo R² may ndcate a better ft of the model, whereas no smple nterpretaton lke for the R² of the lnear regresson model s possble. 35

36 Goodness of ft A hgh value of R² McF does not necessarly ndcate a good ft, however, as R² McF f 0. ln Lˆ( M R² McF ncreases wth addtonal rhs varables. Therefore, an adjusted measure may be approprate: PseudoR adjusted R McF Full ln L ˆ( M ln Lˆ( M K Intercept Further goodness of ft measures: R² of McKelvey and Zavonas, Akake Informaton Crteron (AIC, etc. See also the Stata command ftstat. Full 36

37 Hypothess tests Lkelhood rato test: possblty for hypothess testng, for eample for varable relevance. Basc prncple: Comparson of the log lkelhood functons of the unrestrcted model (ln L U and that of the restrcted model (ln L R Test statstc: λ LR ln (lnl R lnl U χ K λ L L R U 0 λ ( The test statstc follows a χ² dstrbuton wth degrees of freedom equal to the number of restrctons. 37

38 Hypothess tests Null hypothess: All coeffcents ecept that of the ntercept are equal to zero. In the eample: Prob > ch LR χ (3 5,55 Interpretaton: The hypothess that all coeffcents are equal to zero can be rejected at the percent sgnfcance level. 38