Marginal Effects in Probit Models: Interpretation and Testing. 1. Interpreting Probit Coefficients
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1 ECON 5 -- NOE 15 Margnal Effects n Probt Models: Interpretaton and estng hs note ntroduces you to the two types of margnal effects n probt models: margnal ndex effects, and margnal probablty effects. It demonstrates how to calculate these effects for both contnuous and categorcal explanatory varables. A Generc Probt Model 1. Interpretng Probt Coeffcents he conventonal formulaton of a bnary dependent varable model assumes that an unobserved (or latent) dependent varable Y s generated by a classcal lnear regresson model of the form Y x + u u (1) where: k Y a contnuous real-valued ndex varable for observaton that s unobservable, or latent; x ( 1 1 k ), a 1 K row vector of regressor values for observaton ; 0 1 k ) (, a K 1 column vector of regresson coeffcents; x a 1 1 scalar called the ndex functon for observaton ; u an d N(0, σ ) random error term for observaton. k ECON 5 -- Note 15: Flename 5note15_sldes.doc Page 1 of 1 pages
2 he observable outcomes of the bnary choce problem are represented by a bnary ndcator varable Y that s related to the unobserved dependent varable Y as follows: Y 1 f Y 0 f Y > 0 (.1) Y 0 (.) he random ndcator varable Y represents the observed realzatons of a bnomal process wth the followng probabltes: > Pr( Y 1) Pr(Y > 0) Pr(x + u 0) (3.1) Pr( Y 0) Pr(Y 0) Pr(x + u 0) (3.) Probt models analytcally represent the bnomal probabltes (3.1) and (3.) n terms of the standard normal Φ Z as follows: c.d.f. ( ) ( x ) Φ( x ) Pr( Y 1) Pr(Y > 0) Φ (.1) Pr( Y 0) Pr(Y 0) 1 (.) ECON 5 -- Note 15: Flename 5note15_sldes.doc Page of 1 pages
3 Interpretaton of the probt coeffcent vector Under the zero condtonal mean error assumpton, equaton (1) mples that E ( Y x ) E( x x ) + E( u x ) x snce E( u x ) 0. (5) he ndex functon (or regresson functon) x s thus the condtonal mean value of the latent random varable Y for gven values of the regressors. he slope coeffcents j (j 1,, k): If all explanatory varables are contnuous and enter the ndex functon lnearly, the partal dervatves of regresson functon (5) wth respect to the ndvdual regressors are the slope coeffcents j (j 1,, k): E ( Y x ) j j ( j j j + + But f some of the explanatory varables are bnary or enter the ndex functon nonlnearly, the partal dervatves of regresson functon (5) are not so smply nterpreted. k k ) j. ECON 5 -- Note 15: Flename 5note15_sldes.doc Page 3 of 1 pages
4 . wo ypes of Margnal Effects n Probt Models For each explanatory varable, there are two types of margnal effects n bnary dependent varables models. Margnal Index Effects Margnal ndex effects are the partal effects of each explanatory varable on the probt ndex functon x. Case 1: j s a contnuous explanatory varable margnal ndex effect of varable j E ( Y x ) j j Case : j s a bnary explanatory varable (a dummy or ndcator varable) he margnal ndex effect of a bnary explanatory varable equals 1. the value of the ndex functon x when j 1 and the other regressors equal specfed fxed values mnus. the value of the ndex functon x when j 0 and the other regressors equal the same fxed values ECON 5 -- Note 15: Flename 5note15_sldes.doc Page of 1 pages
5 Case : j s a bnary explanatory varable (a dummy or ndcator varable) Formal Defnton: Defne two dfferent vectors of regressor values n whch all explanatory varables except j are held constant at fxed values: 1 0 x any vector of regressor values wth j 1 (and all other explanatory varables equal to fxed values); x the same vector of regressor values but wth j 0. he margnal ndex effect of the bnary (dummy) varable j s: margnal ndex effect of j E( Y 1, h j) E( Y 0, h j) j 1 x0 x h 0h Lmtaton: Margnal ndex effects are dffcult to nterpret because t s dffcult to nterpret and mpossble to measure the latent dependent varable Y. j h 0h ECON 5 -- Note 15: Flename 5note15_sldes.doc Page 5 of 1 pages
6 Margnal Probablty Effects Margnal probablty effects are the partal effects of each explanatory varable on the probablty that the observed dependent varable Y 1, where n probt models Pr( Y 1) Φ( ) x standard normal c.d.f. evaluated at x. Case 1: j s a contnuous explanatory varable margnal probablty effect of varable j Pr ( Y 1 ) Φ( x ) j Usng the chan rule of dfferentaton, we can obtan a general expresson for the margnal probablty effect of a contnuous explanatory varable j : j margnal probablty effect of j where d Φ ( ) ( x ) x Φ ( x ) d Φ( x ) x φ( x ) j d (x ) j x φ the value of the standard normal p.d.f. at x. d (x ) j the margnal ndex effect of j j ECON 5 -- Note 15: Flename 5note15_sldes.doc Page of 1 pages
7 Case : j s a bnary explanatory varable (a dummy or ndcator varable) he margnal probablty effect of a bnary explanatory varable equals 1. the value of Φ( ) mnus. value of Φ( ) x when j 1 and the other explanatory varables h (h j) equal the fxed values 0h x when j 0 and the other explanatory varables h (h j) equal the same fxed values 0h Formal Defnton: Defne two dfferent vectors of regressor values: 1 0 x any vector of regressor values wth j 1; x the same vector of regressor values but wth j 0. he margnal probablty effect of the bnary (dummy) varable j s: margnal probablty effect of j Pr( Y 1 j 1, h 0h h j) Pr( Y 1 j 0, h 0h h j) Φ( x ) Φ( ). 1 x0 ECON 5 -- Note 15: Flename 5note15_sldes.doc Page 7 of 1 pages
8 Relatonshp Between the wo Margnal Effects for Contnuous Varables Compare the margnal ndex effect and margnal probablty effect of a contnuous explanatory varable j. margnal ndex effect of varable j Φ margnal probablty effect of varable j ( x ) ( ) φ x j j Relatonshp: For a contnuous explanatory varable j, the margnal probablty effect s proportonal to the margnal ndex effect of j, where the factor of proportonalty s the standard normal p.d.f. at x : margnal probablty effect of j φ( ) x margnal ndex effect of j j ECON 5 -- Note 15: Flename 5note15_sldes.doc Page 8 of 1 pages
9 A Smple Probt Model 3. Margnal Index and Probablty Effects n Probt Models Y x + u D + D + u where: , and 3 are contnuous explanatory varables D s a bnary (or dummy) explanatory varable defned such that D 1 f observaton exhbts some attrbute, 0 otherwse. he ndex functon s: x D + D 3 1 enters the ndex functon lnearly. enters the ndex functon nonlnearly. 3 enters the ndex functon nonlnearly, nteracted wth D. D enters the ndex functon nonlnearly, nteracted wth 3. he observed bnary dependent varable Y s related to the unobserved dependent varable Y as follows: Y 1 f Y 0 f Y > 0 Y 0 ECON 5 -- Note 15: Flename 5note15_sldes.doc Page 9 of 1 pages
10 he bnomal probabltes Pr( Y 1) and Pr( Y 0) are analytcally represented n probt models n terms of Φ Z : the standard normal c.d.f. ( ) Pr( Y 1) Pr( Y 0) Pr(Y > 0) Φ( x ) Φ ( D + D ) Pr(Y 0) 1 Φ( x ) 1 Φ ( D + D ) ECON 5 -- Note 15: Flename 5note15_sldes.doc Page 10 of 1 pages
11 Margnal Effects of 1 a contnuous varable that enters lnearly x D + D 3 Margnal ndex effect of 1 x margnal ndex effect of Margnal probablty effect of 1 margnal probablty effect of 1 φ ( x ) φ( x ) 1 Margnal Effects of a contnuous varable that enters nonlnearly 1 Margnal ndex effect of x margnal ndex effect of + 3 Margnal probablty effect of margnal probablty effect of φ ( x ) φ( x )( + ) 3 ECON 5 -- Note 15: Flename 5note15_sldes.doc Page 11 of 1 pages
12 Margnal Effects of 3 a contnuous varable that enters nonlnearly x D + D 3 Margnal ndex effect of 3 margnal ndex effect of D + when D 1 when D 0 Margnal probablty effect of 3 margnal probablty effect of 3 φ ( x ) φ( x )( + D ) 3 ( x )( + ) φ when D 1 ( x ) φ when D 0 ECON 5 -- Note 15: Flename 5note15_sldes.doc Page 1 of 1 pages
13 Margnal Effects of D a bnary varable that enters nonlnearly he regressor vector for sample observaton s: x ( 1 D D ) Margnal ndex effect of D Defne two vectors of regressor values that contan the same values 1, and 3 of the other three explanatory varables 1, and 3 : one wth D 1: x 1 ( ) the other wth D 0: ( ) x0 1 3 he correspondng values of the ndex functon are: for D 1: for D 0: x ( + ) x he ndex functon dfference x 1 x he margnal ndex effect of D s therefore: margnal ndex effect of D x 1 x ECON 5 -- Note 15: Flename 5note15_sldes.doc Page 13 of 1 pages
14 x D + D 3 Recall that the probt ndex functons for D 1 and D 0 are gven respectvely by: x ( + ) 3 0 x Margnal probablty effect of bnary dummy varable D wth the other three explanatory varables 1,, and 3 equal respectvely to the fxed values 01, 0, and 03 s: Pr( Y 1 D 1, h 1,, 3) Pr( Y 1 D 0, h 1,, 3) h 0h h 0h Φ( x ) Φ( ) 1 x0 Φ ( ) Φ ( ) Φ ( ( + ) ) Φ ( ) ECON 5 -- Note 15: Flename 5note15_sldes.doc Page 1 of 1 pages
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