Recitation 2. Probits, Logits, and 2SLS. Fall Peter Hull

Transcription

1 Rectaton 2 Probts, Logts, and 2SLS Peter Hull Fall

2 Part 1: Probts, Logts, Tobts, and other Nonlnear CEFs 2

3 Gong Latent (n Bnary): Probts and Logts Scalar bernoull y, vector x. Assume y = x + ν y = 1{y 0} y and ν : latent (unobserved) random varables Whats the CEF (E[y x ])? Depends on the (condtonal) CDF (o ν ): E [y x ] = P(y 0 x ) = P(ν x x ) = 1 F ν ( x ) = F ν (x ) Last lne ollows by CDF symmetry (usually assumed) Probt F ν () =? Logt F ν () =? Must the CEF actually be nonlnear? 3

4 Nonlnear Estmaton Two ways (at least) that s (probably) dented (where "probably" "gven some nnocuous techncal condtons") 1 Maxmum Lkelhood (MLE): MLE = arg max y x (y x,) = arg max F ν (x ) y (1 F ν (x )) 1 y snce P(y = 1 x, ) = F ν (x ) and P(y = 0 x, ) = 1 F ν (x ) 1 Nonlnear Least Squares (NLS) NLS = arg mne (y F ν (x )) snce E [y x ] = F ν (x ) = y = F ν (x ) + ε wth E [ε ] = E [y E [y x ]] = 0 As wth OLS, mnmze expected squared predcton error, E [ε ] 4

5 Maxmum Lkelhood F.O.C.: MLE = arg max = arg max F ν (x ) y (1 F ν (x )) 1 y y ln(f ν (x )) + (1 y )ln(1 F ν (x )) MLE ) MLE ν (x ν (x ) 0 = y x (1 y ) x F MLE ) MLE ν 1 F ν (x (x ) ( ) y (1 y ) = MLE ν (x )x F MLE ) 1 F MLE ν ν (x (x ) (y MLE )) ν (x MLE F ν (x )x = F MLE )(1 F ν (x MLE ν (x )) Plug-n estmator. MLE 5 solves ths n the sample

6 Nonlnear Least Squares NLS F.O.C. (gnorng 2 actor): [ ] = arg mne (y F ν (x )) NLS )) ν (x NLS )x ] 0 =E [(y F ν (x Plug-n estmator -NLS solves ths n the sample Look amlar? 1 0 = - (y NLS F ν (x NLS )) ν (x - )x N 6

7 MLE as Weghted Nonlnear Least Squares Weghted NLS (lke weghted least squares): wnls [ ] = arg mne W (x,y )(y F ν (x )) or some (known) weght uncton W (x,y ). F.O.C.? Recall.. wnls )) ν (x wnls )x 0 = W (x,y )(y F ν (x (y. MLE )) MLE ν (x. F ν (x )x 0 = F. MLE )(1 F ν (x. MLE ν (x )). MLE s a weghted NLLS estmator! But wth what weghts? 7

8 MLE as Weghted Nonlnear Least Squares (cont.) ( ) 1 W MLE (x,y ) = F.MLE )(1 F ν (x. MLE ν (x )) MLE neasble as one-step wnls estmator (. MLE on both rght and let o optmzaton) But recall another neasble estmator ( ) - y x GLS = arg mn V ε (x ) where V GLS ε (x ) s the condtonal varance o ε. (depends on j ) We make GLS easble by takng a rst-step consstent estmate o V ε (x ) (by, say OLS), then solvng. FGLS = arg mn y x. V ε (x ) 8

9 MLE as Weghted Nonlnear Least Squares (cont.) W MLE 1 1 (x,y ) = = F. MLE )(1 F ν (x. MLE ν (x )) V. ν (x ) Because y s bernoull. Can take rst-step consstent estmate o W MLE (x,y ) (by, say NLS) then solvng wnls FOC to get. MLE 1 Use. MLE 1 to get W. MLE 1. MLE E W. MLE E... Iteratng to convergence gves. MLE 9

10 Gong Latent (wth Truncaton): Tobt Assume y = max(0,x + ε ) ε N(0,σ ) Useul normal act: w N(µ,σ ) and c xed, φ E [w w > c] = µ + σ Φ CEF: µ c σ µ c σ φ and E [w w < c] = µ σ Φ c µ σ c µ σ E [y x ] = E [y x,y = 0]P(y = 0 x ) + E [y x,y > 0]P(y > 0 x ) = (x + E [ε x,ε > x ])P(ε > x x ) ( ) φ (x /σ) = x + σ Φ(x /σ) Φ(x /σ) 10 = x Φ(x /σ) + σφ(x /σ)

11 Part E IV/ESLS Facts Part 2: Some Facts about IV and 2SLS 11

12 Part E IV/ESLS Facts Matrx-y IV Setup: n 1 vector Y, n r "endogenous" matrx X 1 n s matrx o "controls" X, n t matrx o "nstruments" Z 1 X 1 = Z 1 π 1 + X π + ν Y = X X + ε (1) (2) Termnology: (1) the rst stage; (2) the second stage. Pluggng (1) nto (2) gves the reduced orm: y = (Z π 1 + X π + ν) 1 + X + ε = Z 1 (π 1 1 ) + X (π 1 + ) + (ν 1 + ε) Model s dented t r (just-dented t = r) Excluson restrcton: E [Z ε] = 0 (weak), E [ε Z] = 0 (strong) 12

13 Matrx-y IV (cont.) Part E IV/ESLS Facts X 1 = Z 1 π 1 + X π + ν Y = X X + ε Dene: [ ] X X 1 X, n (r + s) [ ] Z Z 1 X, n (t + s) Also dene: P Z Z (Z Z ) 1 Z, P X (X X ) 1 X, M I P What's P Z Z =? P Z X =? M X =? P Z P Z =? P Z =? 13

14 Part E IV/ESLS Facts 2SLS s an IV Estmator IV Estmator: j IV (W X ) 1 W Y W ZA where A = (t + s) (r + s) s some (possbly random) matrx. Note that when we're just-dented (t = r) A s (probably) nvertble, so j IV (A Z X ) 1 A Z Y = (Z X ) 1 A 1 A Z Y = (Z X ) 1 Z Y = all IV estmators are (numercally) equvalent when just-d Two-Stage Least Squares sets A (Z Z ) 1 Z X. What's W? 14

15 Part E IV/ESLS Facts 2SLS s a second-stage WLS/OLS regresson Two-Stage Least Squares s. SLS = ((Z(Z Z ) 1 Z X ) X ) 1 (Z (Z Z ) 1 Z X ) Y = ((P Z X ) X ) 1 (P Z X ) Y = (X P Z X ) 1 X P Z Y (3) = ((P Z X ) P Z X ) 1 (P Z X ) Y (4) (some knda) Weghted Least Squares, by (3). What are the weghts dong? (some knda) Ordnary Least Squares, by (4). What are the regressors? 15

16 Part E IV/ESLS Facts Just-I0 IV s "reduced-orm over rst-stage". SLS s OLS o Y on P Z X 1 SLS When r = 1 (one endogenous regressor),. s bvarate OLS o Y on M P Z X. p Cov(y,xˆ 1 ) Cov(y,xˆ 1) SLS 1 = Var(ˆx ) Cov(x 1,xˆ ) When t = r (just-dented), 1 1 so that Var(ˆx 1 ) = π Var(Z 1 1 ) Cov(y,xˆ1 ) = Cov((π 1 1 )Z 1 + X (π 1 + ) + (ν 1 + ε),πz ) = π 1 Var(Z 1 ) π σ Z π σ Z p. SLS RF {}}{ π 1 = = }{{} π 1 FS 16

17 MIT OpenCourseWare Appled Econometrcs: Mostly Harmless Bg Data Fall 2014 For normaton about ctng these materals or our Terms o Use, vst: