SWITCHING REGRESSION MODELS AND ESTIMATION G.S. Maddala Presented by Ying Fei
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Outline Switching Regression Models Model setting Motivation Estimation (Two-stage method) Variations Censored models Models with self-selectivity
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Switching Regression Models — Setting Regime 1: yi = β1' X 1i + u1i
iff
γ ' Z i ≥ ui (1)
Regime 2: yi = β 2' X 2 i + u2i
iff
γ ' Z i < ui (2)
We assume that u1i , u2i , and ui have a trivariate normal distribution, with mean vector zero and covariance matrix σ 12 σ 12 σ 1u 2 = σ σ ∑ 2 2u 1
(3)
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Switching Regression Models — Applications The Union-nonunion-wage model (Lee, 1978) The Housing-demand model (Trost, 1977) Disequilibrium Market model (Fair and Jaffee, 1972) The Labor-supply model (Heckman, 1974) The Labor-supply model (Gronau, 1974) Needs vs. Reluctance model (Polakoff and Sibler, 1967)
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Switching Regression Models — Estimation (1) OLS Estimation φ (γ ' Z i ) E (u1i | ui ≥ γ ' Z i ) = E (σ 1u ui | ui ≤ γ ' Z i ) = −σ 1u φ (γ ' Z i ) E (u2i | ui ≥ γ ' Z i ) = E (σ 2u ui ≥ γ ' Z i ) = σ 2u
φ (γ ' Z i ) 1 − φ (γ ' Z i )
(4)
(5)
OLS estimation is inappropriate.
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Switching Regression Models — Estimation (2) Maximum Likelihood Estimation L( β1 , β 2 , σ 12 , σ 22 ,σ 1u , σ 2 u )
[
γ 'Z
= ∏ ∫−∞ i g ( yi − β1 ' X 1i , ui )dui
] [∫ Ii
∞
γ 'Zi
f ( yi − β 2 ' X 2 i , ui )dui
]
1− Ii
(6)
The ML estimates can be shown to be consistent and asymptotically efficient The estimation can be cumbersome
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Switching Regression Models — Estimation (3) Tobit Models yi = β ' X i + ui
if RHS>0
yi = 0
otherwise
(7)
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Switching Regression Models — Estimation (4)
Two-stage method for Tobit Models E (ui | ui > − β ' X i ) = σ
φi Φi
φi yi = β ' X i + σ + vi Φi
(8)
(9)
where E (vi ) = 0
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Switching Regression Models — Estimation (5) Two-stage method for Tobit Models Stage1: Get the ML estimates of β / σ using probit model, and then get estimated values of unknown variables in the expected value of residuals 1 if yi ≥ 0 (10) Ii = 0 otherwise Likelihood function
L = ∏ [Φ I ]Ii [1 − Φ i ]1− Ii Ii =1
β /σ
(11)
φi , Φ i
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Switching Regression Models — Estimation (6) Two-stage method for Tobit Models Stage 2: Get consistent estimates of β and σ by estimating the original equation by OLS, using φi / Φ i in place of φi / Φ i as an explanatory variable
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Switching Regression Models — Estimation (7) Two-stage method for Tobit Models Or :
E ( yi ) = Pr ob( yi > 0) ⋅ E ( yi | yi > 0) + Pr ob( yi ≤ 0) ⋅ E ( yi | yi ≤ 0) φi = Φi (β ' X i + σ ) + 0 Φi = β ' (Φ i X i ) + σφi (12)
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Switching Regression Models — Estimation (8) Two-stage Estimation Method (Heckman, 1974; Lee, 1976) Essential Features First obtain the expected values of the residuals that are truncated. Estimate the unknown parameters in the expected values by a probit model. Introduce the estimated values of these variables into the original equation and estimate it by proper least squares The two-stage estimates are consistent The estimations is simpler compared with ML The two-stage estimates can be used as initial values for iteration of ML estimation. 12
Switching Regression Models — Variation (2) Censored Models Labor-supply model (Gronau, 1974) Needs vs Reluctance model (Polakoff and Sibler, 1967)
β1' X 1i − β 2' X 2i u2 − u1 γ ' Zi = and u = σ σ
(16)
where, σ 2 = Var (u2 − u1 ) = σ 12 + σ 22 − 2σ 12
(17) 13
Switching Regression Models — Variation (1) Censored Models y1 = β1' X 1 + u1
(13)
y2 = β 2' X 2 + u2
(14)
and we observe y = y1
if y1 ≥ y2
y=0
otherwise
(15)
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Switching Regression Models — Variation (3) Identification Conditions (Nelson, 1975) 1. σ 12 = 0 2. There is at least one variable in X 1 not included in X 2 . (In the context of the labor-supply model, there is at least one explanatory variable in the market-wage function not included in the reservation-wage function.)
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Switching Regression Models — Variation (4) Self-selection Models Occupation decision model (Roy, 1951) Labor Supply by Women (Gronau and Lewis, 1974) Housing demand model (Lee and Trost, 1978) Evaluation of the benefits of social programs
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Switching Regression Models — Variation (5) Self-selection Models (for participants) (18) y1i = X i β i + u1i (for nonparticipants) (19) y2 i = X i β 2 + u2 i (participation decision function) (20) I i* = Z iγ + ε i (21) I i = 1 iff I i* > 0 (22) I i = 0 iff I i* ≤ 0 The observed yi is defined as (23) yi = y1i iff I i = 1 (24) yi = y2 i iff I i = 0 σ 11 σ 12 σ 1ε (25) Cov (u1i , u2 i , ε i ) = σ 12 σ 22 σ 2 ε 1 σ 1ε σ 2 ε 17
Switching Regression Models — Variation (6) Self-selection Models How to measure the benefit of the program?
φ ( Z iγ ) y1i − E ( y2i | I i = 1) = y1i − X i β 2 + σ 2ε Φ ( Z iγ )
(26)
E ( y1i | I i = 1) − E ( y2i | I i = 1) φ ( Z iγ ) = X i ( β1 − β 2 ) + (σ 2ε − σ 2ε ) Φ ( Z iγ )
(27)
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Switching Regression Models — Variation (7) Self-selection Models Total Sample Individual decision to participate Administrator’s decision to select Control group Drop out
Continue
Individual decision not to participate in experiment
Administrator’s decision not to select
Treatment group
Drop out
Continue
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