SWITCHING REGRESSION MODELS AND ESTIMATION
G.S. Maddala Presented by Ying Fei
Outline
Switching Regression Models
Model setting Motivation Estimation (Two-stage method) Variations
Censored models Models with self-selectivity
Regime 1: 
iff 
(1)
Regime 2: 
iff 
(2)
We
assume that 
,
, and 
have a trivariate
normal distribution, with mean vector zero and covariance matrix
(3)
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)
Switching Regression Models —
Estimation (1)
OLS Estimation
(4)
(5)
OLS estimation is
inappropriate.
Switching Regression Models —
Estimation (2)
Maximum Likelihood Estimation
(6)
The ML estimates can be
shown to be consistent and asymptotically efficient
The estimation can be cumbersome
Tobit Models
if RHS>0
otherwise (7)
Two-stage method for Tobit Models
(8)
(9)
where 
Two-stage method for Tobit Models
Stage1: Get the ML estimates of 
using probit model,
and then get estimated values of unknown variables in the expected value of
residuals
(10)
Likelihood function
(11)
Two-stage method for Tobit Models
Stage 2: Get consistent estimates of 
and 
by estimating the
original equation by OLS, using 
in place of 
as an explanatory
variable
Two-stage method for Tobit Models
Or :
(12)
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.
Switching Regression Models —
Variation (2)
Censored Models
Labor-supply model (Gronau, 1974) Needs vs Reluctance model (Polakoff and
Sibler, 1967)
(16)
where,
(17)
Censored Models
(13)
(14)
and we observe
(15)
Identification Conditions (Nelson, 1975)
1.
2.
There
is at least one variable in 
not included in 
. (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.)
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
Switching Regression Models —
Variation (5)
Self-selection Models
(for participants) (18)
(for nonparticipants) (19)
(participation decision
function) (20)
(21)
(22)
The observed 
is defined as
(23)
(24)
(25)
Self-selection Models
How to measure the benefit of the program?
(26)
(27)
Switching Regression Models —
Variation (7)
Self-selection Models
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