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Dummy Endogenous Variables in a Simultaneous Equation System Econometrica, Vol. 46, No. 4 (Jul., 1978), 931-959. James J. Heckman Econ 312, Spring 2019 Heckman 5/29/2019 1. A GENERAL MODEL FOR THE TWO EQUATION CASE Heckman Dummy Endogenous

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SLIDE 1

Dummy Endogenous Variables in a Simultaneous Equation System

Econometrica, Vol. 46, No. 4 (Jul., 1978), 931-959.

James J. Heckman

Econ 312, Spring 2019

Heckman

5/29/2019

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Heckman Dummy Endogenous Variables

  • 1. A GENERAL MODEL FOR THE TWO

EQUATION CASE

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Heckman Dummy Endogenous Variables

  • Pair of simultaneous equations for continuous latent random variables 𝑧1𝑗

βˆ— and y2i βˆ— ,

(1a) 𝑧1𝑗

βˆ— = π‘Œ1𝑗𝛽1 + 𝑒𝑗𝛾1 + 𝑧2𝑗 βˆ— πœ‡1 + 𝑉1𝑗,

(2a) 𝑧2𝑗

βˆ— = π‘Œ2𝑗𝛽2 + 𝑒𝑗𝛾2 + 𝑧1𝑗 βˆ— πœ‡2 + 𝑉2𝑗,

where dummy variable 𝑒𝑗 is defined by (1c) 𝑒𝑗 = 1 iff 𝑧2𝑗

βˆ— > 0,

𝑒𝑗 = 0 otherwise, and 𝐹 𝑉

π‘˜π‘— = 0,

𝐹 𝑉

π‘˜π‘— 2 = 𝜏 π‘˜π‘˜,

𝐹 𝑉1𝑗𝑉2𝑗 = 𝜏12, π‘˜ = 1,2; 𝑗 = 1, … , 𝐽. 𝐹 𝑉

π‘˜π‘—π‘‰π‘˜β€²π‘—β€² = 0, for π‘˜, π‘˜β€² = 1,2; 𝑗 β‰  𝑗′.

β€œπ‘Œ1𝑗” and β€œπ‘Œ2𝑗” are, respectively, 1 Γ— 𝐿1 and 1 Γ— 𝐿2 row vectors of bounded exogenous variables.

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Heckman Dummy Endogenous Variables

  • Equations (1a) and (1b) are identified under standard conditions if 𝛾1 = 𝛾2 =

0 and both 𝑧1𝑗

βˆ— and 𝑧2𝑗 βˆ— are observed for each of the I observations.

  • In this special case, which conforms to the classical simultaneous equation

model, standard methods are available to estimate all of the parameters of the structure.

  • First, note that the model is cast in terms of latent variables 𝑧1𝑗

βˆ— and 𝑧2𝑗 βˆ— which

may or may not be directly observed.

  • Even if 𝑧2𝑗

βˆ— is never observed, the event 𝑧2𝑗 βˆ— > 0 is observed and its occurrence

is recorded by setting a dummy variable, 𝑒𝑗 equal to one.

  • If 𝑧2𝑗

βˆ— < 0, the dummy variable assumes the value zero.

  • Second, note that if 𝑧2𝑗

βˆ— > 0, structural equations (1a) and (1b) are shifted by an

amount 𝛾1and 𝛾2, respectively.

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Heckman Dummy Endogenous Variables

  • To fix ideas, several plausible economic models are discussed that may

be described by equation system (1a)-(1c).

  • First, suppose that both 𝑧1𝑗

βˆ— and 𝑧2𝑗 βˆ— are observed outcomes of a market at

time i, say quantity and price.

  • Equation (1a) is the demand curve while equation (1b) is the supply

curve.

  • If the price exceeds some threshold (zero in inequality (1c), but this can

be readily amended t be any positive constant), the government takes certain actions that shift both the supply curve and the demand curve, say a subsidy to consumers and a per unit subsidy to producers.

  • These actions shift the demand curve and the supply curve by the amount

𝛾1and 𝛾2, respectively.

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Heckman Dummy Endogenous Variables

  • As another example, consider a model of the effect of laws on the status
  • f blacks.
  • Let 𝑧1𝑗

βˆ— be the measured income of blacks in state i while 𝑧2𝑗 βˆ— is an

unmeasured variable that reflects the state’s population sentiment toward blacks.

  • If sentiment for blacks is sufficiently favorable 𝑧2𝑗

βˆ— > 0 the state may

enact antidiscrimination legislation and the presence of such legislation in state i, a variable that can be measured, is denoted by a dummy variable 𝑒𝑗 = 1.

  • In the income equation (1a), both the presence of a law and the

population sentiment towards blacks is assumed to affect the measured income of blacks.

  • The first effect is assumed to operate discretely while the second effect is

assumed to operate in a more continuous fashion.

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Heckman Dummy Endogenous Variables

  • Two conceptually distinct roles for dummy variables:
  • 1. As indicators of latent variables that cross thresholds and
  • 2. As direct shifters of behavioral functions. These two roles must be

carefully distinguished.

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Heckman Dummy Endogenous Variables

  • The model of equations (1a)-(1c) subsumes a wide variety of interesting

econometric models. These special cases are briefly discussed in turn.

  • CASE l: The Classical Simultaneous Equation Model: This model

arises when 𝑧1𝑗

βˆ— and 𝑧2𝑗 βˆ— are observed, and there is no structural shift

in the equations (𝛾1 = 𝛾2 = 0).

  • CASE 2: The Classical Simultaneous Equation Model with

Structural Shift: This model is the same as that of Case 1 except that structural shift is permitted in each equation. It will be shown below that certain restrictions must be imposed on the model in order to generate a sensible statistical structure for this case.

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Heckman Dummy Endogenous Variables

  • CASE 3: The Multivariate Probit Model: This model arises when

𝑧1𝑗

βˆ— and 𝑧2𝑗 βˆ— are not observed but the events 𝑧1𝑗 βˆ— and 𝑧2𝑗 βˆ— are observed

(i.e., one knows whether or not the latent variables have crossed a threshold). The notation of equations (1a)-(1b) must be altered to accommodate two dummy variables but that modification is

  • bvious. No structural shift is permitted (𝛾1 = 𝛾2 = 0). This is the

model of Ashford and Sowden [3], Amemiya [2], and Zellner and Lee [30].

  • CASE 4: The Multivariate Probit Model with Structural Shift: This

model is the same as that of Case 3 except that structural shift is permitted (𝛾1 = 𝛾2 = 0).

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Heckman Dummy Endogenous Variables

  • CASE 5: The Hybrid Model: This model arises when 𝑧1𝑗

βˆ— is

  • bserved and 𝑧2𝑗

βˆ— is not, but the event 𝑧2𝑗 βˆ— β‰·0 is observed. No

structural shift is permitted (𝛾1 = 𝛾2 = 0).

  • CASE 6: The Hybrid Model with Structural Shift: This model is the

same as that of Case 5 except that structural shifts in the equations are permitted.

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Heckman Dummy Endogenous Variables

  • 2. THE HYBRID MODEL WITH STRUCTURAL

SHIFT

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Heckman Dummy Endogenous Variables

  • In this section, a model with one observed continuous random variable,

and one latent random variable is analyzed for the general case of structural shift in the equations.

  • Consider identification only; Heckman (1978) for additional discussion.
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Heckman Dummy Endogenous Variables

  • To facilitate the discussion, equations (1a) and (1b) may be written in

semi-reduced form as

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Heckman Dummy Endogenous Variables

  • In the ensuing analysis it is assumed that exogenous variables included in both

π‘Œ1𝑗 and π‘Œ2𝑗 are allocated to either π‘Œ1𝑗 or π‘Œ2𝑗 but not both.

  • The absence of an asterisk on 𝑧1𝑗 denotes that this variable is observed.
  • β€œπ‘§2𝑗

βˆ— ” is not observed.

  • Random variables 𝑉1𝑗 and 𝑉2𝑗 are assumed to be bivariate normal random

variables.

  • Accordingly, the joint distribution of π‘Š

1𝑗, π‘Š 2𝑗, β„Ž(π‘Š 1𝑗, π‘Š 2𝑗), is a bivariate normal

density fully characterized by the following assumptions: 𝐹 π‘Š

1𝑗 = 0,

𝐹 π‘Š

2𝑗 = 0,

𝐹 π‘Š

1𝑗 2

= πœ•1𝑗, 𝐹 π‘Š

1π‘—π‘Š 2𝑗 = πœ•12,

𝐹 π‘Š

2𝑗 2

= πœ•22.

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Heckman Dummy Endogenous Variables

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Heckman Dummy Endogenous Variables

(i) Conditions for Existence of the Model

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Heckman Dummy Endogenous Variables

  • The first order of business is to determine whether or not the model of equations

(1a)-(1b) as represented in reduced form by equations (3a)-(3b) makes sense.

  • Without imposing a further restriction, it does not.
  • The restriction required is precisely the restriction implicitly assumed in writing

equations (3a) and (3b), i.e., the restriction that permits one to define a unique probability statement for the events 𝑒𝑗 = 1 and 𝑒𝑗 = 0 so that 𝑄𝑗 in fact exists.

  • A necessary and sufficient condition for this to be so is that 𝜌23 = 0, i.e., that the

probability of the event 𝑒𝑗 = 1 is not a determinant of the event.

  • Equivalently, this assumption can be written as the requirement that 𝛿2𝛾1 + 𝛾2 =

0.

  • This condition is critical to the analysis and thus deserves some discussion.
  • The argument supporting this assumption is summarized in the following

proposition.

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Heckman Dummy Endogenous Variables

PROPOSITION: A necessary and sufficient condition for the model of equations (1a)-(1c) or (3a)-(3c) to be defined is that 𝜌23 = 0 = 𝛿2𝛾1 + 𝛾2. This assumption is termed the principal assumption. PROOF: Sufficiency is obvious. Thus, only necessary conditions are discussed. Denote the joint density of π‘Š

2𝑗, 𝑒𝑗 by 𝑒(π‘Š 2𝑗, 𝑒𝑗) which is assumed to be a proper

density in the sense that

𝑒𝑗=0,1 βˆ’βˆž ∞

𝑒(π‘Š

2𝑗, 𝑒𝑗) π‘’π‘Š 2𝑗 = 1.

  • From equations (3b) and (3c), the probability that 𝑧2𝑗

βˆ— β©Ύ 0 given 𝑒𝑗 = 1 must be

unity, so that one may write Pr π‘Š

2𝑗 > π‘šπ‘— 𝑒𝑗 = 1 = 1

where the symbols π‘šπ‘— and π‘šπ‘—

β€² are defined by π‘šπ‘— = βˆ’ π‘Œ1π‘—πœŒ21 + π‘Œ22 + 𝜌23 and π‘šπ‘— β€² =

π‘šπ‘— + 𝜌23.

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Heckman Dummy Endogenous Variables

  • Alternatively, one may write this condition as

(4a)

π‘šπ‘— ∞ 𝑒(π‘Š 2𝑗 , 1) π‘’π‘Š 2𝑗 = 𝑄𝑗

and obviously (4b)

βˆ’βˆž π‘šπ‘— 𝑒 π‘Š 2𝑗, 1 π‘’π‘Š 2𝑗 = 0.

  • Using similar reasoning, one can conclude that

(4c)

βˆ’βˆž π‘šπ‘—

β€²

𝑒 π‘Š

2𝑗, 1 π‘’π‘Š 2𝑗 = 1 βˆ’ 𝑄𝑗

and (4d)

π‘šπ‘—

β€²

∞ 𝑒 π‘Š 2𝑗, 0 π‘’π‘Š 2𝑗 = 0.

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Heckman Dummy Endogenous Variables

  • The sum of the left hand side terms of equations (4a)-(4d) equals the sum of

the right hand side terms which should equal one if the probability of the event 𝑒𝑗 = 1, meaningfully defined.

  • If 𝜌23 = 0, this is the case.
  • But if 𝜌23 < 0, the sum of the left hand side terms falls short of one while if

𝜌23 > 0, this sum exceeds one. Q.E.D.

  • Notice that this argument does not rely on the assumption that π‘Š

2𝑗 is normally

distributed but does rely on the assumption that π‘Š

2𝑗 has positive density at

almost all points on the real line.

  • An intuitive motivation for this condition is possible. Suppose that one

rewrites equations (1a)-(1c) to exclude 𝑒𝑗, i.e., write 𝑧1𝑗

βˆ— = π‘Œ1𝑗𝑏1 + 𝑧2𝑗 βˆ— 𝛿1 + 𝑉1𝑗,

𝑧2𝑗

βˆ— = π‘Œ2𝑗𝑏2 + 𝑧1𝑗 βˆ— 𝛿2 + 𝑉2𝑗,

𝑒𝑗 = 1 iff y2i

βˆ— > 0,

𝑒𝑗 = 0 otherwise.

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Heckman Dummy Endogenous Variables

  • Note that 𝑧1𝑗

βˆ— is an unobserved latent variable.

  • The random variable 𝑧1𝑗 is observed and is defined by the following

equation: 𝑧1𝑗 = 𝑧1𝑗

βˆ— + 𝑒𝑗𝛾1.

  • Making the appropriate substitutions of 𝑧1𝑗 and 𝑧1𝑗

βˆ— in the system given

above, one concludes that 𝑧1𝑗 = π‘Œ1𝑗𝑏1 + 𝑒1𝛾1 + 𝑧2𝑗

βˆ— 𝛿1 + 𝑉1𝑗,

𝑧2𝑗

βˆ— = π‘Œ2𝑗𝑏2 + 𝑧1𝑗 βˆ’ 𝑒𝑗𝛾1 𝛿2 + 𝑉21.

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Heckman Dummy Endogenous Variables

  • Invoking the principal assumption, one reaches equations (1a)-(1c) including

𝑒𝑗,

  • Thus the dummy shift variable 𝑒𝑗𝛾1 may be viewed as a veil that obscures

measurement of the latent variable 𝑧1𝑗

βˆ— .

  • The principal assumption essentially requires that the latent variable 𝑧1𝑗

βˆ— and

not the measured variable 𝑧1𝑗

βˆ— appears in the second structural equation.

  • It is possible to use the latent variable in the second equation because 𝛾1 can

be estimated as will be shown.

  • It is important to note that the principal assumption does not rule out

structural shift in equations (1a) and (1b).

  • It simply restricts the nature of the shift. However, the principal assumption

does exclude any structural shift in the reduced form equation that determines the probability of shift (equation (3b)).