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TWO-PART AND HURDLE MODELS

Econometric Analysis of Cross Section and Panel Data,2e

MIT Press

Jeffrey M. Wooldridge

1. Introduction

2. A General Formulation

3. Truncated Normal Hurdle Model

4. Lognormal Hurdle Model

5. Exponential Type II Tobit Model

1. INTRODUCTION

∙ We consider the case with a corner at zero and a continuous

distribution for strictly positive values.

∙ Why should we move beyond Tobit? It can be too restrictive because

a single mechanism governs the “participation decision” (y  0 versus

y  0) and the “amount decision” (how much y is if it is positive).

∙ Recall that, in a Tobit model, for a continuous variable x

, the partial

effects on Py  0|x and Ey|x,y  0 have the same signs (different

multiples of 

. So, it is impossible for x

to have a positive effect on

Py  0|x and a negative effect on Ey|x,y  0. A similar comment

holds for discrete covariates.

∙ Furthermore, for continuous variables x

and x

∂Py  0|x/∂x







∂Ey|x,y  0/∂x

∙ So, if x

has twice the effect as x

on the participation decision, x

must have twice the effect on the amount decision, too.

∙ Two-part models allow different mechanisms for the participation and

amount decisions. Often, the economic argument centers around fixed

costs from participating in an activity. (For example, labor supply.)

2. A GENERAL FORMULATION

∙ Useful to have a general way to think about two-part models without

specif distributions. Let s be a binary variable that determines whether y

is zero or strictly positive. Let w

∗

be a nonnegative, continuous random

variable. Assume y is generated as

y  s  w

∗

∙ Other than s being binary and w

∗

being continuous, there is another

important difference between s and w

∗

: we effectively observe s

because s is observationally equivalent to the indicator 1y  0

(Pw

∗

 0). But w

∗

is only observed when s  1, in which case

∗

 y.

∙ Generally, we might want to allow s and w

∗

to be dependent, but that

is not as easy as it seems. A useful assumption is that s and w

∗

are

independent conditional on explanatory variables x, which we can write

Dw

∗

|s,x  Dw

∗

|x.

∙ This assumption typically underlies two-part or hurdle models.

∙ One implication is that the expected value of y conditional on x and s

is easy to obtain:

Ey|x,s  s  Ew

∗

|x,s  s  Ew

∗

|x.

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