The double hurdle model analyzes decisions in two stages: (1) the participation decision (a binary choice whether to participate) and (2) the quantity decision (a continuous variable for how much to purchase). It allows for different factors to influence each decision. Three common double hurdle models are the Tobit model, Heckman model, and Cragg model. The Heckman model relaxes the Tobit assumption that the same factors influence both decisions. The Cragg model is more flexible, allowing for zero purchases even after participation. Stata commands like craggit can estimate double hurdle models.

1. Double Hurdle Model
By
Linda Chinenyenwa Familusi

2. Introduction: Double Hurdle Model
• The dependent variable is a dummy variable / binary outcomes /
dichotomous – First stage
• The dependent variable is a continuous variable – Second stage
• Decision 1: Participation
• Decision 2: Quantity sold
• We are running two models in one
1. Choice/Selection model
2. Outcome model
Linda Familusi's Presentation 217/02/2018

3. Introduction: Double Hurdle Model…
1. Participation model:
𝐷𝑖 = 1 𝑖𝑓 𝑍𝑖 𝛿 + 𝑢𝑖 > 0
𝐷𝑖 = 0 𝑖𝑓 𝑍𝑖 𝛿 + 𝑢𝑖 ≤ 0
2. Quantity sold model
𝑌𝑖
∗
= 𝑋𝑖 𝛽 + 𝜀𝑖
𝑌𝑖 = 𝑌𝑖
∗
𝑖𝑓 𝐷𝑖 = 1 𝑎𝑛𝑑 𝑌𝑖
∗
>0
𝑢𝑖 ≈ 𝑁 0,1 ; 𝜀𝑖 ≈ 𝑁(0, 𝜎2)
• 𝑐𝑜𝑟𝑟(𝑢𝑖, 𝜀𝑖)=𝜌 unobserved elements effecting participation may affect
amount of sold
Linda Familusi's Presentation 317/02/2018

4. Types: Double Hurdle Model
• Models that are commonly used include:
1. Tobit model (Tobin, 1958)
2. Heckman model (Heckman, 1979)
3. Cragg double hurdle model (Cragg, 1971),
Linda Familusi's Presentation 417/02/2018

5. Tobit model
• It assumes that the factors explaining the decision to participate in
the market as a seller and how much quantity to sell have the same
effect on these two decisions.
• This model structure cannot handle the situation in which
participation and quantity sold may be a separate decisions, possibly
influenced by different variables or by the same variables but in
different ways.
Linda Familusi's Presentation 517/02/2018

6. Tobit model...
• Tobit model is stated as:
• Yi = Yi
∗
if Yi
∗
> 0
• 𝑌𝑖 = 0 𝑖𝑓 𝑌𝑖
∗
≤ 0
• 𝑌𝑖
∗
= 𝑋𝑖 𝛽 + 𝜀𝑖 and 𝜀𝑖 ≈ 𝑁(0, 𝜎2)
• It has two variables and one model to explain these two variables
Linda Familusi's Presentation 617/02/2018

7. Heckman model
• Heckman argued that an estimation on a selected subsample leads to
sample selection bias which is solved in a two-step estimation procedure
• The difference between the heckit and the Tobit is that- the heckit
observes the process in a two- step or stage decision and then it allows the
use of different sets of explanatory variables in both stages of estimations
where as the tobit uses a one-step procedure and assumes that the factors
(i.e. explanatory variables) affecting the decision to participate in the
market and on quantity level to sale are the same.
• Thus, the heckit is viewed as a ‘generalized version of the Tobit model’
Linda Familusi's Presentation 717/02/2018

8. Heckman model…
• First stage is a Probit estimation
• Error term has a standard normal distribution
• Explanatory variables are independent of the error term
• Second stage is estimated using OLS estimation
• The Two-steps produces efficient estimates of the parameters,
standard errors and a consistent estimator
Linda Familusi's Presentation 817/02/2018

9. Heckman model…
• The residuals of the selection equation are used to construct a
selection bias control factor, which is called Lambda and which is
equivalent to the Inverse Mill's Ratio
• This factor is a summarizing measure which reflects the effects of all
unmeasured characteristics which are related to participation.
• The value of this lambda for each of the respondents is saved and
added to the data file as an additional variable.
Linda Familusi's Presentation 917/02/2018

10. Heckman model…
• The factors that affect the second decision may affect the first but the
factors that affect the first like transportation cost will not affect the
second stage.
• If the same factors affect both decisions, then the estimate of the
regression of the second decision will be biased absent correction for
the first decision stage.
Linda Familusi's Presentation 1017/02/2018

11. Cragg’s Double Hurdle Model
• Households are assumed to first decide whether to participate in the
market or not
• Secondly, they decide how much to offer for sell
• It gives room for these effects to differ, model the decision process in
two steps.
Linda Familusi's Presentation 1117/02/2018

12. Cragg’s Double Hurdle Model....
• Cragg model is a modification of the Tobit model and the Heckman
model because it is more flexible
• The difference between heckit model and Craggs double hurdle
model is that heckits assumes that in the second stage, there will be
no zero observations once the first stage is passed, whereas the
double hurdle still considers that there might be a possibility of a zero
observation which may arise from the individual’s choice (buyers
deliberate choice) or random circumstances
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13. Cragg’s Double Hurdle Models....
• First stage: market entry/participation stage 𝐏(D =1) = 𝐗 ∝ +𝛍
• Second stage: Quantity sold 𝒀∗ = 𝚭𝛃 + 𝛆 ; given that D > 0
• Participation model is estimated using Probit model
• The outcome model is estimated using truncated normal regression
Linda Familusi's Presentation 1317/02/2018

14. Stata Commands for Double hurdle models
• 1. heckman authored by Heckman J. (Probit + OLS)
• 2. craggit authored by William J. Burke (Probit + Truncated normal)
• 3. dblhurdle authored by Bruno García
• 4.dhreg (Probit and Tobit) by Christoph Engel and Peter G. Moffatt
• 5.churdle by Craggs J. (probit + Truncated or log normal for the
second stage)
Linda Familusi's Presentation 1417/02/2018

15. References
• Garcia, Bruno. (2013).Implementation of a double-hurdle model, The Stata
Journal. 13(4). 776-794.
• Briggs,D.C.(2004). Casual inference and the Heckman model.
Journal of Educational and Behavioral Statistics, 29(4), 397-420.
• Tobin, J. (1958). Estimation of Relationships for limited dependent
Variables. Econometrica, 46:1 (24-36).
• Heckman, J. (1979). Sample Selection Bias as a Specification Error.
Econometrica, 47:1 (153-161).
• Cragg, J. (1971). Some Statistical Models for Limited Dependent Variables
with Application to the Demand for Durable Goods. Econometrica, 39:5
(829-844).
• Barrett, C. (2008). Smallholder market participation: Concepts and
evidence from eastern and southern Africa. Food Policy, 33, 299-317.
Linda Familusi's Presentation 1517/02/2018

16. Thank you !!!
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