3.2 Basic Empirical Model
3.2.6 Additional Considerations for Estimation
Multiple Equilibria
The likelihood function, comprised of the probabilities defined in equations 3.5-3.9 and the observed individual outcomes over time, is an “incomplete” discrete econometric model according to the existing econometrics literature (Tamer, 2003). Directly applying a maximum likelihood method produces inconsistent estimates (Heckman 1978; Maddala 1983). In a game theory frame- work, the “incompleteness” is caused by the existence of multiple equilibria in the model. That is, the relationship between the covariates and error terms and the observed outcome is a correspon- dence, not a function (e.g., Bjorn and Vuong 1984; Bresnahan and Reiss 1991; Tamer 2003).
To deal with the multiple equilibria problem, several approaches have been proposed in the existing literature. The first approach is to find a common feature of all equilibria and change the model into one that predicts this feature (e.g., Bresnahan and Reiss 1991). The second approach is to specify a selection rule for the multiple equilibrium (e.g., Bjorn and Vuong 1984; Soetevent and Kooreman 2007). The third approach is to use upper and lower bounds of the choice probability to restrict the parameter estimates to a set and, with this, partially identify the parameters (e.g., Ciliberto and Tamer 2009).
Unlike the above mentioned approaches, we leave the probability of equilibrium selection as an empirical construct to be estimated. The idea is similar to Bajari et al. (2010), who estimate equi- librium selection mechanisms. To illustrate the idea, Figure 3.1 shows the equilibrium pattern for different draws of the smoking i.i.d. error terms (ignoring the evolution of health and death for this example). Supposeβ = (α1, α2, α3, α4, β2, β3, ρS,1)and~Ωit = (1, Ait, Hit, XitS, Hjt, XjtS, ξi, µt)
17The factor loading (ρS) on the permanent individual-level unobserved heterogeneity in the initial smoking proba-
bility (or age of smoking initiation) is normalized to one to satisfy an identification requirement for consistent estima- tion.
where an individual’s social contact’s smoking behavior (yjt) is removed from the information vec- tor and the marginal effect (β1) is similarly removed from the coefficient vector. We see that events
{yit=1andyjt=1}and{yit=−1andyjt=−1}may happen if−βaΩ~it−β1a≤it ≤ −βaΩ~it+β1a and−βbΩ~jt−β1b ≤ jt ≤ −βb~Ωjt +β1b. Therefore, two equilibria, (1,1) and (-1,-1), exist ifSit andS
jt are drawn from this region (i.e., the hashed region in Figure 3.1). We assume that in the multiple equilibria region, outcome (1,1) is selected with probabilityP r(o = (1,1)), and thus out- come (-1,-1) is selected with probability1−P r(o= (1,1)). According to Bajari et al. (2010),β1a,
β1b andP r(o = (1,1)) can be identified if we have exclusion restrictions that shift the smoking behavior ofi(orj) but do not directly shift the smoking behavior ofj (ori).18
Figure 3.1: Equilibrium Pattern for Error Space
−𝛽𝑎Ω 𝑖𝑡 −𝛽𝑎Ω 𝑖𝑡+ 𝛽1𝑎 −𝛽𝑎Ω 𝑖𝑡− 𝛽1𝑎 −𝛽𝑏Ω 𝑗𝑡 −𝛽𝑏Ω 𝑗𝑡− 𝛽1𝑏 −𝛽𝑏Ω 𝑗𝑡+ 𝛽1𝑏 (-1,-1) (-1, 1) (1, -1) (-1, -1) (1, 1) 𝜀𝑖𝑡 𝜀𝑗𝑡 (1, 1) A B
Note: Blue region is the multiple equilibria region.
Identification
To identify β1a, β1b and P r(o = (1,1)), we use different exclusion restrictions for different types of relationship. For spouses, we use factors that shift the wife’s (or the husband’s) smoking status before her (or his) marriage as exclusion restrictions. Specifically, we use the proportion of individuals in the same birth year, same sex cohort of the wife (or the husband) who ever smoked
18P r(o= (1,1))as an additional parameter is only identified if neitherβ1
anorβ1bis zero. One could also allow
the probability of the equilibrium selection to depend on individuals’ observed or unobserved characteristics, but the additional parameters are hard to identify in our model.
before age 19 to capture the effect of cohort-specific factors, such as values toward smoking or the cost of smoking for the specific age-sex cohort. Since the husband and the wife belong to different age-sex groups, the values are different for the husband and the wife, and they separately shift initial smoking status (i.e., ever smoked before age 19) of the husband and the wife, which leads to different smoking stocks that separately shift their smoking behaviors after marriage. We dropped a small portion of the sample who married before 19 to guarantee that the factors shift behavior before marriage.19 For siblings and friends, since it is more likely that they belong to the same cohort, we use smoking status of excluded social contacts as an additional exclusion restriction. The idea is similar to the literature that uses characteristics of excluded peers as instruments to deal with identification problems (Bramoull´e et al. 2009; De Giorgi et al. 2010). Age of the individual also serves as an exclusion restriction, with the assumption that it does not directly shift the social contact’s smoking behavior.
Identification of the effect of health status on smoking behavior requires a variable that alters health events but does not directly shift smoking behavior conditional on the observed health sta- tus. Since our measure for health status is a cardiovascular disease shock, we use an indicator for high-normal blood pressure as an exclusion restriction. The risk of high-normal blood pressure on cardiovascular disease events was not well known at the time most of our data were collected.20 Since individuals with high-normal blood pressure are unaware of their increased risk for an ad- verse cardiovascular health event, we assume that it does not directly affect own smoking behavior.
Distributional Assumptions
We use a full-information maximum likelihood method to jointly estimate the probabilities (in equations 3.5-3.9) of the behaviors and outcomes we observe. We assumeS
it,Sjt (t= 0,1, . . . , T) are i.i.d. idiosyncratic errors conditional on observed exogenous and health variables as well as
19We assume the proportion measure is not correlated with unobserved permanent heterogeneity of the individuals since it represents a population feature (i.e., no homophily for the entire age-sex cohort).
unobserved permanent individual and time-varying environmental factors, and each follows a nor- mal distributionN(0,1). By assumption, the health transitions and mortality error termsH
it, Dit (t=1,2, . . . , T) are i.i.d. and follow standard logistic distributions. These errors are not correlated
with each other and other error terms in the system, conditional on observable characteristics, permanent individual unobserved heterogeneityξi, and common unobserved heterogeneityµt.
Following an approach proposed and used by Heckman and Singer (1984), Mroz and Guilkey (1992) and Mroz (1999), we assume the joint distribution of ξi and ξj can be approximated by a set of discrete mass points and associated weights without imposing a specific distribution. A Monte Carlo analyses demonstrates that this flexible approach performs better in simultaneous equations than assuming, for example, a joint normal distribution for error terms that may not be normally distributed (Mroz 1999). Having been specific about assumptions made for distributions of unobservables, identification, and equilibrium selection, we specify the likelihood function in Appendix C.1.
3.3 Data and Construction of Variables