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Selection Model Methods
One of Heckmans major methodological insights was that selection bias, which involves non-random assignment to a treatment and control group (here IMF participation and non-IMF participation), could be thought of as an omitted variable problem. Selection bias becomes an issue when, say, a researcher tries to assess the relationship between a treatment Z and an outcome Y,
Outcome equation: Y = + X + Z +
under conditions when assignment to Z is non-random
Selection equation: Z = + W +
and so there is correlation between and ; that is, Cov(, ) `" 0. In other words, there is some unobserved factor relating to Z that is also related to Y in a way we have not observed and as a result not corrected for. Generally, if Cov(, ) > 0, or what is not seen that relates to the program Z positively relates to Y, then the estimate associated with Z will be biased upwards, and vice-versa for Cov(, ) < 0. If Cov(, ) = 0, however, the OLS estimates of Z s effect on Y will be unbiased.
In medicine, RCTs provide a random intervention-assignment pattern that can identify the unbiased average effect of an intervention. In observational studies, researchers attempt to simulate such a randomized experiment by using an instrument, or an exogenous identifying variable that can provide a similar random treatment assignment and as a result give rise to unbiased estimates of an interventions average effect (See Heckman and Vytlacil 1999 and Moffitt 1999 for a discussion).
One might ask, why not just include the selection variables in the main equation and estimate the outcome equation as Y = + X + W + Z + where W are some observed determinants of participating in the intervention? We can take this approach if we are able to perfectly predict selection into the IMF program. When we cannot, stacking these additional variables in the main equation can make matters worse: with quasi-experimental data derived from nonrandomized assignments, controlling for additional variables in a regression may worsen the estimate of the treatment effect, even when the additional variables improve the specification. (Achen 1986, p. 27). Achen notes that the unexplained part of participation (s) yields inconsistency in the regression according to Cov(, )/s. Hence, adding variables to the outcome equation could decrease s without decreasing Cov(, ), which would perversely magnify the bias. In other words, the problem is not when selection is on observable factors (a possibility which we have attempted to adjust for in our numerous specification robustness checks) but when selection is on unobservable factors. This is why we specifically mention in the text that we are attempting to control for unobserved selection bias (Despite the robustness tests used in this assessment, and strong evidence supporting the mechanism proposed to explain how IMF conditionalities affect tuberculosis mortality, there may still remain potential for our results to have been driven by some aspect of the changing environment that we have not controlled for, pg. 16)
Consider trying to estimate the effect of a medication on a patients health when a patient is sick. The sicker the patient, the more likely he/she is to take the medicine. The problem is that estimated relationship between the medicine and patients health might be biased by how sick the patient was in the first place. One way to attack the problem is to randomize patients: in treatment and control groups, the patients level of sickness would average out to zero and thus be unrelated to taking the medicine. Another way is to perfectly measure how sick the patient was at the time of treatment and correct for the patients sickness by using standard regression methods. In the context of our study, the patient is the country, the doctor could be thought of as the IMF, and the medicine could be thought of as the IMFs program. We cannot perfectly see how sick the countries are and, as a result, we have to take a third route which uses a selection equation to hold constant how the patients unobserved level of sickness relates to the patients likelihood of taking the medicine so that we can isolate the effect of the medicine.
A strength of our fixed effect modeling approach is that we already hold constant unobserved time-invariant factors. This likely wipes out most of the potential problem of selection on unobservables but does not correct for the possibility that some unobserved time-varying factor (how sick the patient is becoming) is driving countries to take on IMF programs.
We can deal with this sick-patient critique, as described in the manuscript, by using a Heckman selection model or control function approach. The strategy is to model the selection into an IMF program based on a set of unobservable and observable factors (i.e., the patients sickness), and then to correct for this selection in the outcome equation.
To do this, first normalize the probit s predictions of IMF participation () from the selection equation to get the standard normal cumulative distribution function (Z), then calculate the normal distribution (Z) of these predictions using the Gaussian function e-(^2/2)/"(2). According to Heckman (1979), the selection coefficient can be calculated as = (Z)/(1 " (Z)) (also see Maddala 1983, p. 231); since hazard has the general definition H(x) = (Z)/(1 " (Z)), we are simply controlling for the hazard of participation in our models. Some variations in definitions apply depending on whether one models the hazard of participation or the hazard of non-participation. We are modeling the hazard of participation, and thus correct for /$ when under a program and -/1-$ when not under a program. Note that the normal distribution is symmetric, so (Z) = (-Z) and that a property of the cumulative density function is that 1- (Z) = (-Z). In the regression model, the estimated coefficient on the selection variable , or , is incidentally the same as the correlation of the residuals in the two equations (Corr(, ) multiplied by (logged), or the standard error of the residuals in the outcome equation.
Key assumptions for applying the Heckman approach are
1) joint bivariate normality (that is, ~ N(0, ) and ~ N(0, 1)) (See Winship and Mare 1992); and 2) a need for an exclusion restriction/instrumental approach, which means some variables in W are used to identify Z that are not in the set of variables X (See Sartori 2003). Therefore it is important to identify such an instrument for the Heckman selection model. To avoid misspecification, it is important that W contain all variables in X but that the W not in X contain instruments. The criteria for an instrumental variable w to be valid are that first, it has to be correlated with the variable of interest (Cov(w, IMF) `"0), which can be tested, and second it has to be uncorrelated with heterogeneity in the outcome equation (Cov(w, ) = 0), which cannot be directly tested. We know from our models that the first condition holds. A diagnostic test for the second requirementLcd
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We acknowledge that the Heckman-selection strategy features important and pronounced limitations as it has been commonly used, including sensitivity to the model specification, problems of collinearity and reliance on distributional assumptions when the independent variables for selection and the outcome equation are the same. For these reasons, we agree with Winship and Mares critique that Heckmans method is no panacea for selection problems and, when its assumptions are not met, may yield misleading results (1992). For this reason in the article we regard the selection model as an additional robustness check on our basic finding.
Table I. Determinants of Participating in an IMF ProgramCovariateMarginal Probability of Participating in IMF ProgramLag of Percentage Change in Real GDP0.012 (0.050)Log per Capita GDP-0.015 (0.014)***Freedom House Democratization Index-0.001 (0.002)**Military Conflict-0.969 (0.086)***Percentage of Population Working Age-0.002 (0.002)Percentage of Population Urban-0.0003 (0.0003)Lag of Dummy for Non-IMF Lending0.0004 (0.0005)*Size of Largest Titular Nationality0.0006 (0.0007)*Lag of Number of Countries Participating in IMF Programs-0.0001 (0.0004)Lag of Foreign Direct Investment (% of GDP)-0.0001 (0.0005)Time Trend-0.0010 (0.0014)Number of country-years189Number of countries20Pseudo-R20.67Note: Robust standard errors in parentheses clustered by country. - Marginal probability "$/"X presented evaluated at mean values of the covariates.
* = p<0.05, ** = p<0.01, *** = p<0.001
Table II. Frequencies of Actual and Predicted OutcomesActualPredicted01Total03194019140149Total40149189Sensitivity93.96%Specificity77.50%Positive Predictive Value93.96%Negative Predictive Value77.50%Correctly Classified90.48%
References
Achen CH. (1986) The Statistical Analysis of Quasi-Experiments. Berkeley, CA: University of California Press
Heckman J (1979) Sample selection bias as a specification error. Econometrica 41(1): 153-61.
Heckman J and Vytlacil E (1999) Local instrumental variables and latent variable models for identifying and bounding treatment effects. PNAS 96(8): 4730-4.
Maddala GS (1983) Limited-Dependent and Qualitative Variables in Econometrics. Cambridge University Press: United Kingdom.
Moffitt RA (1999). Models of treatment effects when responses are heterogenous. PNAS 96(12): 6575-6.
Sartori AE (2003) An Estimator for Some Binary-Outcome Selection Models Without Exclusion Restrictions. Political Analysis 11:111-138.
Winship C and Mare RD (1992) Models for Sample Selection Bias. Annual Review of Sociology 18:327-50.
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