3.5 Conclusion
4.3.2 Propensity Score Method (PSM)
In accounting and finance, firms face the dilemma of selecting among different alternatives. These alternatives are considered as treatments here. Firms selecting a specific alternative are placed in the treatment group and the others are placed in the control group. The impact of a specific treatmentd(d∈D) on the credit risk-taking behaviour of a bank (i.e. the outcomeR) can be measured by calculating the difference between the state of the world ifdwas chosen and the state of the world if it was not. However, the latter situation is counter-factual as the same bank cannot be observed under both situations. This is the fundamental problem of causal inference (Holland, 1986).
PSM deals with the fundamental problem of causal inference by calculating the proba- bility (πid,t) of receiving the treatment (d) for each unitiat timet. PSM works under two
key assumptions. The first is conditional independence or unconfoundedness assumption. According to Wooldridge (2002), the treatment effect is identified if the information related to observable covariates can be collected and if the treatment is determined by observables only, that is, it is unaffected by unobservables. This assumption is given by:
(Rdi,t+1)⊥⊥Di,t|Xi,t−1,∀X (4.4)
This is a strong assumption as the systemic differences between banks may exist even after conditioning on a set of observable covariates. For instance, such differences may arise because banks operating in different regions face different market conditions thereby affecting their credit policies differently. To overcome this problem, this study focuses on change in the credit risk measured as the difference between the credit risk before and after the issuance of ABS and CBs. The change in the credit risk of a bank is measured as:
ΔRi,t+1=Ri,t+1−Ri,t (4.5)
This is the Difference-in-Differences (DID) or double-differencing approach.17 The im- pact of time-invariant unobservables is controlled in this approach. Focus onΔRi,t+1instead ofRi,t+1helps control the bank fixed effects and removes the bank-level heterogeneity. The
second difference provides the estimate of the impact of ABS and CB issuance on the credit risk-taking behaviour. The assumptions to justify DID are weaker than those required to justify conventional matching under unconfoundedness assumption (Smith & Todd, 2005).
The second assumption is overlap or common support assumption. The fulfilment of this assumption rules out the possibility that pre-treatment covariates can perfectly predict
Dit. This assumption ascertains that units having the same scores can be either in treatment
or control group (Caliendo & Kopeinig, 2008). The Propensity Score (PS) in the presence of this assumption cannot be zero or one, as shown by:
0<pr(Di,t=d|Xi,t−1)<1 (4.6)
PS is conditioned onK-dimensional pre-treatment covariates. The formula for PS is:
πd
i,t=pr(Di,t=d|Xi,t−1) (4.7) Traditionally, the estimated PS are used to match units from the treatment and control group. To estimate the treatment effect, the mean outcome of matched units is compared.
There are two types of treatment effects (τ) estimated in PSM: Average Treatment Effect (ATE) and Average Treatment Effect on the Treated (ATT). ATE is the estimation of difference in the outcome had the entire population been observed under one treatmentdversus the entire population been observed under another treatmentdor no treatment (control group), whereas ATT is defined by the difference in the outcome of units getting treatmentdand that of those actually gettingdhad they received thed(Wooldridge, 2002).
The choice of the treatment effect depends on the research question and target population of the study. ATE is more interesting to calculate if treatment can be offered to all elements of a population. Conversely, if some elements in the population cannot get a particular treatment, ATT becomes the estimand of interest. One of the potential disadvantages of ATT is that it does not provide the information about the relative effect of a treatment program if it is extended beyond its base clientele (McCaffrey et al., 2013). I believe that ATE should be the parameter of interest in this study, as all the banks can issue either ABS or CB but it is matter of personal choice to issue a specific security or not. ATE is given by:
τd,d ate =E(ΔRd i,t+1|Di,t=d,d,Zi,t)−E(ΔRd i,t+1|Di,t=d,d,Zi,t) (4.8)
Covariate Balancing Propensity Scores (CBPS)
This study uses the CBPS methodology. CBPS is a recent development in PSM introduced by Imai and Ratkovic (2014). CBPS is more robust than the traditional propensity score estimation methods as it improves the resulting covariate balance. PS , at the same time, is a covariate balancing score and the conditional probability of assigning the treatment. CBPS exploits this dual characteristic of PS and a single model here determines the mechanism of treatment assignment and the covariate balancing weights. Incorporation of the covariate balance in the estimations is a more robust way of PS estimation that optimises the covariate balance and controls the bias.
Traditionally, the popular method for the estimation of PS is the logistic model:
πβ(Xi) =
exp(XiDβ)
1+exp(XD i β)
(4.9) whereβ∈Θis anL-dimensional vector of unknown parameters andL=K.βis estimated by maximizing the log-likelihood function:
ˆ
βMLE=argmax
β∈Θ N
∑
i=1
Assuming: πβ(.)∈C2(β) =⇒ first order condition 1 N N
∑
i=1 sβ(Di,Xi) =0, sβ(Di,Xi) = Diπβ(Xi) πβ(Xi) − (1−Di)πβ(Xi) 1−πβ(Xi) (4.11) Equation (4.11) shows the condition used to balance the function of covariates. The main problem in this traditional approach is that treatment effects may become largely biased if the treatment model is misspecified (Smith & Todd, 2005). CBPS help overcome this problem by exploiting the above-explained dual characteristics of PS . CBPS operationalises the covariate balancing property by using the inverse propensity score weighting:E= DiX˜i πβ(Xi)− (1−Di)X˜i 1−πβ(Xi) (4.12) whereX˜i= f(Xi), anM-dimensional vector-valued measurable function ofXi. Equation
(4.12) gives more weights to the covariates with a stronger predictive power of the treatment assignment. Maximum Likelihood (MLE) might not balance the covariates if model is misspecified. Hence, settingX˜i=Xiensures that first moment of each covariate is balanced,
even in the presence of model misspecification. Both the first and second moments are balanced whenX˜i= (XiTX2Ti)T. The model estimated through CBPS will be just-identified
whenL=Mand over-identified whenM>L.18 Imai and Ratkovic (2014) found that just identified models perform better than over-identified ones, because the latter fail to detect model misspecification which leads to a significant bias in the estimates.
CBPS has many advantages over traditional approaches of PS estimations. First, it helps mitigate model misspecification by taking in account parameter values that lead to a robust covariate balance. Second, CBPS incorporates all theoretical properties of the GMM and Empirical Likelihood (EL). Third, matching or weighting can be used without any modifications as it already provides improved estimations of PS. Finally, unlike other conventional methods of PS estimations, CBPS can be extended to non-binary treatments and continuous treatments. Moreover, CBPS simplifies the process of balancing covariates and helps avoiding the iterative process of refitting the logistic PS model (Fong & Imai, 2015; Imai & Ratkovic, 2014, 2015).
Another method dealing with multiple treatment cases is the use of Generalised Boosted Models (GBM) by McCaffrey, Ridgeway, and Morral (2004). However, the high dimension-
ality of the pre-treatment covariate vector can become a challenge for GBM. Furthermore, an evaluation of various PS methods conducted by Wyss et al. (2014) shows that CBPS provides more robust estimates than GBM in many instances. In their simulation study, they found that CBPS was the best estimation method. The simulation study of Imai and Ratkovic (2014) also shows that CBPS outperforms GBM.
Inverse Probability of Treatment Weights (IPTW)
Estimation of the treatment effect by the Inverse Probability of Treatment Weighting (IPTW) is an extension of the PSM. The IPTW approach tries to mimic a situation akin to random assignment to the treatment. The specification of the treatment model is more important here as compared to the outcome model (Ellis & Brookhart, 2013). The calculation of treatments weights is based on the computed PS. These weights create a pseudo-population in which the treatment assignment is unrelated to the baseline covariates, given that the treatment model is correctly specified.19 The pseudo-population is formulated in a way that the weight assigned to each element is proportional to the inverse of the element’s probability of getting the treatment. The weights for ATE can be estimated as:
ωate= Di,t−π d i,t(Xi,t−1) πd i,t(Xi,t−1)[1−πid,t(Xi,t−1)] (4.13) As compared to matching, weighting is a more robust approach as exact matching is impossible with estimated propensity scores (Abadie & Imbens, 2011). However, IPTW has a limitation that very large weights can lead to higher variance in the estimated treatment effect. A lower value ofπin the treatment group and higher value in the control group result in large weights. A solution to this problem is using stabilised weights. Stabilisation can be achieved by usingπ (mean ofπ) and 1 -πin the formula used for computing weights. Weights can be trimmed as well if weights after stabilisation are still very large.20 These weights are used in the estimation of treatment effects either through regression or differences in means (t-test or ANOVA). However, calculated weights are already stabilised when CBPS is used for the PS estimations. There is no need to perform separate weight stabilisation or weight trimming while using CBPS.
19CBPS can control the model misspecification. 20See Lee, Lessler, and Stuart (2011) for further details