*3.2. Non-Parametric Regression*

Most previous studies that measured efficiency also attempted to model the determinants of efficiency in a second-stage follow-on regression, often employing the Tobit specification (McDonald 2009), which relies on ad hoc parametric assumptions. Ashraf et al. (2017) measured bank risk-taking behavior with three alternative proxies considered as dependent variables, while taking into account the non-performing loans as measures of risk in the efficiency measurement of banks. In contrast, the present study modeled the determinant of efficiency using non-parametric regression (Hayfield and Racine 2008), which does not rely on arbitrary assumptions. Moreover, it dealt differently with continuous and discrete variables.

$$Y\_{it} = m(Z\_{it}) + \varepsilon\_{i\prime}i = 1,2,\ldots,N;\tag{6}$$

where *Yit* is the dependent variable, i.e., the efficiency score of the bank. Since regressors may be either continuous or discrete, the study defined *Zit* = (*Zcit*, *Zdit*), where *Zcit* refers to the vector of continuous regressors and *Zdit* refers to the vector of discrete regressors, *t* is the time in years, *I* is the bank in emerging economies, and N is the total sample observations (5685).

This non-parametric regression technique yields partial derivatives that are permitted to vary over the domain for the variable in question, in contrast with parametric multivariate linear regression techniques, in which the partial derivative is typically assumed to be constant over its domain (Racine 1997). The np package was used to estimate non-parametric regressions that suggested more robust interpretations (Hayfield and Racine 2008). The kernel bandwidths were selected via least-squares cross-validation and calculated as 3.45σ*n*<sup>−</sup> 15 (Racine 2008). For hypothesis testing in non-parametric regression, the study employed the bootstrapping procedures for continuous variables proposed by Racine (1997) and Racine et al. (2006) for categorical variables.
