Proxy Variable in OECD Database: Application of Parametric Quantile Regression and Median Based Unit Rayleigh Distribution

1.1. Parametric MBUR Quantile Regression Model

The response variable adheres to the Median Based Unit Rayleigh (MBUR) distribution, which adds complexity to the analysis. To uncover a causal relationship between this variable and its influencing covariates, it is essential to define both the parametric function and the link function meticulously. Given that the response variable may exhibit significant skewness and potentially breach the assumptions of normality and homoscedasticity, employing parametric quantile regression could provide a resilient alternative. Nevertheless, it is vital to delve into a myriad of other methodologies to ensure the most accurate and reliable estimation outcomes. The MBUR distribution was discussed by Attia [32]. It has the following PDF, CDF, and quantile function as respectively expressed in equations (1-3). The parametric quantile regression (PQR) depends on the quantile function.

$$f\left(y\right)={\displaystyle \frac{6}{{\alpha }^2}}\left[1-y^{{\displaystyle \frac{1}{{\alpha }^2}}}\right]y^{\left({\displaystyle \frac{2}{{\alpha }^2}}-1\right)},0<y<1,\alpha >0(1)$$

$$F\left(y\right)={3y}^{{\displaystyle \frac{2}{{\alpha }^2}}}-{2y}^{{\displaystyle \frac{3}{{\alpha }^2}}},0<y<1,\alpha >0(2)$$

$$y=F^{-1}\left(y\right)={\left\{-.5\left(cos\left[{\displaystyle \frac{{cos}^{-1}\left(1-2u\right)}{3}}\right]-\sqrt{3}sin\left[{\displaystyle \frac{{cos}^{-1}\left(1-2u\right)}{3}}\right]\right)+.5\right\}}^{{\alpha }^2}(3)$$

Re-parameterize the PDF and CDF of MBUR using the quantile function, $=c^{{\alpha }^2}$, where c is $c=-.5\left(cos\left[{\displaystyle \frac{{cos}^{-1}\left(1-2u\right)}{3}}\right]-\sqrt{3}sin\left[{\displaystyle \frac{{cos}^{-1}\left(1-2u\right)}{3}}\right]\right)+.5,u\in \left(\mathrm{0,1}\right).$ U represents the chosen percentile, if it is the median, so u=0.5, if it is the 25th percentile so u=0.25 so ${\alpha }^2={\displaystyle \frac{ln\left(y\right)}{ln\left(c\right)}}$ .When replacing u=0.5 in c, this gives $ln(.5)=-.6931$ , ${\alpha }^2={\displaystyle \frac{ln\left(y\right)}{ln\left(c\right)}}={\displaystyle \frac{ln\left(median\right)}{ln\left(c\right)}}$ . As y is the median corresponding to u=0.5.

1.2. The Link Function

The linear predictor, $\phi =X^{\prime }B={\beta }_0+{\beta }_1{x}_{i1}+\cdots +{\beta }_k{x}_{ik};i=1,...,n\&k=1,\dots ,k$, can be expressed using different link functions as the logit, clog-log, or the log-log link function. This linear predictor represents the median that should be estimated, where n is the number of cases or observations and k is the number of variables. These link functions of the median are discussed in details by Attia [33].

1.2.1. Logit Link Function

Using the logit function of the median, which will be called $\mu$

$$Logit\left(median\right)=logit\left(\mu \right)=\phi =X^{\prime }B={\beta }_0+{\beta }_1{x}_{i1}+\cdots +{\beta }_k{x}_{ik};i=1,...,n\&k=1,\dots ,k$$

where n is the number of cases or observations and k is the number of variables. Logit median is the linear combination of variables.

So; $\mu ={\displaystyle \frac{e^{X^{\prime }B}}{1+e^{X^{\prime }B}}}={\displaystyle \frac{e^{\phi }}{1+e^{\phi }}}$ & ${\alpha }^2={\displaystyle \frac{ln\left(y\right)}{ln\left(c\right)}}={\displaystyle \frac{ln\left(median\right)}{ln\left(c\right)}}={\displaystyle \frac{ln\left(\frac{e^{X^{\prime }B}}{1+e^{X^{\prime }B}}\right)}{ln\left(c\right)}}={\displaystyle \frac{ln\left(\frac{e^{\phi }}{1+e^{\phi }}\right)}{ln\left(c\right)}},0<y<1,\alpha >0$

1.2.2. Complementary Log-Log Link Function

$$\log \left\{-\log \left(1-median\right)\right\}=X^{\prime }B=\phi ={\beta }_0+{\beta }_1{x}_{i1}+\cdots +{\beta }_k{x}_{ik};i=1,..,n\&k=1,..,k$$

$$median=1-e^{-e^{X^{\prime }\beta }}=1-exp\left(-exp\left(X^{\prime }\beta \right)\right)\&{\alpha }^2={\displaystyle \frac{ln\left(median\right)}{ln\left(c\right)}}={\displaystyle \frac{ln\left(1-e^{-e^{\phi }}\right)}{ln\left(c\right)}}$$

1.2.3. Log-Log Median Link Function

A link function other than the logit can be used. The author used the log-log function

$$\log \left\{-\log \left(median\right)\right\}=X^{\prime }B=\phi ={\beta }_0+{\beta }_1{x}_{i1}+\cdots +{\beta }_k{x}_{ik},i=\mathrm{1,2},\dots ..,n\&k=\mathrm{1,2},\dots ..,k$$

$$median=e^{-e^{X\prime \beta }}=exp\left(-exp\left(X^{\prime }\beta \right)\right)\&{\alpha }^2={\displaystyle \frac{ln\left(median\right)}{ln\left(c\right)}}={\alpha }^2={\displaystyle \frac{ln\left(e^{-e^{X\prime \beta }}\right)}{ln\left(c\right)}}={\displaystyle \frac{-e^{X^{\prime }\beta }}{ln\left(c\right)}}$$

2.1. Diagnostic Tests for Model Specification

Model adequacy, as regards the appropriate response variable distribution and the used link function, can be appraised using two residual-based diagnostics: randomized quantile residuals (RQ)[34] and Cox-Snell residuals (CS)[35]. The RQ residuals follow approximately a standard normal distribution, and the (CS) residuals follow a standard exponential distribution , [36,37].

2.1.1. Randomized Quantile (RQ) Residuals: ${\mathit{r}}_{\mathit{i}}^{\mathit{R}\mathit{Q}}={\mathbf{\Phi }}^{-1}\left(\mathit{F}\left({\mathit{y}}_{\mathit{i}},\widehat{\mathit{\beta }}\right)\right)$

$\mathrm{\Phi }$ is the standard normal CDF. F is the re-parameterized MBUR CDF. $y_i$’s are the observations and $\widehat{\beta }$ are the estimated regression coefficients. These residuals are approximately distributed as standard normal. When the model is correctly specified, these residuals approximately follow the standard normal distribution.

2.1.2. Cox-Snell (CS) Residuals: ${\mathit{r}}_{\mathit{i}}^{\mathit{C}\mathit{S}}=-\mathit{l}\mathit{o}\mathit{g}\left(1-\mathit{F}\left({\mathit{y}}_{\mathit{i}},\widehat{\mathit{\beta }}\right)\right)$

Cox-Snell residuals are approximately distributed as a standard exponential distribution with a scale parameter one. The negative logarithm mentioned above represents the cumulative distribution function (CDF) of the standard exponential distribution with this scale parameter. When the model is correctly specified, these residuals will approximately follow a standard exponential distribution.

2.2. Model Selection Criteria

Model selection is conducted by identifying the options with the minimum values of AIC (Akaike Information Criterion), BIC (Bayesian Information Criterion), and Corrected AIC, ensuring that the chosen models are optimal for analysis. $AIC=-2l\left(\widehat{\theta }\right)+2p$, $CAIC=AIC+{\displaystyle \frac{2p\left(p+1\right)}{n-p-1}}$, $BIC=-2l\left(\widehat{\theta }\right)+p\log \left(n\right)$, where p is the number of parameters and n is the sample size.

In the study conducted by Sánchez, Leiva, Saulo, and colleagues in 2021 [14], they applied a specific methodological approach, $R_M^2$ , that corresponds directly to established practices, usual $R^2$ , in mean regression. This approach is instrumental in yielding reliable estimates and is meticulously defined within the context of regression analysis. Through these criteria and methodologies, researchers can effectively compare and assess the suitability of various models, facilitating improved decision-making in statistical modeling, $R_M^2=1-exp\left({\displaystyle \frac{2}{n}}\left(l\left(\check{\theta }\right)-l\left(\widehat{\theta }\right)\right)\right)$, where $l\left(\check{\theta }\right)andl\left(\widehat{\theta }\right)$ are the maximum log-likelihood for the model without covariates (null model) and the model with all covariates (Full model).