As shown by Fu and Wu in their presentation (https://www.casact.org/education/rpm/2010/handouts/CL1-Fu.pdf), the quantile regression is an appealing approach to model severity measures with high volatilities due to its statistical characteristics, including the robustness to extreme values and no distributional assumptions. Curti and Migueis also pointed out in a research paper (https://www.federalreserve.gov/econresdata/feds/2016/files/2016002r1pap.pdf) that the operational loss is more sensitive to macro-economic drivers at the tail, making the quantile regression an ideal model to capture such relationships.

While the quantile regression can be conveniently estimated in SAS with the QUANTREG procedure, the standard SAS output doesn’t provide goodness-of-fit (GoF) statistics. More importantly, it is noted that the underlying rationale of calculating GoF in a quantile regression is very different from the ones employed in OLS or GLM regressions. For instance, the most popular R-square is not applicable in the quantile regression anymore. Instead, a statistic called “R1” should be used. In addition, AIC and BIC are also defined differently in the quantile regression.

Below is a SAS macro showing how to calculate GoF statistics, including R1 and various information criterion, for a quantile regression.

%macro quant_gof(data = , y = , x = , tau = 0.5); ***********************************************************; * THE MACRO CALCULATES GOODNESS-OF-FIT STATISTICS FOR *; * QUANTILE REGRESSION *; * ------------------------------------------------------- *; * REFERENCE: *; * GOODNESS OF FIT AND RELATED INFERENCE PROCESSES FOR *; * QUANTILE REGRESSION, KOENKER AND MACHADO, 1999 *; ***********************************************************; options nodate nocenter; title; * UNRESTRICTED QUANTILE REGRESSION *; ods select ParameterEstimates ObjFunction; ods output ParameterEstimates = _est; proc quantreg data = &data ci = resampling(nrep = 500); model &y = &x / quantile = &tau nosummary nodiag seed = 1; output out = _full p = _p; run; * RESTRICTED QUANTILE REGRESSION *; ods select none; proc quantreg data = &data ci = none; model &y = / quantile = &tau nosummary nodiag; output out = _null p = _p; run; ods select all; proc sql noprint; select sum(df) into :p from _est; quit; proc iml; use _full; read all var {&y _p} into A; close _full; use _null; read all var {&y _p} into B; close _null; * DEFINE A FUNCTION CALCULATING THE SUM OF ABSOLUTE DEVIATIONS *; start loss(x); r = x[, 1] - x[, 2]; z = j(nrow(r), 1, 0); l = sum(&tau * (r <> z) + (1 - &tau) * (-r <> z)); return(l); finish; r1 = 1 - loss(A) / loss(B); adj_r1 = 1 - ((nrow(A) - 1) * loss(A)) / ((nrow(A) - &p) * loss(B)); aic = 2 * nrow(A) * log(loss(A) / nrow(A)) + 2 * &p; aicc = 2 * nrow(A) * log(loss(A) / nrow(A)) + 2 * &p * nrow(A) / (nrow(A) - &p - 1); bic = 2 * nrow(A) * log(loss(A) / nrow(A)) + &p * log(nrow(A)); l = {"R1" "ADJUSTED R1" "AIC" "AICC" "BIC"}; v = r1 // adj_r1 // aic // aicc // bic; print v[rowname = l format = 20.8 label = "Fit Statistics"]; quit; %mend quant_gof;