Comparison of Least Square Spline and Penalized Spline for Modeling Human Development Index Determinants

Abstract

The Human Development Index (HDI) is a key indicator of national and regional development and supports the achievement of the Sustainable Development Goals (SDGs), particularly Goal 1 (No Poverty) and Goal 4 (Quality Education). However, substantial disparities in HDI remain across Indonesia’s regencies and cities due to complex and nonlinear socioeconomic relationships that cannot always be captured by conventional parametric methods. This study analyzes the determinants of HDI in 514 regencies and cities in Indonesia in 2024 using nonparametric spline regression by comparing Penalized Spline and Least Square Spline estimators. The explanatory variables include mean years of schooling, labor force participation rate, percentage of senior-high-school graduates, and poverty rate. Secondary data from BPS were analyzed through spline basis construction, smoothing parameter selection using Generalized Cross Validation (GCV), parameter estimation, significance testing, and residual diagnostics. The results show that all predictors have nonlinear relationships with HDI. Mean years of schooling and the percentage of senior-high-school graduates positively affect HDI, whereas labor force participation and poverty rate have negative effects. The second-order Penalized Spline model with four knot points achieved the best performance, yielding the lowest GCV (5.388838), the lowest MSE (4.948574), and the highest adjusted R² (86.94%). Residual diagnostics confirmed normality and zero mean but indicated autocorrelation, suggesting spatial dependence. Overall, Penalized Spline regression provides a flexible and accurate approach for modeling HDI determinants and informing evidence-based regional development policy.