Empirical Application Data for "Distinguishing Functional Coefficient Models from Time-Varying Coefficient Models"

1. Research Hypothesis and Data Scope
The study investigates the dynamic pricing mechanism of the European Union Allowance (EUA) market using a monthly time-series dataset spanning December 2008 to February 2025 (N=195), sourced from the Wind database and Climate Data Store. The primary research hypothesis examines the nonlinear relationship between carbon prices and their fundamental drivers. Specifically, it tests whether this relationship is state-dependent (governed by specific economic conditions, known as functional-coefficient models) or time-dependent (evolving structurally over time, known as time-varying coefficient models). The data encompasses EUA futures prices, energy markets (gas, coal), macroeconomic indicators, policy uncertainty, and weather conditions.

2. Data Processing Methodology
To ensure stationarity for rigorous regression analysis, the raw time-series data were transformed using two primary statistical methods. First, Logarithmic Differentiation (Returns) was applied to price level variables to calculate growth rates using the formula R(t) = ln[P(t)] - ln[P(t-1)]. Second, First Differencing was applied to percentage indices or rates using the formula D(t) = V(t) - V(t-1). Additionally, weather data underwent a specific pre-processing step using Fourier series regression to remove seasonal components before differencing, ensuring only non-seasonal temperature anomalies were captured.

3. Detailed Variable Descriptions
The specific variables used in the model are defined as follows. The Dependent Variable is REUA (EUA Returns), representing the monthly logarithmic return of the ICE-traded December EUA futures settlement price (continuous contract). The Explanatory Variables include RGS (Gas Returns), the monthly log-return of NYMEX natural gas futures prices; RCL (Coal Returns), the monthly log-return of ARA (Amsterdam-Rotterdam-Antwerp) coal futures settlement prices; RS (Stock Market Returns), the monthly log-return of the STOXX EUROPE 600 index; DIC (Investor Confidence), the first difference of the Sentix Investor Confidence Index for the Eurozone; DEPU (Policy Uncertainty), the first difference of the European Economic Policy Uncertainty index; and TEMP (Temperature Anomalies), the average temperature across 10 major European cities adjusted for seasonality. Furthermore, six State Variables were compiled to test the state-dependent hypothesis, including DS (Default Spread, the yield difference between Moody’s Baa and Aaa corporate bonds), DGBY (the first difference of EU long-term government bond yields), DCPI (the first difference of EU Harmonized Index of Consumer Prices), DESI (the first difference of Eurostat economic sentiment indicator), DIPI (the first difference of Euro area industrial production index), and GPRI (Global geopolitical risk index).