Likelihood-Based Ergodicity Transformations in Time Series Analysis

TL;DR

The paper tackles non-ergodicity in empirical time series and proposes a likelihood-based method to estimate ergodicity transformations that render increments ergodic. It builds a general framework based on a transformation $F(x_t;\lambda)$ and a profile likelihood $L_{max}(\lambda)$ to estimate $\widehat{\lambda}=\arg\max_\lambda L_{max}(\lambda)$, applicable with Gaussian processes, ARMA, and GARCH models. Simulation studies with geometric Brownian motion (GBM) and arithmetic Brownian motion (ABM) show the method recovers the canonical transformations (e.g., $\widehat{\lambda}=0$ for GBM and $\widehat{\lambda}=1$ for ABM), outperforming Box-Cox. An application to the FRED‑QD macroeconomic panel demonstrates that ergodicity-based transformations produce a more coherent factor structure and can improve forecast accuracy relative to Box-Cox, highlighting practical benefits in economics and finance.

Abstract

Time series often exhibit non-ergodic behaviour that complicates forecasting and inference. This article proposes a likelihood-based approach for estimating ergodicity transformations that addresses such challenges. The method is broadly compatible with standard models, including Gaussian processes, ARMA, and GARCH. A detailed simulation study using geometric and arithmetic Brownian motion demonstrates the ability of the approach to recover known ergodicity transformations. A further case study on the large macroeconomic database FRED-QD shows that incorporating ergodicity transformations can provide meaningful improvements over conventional transformations or naive specifications in applied work.

Figures (7)

• Figure 1: Ten sample trajectories each of geometric and arithmetic Brownian motion with drift $\mu =\qty{0.05}{\per}$, volatility $\sigma = \qty{0.2}{\per}$, and initial condition $X_0 = 1$, simulated over a period of 3 trading years (252 days per trading year).
• Figure 2: Left panel: the Box-Cox procedure applied to each GBM path from Figure \ref{['fig:gbm_paths']}. For each path, the estimated transformation parameter $\widehat{\lambda}$ is indicated by a dashed vertical line. Right panel: the profile log-likelihood $L_{\text{max}}(\lambda)$ defined in \ref{['eq:llf_max']}, together with the corresponding estimate $\widehat{\lambda}$ from \ref{['eq:hat_lambda']}. The true transformation parameter is $\lambda = 0$.
• Figure 3: Same as Figure \ref{['fig:gbm_llfs']}, but for the ABM paths from Figure \ref{['fig:gbm_paths']}. The true transformation parameter is $\lambda = 1$.
• Figure 4: Effects of sample size on the accuracy of the proposed procedure. The bias and standard deviation of $\widehat{\lambda}$ relative to the true values ($\lambda = 0$ for GBM; $\lambda = 1$ for ABM) are reported. Left panel: initial condition, $X_0 = 1$. Right panel: $X_0 = 100$.
• Figure 5: Distribution of estimated power transformation parameters $\widehat{\lambda}$ across FRED-QD series, by order of integration, under the Box-Cox and ergodicity procedures. Benchmark FRED-QD transformations ($\lambda=1$ for levels, $\lambda=0$ for logs) are shown for comparison.