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A GARMA Framework for Unit-Bounded Time Series Based on the Unit-Lindley Distribution with Application to Renewable Energy Data

Guilherme Pumi, Danilo Hiroshi Matsuoka, Taiane Schaedler Prass

TL;DR

This paper develops ULARMA, a GARMA-like framework for unit-bounded time series based on the Unit-Lindley distribution, with Y_t modeled conditionally as UL(mu_t) and a regression-type ARMA-like update for the mean through eta_t = g(mu_t). It provides a complete PMLE-based inference pipeline, including closed-form partial scores and a conditional information matrix, along with large-sample results, hypothesis tests, forecasting, residual diagnostics, and bootstrap-based prediction intervals. Through a Monte Carlo study and a real-data application to the monthly share of hydroelectric power in the US, ULARMA demonstrates competitive out-of-sample forecasting performance and strong parsimony relative to KARMA and beta-ARMA benchmarks. The approach offers a practical, theoretically grounded tool for environmental and energy time series with bounded support, enabling accurate, interpretable forecasts and model selection in applied settings.

Abstract

The Unit-Lindley is a one-parameter family of distributions in $(0,1)$ obtained from an appropriate transformation of the Lindley distribution. In this work, we introduce a class of dynamical time series models for continuous random variables taking values in $(0,1)$ based on the Unit-Lindley distribution. The models pertaining to the proposed class are observation-driven ones for which, conditionally on a set of covariates, the random component is modeled by a Unit-Lindley distribution. The systematic component aims at modeling the conditional mean through a dynamical structure resembling the classical ARMA models. Parameter estimation in conducted using partial maximum likelihood, for which an asymptotic theory is available. Based on asymptotic results, the construction of confidence intervals, hypotheses testing, model selection, and forecasting can be carried on. A Monte Carlo simulation study is conducted to assess the finite sample performance of the proposed partial maximum likelihood approach. Finally, an application considering forecasting of the proportion of net electricity generated by conventional hydroelectric power in the United States is presented. The application show the versatility of the proposed method compared to other benchmarks models in the literature.

A GARMA Framework for Unit-Bounded Time Series Based on the Unit-Lindley Distribution with Application to Renewable Energy Data

TL;DR

This paper develops ULARMA, a GARMA-like framework for unit-bounded time series based on the Unit-Lindley distribution, with Y_t modeled conditionally as UL(mu_t) and a regression-type ARMA-like update for the mean through eta_t = g(mu_t). It provides a complete PMLE-based inference pipeline, including closed-form partial scores and a conditional information matrix, along with large-sample results, hypothesis tests, forecasting, residual diagnostics, and bootstrap-based prediction intervals. Through a Monte Carlo study and a real-data application to the monthly share of hydroelectric power in the US, ULARMA demonstrates competitive out-of-sample forecasting performance and strong parsimony relative to KARMA and beta-ARMA benchmarks. The approach offers a practical, theoretically grounded tool for environmental and energy time series with bounded support, enabling accurate, interpretable forecasts and model selection in applied settings.

Abstract

The Unit-Lindley is a one-parameter family of distributions in obtained from an appropriate transformation of the Lindley distribution. In this work, we introduce a class of dynamical time series models for continuous random variables taking values in based on the Unit-Lindley distribution. The models pertaining to the proposed class are observation-driven ones for which, conditionally on a set of covariates, the random component is modeled by a Unit-Lindley distribution. The systematic component aims at modeling the conditional mean through a dynamical structure resembling the classical ARMA models. Parameter estimation in conducted using partial maximum likelihood, for which an asymptotic theory is available. Based on asymptotic results, the construction of confidence intervals, hypotheses testing, model selection, and forecasting can be carried on. A Monte Carlo simulation study is conducted to assess the finite sample performance of the proposed partial maximum likelihood approach. Finally, an application considering forecasting of the proportion of net electricity generated by conventional hydroelectric power in the United States is presented. The application show the versatility of the proposed method compared to other benchmarks models in the literature.

Paper Structure

This paper contains 18 sections, 1 theorem, 34 equations, 5 figures, 5 tables.

Table of Contents

  1. Introduction
  2. The ULARMA model
  3. Parameter estimation
  4. The partial score vector
  5. Conditional information matrix
  6. Large sample inference
  7. Hypothesis Testing
  8. Forecasting
  9. Residuals and goodness-of-fit
  10. Confidence Intervals and Model Selection
  11. Monte Carlo Simulation
  12. Point estimation exercise
  13. Goodness-of-fit exercise
  14. Joint behavior
  15. Application to Real Data
  16. ...and 3 more sections

Key Result

Lemma 1

Let $Y\sim \mathrm{UL}(\mu)$, $\mu>0$. Then

Figures (5)

  • Figure 1: A typical example of a time series considered in the simulation study. The plot was generated considering $n=500$, $\alpha = 0.5$, $\beta=0.5$, $\phi=-0.4$, $\theta=-0.2$.
  • Figure 2: Boxplots of the simulation results for all parameter for $\alpha=0.5$ (top) and $\alpha=1$ (bottom), with $\beta=0.5$ fixed. Parameter $(\phi,\theta)$ are defined by the scenarios, as follows: scenarios 1 and 5: $(0.2,-0.4)$, scenarios 2 and 6: $(-0.8,0.2)$, scenarios 3 and 7: $(-0.4,-0.2)$, and scenarios 4 and 8: $(0.4,0.2)$.
  • Figure 3: Pairwise joint and marginal behavior of the estimated values for $\alpha=0.5$, $\beta=0.5$, $\phi=0.2$, $\theta=-0.4$. Solid lines in the scatter plot represent the true values.
  • Figure 4: Proportion of net energy generated by hydroelectric data: (a) the observed time series and the corresponding (b) autocorrelation function (ACF), (c) partial autocorrelation (PACF) function and (d) seasonal plot.
  • Figure 5: Observed time series, (a) fitted values (in-sample-forecast) and (b) 12-step ahead (out-of-sample) forecasts obtained with the fitted ULARMA, KARMA and $\beta$ARMA models. (c) Level 10% Bootstrap prediction intervals.

Theorems & Definitions (1)

  • Lemma 1