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Unit-Weibull Autoregressive Moving Average Models

Guilherme Pumi, Taiane Schaedler Prass, Cleiton Guollo Taufemback

Abstract

In this work we introduce the class of unit-Weibull Autoregressive Moving Average models for continuous random variables taking values in $(0,1)$. The proposed model is an observation driven one, for which, conditionally on a set of covariates and the process' history, the random component is assumed to follow a unit-Weibull distribution parameterized through its $ρ$th quantile. The systematic component prescribes an ARMA-like structure to model the conditional $ρ$th quantile by means of a link. Parameter estimation in the proposed model is performed using partial maximum likelihood, for which we provide closed formulas for the score vector and partial information matrix. We also discuss some inferential tools, such as the construction of confidence intervals, hypotheses testing, model selection, and forecasting. A Monte Carlo simulation study is conducted to assess the finite sample performance of the proposed partial maximum likelihood approach. Finally, we examine the prediction power by contrasting our method with others in the literature using the Manufacturing Capacity Utilization from the US.

Unit-Weibull Autoregressive Moving Average Models

Abstract

In this work we introduce the class of unit-Weibull Autoregressive Moving Average models for continuous random variables taking values in . The proposed model is an observation driven one, for which, conditionally on a set of covariates and the process' history, the random component is assumed to follow a unit-Weibull distribution parameterized through its th quantile. The systematic component prescribes an ARMA-like structure to model the conditional th quantile by means of a link. Parameter estimation in the proposed model is performed using partial maximum likelihood, for which we provide closed formulas for the score vector and partial information matrix. We also discuss some inferential tools, such as the construction of confidence intervals, hypotheses testing, model selection, and forecasting. A Monte Carlo simulation study is conducted to assess the finite sample performance of the proposed partial maximum likelihood approach. Finally, we examine the prediction power by contrasting our method with others in the literature using the Manufacturing Capacity Utilization from the US.
Paper Structure (10 sections, 1 theorem, 48 equations, 4 figures, 5 tables)

This paper contains 10 sections, 1 theorem, 48 equations, 4 figures, 5 tables.

Table of Contents

  1. Introduction
  2. The Unit-Weibull distribution and the UWARMA model
  3. Parameter estimation
  4. Partial score vector
  5. Partial information matrix
  6. Inferential tools
  7. Hypothesis tests, confidence intervals and diagnostics
  8. Forecast
  9. Monte Carlo Simulation
  10. Empirical Study

Key Result

Lemma 1

Let $Y\sim \mathrm{UW}(\mu,\lambda;\rho)$ for $\rho,\mu\in(0,1)$ and $\lambda>0$. Then where $\kappa=0.5772156649\dots$ is the Euler-Mascheroni constant GR2007.

Figures (4)

  • Figure 1: Simulated UWARMA$(1,1)$ with $\lambda = 6$, $\phi=0.4$, $\theta=0.6$ and $\rho\in\{0.1,0.5,0.9\}$, produced using the same random seed.
  • Figure 2: Simulation results for $\rho=0.5$, $\lambda=5$, $\phi=0.6$ and $\theta=0.4$. Presented are the scatter plot, and marginal densities and box plot for $n\in\{250,500,1000\}$ considering pairs $(\hat{\phi},\hat{\theta})$ (top-left), $(\hat{\theta},\hat{\lambda})$ (top-right) and $(\hat{\phi},\hat{\lambda})$ (bottom).
  • Figure 3: Example of UWARMA$(1,1)$ with $\phi=0.6$, $\theta=0.4$, $\lambda = 5$ with covariates for different values of $\rho$ produced using the same random seed.
  • Figure 4: Time series plot of Manufacturing Capacity Utilization (percent), Civilian Unemployment Rate (percent), All Employees: Manufacturing (Thousands of Persons), and Total Business: Inventories to Sales Ratio (ratio), Effective Federal Funds Rate (percent), Crude Oil, spliced WTI and Cushing (dollars per barrel), CPI : All Items (index 1982-1984=100), and S&P's Common Stock Price Index: Industrials (index).

Theorems & Definitions (1)

  • Lemma 1