Goodness‐of‐fit tests forβARMA hydrological time series modeling

Abstract

We address the issue of performing portmanteau testing inference using time series data that assume values in the standard unit interval. The motivation involves modeling the time series dynamics of the proportion of stocked hydroelectric energy in the South of Brazil. Our focus lies in the class of beta autoregressive moving average (βARMA) models. In particular, we wish to test the goodness‐of‐fit of such models. We consider several testing criteria that have been proposed for Gaussian time series models and introduce two new tests. We derive the asymptotic null distribution of the two proposed test statistics in two different scenarios, namely, when the tests are applied to an observed time series and when they are applied to the residuals from a fitted βARMA model. It is worth noticing that our results imply the asymptotic validity of standard portmanteau tests in the class of βARMA models that are, under the null hypothesis, asymptotically equivalent to our test statistics. We use Monte Carlo simulation to assess the relative merits of the different portmanteau tests when used with fitted βARMA models. The simulation results we present show that the new tests are typically more powerful than a well‐known test whose test statistic is also based on residual partial autocorrelations. Overall, the tests we propose perform quite well. Finally, we model the dynamics of the proportion of stocked hydroelectric energy in Brazil. The results show that the βARMA model outperforms three alternative models and an exponential smoothing algorithm.

Beta distribution misspecification tests with application to Covid-19 mortality rates in the United States

“…In future research, we will extend the results presented in this paper to cover other univariate laws that are used to model fractional data (e.g., Kumaraswamy and simplex). We will also seek to extend our results to regression settings, in particular to the beta regression model introduced by [ 18 ], and to dynamic beta models, such as the β ARMA model introduced by [ 29 , 30 ]; see also [ 28 , 31 ]. The beta parameterization used in this paper, which is indexed by mean and precision parameters, will be helpful for the aforementioned extensions of our results.…”

Inflated beta autoregressive moving average models

“…Plots of the residuals autocorrelation function (ACF) with horizontal lines at ±1.96/ √ n − m can be used for assessing whether the residuals display white noise behavior (Kedem and Fokianos 2002). It is also possible to test if the residual autocorrelations are equal to zero using, for example, adapted versions of tests given by Ljung and Box (1978) and Monti (1994) (see also Scher et al 2020).…”

“…An IβARMA(2,2) was selected, but the parameters ϕ 1 and θ 1 were not statistically significant. The MAIC and MSIC criteria and the Ljung-Box test (using 20 lags, with corrected degrees of freedom in view of Scher et al (2020)) considering the randomized quantile residuals are also presented. Anyone familiar with ARMA modeling might be surprised with the magnitude of ϕ 2 , but, as we explained before, this is so because the particular form (4) assumes given that the error and AR terms are in the response's level.…”

A dynamic model for double‐bounded time series with chaotic‐driven conditional averages

“…For comparison with other models, 6 months of data, from November 2016 to April 2017, were reserved for out‐of‐sample forecasting, yielding a sample of size n =190 for fitting purposes. This data was first analyzed in Scher et al (2019) where the authors fit a βARMA(1,1) model to the data and compare its forecasting capabilities with four other models: The KARMA(1,1) of Bayer et al (2017), Gaussian ARMA(1,1), and AR(2) models and also with the Holt exponential smoothing algorithm. Our goal is to fit the proposed βARC model models and compare to the results reported in Scher et al (2019).…”

“…As mentioned before, Scher et al (2019) also considered the same data and modeled it using five different models (βARMA(1,1), KARMA(1,1), Gaussian ARMA(1,1), Gaussian AR(2) models, and the Holt exponential smoothing algorithm). Among these models, the authors report that the βARMA(1,1) presented the smallest AIC (‐307.9635) and also the best out‐of‐sample forecasting performance with an MAE of 0.1839 for the same data considered here (other forecasting accuracy measures were not reported).…”