Goodness‐of‐fit tests for βARMA hydrological time series modeling

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.

[1]  G. Pumi,et al.  Beta autoregressive fractionally integrated moving average models , 2018, Journal of Statistical Planning and Inference.

[2]  Francisco Cribari‐Neto,et al.  Beta seasonal autoregressive moving average models , 2018, Journal of Statistical Computation and Simulation.

[3]  Guilherme Pumi,et al.  Kumaraswamy autoregressive moving average models for double bounded environmental data , 2017, 1710.05069.

[4]  Rong Chen,et al.  Dirichlet ARMA models for compositional time series , 2017, J. Multivar. Anal..

[5]  Francisco Cribari-Neto,et al.  Erratum to: Beta autoregressive moving average models , 2017 .

[6]  Fábio M. Bayer,et al.  Bootstrap-based inferential improvements in beta autoregressive moving average model , 2017, Commun. Stat. Simul. Comput..

[7]  R. Casarin,et al.  Bayesian Model Selection for Beta Autoregressive Processes , 2010, 1008.0121.

[8]  A. Zeileis,et al.  Beta Regression in R , 2010 .

[9]  Andréa V. Rocha,et al.  Beta autoregressive moving average models , 2009 .

[10]  Rob J Hyndman,et al.  Automatic Time Series Forecasting: The forecast Package for R , 2008 .

[11]  J. Randall Brown,et al.  Rational Arithmetic Mathematica Functions to Evaluate the One-sided One-sample K-S Cumulative Sample Distribution , 2007 .

[12]  A. Ian McLeod,et al.  Improved Pena-Rodriguez portmanteau test , 2006, Comput. Stat. Data Anal..

[13]  M. Chiogna,et al.  Mining epidemiological time series: an approach based on dynamic regression , 2005 .

[14]  Andy C. C. Kwan,et al.  A comparative study of the finite-sample performance of some portmanteau tests for randomness of a time series , 2005, Comput. Stat. Data Anal..

[15]  S. Ferrari,et al.  Beta Regression for Modelling Rates and Proportions , 2004 .

[16]  Benjamin Kedem,et al.  Partial Likelihood Inference For Time Series Following Generalized Linear Models , 2004 .

[17]  R. Rigby,et al.  Generalized Autoregressive Moving Average Models , 2003 .

[18]  Julio Rodríguez,et al.  A Powerful Portmanteau Test of Lack of Fit for Time Series , 2002 .

[19]  Andy C. C. Kwan,et al.  On the finite-sample distribution of modified portmanteau tests for randomness of a Gaussian time series , 1996 .

[20]  Anna Clara Monti.  A proposal for a residual autocorrelation test in linear models , 1994 .

[21]  W. Li,et al.  Time series models based on generalized linear models: some further results. , 1994, Biometrics.

[22]  J. Shao.  Asymptotic theory in generalized ilinear models with nuisance scale parameters , 1992 .

[23]  G. Ljung,et al.  Diagnostic testing of univariate time series models , 1986 .

[24]  L. Fahrmeir,et al.  Correction: Consistency and Asymptotic Normality of the Maximum Likelihood Estimator in Generalized Linear Models , 1985 .

[25]  C. Jiménez,et al.  Nonnegative Definiteness of the Sample Autocovariance Function , 1984 .

[26]  G. Box,et al.  On a measure of lack of fit in time series models , 1978 .

[27]  A. I. McLeod,et al.  On the Distribution of Residual Autocorrelations in Box–Jenkins Models , 1978 .

[28]  N. Davies,et al.  Significance levels of the Box-Pierce portmanteau statistic in finite samples , 1977 .

[29]  H. Akaike.  A new look at the statistical model identification , 1974 .

[30]  David A. Pierce,et al.  Residual Correlations and Diagnostic Checking in Dynamic-Disturbance Time Series Models , 1972 .

[31]  G. Box,et al.  Distribution of Residual Autocorrelations in Autoregressive-Integrated Moving Average Time Series Models , 1970 .

[32]  E. B. Andersen,et al.  Asymptotic Properties of Conditional Maximum‐Likelihood Estimators , 1970 .

[33]  G. M. Jenkins,et al.  An angular transformation for the serial correlation coefficient , 1954 .

[34]  H. Hotelling.  New Light on the Correlation Coefficient and its Transforms , 1953 .

[35]  J. Wishart.  Probable Error , 1932, The Mathematical Gazette.

[36]  R Core Team,et al.  R: A language and environment for statistical computing. , 2014 .

[37]  Mikis D. Stasinopoulos,et al.  Fitting Non-Gaussian Time Series Models , 1998, COMPSTAT.

[38]  Takuji Nishimura,et al.  Mersenne twister: a 623-dimensionally equidistributed uniform pseudo-random number generator , 1998, TOMC.

[39]  Andy C. C. Kwan,et al.  Portmanteau tests of randomness and Jenkins' variance-stabilizing transformation , 1996 .

[40]  Ales Linka,et al.  On transformations of multivariate ARMA processes , 1988, Kybernetika.

[41]  S. Zeger,et al.  Markov regression models for time series: a quasi-likelihood approach. , 1988, Biometrics.

[42]  Jean-Marie Dufour,et al.  Generalized Portmanteau Statistics and Tests of Randomness , 1985 .

[43]  R. Fisher.  014: On the "Probable Error" of a Coefficient of Correlation Deduced from a Small Sample. , 1921 .