A note on inﬂuence diagnostics in AR(1) time series models Mauricio Zevallos a,n , Bruno Santos b , Luiz K. Hotta a a Department of Statistics, University of Campinas, IMECC-UNICAMP, Rua Se ´rgio Buarque de Holanda 651, Cidade Universita ´ria, Bar ~ ao Geraldo, CEP 13083-859, Campinas, SP, Brazil b Equity Research Division, Votorantim Corretora, Brazil article info Article history: Received 20 July 2010 Received in revised form 16 May 2012 Accepted 17 May 2012 Available online 24 May 2012 Keywords: Slope Curvature Local inﬂuence Outliers abstract The purpose of this paper is to develop inﬂuence diagnostics for AR(1) models under the innovative and the data perturbation schemes. There are four main contributions. First, we derive analytical expressions for the slope and curvature statistics. Second, we establish a relationship between the slope and curvature showing that the standardised slope and standardised curvature are equal for the innovative perturbation scheme, and these vectors are nearly identical for several values of the autoregressive parameter, for the data perturbation scheme. Third, we present a connection between the inﬂuence statistics and the tests for outlier detection. Fourth, for the innovative perturbation scheme, we derive the asymptotic distribution of a new inﬂuence statistic, whereas for the data perturbation scheme, the distribution of the inﬂuence statistics is obtained via Monte Carlo simulation. We additionally discuss practical guidelines for the use of local inﬂuence statistics, which are illustrated on a chemical process data set. & 2012 Elsevier B.V. All rights reserved. 1. Introduction Since the seminal work of Cook (1986), local inﬂuence methodology has been a useful paradigm for detecting observations that have a strong effect on a model. (See Zhu et al., 2007 and references therein.) Cook (1986) introduced local inﬂuence methods to assess the effect of small perturbations based on the concepts of differential geometry. Later, Billor and Loynes (1993) suggested the use of the slope of a modiﬁed inﬂuence graph for the same task. There are some investigations of local inﬂuence diagnostics in time series. Contributions for regression models with AR(1) errors include those of Tsai and Wu (1992), Kim and Huggins (1998) and Paula et al. (2009). All of these researches are focused on the inﬂuence on the regression parameters. In addition, Schall and Dunne (1991) discussed inﬂuence diagnostics for ARMA models. Zhang (2004), Liu (2004), Zhang and King (2005), Dark et al. (2010) and Zevallos and Hotta (in press) investigated the inﬂuence diagnostics for GARCH models. Although the curvature inﬂuence diagnostics were studied on ARMA models by Schall and Dunne (1991), those authors do not provide analytic expressions for the curvature statistics. In addition, several issues about practical guidelines for the use of local inﬂuence statistics in these types of models have not been discussed. The aim of this paper is to ﬁll these gaps by studying one of the simplest models in the ARMA family, the AR(1) model. The purpose of the paper is to develop inﬂuence diagnostics for AR(1) models under the innovative and the data perturbation schemes. The main contributions are to present analytic expressions for both the slope and curvature statistics, explore the differences between slope and curvature diagnostics, study the relationships among the inﬂuence statistics and outlier detection tests, discuss the statistical aspects of inﬂuential diagnostics and give practical guidelines for data analysis. Contents lists available at SciVerse ScienceDirect journal homepage: www.elsevier.com/locate/jspi Journal of Statistical Planning and Inference 0378-3758/$ - see front matter & 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.jspi.2012.05.004 n Corresponding author. E-mail addresses: amadeus@ime.unicamp.br (M. Zevallos), brunoreiss@gmail.com (B. Santos), hotta@ime.unicamp.br (L.K. Hotta). Journal of Statistical Planning and Inference 142 (2012) 2999–3007