A Distance Measure for Classifying Arima Models

Abstract

In a number of practical problems where clustering or choosing from a set of dynamic structures is needed, the introduction of a distance between the data is an early step in the application of multivariate statistical methods. In this paper a parametric approach is proposed in order to introduce a well-defined metric on the class of autoregressive integrated moving-average (ARIMA) invertible models as the Euclidean distance between their autoregressive expansions. Two case studies for clustering economic time series and for assessing the consistency of seasonal adjustment procedures are discussed. Finally, some related proposals are surveyed and some suggestions for further research are made.

Clustering heteroskedastic time series by model-based procedures

“…Piccolo (1990) proposed a metrics measuring the distance between two ARMA models, based on the comparison of the coefficients of their AR(∞) representation. This tool has had a large success in several fields; a review of its properties and applications with several references can be found in Piccolo (2007) and Corduas and Piccolo (2008).…”

“…In particular, he distinguishes three major categories of approaches to time series clustering: 1) raw-data-based approaches, in which the series compared are considered as normally sampled at the same interval; 2) features-based approaches, in which the series are compared using some selected features; 3) model-based methods, where the time series are considered similar when the models characterizing them are similar. The approach proposed in this work belongs to the third category; in particular it follows the tradition of AR processes to capture the similarity among time series, as in Piccolo (1990), Maharaj (1996Maharaj ( , 1999Maharaj ( , 2000, Xiong and Yeung (2002) (see Piccolo, 2007, and Corduas and Piccolo, 2008, for a review). Most of these studies are devoted to capturing the structure of the mean of the process hypothesized as generator of the data, whereas little attention was put on the variance.…”

“…Most of these studies are devoted to capturing the structure of the mean of the process hypothesized as generator of the data, whereas little attention was put on the variance. This is a correct approach when dealing with classifications based on ARMA models and in presence of homoskedastic variance (for example, the clustering methods based on the AR metrics proposed by Piccolo, 1990); in fact, in this case, the variance is a function of the process parameters, so that it is implicitly considered in the classification. Dealing with heteroskedastic time series, in which the (conditional) variance follows a stochastic process (typically a GARCH process; Engle, 1982, Bollerslev, 1986, the comparison of the dynamics of the variances is fundamental.…”

“…From this model we can separate the volatility in a constant part and a time-varying part; this subdivision can have an appealing interpretation, in particular when we use the volatility to represent the risk of the asset. The constant part of the volatility is measured in a natural way, whereas we measure the time-varying part extending the idea of distance between AR models (Piccolo, 1990) to the GARCH family. The constant part could be interpreted as the lower bound of volatility (minimum expected risk) of a certain series, whereas the unconditional volatility represents the expected risk and seems more interesting for the investor.…”

Clustering data with measurement errors

“…In the above example, when online users are modelled as mixture of Markov chains, each user is represented by a vector consisting of the transition probabilities of the underlying Markov chain, and then the clusters can be obtained based on similarity between the estimates of transition probabilities for different users [2,7]. Similarly, when the sales pattern of an item is modelled as a mixture of time series, the clusters of items can be obtained based on similarity between the estimates of the time series parameters for each item [22,24]. Another example is clustering of similar stocks based on their β values that can be obtained by using ordinary linear regression (the Capital Asset Pricing Model) on the observed stock prices [16].…”