martinsewell.com

# Time Series

## Definitions

time series, n. (Statistics) a sequence of data indexed by time, often comprising uniformly spaced observations.’
Borowski and Borwein (Collins Dictionary of Mathematics, 1989)

time series n. Statistics. a series of values of a variable taken in successive periods of time.’
Collins English Dictionary (1991)

time-series analysis Techniques used in statistics to study data, often economic, collected over a period of time, e.g. quarterly sales figures for cars over a number of years. Time series are used mainly for prediction purposes. To make useful predictions, allowance must be made for seasonal variation; this gives seasonally adjusted data. Interest then centres on any trend in the series. Typically, for car sales there may be a steady, or accelerating, increase or decrease in demand. Properties such as a five-year periodicity in sales may also be of interest.’
Daintith and Nelson (The Penguin Dictionary of Mathematics, 1989)

‘A time series is a set of data collected at successive points in time or over successive periods of time. A sequence of monthly data on new housing starts and a sequence of weekly data on product sales are examples of time series. Usually the data in a time series are collected at equally spaced periods of time, such as hour, day, week, month, or year.’
Encyclopædia Britannica (2006)

Time series: Values of a variable recorded, usually at a regular interval, over a long period of time. The observed movement and fluctuations of many such series are composed of four different components, secular trend, seasonal variation, cyclical variation, and irregular variation. An example from medicine is the incidence of a disease recorded yearly over several decades (see Fig. 125). Such data usually require special methods for their analysis because of the presence of serial correlation between the separate observations.’
Everitt (The Cambridge Dictionary of Statistics, 1998)

‘Time series analysis deals with records that are collected over time.’
Fan and Yao (2003)

‘A time series is a set of ordered observations on a quantitative characteristic of an individual or collective phenomenon taken at different points of time. Although it is not essential, it is common for these points to be equidistant in time. The essential quality of the series is the order of the observations according to the time variable, as distinct from those which are not ordered at all, e.g. in a random sample chosen simultaneously or are ordered according to their internal properties e.g. a sat arranged in order of magnitude.’
Marriott (A Dictionary of Statistical Terms, 1990)

time series noun Statistics a series of values of a quantity obtained at successive times, often with equal intervals between them.’
The New Oxford Dictionary of English (1998)

‘A time series is a series of observations taken sequentially over time. In a standard regression model the order in which observations are included in the data set is irrelevant: any ordering is equally satisfactory as far as the analysis is concerned. It is the order property that is crucial to time series and that distinguishes time series from non-time-series data. Actions taken at some time have consequences and effects that are experienced at some later time. Time itself, through the mechanism of causality, imparts structure into a time series.’
Pole, West and Harrison (1994)

time series A series of measurements over time, usually at regular intervals, of a random variable. A prime concern is the forecasting of future values using methods such as exponential smoothing, Holt-Winters forecasting, or Box-Jenkins methods. Models fitted include autoregressive models and moving average models. It is often necessary to deseasonalize the data and to remove any underlying trend before undertaking the analysis.’
Upton and Cook (Oxford Dictionary of Statistics, 2002)

‘In statistics, signal processing, and many other fields, a time series is a sequence of data points, measured typically at successive times, spaced at (often uniform) time intervals.’
Wikipedia (2009)

## Introduction

A time series is a chronologically-indexed sequence of data points. The task is generally to describe the stochastic process generating the time series. One can never generalize beyond one’s data without making subjective assumptions (Hume 1739–40; Mitchell 1980; Schaffer 1994; Wolpert 1996), so let us assume that the time series is stationary, with zero mean, and that the next point in a time series is linearly dependent in some way on the previous n points. Assume also that the value of each data point is determined by an algorithm plus white noise.

autoregressive (AR) model
Each point depends on the previous n points, plus white noise.
moving average (MA) model
Each point depends on the previous n instances of white noise, plus its own white noise.

In the AR model, as each point ultimately depends on all of the previous points, it is essentially an infinite impulse response filter. In the MA model, each instance of white noise is independent, so it is essentially a finite impulse response filter. An ARMA process is an AR process plus an MA process.

## Books

The fourth edition of Time Series Analysis: Forecasting and Control (Box et al., 2008) is a revision of the classic 1970 book, Hamilton (1994)’s tome is the bible, whilst Weigend and Gershenfeld (1994) covers more recent innovations in time series prediction. To search Amazon.com for books on time series, click here.

## Glossary

AR (autoregressive) model
Models for a time series where the next point is dependent on the previous n points: AR(n). (AR(1) is a Markov chain.)
ARCH (autoregressive conditional heteroskedasticity)
In econometrics, ARCH (Engle 1982) is a model used for forecasting volatility which captures the conditional heteroskedasticity (serial correlation of volatility) of financial returns. Today’s conditional variance is a weighted average of past squared unexpected returns. ARCH is an AR process for the variance.
ARIMA (autoregressive integrated moving average) models
Models for time series which resemble ARMA models except in that it is presumed the time series has a steady underlying trend. The models therefore work with the differences between the successive observed values, instead of the values themselves. To retrieve the original data from the differences requires a form of integration and the models are therefore called autoregressive integrated moving average models.
ARMA (autoregressive moving average) models
Models for a time series with no trend (the constant mean is taken as 0). They incorporate the terms in both an autoregressive (AR) model and a moving average (MA) model.
autocorrelation
A measure of the linear relationship between two separate instances of the same random variable.
Box-Jenkins procedure
A general strategy for the analysis of time series based on the use of ARIMA models or, for seasonal data, SARIMA models. The procedure was set out by Box and Jenkins in their classic 1970 book (the current edition is Box, Jenkins and Reinsel 2008). The first stage consists of removing trends or cycles from the data. An appropriate type of model must then be identified and its parameters estimated. The estimated model is then compared with the original data and adjustments are made if necessary.
deseasonalize
To remove regular seasonal fluctuations from a time series for the purposes of analysis (for example, to estimate an underlying trend).
GARCH (generalized autoregressive conditional heteroskedasticity)
GARCH (Bollerslev 1986) generalizes the ARCH model. Today’s conditional variance is a function of past squared unexpected returns and its own past values. The model is an infinite weighted average of all past squared forecast errors, with weights that are constrained to be geometrically declining. GARCH is an ARMA(p,q) process in the variance.
Holt-Winters forecasting
An application of exponential smoothing to a time series that displays a trend and seasonality.
MA (moving average) models
Models for a time series with constant mean (taken as 0) where the next point is dependent on the previous n errors: MA(n).
serial correlation
See autocorrelation.
trend
If the mean of a time series changes steadily over time then it is said to exhibit a trend.
unit root
In autoregressive models in econometrics, a unit root is present if yt = yt-1 + c + εt-1.