Definition Of A Time Series

1. Core Concept Overview & Intuition

A time series is the natural statistical object for data whose defining feature is order in time. Unlike a cross-section, where the index of an observation is arbitrary, a time series carries information in the sequence itself: the value recorded at one instant is related to values recorded before and after it. The central intuition is that time is one of the two variables involved. One variable is time, denoted $t$, and the other is the quantity observed at that time, denoted $X_t$. For instance, the population $X_t$ observed in different years $t$ forms a time series.

The purpose of studying such data is to move from a raw sequence of numbers to an understanding of the mechanism that generated it. A plot of the observations against time is the first descriptive step, because it exposes the broad tendency of the data, recurring movements, and unusual observations that do not fit the rest of the record. From this descriptive base, analysis proceeds to explanation, prediction, and control.

Two distinctions organise the whole subject. The first concerns how time is recorded: observations may be taken continuously with time, or only at specific times, usually equally spaced. The second concerns predictability: a series may be fully predictable from its past, or only partly so. The course restricts attention to the discrete and stochastic case, which is the realistic setting for economic, physical, marketing, and demographic data.

2. Mathematical Formalization & Core Definitions

> Definition (Time Series). A time series is a collection of observations made sequentially with time. Equivalently, it is an arrangement of statistical data in accordance with the occurrence of time.

A time series therefore involves two variables, one of which is time. Writing $X_t$ for the observation at time $t$, the series is the ordered collection of pairs $(t, X_t)$.

> Definition (Continuous Time Series). A time series is continuous when the observations are made continuously with time.

> Definition (Discrete Time Series). A time series is discrete when observations are taken only at specific times, usually equally spaced.

The discrete case is the one treated in this course.

> Definition (Deterministic Time Series). A deterministic time series is a series that can be predicted exactly; future values may be predicted fully from past observations.

> Definition (Stochastic Time Series). A stochastic time series is one in which the future is partly determined by past values, so that exact predictions are impossible.

Most time series are stochastic. The course considers discrete and stochastic time series.

2.1 The Four Types of Time Series

The classification below is by field of application.

2.2 Objectives of Time Series Analysis

The objectives of analysing a time series are Description, Explanation, Prediction, and Control.

> Description. The first step is usually to plot the data and obtain simple descriptive measures of the main properties of the series. Features such as trend and seasonal effects can sometimes be seen directly from such a plot, and unusual observations or outliers inconsistent with the rest of the data can be identified.

> Explanation. When observations are taken on two or more variables, the variation in one time series may be used to explain the variation in another, leading to a deeper understanding of the process that generates the series.

> Prediction. Given an observed series, one may forecast its future values. This is important in sales forecasting and in the analysis of economic and industrial time series.

> Control. When a series measures the quality of a manufacturing process, the aim may be to control the process: a stochastic model is fitted, future values are predicted, and the input process variables are adjusted to keep the process on target.

2.3 Goals of the Analysis

The main problem in time series analysis is twofold:

1. to identify the component at work, and
2. to isolate, study, analyse, and measure those components independently.

2.4 Models for a Time Series

Two model forms are used to represent how the components combine.

> Additive Model. A time series can be expressed as

$$
X_t = T_t + S_t + C_t + e_t
$$

> where $X_t$ is the time series value at time $t$, $T_t$ the trend value at time $t$, $S_t$ the seasonal value at time $t$, $C_t$ the cyclic value at time $t$, and $e_t$ the random value at time $t$.

The additive model implies that seasonal forces in different years, cyclic forces in different cycles, and random forces over long periods operate with equal absolute effects, irrespective of the level of the trend value.

> Multiplicative Model. A time series can be expressed as

$$
X_t = T_t \cdot S_t \cdot C_t \cdot e_t
$$

> This holds when the various components operate proportionally to the general level of the series.

Two remarks accompany the multiplicative model. First, $S_t$, $C_t$, and $e_t$ do not assume positive and negative values; instead they are indices fluctuating above or below unity, and the geometric means of $S_t$ within a year, of $C_t$ within a cycle, and of $e_t$ over a long period are each unity. Second, the multiplicative model converts to an additive model in the logarithms of the original series:

$$
\ln X_t = \ln T_t + \ln S_t + \ln C_t + \ln e_t
$$

The time plot should be examined to judge which model is likely to give the better description.