Time Series Econometrics

A Concise Introduction

Terence C. Mills

• Verlag: Palgrave Macmillan
• Jahr: 2015
• ISBN:
  9781137525338
• Thema: Économie
• Thema: Économétrie

This book provides an introductory treatment of time series econometrics, a subject that is of key importance to both students and practitioners of economics. It contains material that any serious student of economics and finance should be acquainted with if they are seeking to gain an understanding of a real functioning economy.

1

Introduction

About the book

1.1   The aim of this book is to provide an introductory treatment of time series econometrics that builds upon the basic statistical and regression techniques contained in my Analysing Economic Data: A Concise Introduction.¹ It is written from the perspective that the econometric analysis of economic and financial time series is of key importance to both students and practitioners of economics and should therefore be a core component of applied economics and of economic policy making. What I wrote in the introduction of Analysing Economic Data thus bears repeating in the present context: this book contains material that I think any serious student of economics and finance should be acquainted with if they are seeking to gain an understanding of a real functioning economy rather than having just a working knowledge of a set of academically constructed models of some abstract aspects of an artificial economy.

1.2   After this introductory chapter the basic concepts of stochastic processes, stationarity and autocorrelation are introduced in Chapter 2 and the class of autoregressive-moving average (ARMA) models are developed. How these models may be fitted to an observed time series is illustrated by way of a sequence of examples.

Many economic time series, however, are not stationary, but may often be transformed to stationarity by the simple operation of differencing. Chapter 3 examines some informal methods of dealing with non-stationary data and consequently introduces the key concept of an integrated process, of which the random walk is a special case, so leading to the class of autoregressive-integrated-moving average (ARIMA) models. As is demonstrated by way of examples, although the informal methods proposed in Chapter 3 often work well in practice, it is important that formal means of testing whether a series is integrated or not and, if it is, of testing what order of integration it might be, are available. Chapter 4 thus develops the theory and practice of testing for one or more unit roots, the presence of which is the manifestation of ‘integratedness’ of a time series. An alternative to differencing as a means of inducing stationarity is to detrend the series using a polynomial, typically a linear, function of time. How to distinguish between these two methods of inducing stationarity by way of generalised unit root tests and the differing implications of the two methods for the way the series reacts to shocks are also discussed in this chapter.

Up to this point we have assumed that the errors, or innovations, in the various models have constant variance. For many economic time series, particularly financial ones observed at relatively high frequencies, this assumption is untenable, for it is well known that financial markets go through periods of excessive turbulence followed by periods of calm, a phenomenon that goes under the general term ‘volatility’. The manifestation of market volatility is that error variances change over time, being dependent upon past behaviour. Chapter 5 therefore introduces the class of autoregressive conditionally heteroskedastic (ARCH) processes. These are designed to incorporate volatility into models and, indeed, to provide estimates of such volatility.

An important aspect of time series modelling is to forecast future observations of the series being analysed. Chapter 6 develops a theory of forecasting for all the models introduced so far, emphasising how the properties of the forecasts depend in important ways on the model used to fit the data.

Only individual time series have been analysed so far, and hence the models have all been univariate in nature. Chapter 7 extends the analysis to consider a set of stationary time series, brought together as a vector and modelled as an autoregressive process, thus introducing the vector autoregression (VAR). With a vector of time series, the multivariate linkages between the individual series need to be investigated, so leading to the concept of Granger-causality, impulse response analysis and innovation accounting, all of which are discussed in this chapter.

Of course, assuming that the vector of time series is stationary is far too restrictive, but allowing the individual series to be integrated raises some interesting modelling issues for, as we demonstrate, it is possible for a linear combination of two or more integrated time series to be stationary, a concept known as cointegration. Cointegration is related to the idea of a dynamic equilibrium existing between two or more variables and, if it exists, it enables multivariate models to be expressed not only in terms of the usual differences of the series but also by the extent to which the series lie away from equilibrium: incorporating this ‘equilibrium error’ leads to the class of vector error correction models (VECMs).

Chapter 8 thus focuses on the consequences for conventional regression analysis when the variables in a regression are non-stationary, thus introducing the idea of spurious regression, before considering the implications of the variables in the regression being cointegrated. Tests for cointegration and estimation under cointegration are then discussed. Chapter 9 explicitly considers VECMs and how to test for and model cointegration within a VAR framework.

The final chapter, Chapter 10, explicitly recognises that this is only an introductory text on time series econometrics and so briefly discusses several extensions that more advanced researchers in the modelling of economic and financial time series would need to become familiar with. To keep within the remit of a ‘concise introduction’, however, no mention is made of the increasingly important subject of panel data econometrics, which combines time series with cross-sectional data, for which several textbooks are available.²

Mathematical level, focus and empirical exercises

1.3   As well as knowledge of basic statistics and econometrics, at the level provided by Analysing Economic Data, essentially all that is required to understand the material up to Chapter 7 is a good grounding in basic algebra, with some knowledge of solving equations and linear algebra plus some concepts of difference equations. Chapters 7 to 9, however, also require a basic knowledge of matrix algebra. Some technical material is placed in the notes that accompany each chapter, where key references, historical perspective and related discussion may also be found.³

Several examples using actual data, typically from the UK, are developed throughout the book. The content is thus suitable for final year undergraduate and postgraduate students of economics and finance wishing to undertake an initial foray into handling time series data.

1.4   Empirical exercises accompany most chapters. These are based on the software package Econometric Views (or EViews), now the industrial standard for econometric time series software, and illustrate how all the examples used in the book may be calculated and suggest how they might be extended. The data are available in an EViews workfile available for download.⁴ It is assumed that readers already have a basic working knowledge of EViews or are prepared to obtain this knowledge via the extensive online help facility accompanying the package.⁵

1.5   A brief word on notation: as can be seen, chapter sections are denoted x.y, where x is the chapter and y is the section. This enables the latter to be cross-referenced as §x.y. Matrices and vectors are also written in bold font, upper case for matrices, lower case for vectors, the latter being regarded as column vectors unless otherwise stated: thus A is a matrix and a is a vector.

Notes

1.Terence C. Mills, Analysing Economic Data: A Concise Introduction (Palgrave Macmillan, 2014).

2.A very popular text is Badi H. Baltagi, Econometric Analysis of Panel Data, 5th edition (Wiley, 2013).

3.A convenient presentation of the matrix algebra required is Mills, Matrix Representation of Regression Models: a Primer (Lulu Press, 2013), chapter 2. Key references in time series econometrics are gathered together in Mills, Time Series Econometrics (Routledge, 2015).

4.At http://www.palgrave.com//resources/Product-Page-Downloads/M/Mills%20-%20Time%20Series%20Econometrics/Resources.zip

5.EViews 8 is used throughout: see EViews 8 (Quantitative Micro Software, LLC, Irving CA: www.eviews.com).

2

Modelling Stationary Time Series: the ARMA Approach

Stochastic processes, ergodicity and stationarity

2.1 When analysing a time series using formal statistical methods, it is often useful to regard the observations (x₁,x₂,…,x_T) on the series, which we shall denote generically as x_t, as a particular realisation of a stochastic process.¹ In general, a stochastic process can be described by a T-dimensional probability distribution, so that the relationship between a realisation and a stochastic process is analogous to that between the sample and population in classical statistics. Specif...