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Über dieses Buch

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

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.

Terence C. Mills

2. Modelling Stationary Time Series: the ARMA Approach

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. Specifying the complete form of the probability distribution, however, will typically be too ambitious a task and we usually content ourselves with concentrating attention on the first and second moments: the T means <math display='block'> <mrow> <mi>E</mi><mo stretchy='false'>(</mo><msub> <mi>x</mi> <mn>1</mn> </msub> <mo stretchy='false'>)</mo><mo>,</mo><mi>E</mi><mo stretchy='false'>(</mo><msub> <mi>x</mi> <mn>2</mn> </msub> <mo stretchy='false'>)</mo><mo>,</mo><mo>&#x2026;</mo><mo>,</mo><mi>E</mi><mo stretchy='false'>(</mo><msub> <mi>x</mi> <mi>T</mi> </msub> <mo stretchy='false'>)</mo> </mrow> </math>$$E({x_1}),E({x_2}), \ldots ,E({x_T})$$T variances <math display='block'> <mrow> <mi>V</mi><mo stretchy='false'>(</mo><msub> <mi>x</mi> <mn>1</mn> </msub> <mo stretchy='false'>)</mo><mo>,</mo><mi>V</mi><mo stretchy='false'>(</mo><msub> <mi>x</mi> <mn>2</mn> </msub> <mo stretchy='false'>)</mo><mo>,</mo><mo>&#x2026;</mo><mo>,</mo><mi>V</mi><mo stretchy='false'>(</mo><msub> <mi>x</mi> <mi>T</mi> </msub> <mo stretchy='false'>)</mo> </mrow> </math>$$V({x_1}),V({x_2}), \ldots ,V({x_T})$$ and T(T-1)/2 covariances <math display='block'> <mrow> <mi>C</mi><mi>o</mi><mi>v</mi><mo stretchy='false'>(</mo><msub> <mi>x</mi> <mi>i</mi> </msub> <mo>,</mo><msub> <mi>x</mi> <mi>j</mi> </msub> <mo stretchy='false'>)</mo><mo>,</mo><mtext>&#x2003;</mtext><mi>i</mi><mo>&#x003C;</mo><mi>j</mi> </mrow> </math>$$Cov({x_i},{x_j}),\quad i < j$$ If we could assume joint normality of the distribution, this set of expectations would then completely characterise the properties of the stochastic process. Such an assumption, however, is unlikely to be appropriate for every economic and financial series we might wish to analyse.

Terence C. Mills

3. Non-stationary Time Series: Differencing and ARIMA Modelling

The class of ARMA models developed in the previous chapter relies on the assumption that the underlying process is weakly stationary, thus implying that the mean, variance and autocovariances of the process are invariant under time shifts. As we have seen, this restricts the mean and variance to be constant and requires the autocovariances to depend only on the time lag. Many economic and financial time series, however, are certainly not stationary and, in particular, have a tendency to exhibit time-changing means and/or variances.

Terence C. Mills

4. Unit Roots and Related Topics

As we have shown in §§3.5–3.12, the order of integration, d, is a crucial determinant of the properties that a time series exhibits. If we restrict ourselves to the most common values of 0 and 1 for d, so that xt is either I(0) or I(1), then it is useful to bring together the properties of such processes.

Terence C. Mills

5. Modelling Volatility using GARCH Processes

Following the initial work on portfolio theory in the 1950s, volatility has become an extremely important concept in finance, appearing regularly in models of, for example, asset pricing and risk management. Much of the interest in volatility has to do with it not being directly observable, and several alternative measures have been developed to approximate it empirically. The most common measure of volatility has been the unconditional standard deviation of historical returns. The use of this measure, however, is severely limited by it not necessarily being an appropriate representation of financial risk and by the fact that returns tend not to be independent and identically distributed, so making the standard deviation a potentially poor estimate of underlying volatility.

Terence C. Mills

6. Forecasting with Univariate Models

Terence C. Mills

7. Modelling Multivariate Time Series: Vector Autoregressions and Granger Causality

So far our focus has just been on modelling individual time series but we now extend the analysis to multivariate models. To develop methods of modelling a vector of time series, consider again the AR(1) process, now written for the stationary series y t and with a slightly different notation to that used before: 7.1<math display='block'> <mrow> <msub> <mi>y</mi> <mi>t</mi> </msub> <mo>=</mo><mi>&#x03B8;</mi><mo>+</mo><mi>&#x03D5;</mi><msub> <mi>y</mi> <mrow> <mi>t</mi><mo>&#x2212;</mo><mn>1</mn> </mrow> </msub> <mo>+</mo><msub> <mi>a</mi> <mi>t</mi> </msub> </mrow> </math>]]</EquationSource> <EquationSource Format="TEX"><![CDATA[$${y_t} = \theta + \phi {y_{t - 1}} + {a_t}$$ The standard dynamic regression model adds exogenous variables, perhaps with lags, to the right-hand side of (7.1); to take the simplest example of a single exogenous variable x t having a single lag, consider 7.2<math display='block'> <mrow> <msub> <mi>y</mi> <mi>t</mi> </msub> <mo>=</mo><mi>c</mi><mo>+</mo><mi>a</mi><msub> <mi>y</mi> <mrow> <mi>t</mi><mo>&#x2212;</mo><mn>1</mn> </mrow> </msub> <mo>+</mo><msub> <mi>b</mi> <mi>a</mi> </msub> <mo>+</mo><msub> <mi>b</mi> <mn>1</mn> </msub> <msub> <mi>x</mi> <mrow> <mi>t</mi><mo>&#x2212;</mo><mn>1</mn> </mrow> </msub> <mo>+</mo><msub> <mi>e</mi> <mi>t</mi> </msub> </mrow> </math> ]]</EquationSource><EquationSource Format="TEX"><![CDATA[$${y_t} = c + a{y_{t - 1}} + {b_a} + {b_1}{x_{t - 1}} + {e_t}$$ Again, note the change of notation as coefficients and innovations will not, in general, be the same across (7.1) and (7.2): c and a will differ from θ and ϕ, as will the variance of e t , σ2 e , differ from that of at,σ2a with, typically σ2e < σ2 a if the additional coefficients b0and b1 are non-zero.

Terence C. Mills

8. Cointegration in Single Equations

The VAR framework of the previous chapter requires that all the time series contained in the model be stationary. Whilst stationarity can be achieved, if necessary, by differencing each of the individual series, is this always an appropriate approach to take when working within an explicitly multivariate framework? We begin our answer to this question by introducing the simulation example considered by Clive Granger and Paul Newbold in an important article examining some of the likely empirical consequences of nonsense, or spurious, regressions in econometrics.1

Terence C. Mills

9. Cointegration in Systems of Equations

Terence C. Mills

10. Extensions and Developments

As we emphasised in Chapter 1, this text book is very much an introduction to time series econometrics: consequently, some topics have not been covered either because they are too peripheral to the main themes or are too advanced. Three areas not covered, but which nevertheless can be important when analysing time series data, are seasonality, non-linearities and breaks.

Terence C. Mills


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