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Über dieses Buch

This book presents modern developments in time series econometrics that are applied to macroeconomic and financial time series, bridging the gap between methods and realistic applications. It presents the most important approaches to the analysis of time series, which may be stationary or nonstationary. Modelling and forecasting univariate time series is the starting point. For multiple stationary time series, Granger causality tests and vector autogressive models are presented. As the modelling of nonstationary uni- or multivariate time series is most important for real applied work, unit root and cointegration analysis as well as vector error correction models are a central topic. Tools for analysing nonstationary data are then transferred to the panel framework. Modelling the (multivariate) volatility of financial time series with autogressive conditional heteroskedastic models is also treated.

Inhaltsverzeichnis

Frontmatter

1. Introduction and Basics

A time series is defined as a set of quantitative observations arranged in chronological order. We generally assume that time is a discrete variable. Time series have always been used in the field of econometrics. Already at the outset, JAN TINBERGEN (1939) constructed the first econometric model for the United States and thus started the scientific research programme of empirical econometrics. At that time, however, it was hardly taken into account that chronologically ordered observations might depend on each other. The prevailing assumption was that, according to the classical linear regression model, the residuals of the estimated equations are stochastically independent from each other. For this reason, procedures were applied which are also suited for cross section or experimental data without any time dependence.

Gebhard Kirchgässner, Jürgen Wolters, Uwe Hassler

2. Univariate Stationary Processes

As mentioned in the introduction, the publication of the textbook by GEORGE E.P. BOX and GWILYM M. JENKINS in 1970 opened a new road to the analysis of economic time series. This chapter presents the Box-Jenkins Approach, its different models and their basic properties in a rather elementary and heuristic way. These models have become an indispensable tool for short-run forecasts. We first present the most important approaches for statistical modelling of time series. These are autoregressive (AR) processes (

Section 2.1

) and moving average (MA) processes (

Section 2.2

), as well as a combination of both types, the so-called ARMA processes (

Section 2.3

). In

Section 2.4

we show how this class of models can be used for predicting the future development of a time series in an optimal way. Finally, we conclude this chapter with some remarks on the relation between the univariate time series models described in this chapter and the simultaneous equations systems of traditional econometrics (

Section 2.5

).

Gebhard Kirchgässner, Jürgen Wolters, Uwe Hassler

3. Granger Causality

So far, we have only considered single stationary time series. We analysed their (linear) structure, estimated linear models and performed forecasts based on these models. However, the world does not consist of independent stochastic processes. Just the contrary: in accordance with general equilibrium theory, economists usually assume that everything depends on everything else. Therefore, the next question that arises is about (causal) relationships between different time series.

Gebhard Kirchgässner, Jürgen Wolters, Uwe Hassler

4. Vector Autoregressive Processes

The previous chapter presented a statistical approach to analyse the relations between time series: starting with univariate models, we asked for relations that might exist between two time series. Subsequently, the approach was extended to situations with more than two time series. Such a procedure where models are developed bottom up to describe relations is hardly compatible with the economic approach of theorising where – at least in principle – all relevant variables of a system are treated jointly. For example, starting out from the general equilibrium theory as the core of economic theory, all quantities and prices in a market are simultaneously determined. This implies that, apart from the starting conditions, everything depends on everything, i.e. there are only endogenous variables. For example, if we consider a single market, supply and demand functions simultaneously determine the equilibrium quantity and price.

Gebhard Kirchgässner, Jürgen Wolters, Uwe Hassler

5. Nonstationary Processes

So far, we have only considered stationary time series. As a matter of fact, however, most economic time series are trending, like, for example, the GDP series investigated in

Chapter 1

. We tried to eliminate the trend by using first differences or growth rates. These filtered series can be investigated by employing the concepts that were developed for the analysis of stationary time series.

Gebhard Kirchgässner, Jürgen Wolters, Uwe Hassler

6. Cointegration

In the preceding chapter, we used stochastic trends to model nonstationary behaviour of time series, i.e. the variance of the data generating process increases over time, the series exhibits persistent behaviour and its first difference is stationary. For many economic time series, such a data generating process is a sufficient approximation, so that, in the following, we only consider processes which are integrated of order one (I(1)).

Gebhard Kirchgässner, Jürgen Wolters, Uwe Hassler

7. Nonstationary Panel Data

In

Chapter 4

we introduced an approach to analyse vectors of stationary time series, while

Chapter 6

was devoted to the nonstationary case. With yth we denote the ith component at time t, t = 1, …, T. In typical time series applications the dimension of the vector is small (for instance equal to 3 in

Examples 4.4

. or

6.8

), while the time dimension is rather large (T > 100). In a panel situation the number of components or units, denoted by N, is large as well, i = 1, …, N. There may be N price indices, N exchange rates or generally N countries or units. The unrestricted VAR(p) model from equation (4.1) allows each component to depend on its own lagged values and on the past of all other components. Hence, (4.1) includes p•N2 + N parameters when modelling time series from N units, a number growing fast with the dimension N. Already with N = 10 there would be hundreds of parameters to estimate. Therefore, the VAR approach is not applicable unless the cross-sectional dimension is rather small.

Gebhard Kirchgässner, Jürgen Wolters, Uwe Hassler

8. Autoregressive Conditional Heteroscedasticity

All models discussed so far use the conditional expectation to describe the mean development of one or more time series. The optimal forecast, in the sense that the variance of the forecast errors will be minimised, is given by the conditional mean of the underlying model. Here, it is assumed that the residuals are not only uncorrelated but also homoscedastic, i.e. that the unexplained fluctuations have no dependencies in the second moments. However, BENOIT MANDELBROT (1963) already showed that financial market data have more outliers than would be compatible with the (usually assumed) normal distribution and that there are ‘volatility clusters’: small (large) shocks are again followed by small (large) shocks. This may lead to ‘leptokurtic distributions‘, which – as compared to a normal distribution – exhibit more mass at the centre and at the tails of the distribution. This results in ‘excess kurtosis’, i.e. the values of the kurtosis are above three.

Gebhard Kirchgässner, Jürgen Wolters, Uwe Hassler

Backmatter

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