Lütkepohl is the author of Introduction to Multiple Time Series Analysis () and .. JMulTi software and the applied time series econometrics text describing the . pdf probability density function. QML quasi-maximum likelihood. RESET. Edited by Helmut Lütkepohl, European University Institute, Florence, Markus Krätzig, Humboldt-Universität zu Berlin. By Helmut Lütkepohl, Markus Krätzig, San Domenico di Fiesole and Berlin. By Jörg Breitung, Ralf Brüggemann, Helmut Lütkepohl. applied econometric work than, for example, vector autoregressive much of the contents of the corresponding chapter in Lütkepohl (). Some Although multiple time series analysis is applied in many disciplines, I have.
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APPLIED TIME SERIES. ECONOMETRICS. Edited by. HELMUT LÜTKEPOHL. European University Institute, Florence. MARKUS KRÄTZIG. Humboldt University . PDF | On Mar 1, , Maryam Moghaddas Bayat and others published Applied Time Series Econometrics: written by Lutkepohl; (translated. Request PDF on ResearchGate | In Applied Time Series Econometrics | Time 8 As discussed in Lütkepohl and Krätzig (), starting with a large order, and.
West, M. Harrison Bayesian Forecasting and Dynamic Models. New York: Springer- Verlag. White, H. Whittle, P. Oxford: Blackwell. Wooldridge, J. Yaglom, A. New York: Dover. Newman, M. Milgate and J. Eatwell, eds. Phillips, P. Econometrica, 66, B Journal of Econometrics, 11, Hansen Loretan Saikkonen, P. Stock, J. Watson Hartigan, J. Bayes Theory. New York: Springer-Verlag. Johnstone Kim, J. Kim J. Le Cam, L. Yang Asymptotics in Statistics: Some Basic Concepts.
Ploberger Ploberger W.
Sweeting, T. Adekola Leadbetter Stationary and Related Stochastic Processes. Rozanov, Y. Walters, P. Projections and the Wold Decomposition Anderson op. Whittle op. Gallant A. New York: Basil Blackwell. Ibragimov, I. Groningen: Wolters-Noordhoff. Potscher B. Prucha op. Nelson Doob, J. Heyde Martingale Limit Theory and its Application. McLeish, D. Solo op. Andrews, D. Den Haan, W. Levin, , "A practitioner's guide to robust covariance matrix estimation," in Handbook of Statistics 15, G.
Maddala and C. Rao, eds. Lee, C. Newey, W. Parzen, E. Sun and S. Robinson, P. Sul, D. Sun, Y. Phillips, and S. Spectral Regression Theory Corbae, D. Ouliaris and P. Econometrica, 70, Rosenblatt Ed. Xiao, Z.
Phillips, We thank all of them for their cooperation. Kirstin Hubrich. We would like to mention the following contributions specifically. Aaron Mehrotra. Dmitri Boreiko. On the other hand. Of course. Stefan Lundbergh. Acknowledgments There are also many further people who contributed significantly to this book and JMulTi in one way or other.
Maria Eleftheriou. We thank all the contributors for their good cooperation and help in finalizing the book. It became apparent that such a text might be useful to have for the students. Rolf Tschernig. With a quite powerful software for time series econometrics at hand. These models have not become as popular in empirical work as some of the methods that are included in JMulTi.
Franz Palm. Carsten Trenkler. Stefan Kruchen. Preface xvii Herwartz. Because some of the people who have written the software components for JMulTi clearly have more expert knowledge on their methods than we do.
San Domenico di Fiesole and Berlin. SFB Notation and Abbreviations General Symbols: FPE final prediction error criterion HQ. Herwartz stat-econ. Tschernig KE. Terasvirta hhs. ITALY email: Many economic problems can be analyzed using time series data. They can determine in part which models and statistical tools are suitable. For exam- ple. Another important goal is understanding the relations between a set of possibly related variables or uncovering the ongoings within an economic system or a specific market.
Forecasting the future economic conditions is one important objective of many analyses. When a time series data set has been created. Before engaging in an econometric time series analysis it is a good idea to be clear about the objectives of the analysis.
The discussion is presented in two separate sections because it is one thing to find data in some suitable data source and another issue to prepare the data for the project of interest.
The next step is getting a good data set to work with. Some discussion of this step is provided in Sections 1. This is the stage at which the actual econometric analysis begins. A brief discussion of this initial stage of a project follows in Section 1.
To be even more specific. The observations have a natural ordering in time. For instance. The data usually have features that are not well explained or described by economic theory. Several statistical tools will be presented in the following chapters that can be used for checking the adequacy of a model.
In that case a forecast of a specific variable is desired. Sometimes the objectives are formulated in a less precise way. What are the implications for the income distribution of the households of the target economy?
In short. In economics it is clear that many variables interact more or less strongly. A brief overview of the topics considered in this book is given in the final section of this chapter. For a proper econometric analysis they still have to be captured in the model for the DGP.
Such theories are useful in different respects. This aspect provides a second important ingredient for the analysis that comes from economic theory. These objectives may be formulated by a customer who is interested in specific results or the solution of a particular problem. Otherwise a meaningful statistical analysis is not possible on the basis of the given data information.
When the models and statistical tools for an econometric time series analysis are discussed in subsequent chapters. Here economic theory has an important part to play. In addition. Often alternative theories exist that have something to say on a particular problem.
When an econometric model has been constructed for the DGP. A more precise question in this context would be. When the objectives of the analysis are specified. The basket of goods is typically adjusted every few years. The quoted value may be the closing price at some specific stock exchange. It is not always easy to determine the exact definition or construction pro- cedure of a particular time series. Nevertheless it should be clear that a good background knowledge about the data can be central for a good analysis.
We have already mentioned the frequent adjust- ments of the basket of goods underlying CPI data. There are different possibilities to define the price associated with a specific day. Otherwise some other theory may have been a better basis for the choice of variables. That stage is discussed briefly in the next section. Some of them refer to West Germany only. In turn. In any case. Initial Tasks and Overview 3 theory is compatible with the data may just be the main objective of an analysis. The problem of nonuniqueness and ambiguity of the definitions of the vari- ables is not limited to macroeconomic data by the way.
As another example consider German macroeconomic variables. A problem arises. How is the CPI constructed? That depends. When the set of potentially most relevant variables is specified. Is it preferable to measure it in terms of consumer prices using. It is also possible that the definition of a variable will change over time. Ignoring them can lead to distor- tions of the relation with other series that have seasonal components. In that case. Another problem with the data offered in many databases is that they have been adjusted.
Such an approach. Seasonal adjustment is. Should one use the value of the last month of each quarter as the quarterly value or should an average of the values of the three months of each quarter be used?
If it is not clear which variable best reflects the quantity one would like to include in the model. For ex- ample. The reason is that defining and determining the seasonal component of a series are not easy tasks. Often the series of interest have different frequencies of observation. Aggregation is another issue of importance in setting up a suitable data set.
We will briefly touch on such procedures in Chapter 2. Although it is in principle possible to interpolate missing values of a time series. Suppose that a monthly interest rate series is given. The required operations for making the data more homogenous are often easy to perform with the software tool available.
In other words. The tools available for univariate analysis are presented in Chapter 2. With respect to the models for describing univariate DGPs.
Even when the objective is a joint analysis of a set of time series. More details on data handling with the software JMulTi frequently referred to in this volume are discussed in Chapter 8. A careful examination of the data definitions and specifications is therefore advisable at an early stage of an analysis.
In that chapter. Some tools for this stage of the analysis are presented in the following chapters. Still it may be useful to make adjustments before the econometric analysis begins.
Data formats and codings can be — and often are — different when the data come from different sources. Initial Tasks and Overview 5 In conclusion. Data from different sources may be collected or constructed in markedly different ways even if they refer to the same variable. In this analysis the results of preliminary unit root tests are of some importance.
At the multivariate level. Chapter 3 emphasizes modeling of cointegrated series. Some models. More generally.
Nowadays impulse responses and forecast error variance decompositions are used as tools for analyzing the relations between the variables in a dynamic econometric model. If sufficient information is available in the data to make an analysis of non- linearities and higher order moment properties desirable or possible. The recent empirical literature has found it useful to distinguish between the short.
These tools are considered in Chapter 4. It turns out. For multivariate systems.
In a univariate context. Once a model for the joint DGP of a set of time series of interest has been found. The objective of such an analysis may be an investigation of the adequacy of a particular theory or theoretical argument.
These parts are conveniently separated in a VECM by paying particular attention to a detailed modeling of the cointegration properties of the variables. An important extension that is often of interest for financial market data is to model the conditional second moments. Given that data sets are often quite limited and that even linear models can contain substantial numbers of parameters.
The soft- ware JMulTi introduced in Chapter 8 is supposed to be able to decrease the time gap between the development of new methods and their availability in user-friendly form. Readers may therefore find it useful to familiarize themselves with the software as they go through the various chapters of the book.
The smoothness of the transition from one extreme regime to the other accounts for the name of this model. In the last chapter of this volume. In modern applied time series econometrics the computer is a vital tool for carrying out the analysis. Estimation of the nonlinear functions is done nonparametrically using suitable local approximations that can describe general nonlinear functions in a very flexible way.
Initial Tasks and Overview 7 estimation methods. A more general approach. The discussion in Chapter 6 also covers purely univariate smooth transition autoregressive STAR models that have been frequently fitted to economic and other time series. The modeling strategy described in Chapter 6 is only applicable to single-equation models. Chapter 6 contains a description of the parametric smooth transition STR model.
The drawback of the additional flexibility is. Nonlinear modeling of the conditional mean is considered in Chapters 6 and 7. This software provides a flexible framework for checking new methods and algorithms quickly. The first series consists of changes in seasonally adjusted U. In addition there is a level shift in the third quarter of It moves around a fixed mean value.
It appears to fluctuate randomly around a con- stant mean. They may evolve regularly around a fixed value. It appears to evolve around a determin- istic polynomial trend. Some important characteristics of time series can be seen in the example series plotted in Figure 2. Although German re- unification took place officially in October The sluggish.
This shift is due to a redefinition of the series. The third series represents German gross national product GNP. To summarize. In contrast. The variability is quite dissimilar in different parts of the sample period.
Such an unusual value is sometimes referred to as an outlier. Such an analysis is important because the properties of the individual series have to be taken into account in modeling the data generation process DGP of a system of potentially related variables. Some correlation between consecutive values seems possible. Example time series. Univariate Time Series Analysis 9 a quarterly changes in U. They will not be discussed here. The specific characteristics of the series may be an integral part of the relationship of interest.
The fact that quarters are not of identical length will be ignored.
Time Series Analysis with R
Roughly speaking. All these characteristics have to be taken into account in constructing models for a set of related time series variables. In this volume. Some important characteristics of the DGPs of time series will be described more formally in this chapter.. Each time series observation is assumed to be generated by a different member of the stochastic process. The subscripts t are usually thought of as representing time or.
There are methods for dealing explicitly with missing observations. The associated random variables assumed to have generated the time series observations will usually be denoted by the same symbols as the observations. Some of the characteristics may have an impact on the statistical inference procedures used in modeling and analyzing the underlying economic system. In that case the range of the subscript is either not important or it is understood from the context.
Such an assumption is convenient for theoretical discussions. Our notation is also meant to imply that the means. The first condition means that all members of a stationary stochastic process have the same constant mean. Univariate Time Series Analysis 11 time periods. In this chapter many concepts.
Priestley To simplify the notation further we sometimes use it to denote the full stochastic process or the related time series. The second condition ensures that the variances are also time invariant because. Note that the DGP may begin before the first time series value is observed. Examples are Fuller It will be obvious from the context whether the symbol yt refers to an observed value or the underlying random variable. Several time series textbooks are available with a more in-depth treatment that may be consulted for further details and discussions.
Often T is the set of all integers or all nonnegative integers. Sometimes a process satisfying this condition is described as being strictly stationary. Such a time series is sometimes referred to as a stationary time series for simplicity of terminology.
This terminology will not be used here. If the process starts in some fixed time period e. From our examples it may seem that stationarity is a rare property of economic time series.
This happens often if the process can be made station- ary by modifying the initial members of the process. Before we go on with our discussion of stationary processes. We will not always distinguish be- tween asymptotic stationarity and stationarity but will call a process stationary if stationarity can be achieved by modifying some initial variables.
Although there is some truth to this impression. Some of them will be discussed shortly. Some authors call a process with time-invariant first and second moments covariance stationary. These models are discussed in more detail in Section 2.
A stationary process for which all autocorrelations are zero is called white noise or a white noise process. If it is purely random.
In this case they all tend to approach small values quickly for increasing h. In Figure 2.
Applied Time Series Econometrics
We will see later that. The dashed lines in Figure 2. Partial autocorrelations PACs are also quantities that may convey useful information on the properties of the DGP of a given time series.
We will discuss formal statistical tests for stationarity later on in Section 2. For a series with stationary DGP.
For stationary processes. Notice that the sample autocorrelations are estimates of the actual autocorrela- tions if the process is stationary. Autocorrelation functions and partial autocorrelation functions of U. The following examples of spectral windows. This property is a result of the increasing number of sample autocovariances included in the periodogram with growing sample size.
The autocorrelations of a stationary stochastic process may be summarized compactly in the spectral density function. It is obtained by replacing the autocorrelations by estimators.
MT represent the so-called spectral window and MT is the truncation point. Univariate Time Series Analysis 15 autocorrelation functions and partial autocorrelation functions can give useful information on specific properties of a DGP other than stationarity. A possible estimator of the spectral density is the periodogram.
Periodogram and spectral density of log income series.
The series has an upward trend and a distinct seasonal pattern. In such a situation. Because the spike is so large. The trend is reflected as a spike near zero in the periodogram. The weights decrease with increasing j.
Univariate Time Series Analysis 17 The logarithm is a monoton transformation and therefore ensures that larger values remain larger than smaller ones. The observation that much of the spectral mass is concentrated near the zero frequency and further peaks occur around the seasonal frequencies is quite common for macroeco- nomic time series. Larger values lead to more volatile function estimates with larger variances.
In such a case. It describes the fact that the series is trending with long-term movements. An important problem in estimating the spectral density of a time series is the choice of the window size MT. The relative size is reduced. For descriptive purposes. This is clearly seen in the third panel of Figure 2. Because the frequency is measured in radians.
Now it is obvious that the variability in the periodogram estimates is quite large. For the example series this is clearly reflected in a second peak at the right end of the graph. Chapter 7 ]. This can be seen in Figure 2. Although the log income series is hardly stationary — and. As we have seen. Note that. In that graph the series is seen to be dominated by very low frequency cycles trend and seasonal cycles.
There may be further peaks at mul- tiples of the seasonal frequency because more than one cycle may be completed within a year. For the present example series a pronounced seasonal pattern remains in the rates of change. The last plot in Figure 2. Such a transformation has implica- tions for the distribution of the DGP. Quarterly West German real per capita personal disposable income and its transformations for the period — Taking now first differences of the logs.
Because the first differences of the logs are roughly the rates of change of the series. The series clearly has important characteristics of a stationary series. For the income series. They are plotted in the right-hand panel in the second row of Figure 2. As an alternative. Although the series fluctuates around a constant mean and appears to have a rather stable variance.
In this case. In practice many filters are linear functions. Formal statistical tests to help in deciding which transformation to use will be considered in Section 2. To see this more clearly. Univariate Time Series Analysis 19 It is not uncommon. This filter may remove seasonal variation from a quarterly series yt. As an example. This kind of nonstationarity may.
Time series are often filtered to extract or eliminate special features or components. There are also other transformations that make economic time series look like stationary series.
For the example series the result of subtracting the seasonal means can also be seen in Figure 2. We will encounter several special filters in the following sections. This filter may be defined indirectly by specifying the trend of the series y1. Filtering is often used for seasonal adjustment. This feature is sometimes undesirable — especially if. In business cycle analysis it is sometimes desirable to extract the trend from a series to get a better understanding of the business cycle fluctuations.
They may also distort important features of a given series in addition to removing the seasonality. Using this notation. In practice. Thereby complicated nonlinear filters may result.
Because this trans- formation has been used successfully for many economic time series. A nonstationary stochastic process that can be made stationary by considering first differences is said to be integrated of order one I 1.
Taking first differences is a useful device for removing a series trend — either stochastic or deterministic or both. Integrated processes. Univariate Time Series Analysis 21 Figure 2. If s denotes the periodicity of the season e. Again the origin of this terminology will be explained in the next section. For rea- sons that will become clear later. A stationary process yt is sometimes called I 0. To simplify the notation it is some- times helpful to use the lag operator L.
Some of them will be discussed in Section 2. In this case they represent annual rates of change and look even more stationary than quarterly rates of change. Because most economic time series exhibit serial correlation. There are some simple parametric models. Using the lag operator. In this section we will briefly discuss autoregressive AR processes.
If this operator removes the nonstationarity of a process. Assuming that T is the set of all integers and. It sat- isfies the condition 2. For the special case in which the AR operator has a unit root. Autocorrelation functions. Notice that the left-hand process has positive autocorrelation and is therefore less volatile than the right-hand one for which consecutive members are negatively corre- lated.
For the previously con- sidered AR 1 example process we obtain. This relation between the unit roots of the AR operator and the integrat- edness of the process explains why an integrated process is sometimes called a unit root process. For seasonal processes. The process is stationary. Uniqueness of the MA representation requires restrictions on the coefficients. These roots are sometimes called seasonal unit roots.
Uniqueness is guaranteed. As a special case we obtain a constant spectral density of a white noise process. The autocorrelation functions.
For processes with distinct sea- sonality. Such operators can some- times result in a more parsimonious parameterization of a complex seasonal serial dependence structure than a regular nonseasonal operator.
The spectral density of the ARMA process 2. If the process is stable. For a series with seasonal periodicity s e. In compact lag operator notation. Univariate Time Series Analysis 27 processes can be very similar. For mixed processes with nontrivial AR and MA parts. Notice that also a seasonal differencing operator may be included. We have fitted an AR 4 model to the data.
In par- ticular. By now the acronym ARCH stands for a wide range of models for changing conditional volatility. A more extensive introduction to modeling conditional heteroskedasticity is given in Chapter 5.
Consider the univariate AR p model 2. The German long-term interest rate series is considered as an example. The residuals u t of this model are said to follow an autoregressive conditionally heteroskedastic process of order q ARCH q if the conditional distribution of u t.
Although the conditional distribution is normal. Bollerslev and Taylor have proposed gain- ing greater parsimony by extending the model in a way similar to the approach. Engle observed that. Although the autocorrelation function is consistent with a white noise process. Even with this special distributional assumption the model is capable of generating series with characteristics similar to those of many observed financial time series.
It was observed by some researchers that. Engle in his seminal paper on ARCH models assumed the conditional distribution to be normal. Whereas the variance is low at the end. The residual series shows some variation in its variability. Plots of residuals from AR 4 model for German long-term interest rate. Figure 2. For the remainder of this chapter.
If the deterministic. Phillips, P. Econometrica, 66, B Journal of Econometrics, 11, Hansen Loretan Saikkonen, P. Stock, J. Watson Hartigan, J. Bayes Theory. New York: Springer-Verlag. Johnstone Kim, J. Kim J. Le Cam, L. Yang Asymptotics in Statistics: Some Basic Concepts. Ploberger Ploberger W. Sweeting, T. Adekola Leadbetter Stationary and Related Stochastic Processes. Rozanov, Y.
New Introduction to Multiple Time Series Analysis
Walters, P. Projections and the Wold Decomposition Anderson op. Whittle op. Gallant A. New York: Basil Blackwell. Ibragimov, I. Groningen: Wolters-Noordhoff.
Potscher B. Prucha op. Nelson Doob, J. Heyde Martingale Limit Theory and its Application. McLeish, D. Solo op. Andrews, D. Den Haan, W. Levin, , "A practitioner's guide to robust covariance matrix estimation," in Handbook of Statistics 15, G. Maddala and C. Rao, eds. Lee, C. Newey, W. Parzen, E. Sun and S. Robinson, P. Sul, D. Sun, Y. Phillips, and S. Spectral Regression Theory Corbae, D.
Ouliaris and P. Econometrica, 70, Rosenblatt Ed. Xiao, Z. Phillips, Litterman, R. Data," Econometrica, Runkle, D. Todd, R.
Zellner, A. Journal of Econometrics, 73, Baillie, R. Bollerslev Journal of International Money and Finance, 13, Geweke J. Journal of Time Series Analysis, 4, Journal of Econometrics, 14, Parallel to writing up the statistical theory contained in that book a menu-driven program based on GAUSS was already developed under the name MulTi [see Haase. Doob, J. Heyde Martingale Limit Theory and its Application. New York: Springer Verlag. Because it is not fully clear where the shift actually occurred.
Another issue worth mentioning perhaps is that including lagged differences in the model up to lag order 3 corresponds to a levels AR model of order 4. SFB The squared residuals show a fairly homogeneous variability of the series.
Initial Tasks and Overview 5 In conclusion.