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Über dieses Buch

This study is concerned with forecasting time series variables and the impact of the level of aggregation on the efficiency of the forecasts. Since temporally and contemporaneously disaggregated data at various levels have become available for many countries, regions, and variables during the last decades the question which data and procedures to use for prediction has become increasingly important in recent years. This study aims at pointing out some of the problems involved and at pro­ viding some suggestions how to proceed in particular situations. Many of the results have been circulated as working papers, some have been published as journal articles, and some have been presented at conferences and in seminars. I express my gratitude to all those who have commented on parts of this study. They are too numerous to be listed here and many of them are anonymous referees and are therefore unknown to me. Some early results related to the present study are contained in my monograph "Prognose aggregierter Zeitreihen" (Lutkepohl (1986a)) which was essentially completed in 1983. The present study contains major extensions of that research and also summarizes the earlier results to the extent they are of interest in the context of this study.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Prologue

Abstract
In making choices between alternative courses of action, decision makers at all structural levels often need predictions of aggregated variables. For example, in the process of planning a government budget, forecasts of annual tax revenues may be required. If quarterly or monthly figures of previous revenues are available then a time series model may be constructed for the generation process of the quarterly or monthly data. This model can then be used to obtain predictions for the next quarters or months and these forecasts can be aggregated to obtain annual forecasts of the tax revenues. Alternatively, the available monthly or quarterly data may be aggregated to obtain an annual series of tax revenues. Based on this series a model may be constructed to generate annual forecasts.
Helmut Lütkepohl

Chapter 2. Vector Stochastic Processes

Abstract
In this chapter some notation is introduced and assumptions are discussed that will be used in later parts of the book. As explained in Chapter 1 the main concern of this study will be with forecasting economic variables of interest. Assuming that T observations xkt, t = 1,…,T; k = 1,…,K; for each of the K time series variables x1,…,xk are available one possible approach is to construct a model for the generation process of the multiple time series x t = (x1t,…,xkt)', t = 1,…T, and use that model for predicting future values of the variables x1,…xk. The model for the data generating process is chosen from the class of stochastic processes. In the following sections specific stochastic processes are introduced that are often suitable for modelling economic data. In Section 2.1 nondeterministic stationary processes are considered, in Section 2.2 an extension to certain types of nonstationary processes is provided and in Section 2.3 the special class of vector ARMA (autoregressive moving average) processes is discussed.
Helmut Lütkepohl

Chapter 3. Forecasting Vector Stochastic Processes

Abstract
In this chapter prediction formulas will be given and the corresponding mean squared errors (MSEs) will be derived for the processes discussed in Chapter 2. The general strategy for forecasting or prediction (the two terms will be used interchangeably) is to assume that a model for the generation process of a set of variables of interest x = (x1,…,xk)' and an information set, say Jt, containing the available information up to time t are given. For example, the data generation process could be an AR or MA process as introduced in Chapter 2 and Jt may contain all current and previous values of the x variables, i.e., Jt = x ss ≤ t. If forecasts are desired for a particular purpose a specific cost function may be associated with the possible forecast errors. Then a forecast will be optimal if it minimizes the cost or expected cost. Thus, it will depend on the particular cost function which forecast is optimal.
Helmut Lütkepohl

Chapter 4. Forecasting Contemporaneously Aggregated Known Processes

Abstract
A contemporaneous aggregate of the variables x1t,…,xKt at time t is their sum or weighted sum, yt = f1x1t + … + fkxKt, where the fk, k = 1,…,K, are the aggregation weights. Examples are numerous in economics. For instance, the Gross National Product (y) is the sum of Private Consumption Expenditures (x1), Gross Private Domestic Investment (x2), Government Expenditures (x3) and Net Exports (x4), that is, in this case the aggregation weights are all equal to one, f1 = f2 = f3 = f4 = 1. Also price indices are weighted sums of prices of different commodities. In fact, practically all macroeconomic variables and many micro economic variables as well are contemporaneous aggregates of some sort. Therefore systems of contemporaneous aggregates are often of interest. Such a system, say y = (y1,…,yM)', can be written as a linear transformation of the disaggregate components x = (x1,…,xk)', that is y = Fx where F is a suitable (M x K) transformation matrix. Thus, if analyzing a contemporaneously aggregated vector stochastic process xt is the aim, linear transformations y t = Fx t must be considered. Therefore linear transformations of vector stochastic processes are the subject of this chapter. It will be assumed that the aggregation matrix F is the same for all periods, that is, F does not depend on t.
Helmut Lütkepohl

Chapter 5. Forecasting Contemporaneously Aggregated Estimated Processes

Abstract
In the previous chapter three predictors have been compared for a contemporaneously aggregated vector stochastic process. The first predictor is obtained by aggregating the forecasts based on the original process. The second predictor results from forecasting the aggregate process directly based on the aggregate variables. Finally, the third predictor is obtained from aggregating univariate forecasts of the individual components of the disaggregate process. The comparison has been carried out under the assumption that the predictors are based on known processes.
Helmut Lütkepohl

Chapter 6. Forecasting Temporally and Contemporaneously Aggregated Known Processes

Abstract
A temporal aggregate is a variable obtained by aggregating over time, that is, weighted values associated with consecutive time periods are added up. For instance, the annual Gross National Product (GNP) is the sum of quarterly or monthly GNP values. As another example, the monthly number of unemployed persons may be computed as the average of the daily unemployment figures or, alternatively, the value of a particular key day, e.g., the last working day of the month, may be used. In both cases the unemployment figure is a weighted sum of the daily numbers. While equal weights are associated with each day in the first alternative, the last day gets a weight of one and all other days get zero weights in the second alternative.
Helmut Lütkepohl

Chapter 7. Temporal Aggregation of Stock Variables — Systematically Missing Observations

Abstract
The subject of this chapter is temporal aggregation of stock variables where the aggregate consists of every m-th variable (or vector of variables) of the original, full process. In other words, forecasting time series with systematically (or periodically) missing observations will be discussed. Treating this special form of aggregation separately is useful in order to demonstrate the implications of the general results of the previous chapter for this special case.
Helmut Lütkepohl

Chapter 8. Temporal Aggregation of Flow Variables

Abstract
Adding the values of a variable associated with consecutive time periods is the type of temporal aggregation considered in this chapter. Examples of flow variables where this form of aggregation is common are income, consumption expenditures, sales, imports, etc.. For instance, the annual income is the sum of monthly or quarterly incomes. In this chapter forecasts of the aggregate will be investigated.
Helmut Lütkepohl

Chapter 9. Joint Temporal and Contemporaneous Aggregation

Abstract
After having considered the consequences of forecasting with estimated processes for contemporaneous aggregation in Chapter 5 and for temporal aggregation in Chapters 7 and 8 the two types of aggregation will be treated jointly in this chapter. In practice the two types of aggregation are often both present. For instance, the annual tax revenues of a country that consists of a number of states, is the sum of the monthly revenues in each state. Thus, the total figure is obtained by aggregating over time (temporally) and over regions (contemporaneously). In this chapter the particular type of temporal and contemporaneous aggregation is considered where all the disaggregate figures are simply added as in the foregoing example. This simplification is justified because it suffices to illustrate the pertinent problems. On the other hand, it helps to simplify some of the arguments.
Helmut Lütkepohl

Chapter 10. Epilogue

Abstract
In this study a number of predictors for temporal and/or contemporaneous aggregates have been compared using the MSE as measure of forecast precision. In particular, the impact of the level of aggregation of the data used for prediction has been investigated. The following results have been obtained in this study.
Helmut Lütkepohl

Backmatter

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