TimeSeriesBook.pdf

Random walk figure 17 realization of a random walk

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random walk Figure 1.7: Realization of a Random Walk The simple branching process is defined through the recursion X t +1 = X t X j =1 Z t,j with starting value: X 0 = x 0 . In this example X t represents the size of a population where each mem- ber lives just one period and reproduces itself with some probability. Z t,j thereby denotes the number of offsprings of the j -th member of the population in period t . In the simplest case { Z t,j } is nonnegative integer valued and identically and independently distributed. A real- ization with X 0 = 100 and with probabilities of one third each that the member has no, one, or two offsprings is shown as an example in Figure 1.8. 1.3 Stationarity An important insight in time series analysis is that the realizations in dif- ferent periods are related with each other. The value of GDP in some year obviously depends on the values from previous years. This temporal depen- dence can be represented either by an explicit model or, in a descriptive way, by covariances, respectively correlations. Because the realization of X t in some year t may depend, in principle, on all past realizations X t - 1 , X t - 2 , . . . ,
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1.3. STATIONARITY 11 0 10 20 30 40 50 60 70 80 90 100 40 60 80 100 120 140 160 180 time population size Figure 1.8: Realization of a Branching process we do not have to specify just a finite number of covariances, but infinitely many covariances. This leads to the concept of the covariance function . The covariance function is not only a tool for summarizing the statistical prop- erties of a time series, but is also instrumental in the derivation of forecasts (Chapter 3), in the estimation of ARMA models, the most important class of models (Chapter 5), and in the Wold representation (Section 3.2 in Chap- ter 3). It is therefore of utmost importance to get a thorough understanding of the meaning and properties of the covariance function. Definition 1.4 (Autocovariance function) . Let { X t } be a stochastic process with V X t < for all t Z then the function which assigns to any two time periods t and s , t, s Z , the covariance between X t and X s is called the autocovariance function of { X t } . The autocovariance function is denoted by γ X ( t, s ) . Formally this function is given by γ X ( t, s ) = cov( X t , X s ) = E [( X t - E X t )( X s - E X s )] = E X t X s - E X t E X s . Remark 1.3. The acronym auto emphasizes that the covariance is computed with respect to the same variable taken at different points in time. Alterna- tively, one may use the term covariance function for short. Definition 1.5 (Stationarity) . A stochastic process { X t } is called stationary if and only if for all integers r , s and t the following properties hold: (i) E X t = μ constant;
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12 CHAPTER 1. INTRODUCTION (ii) V X t < ; (iii) γ X ( t, s ) = γ X ( t + r, s + r ) . Remark 1.4. Processes with these properties are often called weakly station- ary, wide-sense stationary, covariance stationary, or second order stationary.
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