TimeSeriesBook.pdf

# Definition 18 gaussian process a stochastic process x

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Definition 1.8 (Gaussian process) . A stochastic process { X t } is called a Gaussian process if all finite dimensional distributions of { X t } are multi- variate normally distributed. Remark 1.9. A Gaussian process is strictly stationary. For all n, h, t 1 , . . . , t n , ( X t 1 , . . . , X t n ) and ( X t 1 + h , . . . , X t n + h ) have the same mean and the same co- variance matrix. At this point we will not delve into the relation between stationarity, strict stationarity and Gaussian processes, rather some of these issues will be further discussed in Chapter 8. 6 An example of a process which is strictly stationary, but not stationary, is given by the IGARCH process (see Section 8.1.4). This process is strictly stationary with infinite variance.

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14 CHAPTER 1. INTRODUCTION 1.4 Construction of stochastic processes One important idea in time series analysis is to build up more complicated process from simple ones. The simplest building block is a process with no autocorrelation called a white noise process which is introduced below. Taking moving-averages from this process or using it in a recursion gives rise to more sophisticated process with more elaborated autocovariance functions. 1.4.1 White noise The simplest building block is a process with no autocorrelation called a white noise process. Definition 1.9 (White noise) . { Z t } is called a white noise process if { Z t } satisfies: E Z t = 0 γ Z ( h ) = ( σ 2 h = 0; 0 h 6 = 0 . We denote this by Z t WN(0 , σ 2 ) . The white noise process is therefore stationary and temporally uncorre- lated, i.e. the ACF is always equal to zero, except for h = 0 where it is equal to one. As the ACF possesses no structure it is impossible to draw inferences from past observations to its future development, at least in a least square setting with linear forecasting functions (see Chapter 3). Therefore one can say that a white noise process has no memory. If { Z t } is not only temporally uncorrelated, but also independently and identically distributed we write Z t IID(0 , σ 2 ). If in addition Z t is normally distributed, we write Z t IIN(0 , σ 2 ). An IID(0 , σ 2 ) process is always a white noise process. The converse is, however, not true as will be shown in Chapter 8. 1.4.2 Construction of stochastic processes: some ex- amples We will now illustrate how complex stationary processes can be constructed by manipulating of a white noise process. In Table 1.1 we report in column 2 the first 6 realizations of a white noise process { Z t } . Figure 1.4.2 plots the first 100 observations. We can now construct a new process { X (MA) t }
1.4. CONSTRUCTION OF STOCHASTIC PROCESSES 15 by taking moving-averages over adjacent periods. More specifically, we take X t = Z t + 0 . 9 Z t - 1 , t = 2 , 3 , . . . . Thus, the realization of { X (MA) t } in period 2 is { x (MA) 2 } = - 0 . 6387 = - 0 . 8718 + 0 . 9 × 0 . 2590. 7 The realization in period 3 is { x (MA) 3 } = - 1 . 5726 = - 0 . 7879 + 0 . 9 × - 0 . 6387, and so on. The resulting realizations of { X (MA) t } for t = 2 , . . . , 6 are reported in the third column of Table 1.1 and the plot is shown in Figure 1.4.2. On can see that the averaging makes the series more smooth. In section 1.4.3 we will provide a more detailed analysis of this moving-average process.

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