TimeSeriesBook.pdf

3 cross correlations between real growth of gdp and

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Figure 11.3: Cross-correlations between real growth of GDP and the Con- sumer Sentiment Index instead we filter both time series by an AR(8) model and investigate the cross-correlations of the residuals. 6 The order of the AR model was chosen deliberately high to account for all autocorrelations. The cross-correlations of the filtered data are displayed in the lower panel of Figure 11.3. As it turns out, only the cross-correlation which is significantly different from zero is for h = 1. Thus the Consumer Sentiment Index is leading the growth rate in GDP. In other words, an unexpected higher consumer sentiment is reflected in a positive change in the GDP growth rate of next quarter. 7 6 With quarterly data it is wise to set to order as a multiple of four to account for possible seasonal movements. As it turns out p = 8 is more than enough to obtain white noise residuals. 7 During the interpretation of the cross-correlations be aware of the ordering of the variables because ρ 12 (1) = ρ 21 ( - 1) 6 = ρ 21 (1).
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230 CHAPTER 11. ESTIMATION OF COVARIANCE FUNCTION
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Chapter 12 Stationary Time Series Models: Vector Autoregressive Moving-average Processes (VARMA Processes) The most important class of models is obtained by requiring { X t } to be the solution of a linear stochastic difference equation with constant coeffi- cients. In analogy to the univariate case, this leads to the theory of vector autoregressive moving-average processes (VARMA processes or just ARMA processes). Definition 12.1 (VARMA process) . A multivariate stochastic process { X t } is a vector autoregressive moving-average process of order ( p, q ) , VARMA ( p, q ) process, if it is stationary and fulfills the stochastic difference equation X t - Φ 1 X t - 1 - . . . - Φ p X t - p = Z t + Θ 1 Z t - 1 + . . . + Θ q Z t - q (12.1) where Φ p 6 = 0 , Θ q 6 = 0 and Z t WN(0 , Σ) . { X t } is called a VARMA ( p, q ) process with mean μ if { X t - μ } is a VARMA ( p, q ) process. With the aid of the lag operator we can write the difference equation more compactly as Φ(L) X t = Θ(L) Z t where Φ(L) = I n - Φ 1 L - . . . - Φ p L p and Θ(L) = I n 1 L+ . . . q L q . Φ(L) and Θ(L) are n × n matrices whose elements are lag polynomials of order smaller or equal to p , respectively q . If q = 0, Θ(L) = I n so that there is no moving-average part. The process is then a purely autoregressive one which is simply called a VAR(p) process. Similarly if p = 0, Φ(L) = I n and there 231
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232 CHAPTER 12. VARMA PROCESSES is no autoregressive part. The process is then a purely moving-average one and simply called a VMA(q) process. The importance of VARMA processes stems from the fact that every stationary process can be arbitrarily well approximated by a VARMA process, VAR process, or VMA process. 12.1 The VAR(1) Process we start our discussion by analyzing the properties of the VAR(1) process which is defined as the solution the following stochastic difference equation: X t = Φ X t - 1 + Z t with Z t WN(0 , Σ) .
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