TimeSeriesBook.pdf

2 both methods produce similar estimates they clearly

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same data as in Figure 6.2. Both methods produce similar estimates. They clearly show waves of periodicity of half a year and a year, corresponding to frequencies π 2 and π . The nonparametric estimate is, however, more volatile in the frequency band [0 . 6 , 1] and around 2 . 5. 6.4 Linear Time-invariant Filters Time-invariant linear filters are an indispensable tool in time series analysis. Their objective is to eliminate or amplify waves of a particular periodicity. For example, they may be used to purge a series from seasonal movements to get a seasonally adjusted time series which should reflect the cyclical movements more strongly. The spectral analysis provides just the right tools to construct and analyze such filters.
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130 CHAPTER 6. SPECTRAL ANALYSIS AND LINEAR FILTERS 10 -1 10 0 10 -4 10 -3 10 -2 nonparametric estimate parametric estimate using an AR(4) model π /2 Figure 6.3: Comparison of nonparametric and parametric estimates of the spectral density of the growth rate of investment in the construction sector Definition 6.3. { Y t } is the output of the linear time-invariant filter (LTF) Ψ = { ψ j , j = 0 , ± 1 , ± 2 , . . . } applied to the input { X t } if Y t = Ψ( L ) X t = X j = -∞ ψ j X t - j with X j = -∞ | ψ j | < . The filter is called causal or one-sided if ψ j = 0 for j < 0 ; otherwise it is called two-sided. Remark 6.5. Time-invariance in this context means that the lagged process { Y t - s } is obtained for all s Z from { X t - s } by applying the same filter Ψ . Remark 6.6. MA processes, causal AR processes and causal ARMA pro- cesses can be viewed as filtered white noise processes. It is important to recognize that the application of a filter systematically changes the autocorrelation properties of the original time series. This may be warranted in some cases, but may lead to the “discovery” of spurious regularities which just reflect the properties of the filter. See the example of the Kuznets filter below. Theorem 6.4 (Autocovariance function of filtered process) . Let { X t } be a mean-zero stationary process with autocovariance function γ X . Then the
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6.4. LINEAR TIME-INVARIANT FILTERS 131 filtered process { Y t } defined as Y t = X j = -∞ ψ j X t - j = Ψ(L) X t with j = -∞ | ψ j | < is also a mean-zero stationary process with autoco- variance function γ Y . Thereby the two autocovariance functions are related as follows: γ Y ( h ) = X j = -∞ X k = -∞ ψ j ψ k γ X ( h + k - j ) , h = 0 , ± 1 , ± 2 , . . . Proof. We first show the existence of the output process { Y t } . For this end, consider the sequence of random variables { Y ( m ) t } m =1 , 2 ,... defined as Y ( m ) t = m X j = - m ψ j X t - j . To show that the limit for m → ∞ exists in the mean square sense, it is, according to Theorem C.6, enough to verify the Cauchy criterion E Y ( m ) t - Y ( n ) t 2 -→ 0 , for m, n → ∞ . Taking without loss of generality m > n , Minkowski’s inequality (see Theo- rem C.2 or triangular inequality) leads to E m X j = - m ψ j X t - j - n X j = - n ψ j X t - j 2 1 / 2 E m X j = n +1 ψ j X t - j 2 1 / 2 + E - m X j = - n - 1 ψ j X t - j 2 1 / 2 .
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