4.23 Suppose xt is a stationary series, and we apply two filtering operations in succession, say, and

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4.23 Suppose xt is a stationary series, and we apply two filtering operations in succession, say,

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and then

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(a) Show the spectrum of the output is fz(ω) = |A(ω)|2|B(ω)|2fx(ω), where A(ω) and B(ω) are the Fourier transforms of the filter sequences at and bt, respectively.

(b) What would be the effect of applying the filter ut = xt − xt−1 followed by vt = ut − ut−12 to a time series?

(c) Plot the predicted frequency responses of the simple difference filter and of the seasonal difference of the first difference. Filters like these are called seasonal adjustment filters in economics because they tend to attenuate frequencies at multiples of the monthly periods. The difference filter tends to attenuate low-frequency trends.

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