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2.7. FILTERING 67where a 21 = − ³ b 11 + βb 21 + b 12+ b β 22´, a22 =1+r 11 − βr 21 + b 22 + b 12, βand a 23 = − ³ b 22 + b 12´.β Together you have the VAR(2)"∆qtz t#="a11 a 12+a 21"#" #∆qt−1+a 22 z t−1" #" #0 a13 ∆qt−20 a 23 z t−2#u qt. (2.98)u qt − βu ft(2.98) is easier to estimate than the VECM and the standard forecastingformulae for VARs can be employed without modiÞcation.2.7 FilteringMany international macroeconomic time-series contain a trend. Thetrend may be deterministic or stochastic (i.e., aunitrootprocess).Real business cycle (RBC) theories are designed to study the cyclicalfeatures of the data, not the trends. So in RBC research, the data thatis being studied is usually passed through a linear Þlter to remove thelow-frequency or trend component of the data. To understand whatÞltering does to the data you need to have some understanding of thefrequency or spectral representation of time series where we think ofthe observations as being built up from individual subprocesses thatexhibit cycles over different frequencies.Linear Þlters take a possibly two-sided moving average of an originalset of observations q t to create a new series ˜q t˜q t =∞Xj=−∞a j q t−j , (2.99)where the weights are summable, P ∞j=−∞ |a j | < ∞. One way to assesshow the Þlter transforms the properties of the original data is to seewhich frequency components from the original data that are allowed topass through and how these frequency components are weighted—thatis, are the particular frequency components that are allowed throughrelatively more or less important than they were in the original data.