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# psd - Comparing models and data Suppose we would like to...

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Comparing models and data Suppose we would like to see whether output is more or less volatile than investment We want a measure of volatility that is invariant to scale Simple variance or standard deviation is not scale invariant Suppose for example that investment is a constant fraction λ of GDP x = λy Now sd ( x ) = λsd ( y ) A simple remedy would be rescale both variables by their mean prior to computing the standard deviation sd ( x µ x ) = 1 µ x sd ( x ) = 1 λµ y λsd ( y ) = sd ( y µ y ) Now if we have variables that are non-stationary we need to apply some sort of fi lter prior to computing measures of dispersion. Suppose x gt is the value of the growth component at t, and x ct = x t x gt is the value of cyclical component Now rather than dividing by the mean, we should divide by the trend prior to computing the standard deviation The percentage standard deviation of the fi ltered series is given by % sd ( x ct ) = 100 × sd μ x ct x gt In our simple example, if our fi lter has the property that x gt = λy gt ,

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psd - Comparing models and data Suppose we would like to...

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