Filtering the data

Detrending

• Economic time series are a superposition of various phenomena

• If there exists a « business cycle », we need to insulate it from other phenomena.

• Low frequency: long-term growth, long waves, demography…

• High frequency: seasonal fluctuations, sales, football matches…

Linear detrending is not enough

• Trend growth may change, and introducing structural breaks is arbitrary

• High frequency fluctuations are not filtered.

• To insulate business cycles, one uses pass-band filters

• To understand pass-band filters, we need to study the spectrum of stochastic time-series

Consider a stationary time series

Fourier transforms

• If a sequence of numbers is deterministic, we can decompose it (in C) into a sum of deterministic cycles of all frequencies

• The weight on each frequency is computed as the Fourier transform of the original sequence

• The initial series is recovered from FT by applying the inverse Fourier transform, which proves the decomposition

Definition:

Ex.:

• The lowest frequency component:

• The highest frequency component:

Properties of the Fourier transform

• The FT is linear• The FT preserves the norm

Can we extend it to a stochastic time series?

• We can define periodicity as the average length of a shock

• Shocks only last one period: highest frequency

• Shocks last long: low frequency

• To measure the length of shocks we define the correlogram

The correlogram

Example: White Noise

Example: AR1

The spectrum

• By definition, it is the Fourier transform of the correlogram.

• Because the correlogram is symmetrical, the spectrum is real.

Example: White Noise

• A white noise has all frequencies with the same weight

Example: AR1

• An AR1 has more weight on low frequencies, more so, the more persistent it is (the higher ro)

The spectrum as a variance decomposition

• Using the inverse FT and the definition of the correlogram we get

Computing the spectrum: the covariance-generating function

Filtering

• In the time space, filter characterized by a lag polynomial applied to the series

• In the frequency space, characterized by its spectrum, i.e. the proportions in which each frequency appears

• The inverse FT transform allows to get the coefficients from the filter’s spectrum

The pass- band filter

The Hodrick-Prescott filter

• Minimize a loss function which– Increases when the trend differs more from the series– Increases when the trend accelerates or decelerates

more

Unit roots and filtering

• I(1) series are not stationary and have no MA representation

• Their correlogram has no norm and their spectrum is not defined

• To make them stationary, a filter must satisfy B(1)=0

• Consequently, it must eliminate zero frequencies

U.S. Business cycles

Stylized facts I

• All GDP components move together• Employment in all sectors is pro-cyclical• « Tension » variables are pro-cyclical:

hours, capacity utilization, employment rate

• The vacancy rate is pro-cyclical and a leading indicator

• The job loss rate is counter-cyclical and a leading indicator

Stylized facts II

• Stock prices are pro-cyclical and lead output

• The price level (detrended) is counter-cyclical

• The price level is a leading indicator

• Inflation is pro-cyclical and lagging

Stylized facts III

• Nominal wages move like prices

• Real wages are a-cyclical

• Nominal interest rates are pro-cyclical and leading

• The nominal money stock is pro-cyclical and leading