Time Series – State Space Formulation of Hodrick-Prescott Filter Explained

filterkalman filtertime seriestrend

I would like to apply the Kalman filter in order to get a causal Hodrick-Prescott ﬁlter.
The Hodrick-Prescott ﬁlter models a time series $(y_t)_{t=0}^T$ as
$$
y_t = \tau_t + c_t
$$
where $\tau_t$ is a trend component and $c_t$ is a cyclical component.

This reference defines a state space formulation of the form
$$
y_t = \tau_t + c_t
$$
as the measurement equation and
$$
\tau_t = 2 \tau_{t−1} − \tau_{t−2} + \epsilon_t
$$
for the unobservable trend.

I have three questions on this:

A) $c_t$ is assumed to be a random error here, right? Normally distributed with constant variance. This seems a difficult assumption to me.

B) What's the logic behind the equation for the trend?

C) Does anybody know a different state space formulation for this problem? Or a nother reference?

Thanks!

Best Answer

A) Actually, your reference says "a stationary residual or cyclical component, $c_t$", while this calls $c_t$ a 'cyclical component'.

Maravall and del Rio (2001) say "a residual, $c_t$, to be called “cycle”."

B) This formulation follows from the form of $A$; the second difference of $\tau$ is the error term.

C) some other references you may find helpful:

http://www.reservebank.govt.nz/research/discusspapers/dp03_02.pdf

http://www.stat.itam.mx/seminarios/Resumens/p136.pdf (see p6 which is relevant to your B, but they appear to have a sign in the A-matrix wrong - c.f. your link, p13)

http://www3.istat.it/dati/pubbsci/contributi/Contributi/contr_2005/2005_07.pdf

http://www.terrapub.co.jp/journals/jjss/pdf/3801/38010041.pdf

Related Solutions

Solved – Formula for one-sided Hodrick-Prescott filter

An implementation in R using the dlm package is shown below. It's a bit hackish, but it does the job nonetheless. You may be able to take it and adjust it to your need. I may come back and tidy this up and put the corresponding equations to help.

library(dlm)

filterHP <- function(series,lambda=1600){
  # Function description: HP filter using DLM package.
  if(!"ts" %in% class(series)) stop("series is not a \"ts\" object.")

  # Set state priors
  level <- series[1]
  slope <- mean(diff(series),na.rm=TRUE)

  # Set up HP filter in a DLM model
  model <- function(param){
    trend <- dlmModPoly(dV = 1,
                        dW = c(0,1/lambda),
                        m0 = c(level,slope),
                        C0 = 2 * diag(2))

    # AR(2) model
    cycle <- dlmModARMA(ar     = ARtransPars(c(0,0)),
                        sigma2 = 1e-07)      

    hp_filter_dlm <- trend  + cycle

    return(hp_filter_dlm)
  }

  # MLE Estimation
  MLE       <- dlmMLE(series,c(0.5,0.4),model)
  # Estimated parameters
  EstParams <- model(MLE$par)
  # Smoothed series
  Smooth_Estimates <- dlmSmooth(series,EstParams)

  # Trend and Cycle
  trend <- Smooth_Estimates$s[,1]
  cycle <- series - trend

  # Plot the data ---
  par(mfrow = c(2,1),
      oma = c(1,3,0,0) + 0.1,
      mar = c(1.5,1,1,1) + 0.1)
  plot(series,las=1,col="black")
  lines(trend,col="blue")
  legend("topleft",legend=c("Observed","Trend"),border=FALSE,bty="n",col=c("black","blue"),lwd=1)
  title(main="HP Filter - Trend")
  par(mar = c(1,1,1.5,1) + 0.1)
  plot(cycle,las=1,col="red")
  title(main="HP Filter - Cycle")
  abline(h=0)
  par(mfrow = c(1,1),
      oma = c(0,0,0,0),
      mar = c(5.1,4.1,4.1,2.1))  

  # Return the data
  data <- ts.union(series,trend,cycle)
  return(data)
}

# Load some data
data(USecon)

# Compare two implementations
filterHP(USecon[,"M1"],lambda=1600)

library(mFilter)
plot(hpfilter(USecon[,"M1"],freq=1600))

# The data used is annual, so arguably lambda = 1600 should really be 6.25.

Best Answer

Related Solutions

Solved – Formula for one-sided Hodrick-Prescott filter

Related Question