Formula for one-sided Hodrick-Prescott filter I am not very familiar with filters. The Hodrick-Prescott filter as one can find it e.g. in wikipedia is two-sided. I also found an R implementation for this in the R package mFilter.
There the filter is given as: find $(\tau_t)_{t=1}^T$ such that
$$
\left(\sum_{t=1}^T (y_t - \tau_t)^2 + \lambda \sum_{t=2}^{T-1} (\tau_{t+1}-2 \tau_{t} + \tau_{t-1} )^2\right) \rightarrow Min.
$$
If I try to formulate a one-sided version of it myself, then I would take backward looking second order differences. I.e. find $(\tau_t)_{t=1}^T$ such that
$$
\left(\sum_{t=1}^T (y_t - \tau_t)^2 + \lambda \sum_{t=3}^{T} (\tau_{t-2}-2 \tau_{t-1} + \tau_t )^2\right) \rightarrow Min.
$$
How is the usual formulation of a one-sided Hodrick-Prescott filter and does there exist a robust R implementation?
 A: 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.

A: It turns out that the Kalman filter representation as one can find here:
State Space formulation of Hodrick-Prescott ﬁlter
yields a solution.
Having formulated the equations one can use the package SSPiR to estimate the solution.
A: This is the one sided version:
 Stock; Watson (1999). "Forecasting Inflation". Journal of Monetary Economics 44: 293–335. doi:10.1016/s0304-3932(99)00027-6.
