Estimating State Space Model Parameters

Question

I'm having a bit of difficulty estimating parameters in DLM in R and I was wondering if I could get a bit of help with it. I have a system of equations given as:

$p_{t} = m_{t} + s_{t}$

$m_{t} = m_{t-1} + w_{t}$

$s_{t} = \phi_{1}s_{t-1} + \phi_{2}s_{t-2} + \epsilon_{t}$

which can be written in state space form as follows:

Observation Equation:

$p_{t} = \begin{bmatrix} 1 & 1 & 0 \end{bmatrix} \begin{bmatrix} m_{t}\\ s_{t} \\ s_{t-1}\end{bmatrix}$

State Equation:

$ \begin{bmatrix} m_{t}\\ s_{t} \\ s_{t-1}\end{bmatrix} = \begin{bmatrix} 1 & 0 & 0 \\ 0 & \phi_{1} & \phi_{2}\\ 0 & 1 & 0\end{bmatrix}\begin{bmatrix} m_{t-1}\\ s_{t-1} \\ s_{t-2}\end{bmatrix} + \begin{bmatrix} 1 & 0\\ 0 & 1 \\ 0 & 0\end{bmatrix}\begin{bmatrix} w_{t} \\ \epsilon_{t}\end{bmatrix}$

And I would like to find the parameters $\phi_{1}, \phi_{2}, \sigma^{2}_{w}, \sigma^{2}_\epsilon$

To this end, I've written the following code in DLM package in R:

buildModelDLM = function(p) {

     # This part evaluates the four parameters

     phi1 <- p[[1]] # This evaluates phi values in T matrix
     phi2 <- p[[2]] # This evaluates phi values in T matrix
     dW1 <- exp(p[3]) # This evaluates the variance of the first error term
     dW2 <- exp(p[4]) # This evaluates the variance of the second error term

     # This part defines the remaining matrices and initial conditions

     GG = matrix(c(1,0,0,0,phi1,phi2,0,1,0),3,3,byrow=T)
     W = diag(c(dW1, dW2))
     FF = matrix(c(1,1,0),1,3)
     C0 = 10^7 *diag(3)

     return( list( m0=rep(0,4),C0=C0, FF=FF, GG=GG, W=W, V=0) )
}


mle <- dlmMLE(HFTsubset$lmid, parm=rep(0,4), build = buildModelDLM, method="BFGS")

However, the maximum likelihood estimator only seems to be estimating one parameter and I'm not sure what I'm doing wrong, so if someone could help me with this I'd really appreciate it, thanks in advance. I also don't mind if there's another way to do this in another R package, but I would prefer to work with DLM (or maybe MARSS) if possible

Here's a test dataset:

c(0.0511682865744,0.0222506089348,0.0976710271734,0.0511682865744,0.1548647105609,0.0222506089348,0.1021052229372,0.1333063673082,0.0464063728142,0.0976710271734,0.0700856075667,0.0511682865744,0.0560021901153,0.1200027923947,0.1548647105609,0.1021052229372,0.0511682865744,0.1548647105609,0.0747363461140,0.1333063673082,0.0464063728142,0.0174469136037,0.1066996186591,0.0222506089348,0.0560021901153,0.0271286673883,0.0511682865744,0.0606246218164,0.1506588762851,0.0464063728142,0.0511682865744,0.0954010847633,0.0653194661206,0.0319830458531,0.0271286673883,0.0511682865744,0.0700856075667,0.0560021901153,0.1419334958887,0.0464063728142,0.0793655553807,0.0606246218164,0.0976710271734,0.1506588762851,0.0222506089348,0.0222506089348,0.1021052229372,0.0747363461140,0.0700856075667,0.0074720148387,0.0536355491659,0.0464063728142,0.0560021901153,0.0464063728142,0.0416216746908,0.1506588762851,0.1462625069783,0.0838814839807,0.0606246218164,0.0606246218164,0.0271286673883,0.0511682865744,0.0653194661206,0.1506588762851,0.0369103540201,0.1548647105609,0.0464063728142,0.0511682865744,0.0885601769795,0.0271286673883,0.1419334958887,0.0724136805090,0.0511682865744,0.1066996186591,0.0124225199986,0.0511682865744,0.0511682865744,0.0653194661206,0.1200027923947,0.1066996186591,0.0747363461140,0.0271286673883,0.0511682865744,0.0222506089348,0.0606246218164,0.0998453349697,0.0511682865744,0.0511682865744,0.0464063728142,0.0464063728142,0.1021052229372,0.1333063673082,0.0653194661206,0.0511682865744,0.1506588762851,0.0931259779895,0.0369103540201,0.0606246218164,0.1548647105609,0.0222506089348)

Answer 1

Try this. You had a few matrices of the wrong size.

buildModelDLM <- function(p) {
  phi1 <- p[1]
  phi2 <- p[2]
  dW1 <- exp(p[3])
  dW2 <- exp(p[4])

  GG <- matrix(c(1,0,0,0,phi1,phi2,0,1,0),3,3,byrow=T)
  W <- diag(c(dW1, dW2, 0))
  FF <- matrix(c(1,1,0),1,3)
  C0 <- 10^7 *diag(3)

  return( list(m0=rep(0,3),C0=C0, FF=FF, GG=GG, W=W, V=0) )
}

Warning: The function dlmMLE calls dlmLL to evaluate the log-likelihood. The documentation to dlmLL warns: "[t]he observation variance V in mod must be nonsingular." Yours is singular, so you should use another function.