# Calculating the likelihood of time series data when there are missing data

I am trying to calculate the log-likelihood of some time series data given parameter sets estimated in BUGS. I can not figure out how to handle some missing values at random points in time.

For the complete data situation, such as $Y=(0.1,0.3,0.5,0.4,0.2,0.1)$, (real data is much longer) I have fitted a time series model assuming errors are normally distributed. For example, my BUGS code is something like:

for(t in 2:6){
y[t] ~ dnorm(y.mean[t], tau)
y.mean[t] <- phi0 + phi1*y[t-1]
}


i.e., the data is assumed to follow a normal distribution: $y_t \sim N(\phi_0+\phi_1 y_{t-1}, \sigma^2), 2<t<6$, where $\sigma$ is the standard deviation to the tolerance tau in the BUGS code. In R I can derive the log-likelihood of data,

$l(y_t|\phi_0,\phi_1,\sigma,y_{t-1})=\sum_{t=2}^{t=6}P(Y_t=y_t)$

where $P(Y_t=y_t)$ is a normal probability density function, given a single MCMC sample of parameters (for example, $\phi_0=0.25$, $\phi_1=0.55$ and $\sigma=0.35$) as such:

> y <-c(0.1,0.3,0.5,0.4,0.2,0.1)
> phi0 <- 0.25
> phi1 <- 0.55
> sigma <- 0.35
>
> ymean <- phi0+phi1*y[1:5]
> ll <- sum(dnorm(y[2:6], mean = ymean , sd = sigma, log = TRUE))
> ll
[1] -0.01241878


However, I am stuck when it comes to performing the correct calculation of the log-likelihood when there are missing data, say $Y=(0.1,0.3,0.5,NA,0.2,0.1)$ and $NA$ is missing? I believe that y[4] has to dropped in the R code/likelihood calculation. I am not sure how (or if) to estimate ymean[5], which is dependent on a missing $y_4$? BUGS of course provides a MCMC sample(s) for this missing data point, but should I use it, or do I keep the R code as is, adjusting for NA in the ymean[5] with na.rm=TRUE when summing over probability density functions:

> y[4]<-NA
> ymean<-phi0+phi1*y[1:5]
> ymean
[1] 0.305 0.415 0.525    NA 0.360
> ll <- sum(dnorm(y[2:6], mean = ymean , sd = sigma, log = TRUE), na.rm=TRUE)
> ll
[1] 0.08714057

-

Thanks again, sorry to push you, but what is the justification for including the posterior values of $y_4$? I thought the likelihood function was equal to the probability of the observed data, given the parameter values...$l(Y|\phi_0,\phi_1,\sigma)$. Do the posterior values of $y_4$ generated by BUGS qualify as parameters...$l(Y|\phi_0,\phi_1,\sigma,y_4)$? –  gjabel Jan 17 '13 at 9:51