Estimating AR(1) coefficient using metropolis-Hastings algorithm (MCMC) in R - r

Question

I am trying to write a program to estimate AR(1) coefficients using metropolis-hastings algorithm. My R code is as following,
set.seed(101)
#loglikelihood
logl <- function(b,data) {
ly = length(data)
-sum(dnorm(data[-1],b[1]+b[2]*data[1:(ly-1)],(b[3])^2,log=TRUE))
}
#proposal function
proposalfunction <- function(param,s){
return(rnorm(3,mean = param, sd= s))
}
#MH sampler
MCMC <- function(startvalue, iterations,data,s){
i=1
chain = array(dim = c(iterations+1,3))
chain[i,] = startvalue
while (i <= iterations){
proposal = proposalfunction(chain[i,],s)
probab = exp(logl(proposal,data = data) - logl(chain[i,],data = data))
if(!is.na(probab)){
if (runif(1) <= min(1,probab)){
chain[i+1,] = proposal
}else{
chain[i+1,] = chain[i,]
}
i=i+1
}else{
cat('\r !')
}
}
acceptance = round((1-mean(duplicated(chain)))*100,1)
print(acceptance)
return(chain)
}
#example
#generating data
data <- arima.sim(list(order = c(1,0,0), ar = 0.7), n = 2000,sd = sqrt(1))
r=MCMC(c(0,.7,1),50000,data,s=.00085)
In the example, I must get zero for the mean and 0.7 for the coefficient and 1 for error variance. but everytime I run this code I get completely different values. I tried to adjust proposal scale but still I get the results that are far from the true values. Figure below shows the results.

Answer 1

You've flipped the sign in your log-likelihood function. This is an easy mistake to make because maximum likelihood estimation usually proceeds by minimizing the negative log-likelihood, but the requirement in MCMC is to be working with the likelihood itself (not its inverse).
Also:
dnorm() takes the standard deviation as its third argument, not the variance. You can simplify your code slightly by using head(data,-1) to get all but the last element in a vector. So your log-likelihood would be:
sum(dnorm(data[-1],b[1]+b[2]*head(data,-1),b[3],log=TRUE))
you're probably hurting yourself by fixing the candidate distribution to be independent Normal with equal SDs for all variables; allowing them to differ (although the posterior SDs are not as different as I thought they might be - about {0.02,0.016,0.0077} - so this might not be such a big problem.