Manual Maximum-Likelihood Estimation of an AR-Model in R - r

Question

I am trying to estimate a simple AR(1) model in R of the form y[t] = alpha + beta * y[t-1] + u[t] with u[t] being normally distributed with mean zero and standard deviation sigma.
I have simulated an AR(1) model with alpha = 10 and beta = 0.1:
library(stats)
data<-arima.sim(n=1000,list(ar=0.1),mean=10)
First check: OLS yields the following results:
lm(data~c(NA,data[1:length(data)-1]))
Call:
lm(formula = data ~ c(NA, data[1:length(data) - 1]))
Coefficients:
(Intercept) c(NA, data[1:length(data) - 1])
10.02253 0.09669
But my goal is to estimate the coefficients with ML. My negative log-likelihood function is:
logl<-function(sigma,alpha,beta){
-sum(log((1/(sqrt(2*pi)*sigma)) * exp(-((data-alpha-beta*c(NA,data[1:length(data)-1]))^2)/(2*sigma^2))))
}
that is, the sum of all log-single observation normal distributions, that are transformed by u[t] = y[t] - alpha - beta*y[t-1]. The lag has been created (just like in the OLS estimation above) by c(NA,data[1:length(data)-1]).
When I try to put it at work I get the following error:
library(stats4)
mle(logl,start=list(sigma=1,alpha=5,beta=0.05),method="L-BFGS-B")
Error in optim(start, f, method = method, hessian = TRUE, ...) :
L-BFGS-B needs finite values of 'fn'
My log-likelihood function must be correct, when I try to estimate a linear model of the form y[t] = alpha + beta * x[t] + u[t] it works perfectly.
I just do not see how my initial values lead to a non-finite result? Trying any other initial values does not solve the problem.
Any help is highly appreciated!

Answer 1

This works for me -- basically what you've done but leaving out the first element of the response, since we can't predict it with an AR model anyway.
Simulate:
library(stats)
set.seed(101)
data <- arima.sim(n=1000,list(ar=0.1),mean=10)
Negative log-likelihood:
logl <- function(sigma,alpha,beta) {
-sum(dnorm(data[-1],alpha+beta*data[1:length(data)-1],sigma,log=TRUE))
}
Fit:
library(stats4)
mle(logl,start=list(sigma=1,alpha=5,beta=0.05),method="L-BFGS-B")
## Call:
## mle(minuslogl = logl, start = list(sigma = 1, alpha = 5, beta = 0.05),
## method = "L-BFGS-B")
##
## Coefficients:
## 0.96150573 10.02658632 0.09437847
Alternatively:
df <- data.frame(y=data[-1],ylag1=head(data,-1))
library(bbmle)
mle2(y~dnorm(alpha+beta*ylag1,sigma),
start=list(sigma=1,alpha=5,beta=0.05),
data=df,method="L-BFGS-B")