Log-likelihood function

we create the log-likelihood function

>  LL = function(params , rets ) {alpha = params [ 1 ] ; sigsq = params [ 2 ]

+     logf = −log(sqrt (2*pi*sigsq))

+     − ( rets−alpha )^2/(2*sigsq )

+      LL = −sum( logf ) }

We then go ahead and do the MLE using the nlm (non-linear minimization) package in R. It uses a Newton-type algorithm.

# create starting guess for par ameters

 params = c(0.001, 0.001 ) 

> res = nlm( LL,params, rets )

> params = c(0.001, 0.001 )

> res = nlm( LL,params, rets )

There were 50 or more warnings (use warnings() to see the first 50)

> res

$minimum

[1] -6.591566

$estimate

[1] 1.000000e-03 2.995424e-07

$gradient

[1]      0.0 733755.7

$code

[1] 2

$iterations

[1] 7

We now pick off the results and manipulate them to get the annualized parameters {µ, σ}.

>  alpha = res$estimate [ 1 ]

>  sigsq = res$estimate [ 2 ]

>  sigma = sqrt( sigsq/h )

> sigma

[1] 0.008688192

> mu = alpha/h+0.5*sigma^2 

> mu

[1] 0.2520377