Maximum likelihood estimation in ARMA Models

  Data: Annual number of lynx trappings in the Mackenzie River
     District for the period 1821 to 1934.

  Previously we used the Yule-Walker equations to fit an AR(1) and AR(2)
model to the data.

  In Splus, we find the MLE's using the command arima.mle

  NOTE: The command arima.mle does not explictly allow for a non-zero
  mean of the time series. There are two ways for us to account for
  this, which will be illustrated in the examples:

   1. Subtract the mean from the series

   2. Tell Splus to estimate a constant term in our model, i.e.

    x_t = c + phi_1 * x_(t-1) + ...

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 (a) AR(2). We asking Splus to find the MLE, we need to give it a
 starting "guess" at the estimates.  

  First, do this by subtracting mean.

> lynx.ar2mle <- arima.mle(lynx - mean(lynx), model = list(ar=c(.2, .2)))   

   Some output deleted:

 > lynx.ar2mle

Call: arima.mle(x = lynx - mean(lynx), model = list(ar = c(0.2, 0.2)))
Method:  Maximum Likelihood
Model :  2 0 0
 
Coefficients:
        AR : 1.15256 -0.60646
 
Variance-Covariance Matrix:
             ar(1)        ar(2)
ar(1)  0.005644742 -0.004049843
ar(2) -0.004049843  0.005644742 

  Second, use the xreg option to introduce a constant term:

> lynx.ar2mlexr <- arima.mle(lynx,  model = list(ar=c(.2, .2)), xreg=1) 

> lynx.ar2mlexr

Coefficients:
        AR : 1.15244 -0.60623
 
Variance-Covariance Matrix:
             ar(1)        ar(2)
ar(1)  0.005647161 -0.004051740
ar(2) -0.004051740  0.005647161

Coeffficients for regressor(s): intercept
[1] 1564.778

########################################
 
 (b) MA(1). 

 > lynx.ma1mle <- arima.mle(lynx - mean(lynx), model = list(ma=.2))   

 > lynx.ma1mle
 
Coefficients:
        MA : -0.79335

###################################################

 (c)  ARMA(1,1).

> lynx.arma <- arima.mle(lynx, model = list(ar = .2, ma=.2), xreg=1)

> lynx.arma

Coefficients:
        AR : 0.54776
        MA : -0.61034
 
Variance-Covariance Matrix:
            ar(1)       ma(1)
ar(1) 0.008222784 0.003866884
ma(1) 0.003866884 0.007371418

Coeffficients for regressor(s): intercept
[1] 1587.058