AIC, Model Selection and Diagnostics : Lynx Data 

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

  Let's see what the AIC tells us about our potential models. We can
get Splus to find the AIC when doing ML estimation and when using the
Yule-Walker estimates.

 First, we'll look at the Yule-Walker estimates. In this case, Splus
will tell us it's "preferred" model, based on minimizing AIC.

 WARNING: This may give a model that is more complicated than we like.
As in all model selection, we want to reach a balance between a "good"
model and one that isn't too complicated.

  > lynx.ywaic <- ar.yw(lynx)  

 # The summary command tells us what results are saved in 
 # lynx.ywaic

 >  summary(lynx.ywaic)
           Length Class      Mode
     order   1          numeric
        ar   8          numeric
  var.pred   1          numeric
       aic  21          numeric
    n.used   1          numeric
 order.max   1          numeric
partialacf  20          numeric
     resid 114    rts   numeric
    method   1          character
    series   1          character

> lynx.ywaic$order # This tell us that, based on AIC, an AR(8) 
[1] 8              # model is preferred

  What model(s) did our plots suggest?

  # Print the AIC values for AR(1) through AR(20). Note that Splus
  # transforms the AIC values by subtracting the minimum AIC from each
  # So the smallest AIC listed will always be 0.

> lynx.ywaic$aic
 [1] 133.867676  55.642578   9.299805  11.125488   5.794189   6.774414
 [7]   8.464844   8.843018   0.000000   1.660156   2.906738   4.000488
[13]   5.820068   6.699463   8.676270   9.203613   9.869385  11.865234
[19]  12.316406  14.285889  16.250732


  Next, let's look at some of the AIC values from ML estimation, which
were found on a previous handout:

> lynx.ar2mle$aic # AR(2)
[1] 1840.895     

> lynx.ma1mle$aic  # MA(1)
[1] 1913.947

> lynx.arma$aic  # ARMA(1, 1)
[1] 1872.079

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

  Next, we'll conduct some model diagnostics, using  the 1-step 
ahead prediction residuals.

  If we have done ML estimation, we can use the commands "arima.diag" 
and "arima.diag.plot"


 > lynxar2.diag <- arima.diag(lynx.ar2mle) 

 This command gives the attached plot.


 What do these plots suggest?



 What is contained in lynxar2.diag?

> summary(lynxar2.diag)
            Length Class      Mode
   acf.list   7    acf   list          # ACF of residuals
  pacf.list   7    acf   list          # PACF of residuals
     acf.se  21          numeric       # Standard error of ACF values 
        gof   4          list
  std.resid 114    ts    numeric       # Standardized residuals
     series   1          character
model.order   3          numeric
   reg.coef   0          NULL

 Another way to get the plot is

> lynxar2.diag2 <- arima.diag(lynx.ar2mle, plot = F)
> arima.diag.plot(lynxar2.diag2)

 Also attached are the residual plots from the MA(1) and ARMA(1, 1)
cases. What do these plots suggest?