SASCODE 8 
 Transfer Function Simulation 


*  For this homework we will try some simulation. 

Create an AR(1) series X(t); 

Data GenX; 
  Z=0; do t = 1 to 100; Z=.8*Z+normal(1827655); 
  X= 100 + 3*Z; output; end; 
proc gplot; plot X*T; 
symbol1 v=none i=join; title "X Series";

* Estimate an AR(2) model.  Is the second order autoregressive 
  coefficient significant?  Should it be?  

  Estimate an AR(1) model for the data.  How many standard errors 
  away from the true value is the estimated mean?   How many 
  standard errors away from the true value is the autoregressive 
  coefficient?   

  Generate a series 
     Y(t) = -350 + 3*X(t-1) + 2*X(t-2) + e(t) + .6*e(t-1); 

Data GenY; set GenX; e1=e; e=normal(12345); 
if _n_=1 then e1=normal(8523609); 
Y = -350 +3*X1 + 2*X2 +e + .6*e1; X2=X1; X1=X; retain X1 X2 e;
PROC GPLOT; PLOT (X Y)*t/overlay; 

* What is the theoretical correlation between 
Y(t) and X(t)?  Compute the theoretical correlation 
between Y(t)-.8 Y(t-1) and X(t-j) - .8 X(t-j-1)  for 
j=0, 1, 2, 3, 4. Compute the theoretical mean of Y. 
 
  Notice that the transformation from Y(t) to Y(t)-.8 Y(t-1)
and similarly for X gives cross correlations that show 
the true relationship between Y and X.  Why is .8 used 
here?  What sort of time series model describes the 
tranformed series X(t)-.8 X(t-1)?

  Compute the cross correlations between X(t) and Y(t) as 
follows; 

PROC ARIMA data=GENY; 
  IDENTIFY VAR=Y CROSSCOR=(X); 


*  Explain why these cross correlations, even though they 
  are computed correctly are not as informative as one might 
  like.

  Compute the cross-correlations between the transformed Y and X. 
  This is easy in PROC ARIMA.  If you have a model for X before  
  your Y identify statement, SAS will automatically remember
  your X model and use it to "prewhiten" (transform) both 
  X and Y for the cross-correlations.  Which of these SHOULD be 
  significant?  Does the simulation match theory reasonably well? 
  Just comment on the estimated cross-correlations. 

  The ESTIMATE statement has an INPUT option.  For example
        ESTIMATE INPUT = (X1 X2) p=2 q=1
  fits a regression of Y, say, on X1 and X2 with an ARMA(2,1) 
  error series.  The statement 
        ESTIMATE INPUT = (X1 X2) PLOT
  will produce the ACF etc. of the regression residuals. 
  
  Any INPUT variable can be preceded by a backshift operator. 
The form is s$(n1, n2, ..., nq)/(d1, d2, ..., dp)
where s is a pure delay, n1, n2 etc. are numerator lags and
d1, d2, ... are denominator lags. 

Examples
 3$(1,2) indicates  (A0 - A1 B - A2 B**2)X(t-3)  
 2$ / (1) indicates A0/(1 - A1 B)X(t-2) 
    = A0 (X(t-2) + A1 X(t-3) + A1**2 X(t-4) + A1**3 X(t-5) + ...)
 and 
 $(1)/(1) indicates (A0 - A1 B)/(1 - A2 B) X(t)

where A0, A1, etc. are parameters to be estimated 

  With these facts in mind, fit the input model suggested by the 
  crosscorrelations.  Use the PLOT option and explain whether or
  not the estimated ACF etc. have the shapes they should based on 
  the true error model used to generate the data. 

  Now refit with the appropriate error model, forecast 10 
  periods ahead and plot the original series with the forecasts
  and upper and lower 95% prediction limits coming off the end
  of the historic data. Note: In PROC ARIMA, if you model X
  then fit a transfer function asking for forecasts, ARIMA 
  will automatically generate enough X forecasts to allow the 
  requested Y forecasts to be computed.  The error in forecasting 
  X will be incorporated into the Y forecast intervals.  Why are
  the first couple of forecast intervals so much narrower than 
  the others? 

  If you SUPPLY future values of X they are treated as known in 
  advance, that is, no error in future Xs is assumed.  This is 
  like PROC AUTOREG where future values of the inputs MUST 
  be supplied and are assumed to be known without error. With 
  this in mind you might want to 
   (Optional - not graded) 
  append the forecasts of X into the X data set and rerun the 
  transfer function model.  Your forecasts should match those 
  above but the intervals should be narrower as suggested in 
  the above paragraph. 

  ;