Homework 6 
 American Airlines 

  On Sept. 11, 2001 terrorist attacks on the U.S. involving, among others, 
American Airlines jets produced drastic effects on many aspects of life in 
the U.S. including effects on the stock market.  AMR is the parent company 
for American Airlines and the data below have information about stock in 
that company. 

 Questions 

 
Explain why there is a colon :date7. in the informat for date. 
What happens if it is omitted?  Explain. 

 
Plot opening, high, low, and closing prices versus date. Put a vertical 
reference line at Sept. 11. 

  
Plot volume versus date with the same reference line, then plot volume
versus date only for dates before Sept. 11. Note the interesting upturn in 
volume prior to the terrorist attacks.  
[Food for thought, no answer expected: Would you have considered the doubling 
of volume Friday prior to the attack unusual enough to have caught your 
attention???

  
Look at the PROC ARIMA output. Does the ARMA(1,1) model seem 
to fit well? Write down the model, and the theoretical ACF you would 
get if the estimated parameters were taken as true values. 
Note the "where" statement. 

  
For each of these volume-related variables, compute and 
plot P_0j/(4*3.14159) and S_0j versus frequency, overlaid.  
Use only the pre-attack data. Here P_0j is the 
periodogram and S_0j the smoothed spectral estimate using 
weights 1 2 3 4 5 4 3 2 1 for the jth variable:
For Volume
for first differences:  D = volume - lag(volume) 
   [This looks like overkill in that it wipes out the low 
   frequency components]
for D8 = volume - rho*lag(volume) 
   where rho is the AR coefficient from our ARMA(1,1) model. 
   [This uses the AR "filter" from your ARIMA model and 
    should resemble an MA(1) spectrum]
For R = residuals from your PROC ARIMA model 
I suggest symbol1 v=none i=join c=green; for S_0j and 
          symbol2 v=dot i=none c=red h=.6; for P_0j. 
You wonder if there is some sort of weekly periodicity in volume. 
This is 1 year of data, but the stock market is not open every day.  
Divide the number of days in the (complete) dataset by 52 to get the 
(average) number _____ of observations per week. Record this in your 
homework answers as well as the frequency ____  associated with this 
number.  Put a vertical reference line in your graphs at this frequency.  
Note the locally high ordinates there, suggesting a mild weekly cycle.   
Note also that our ARIMA model, while it seemed to fit well, did not 
pick up this cycle.  



 Optional exercises (Not graded - don't hand in)

Test the pre-attack data for stationarity.
In the full dataset, insert a dummy variable for post-attack 
data, run a regression of volume on this and check the Durbin-Watson 
statistic. Analyse these regression residuals. 
(We will be discussing nicer ways of modeling such "intervention" effects)
Compute a histogram of volume on teh pre-attack data. 
Does it seem skewed?  Does taking logarithms help? 
Rerun your plots on the logarithms. 

SAS Code