Solutions for Homework #9

  *** see hwk059s.sas for code ***

 a) Plot the data (of course!) 
     Although the trend is hardly a straight line, there
     is strong growth in djia over the time period (1950-1985).

 b) Fit linear trend with AR(1) errors using PROC AUTOREG
    Regression model gives
      adjclose(t) = 273.31 + 2.241*t + e(t)
                   (61.87)  (.2411)
    where t is monthly time counter and e(t) is AR(1)
    with coefficient .9577 (.0139) and noise variance estimate
    of 942.14 with 429 df.  (root mse is 30.7)
 
 c) Yes, slope is statistically significant (t = 9.30 and tiny p-value).
    Short term, this trend is not very big: a little more than 2
    when each month's noise sqrt(Var(y(t))) is over 100.  But long 
    term, this was historically significant growth -- from around 200
    post WWII to over 1500 in 1985.
  
 d) For Augmented Dickey-Fuller test, we should be looking at the
    trend alternative and no lags.  We get tau-hat = -1.73 well above
    the .05 critical value of -3.42 (we reject unit root if tau-hat 
    is too small) or a p-value over .73, so do NOT reject hypothesis 
    of unit root.

 e) An appropriate ARMA model for the differenced data is just
    ARMA(0,0), or 
      adjclose(t) = drift + adjclose(t-1) + a(t)
    since the smallest p-value for the Ljung-Box stats for the 
    differenced series is 0.1282.

 f) The estimated mean (or drift) is 3.12037 (1.476)
 g)  which should be interpreted as a drift in this random walk model,
     and compared to the slope estimated previously as 2.241.