ST 518  
D. A. Dickey, Fall 1993
Homework DAD_1
Due Tues. Sept. 9

1.  For the time series  Y  =  3 + .6 Y     +e  with  e    N(0,5)

    A. Find the mean and variance of Y.
    B. Notice that the average of 100 consecutive values of
                                                 _        _ 
        Y  - .6 Y   is almost the same as (1-.6) Y  where Y is 
       the average of 100 consecutive Y's.  Using this, find a value
                                  _
       which would be exceeded by Y only 10% of the time.  
    C. Suppose I observe Y=8 for this series.  Give me a two step ahead
       forecast with upper and lower 95% forecast error bounds.

2.  For the series Y =   e  +  2 e    with e    N(0,5), find the 
    autocovariance sequence. 
    
    If you saw this autocovariance sequence, what would you guess as the
    appropriate model?   What is the innovations variance for your model?

    To see why we would prefer this second model to the first, suppose
    you observe 10 Y's from this series and you guess that e  = 2.  

    t=  1   2   3   4   5   6   7   8   9   10
    Y  -1  -3  -1   3   1   5   2   0  -3   -2
    e   2
        
    Forecast the 11 th Y.  Now perturb your guess of the first e to, say, 3.
    Now what is the forecast?

3.  The model Y  = 100 + 1.1(Y   -100) - .3(Y   -100) + e   can be 
    reexpressed as 100 plus an infinite weighted sum of e's.  Show the 
    leading 4 terms in this representation and tell how you know the
    coefficients will die off exponentially. 

    If the variance of e is 1, what is the variance of the 1, 2 and 3
    step ahead forecast errors?  

    What does the variance of the n-step ahead forecast error variance approach 
    as n gets arbitrarily large?  What does the forecast itself approach?