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can someone helps me to solve problem 2 and 3??? thanks a lot...(problem 3 is in the book chapter 3 No.14)


Stat 153 Problem Set 3 (due March 2, 2016)

 

2

 

In the following problems, wt ? WN(0, ?w ).

 

?

 


 

1. We saw that xt = j=0 ?j wt?j is the unique stationary solution to the

 

AR(1) recursion xt ? ?xt?1 = wt for |?| < 1. However, there are many

 

?

 

non-stationary solutions. Check that yt = c?t + j=0 ?j wt?j is a solution

 

to the AR(1) recursion for any c ? R. Show that {yt } is not stationary

 

for c = 0.

 

2. Consider the ARMA(2,1) process that satis?es xt ? xt?1 + 1 xt?2 = wt +

 

2

 

1

 

2 wt?1 . Show that there is a causal stationary solution and compute its

 

ACF.

 

3. tsa3 3.14

 

4. Let ? ? Rn be the coe?cients of the best linear forecast of xn+1 given

 

xn:1 :

 

P (xn+1 | x1:n ) = ?T xn:1 =

 

?j xn?j+1 .

 

j?[n]

 


 

Show that the coe?cents of the best linear backcast of x0 given x1:n is

 

?

 

? = P ?, where ? ? Rn are the coe?cients of the MMSE linear forecast of

 

xn+1 given x1:n and P ? Rn×n is the anti-identity matrix. That is, show

 

that

 

?

 

P (x0 | x1:n ) = ?T xn:1 =

 

?j xj .

 

j?[n]

 


 

5. Download the data on sun spot activity. It consists of n = 285 observations

 

of sun spot activity from 1700 to 1984. We study the square root of the

 

data.

 

? Compute the sample ACF and sample PACF.

 

? Which of the following models is most appropriate for the data: AR(1),

 

A(2), MA(1), or MA(2)?

 

? Use the data to estimate the parameters of the model that you chose.

 

n

 

? Using your ?tted model, compute the forecasts Xn+h for h = 1, 2, 3, 4.

 

Give a 95% prediction interval (assuming the noise is Gaussian)

 

? The number of sun spots for the years 1985 to 1988 are 17.90, 13.40,

 

29.20, 100.20 respectively. Plot the data (including the 1985 to 1988

 

data), your forecasts, and your prediction intervals.

 


 

1

 


 

 


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