Homework assignment 1
Problem 1.
1. Let Xt = A cos(ω0 t + φ), where φ is a random variable distributed uniformly on [0, 2π]. Draw two
different sample paths of the process {Xt }. Compute the autocovariance function. Is the process
stationary?
2. Let ξ ∼ N (0, 1), and defined Xt = ξ, ∀t. Draw two different sample paths and compute the autocovariance function. Is the process stationary?
Problem 2.
Let X and Y be two random variables, and E(Y ) = µ and EY 2 < ∞.
1. Show that the constant c that minimizes E(Y − c)2 is c = µ.
2. Deduce that the random variable f (X) that minimizes E[(Y − f (X))2 |X] is
f (X) = E[Y |X].
3. Deduce that the random variable f (X) that minimizes E[Y − f (X)]2 is
f (X) = E[Y |X].
Problem 3. (generalization of Problem 2).
Suppose X1 , X2 , . . . is a sequence of random variables
with E(Xt2 ) < ∞ and E(Xt ) = µ.
1. Show that the random variable f (X1 , . . . , Xn ) that minimizes
E[(Xn+1 − f (X1 , . . . , Xn ))2 |X1 , . . . , Xn ] is
f (X1 , . . . , Xn ) = E[Xn+1 |X1 , . . . , Xn ].
2. Deduce that the random variable that minimizes E(Xn+1 − f (x1 , . . . , Xn ))2 is also
f (X1 , . . . , Xn ) = E[Xn+1 |X1 , . . . , Xn ].
3. If X1 , . . . , Xn is independent identically distributed with E(Xt )2 < ∞ and E(Xt ) = µ, where µ is
known, what is the minimum mean squared error predictor of Xn+1 in terms of X1 , . . . , Xn ?
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Let {Zt } be a sequence of independent normal random variables, each with mean 0 and
Problem 4.
variance σ 2 , and let a, b, c and ω are constants. Which, if any, of the following processes are stationary?
For each process specify the mean and autocovariance function.
a. Xt = a + bZt−1 + cZt−3
b. Xt = Z1 cos(ωt) + Z2 sin(ωt)
c. Xt = Zt cos(ωt) + Zt−1 sin(ωt)
d. Xt = a + bZ0
e. Xt = Z0 cos(ωt)
f. Xt = Zt Zt−2
Let Zt be independent identically distributed N (0, 1) random variables. Define
Zt ,
t is even
√
Xt =
2
(Zt−1
− 1)/ 2, t is odd.
Problem 5.
a. Show that {Xt } is white noise W N (0, 1) but not IID noise.
b. Find E[Xn+1 |X1 , . . . , Xn ] for even and odd n and compare results.
Suppose that we want to predict a stationary series {Xt } with zero mean and autocovariance
Problem 6.
and autocorrelation functions γX (h), ρX (h) at some future time, say, n + k.
a. If we predict using only Xn and use a linear predictor X̂n+k = aXn with some scalar a, show that the
mean squared error MSE(a) = E(Xn+k − aXn )2 is minimized when a = a∗ = ρX (k).
b. Show that the minimal mean–squared error is M SE(a∗ ) = γX (0)[1 − ρX (k)].
c. If Xn+k = aXn then show that ρX (k) = 1 if a > 0 and ρX (k) = −1 if a < 0.
Problem 7.
If mt =
p+1
hence that ∆
Problem 8.
Pp
k=0 ck t
k
, t = 0, ±1, . . ., show that ∆mt is a polynomial of degree p − 1 in t and
mt = 0.
If Xt = a+bt+st +Yt , where st is a seasonal component with period 12, and Yt is a stationary
process, show that ∆∆12 Xt = (1 − B)(1 − B 12 )Xt is stationary and express its autocovariance function in
terms of that of {Yt }.
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Problem 9.
Let Xt = a+bt+Yt , where {Yt , t = 0, ±1, ±2, . . .} is an independent and identically distributed
sequence of random variables with mean 0 and variance σ 2 , and a and b are constants. Define
Wt =
q
X
1
Xt+j .
2q + 1 j=−q
Compute mean and autocovariance function of {Wt }. Is {Wt } stationary?
Problem 10.
Let {Zt } is Gaussian white noise with variance σ 2 = 1.
a. Using R software simulate and plot n = 200 observations from the following model:
Yt = St + Zt + 0.5Zt−1 ,
t = 1, . . . , 200,
where
St =
0,
10 exp{−(t − 100)/200} cos(2πt/4),
t = 1, . . . , 100,
t = 101, 102, . . . , 200.
b. Compare visual appearance of the series with the earthquake data displayed as an example in the first
lecture slides. Is it the sequence {Xt } stationary?
c. What is the autocovariance function of {Xt }? Is it informative? Explain.
d. Compute and plot the sample autocovariance function.
Your solution should contain all relevant graphs and corresponding R commands that produced the simulated
series and graphs.
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