Suppose Yt = 1 + 10t + t2 + Xt where {Xt} is a zero-mean
stationary...
Suppose Yt = 1 + 10t + t2 + Xt where {Xt} is a zero-mean
stationary series with autocovariance function γk. Show that {Yt}
is not stationary but that Zt = Wt − Wt−1 where Wt = Yt − Yt−1 is
stationary.
Suppose we have the process (time series)
xt = 0.5wt-1 + wt;
where wt is white noise with mean zero and variance sigma squared
w.
Find the mean, variance, autocovariance, and ACF of xt.
EvaluateRC F·dr where F = hyzexz,exz,xyexzi and C : r = ht2 +
1,t2 −1,t2 −2ti, 0 ≤ t ≤ 2. Hint: Check whether F is conservative.
If so, the Fundamental Theorem for Line Integrals might be
useful
Suppose that xt = wt + kwt−1 + kwt−2 + kwt−3 + · · · + kw0, for
t > 0, k constant, and wi iid N(0, σ2w).
(a) Derive the mean and autocovariance function for {xt}. Is
{xt} stationary?
(b) Derive the mean and autocovariance function for {∇xt}. Is
{∇xt} stationary?
1. Let Ct be consumption and Xt be a predictor of consumption.
Suppose you have quarterly data on C and X. Let D1t , D2t , D3t ,
and D4t be dummy variables such that D1t takes the value 1 in
quarter 1 and 0 otherwise, D2t takes the value 1 in quarter 2 and 0
otherwise, etc. Which of the following, if any, suffer from perfect
multicollinearity and why? a) Ct = α + βXt + γ1XtD1t + γ2XtD2t...
Consider the time series Xt = 4t + Wt + 0.9Wt−1, where Wt ∼ N(0,
σ2 ).
(i)What are the mean function and the variance function of this
time series? Is this time series stationary? Justify your
answer
(ii). Consider the first differences of the time series above,
that is, consider Yt = Xt − Xt−1. What are the mean function and
autocovariance function of this time series? Is this time series
stationary? Justify your answer
Consider the time
series given by yt =
a1yt-1 +
a2yt-2 +
εt. Where
εt is independent white noise and
yt is stationary.
A. Compute the mean of
yt.
E(yt)
B. Compute the variance of
yt.
E[yt −
E(yt)]2
C. Compute the first three
autocovariances for yt.
(E[(yt
−E(yt))(yt−i
−E(yt−i))]
i=1,2,3).
Suppose X is a normal with zero mean and standard deviation of
$10 million.
a) Find the value at risk for X for the risk tolerances h=0.01,
0.02, 0.05, 0.10, 0.50, 0.60, and 0.95.
b) Is there a relation between VaR for values of h <= 0.50
and values for h>= 0.50?
Consider the model Yt = ΦYt−3 + et − θet−1, where et has
variance σ 2 .
(a) Identify Yt as a certain SARIMA(p, d, q) × (P, D, Q)s model.
That is, specify each of p, d, q, P, D, Q, and s. You may assume
that Φ < 1.
(b) Find the variance of Yt .
(c) What are the forecasts for Yt+1 and Yt+4?
(d) What are the error variances for your forecasts above?
(e) If σ...