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
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?
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.
1. Consider the process {Xt} in which Xt =
Zt + 0.5Zt-1 - 2Zt-2. Investigate
the
stationarity of the process under the following conditions.
Calculate the ACF for the
stationary models.
(a) Zt ~ WN(0,(sigma)2) ; (sigma)2 <
infinity
(b) {Zt } is a sequence of i.i.d random variables with
the following distribution:
fzt(z) = 2/z3 ; z > 1
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...
{Zt} is a Gaussian White Noise in time series. Yt = Zt^2. Can we
conclude Yt as a non-Gaussian White Noise distribution by the
definition of White Noise? Why?
We have a solution of H2O (0.1 wt%) and
H2SO4 (0.9 wt%).
Find the mass density (Ibm/ft3) and molar
concentration (Ib-mole/ft3) of water in this
mixture?
Consider a regression model of monthly
time series data where we model the price of petrol which is
dependent on the Crude Oil price and Exchange rate (against US$).
Data for the three variables were collected over a 50 month period.
Suppose the estimation results showed that the Durbin-Watson (DW)
test value d is 1.38. Perform the DW test for first order positive
autocorrelation of the error terms at the 5% level of
significance.
Model: et = r...
Consider a process where 15% of the parts produced
have a defect.
If we have a sample of 250 parts, we want to find the probability
that there are between
30 and 45 defective parts in this sample.
i) Calculate the probability that there are exactly 30 defective
parts in the sample.
ii) Write out (but don't calculate) the expression for finding the
probability that between
30 and 45 parts are defective. Hint: use the binomial
distribution.
iii) use the...