Stat 4443/6443 Time Series Assignment 2
时间序列作业代写 Problem 1 [4*4=16 points]. The data of monthly numbers of sunspots recorded from 1964 till 1983 are in the file Sunspots
(Due in class on February 8, 2018)
Part I Practical problems 时间序列作业代写
Problem 1 [4*4=16 points]. The data of monthly numbers of sunspots recorded from 1964 till 1983 are in the file Sunspots1964_1983.mtw in the D2L.
a)Do a time series plot of the monthly numbers of sunspots recorded from 1964 till 1983. Comment on trend, seasonal effects and variations.
b)Take square root of the monthly numbers of sunspots recorded from 1964 till 1983. Do a time series plot of this square root transformed sunspot data. Comment on trend, seasonal effects and variations.时间序列作业代写
c)Take the difference of lag 1 on the square root transformed sunspot data. Comment on trend, seasonal effects and
d)Plot the ACF and PACF functions of the difference data in part c) and comment onpossible autocorrelations.
Part II. Theoretical problems 时间序列作业代写
Problem 2. [10 points each, 2*10 = 20 points]
Let { Zt } be a sequence of independent normal random variables, each with mean 0 and variance σ2, and let a, b, and c be constants. Which, if any, of the following processes are stationary? For each stationary process specify the mean and autocovariance function.
- Xt = a + bZt +cZt−2
- Xt = Z1 cos(ct) + Z2sin(ct)
- [Hint: cos(φ – θ) = sin(φ)sin(θ) + cos(φ)cos(θ)]
Problem 3. [15 + 15 = 30 points] 时间序列作业代写
Given a seasonal series of monthly observations with a linear trend m t = a + bt and seasonal effect s t = s t -12 for all t. In addition, {Yt} is a zero mean white noise. Show that
A)for an additive model X t = m t + s t + Yt , the new time series Ñ12 X t = X t – X t –12 is weakly stationary.时间序列作业代写
B)for a multiplicative model X t = m ts t + Yt , the new time series Ñ12 X t = X t – X t –12 is notweakly stationary.
Problem 4. [10 + 16 = 26 points]
A)Let Wt be a white noise process { Wt } ~WN(0, s2). Derive the autocovariance Cov(Vt, Vt+2) for the 3-point moving average process Vt = 0.2Wt-1 + 0.5Wt + 3Wt+1.
B)Let Xt be a random walk with a constant drift dsuch that Xt = d + Xt-1 + Wt where Wt is a white noise { Wt } ~ WN(0, s2).时间序列作业代写
i)Derive the expectation E(Xt) and covariance Cov(Xt, Xs) =min{s,t}s2.
ii)Derive the autocovariance Cov( ÑXt, ÑXt-1 ) for the difference process ÑXt = Xt -Xt-1.
Problem 5. [16 points]
Which of a) – d) is correct? Prove your answer carefully.
Part III. [for Master Students in Statistics, bonus for other students]时间序列作业代写
Problem 6. [10 points each, 2*10 = 20 points]
Let { Zt } be a sequence of independent normal random variables, each with mean 0 and variance σ2, and let a, b, and c be constants. Which, if any, of the following processes are stationary? For each stationary process specify the mean and autocovariance function.时间序列作业代写
A)Xt = Zt cos(ct) + Zt−1sin(ct)
B)Xt =ZtZt−1
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