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Stat1代写 Time Series代写 functions代写 additive model代写

2020-11-27 17:49 星期五 所属: 作业代写 浏览:333

Stat1代写

Stat 4443/6443 Time Series Assignment 2

Stat1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 Stat1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.Stat1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.

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Stat1代写

Part II. Theoretical problems Stat1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.

  1. Xt = a + bZt +cZt−2
  2. Xt = Z1 cos(ct) + Z2sin(ct)
  3. [Hint: cos(φ – θ) = sin(φ)sin(θ) + cos(φ)cos(θ)]

Problem 3. [15 + 15 = 30 points] Stat1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.Stat1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).Stat1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]

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Stat1代写

Which of a) – d) is correct? Prove your answer carefully.

Part III. [for Master Students in Statistics, bonus for other students]Stat1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.Stat1代写

A)Xt = Zt cos(ct) + Zt−1sin(ct)

B)Xt =ZtZt−1

Stat1代写
Stat1代写

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