Bayesian Statistics: Time Series Analysis Coursera Quiz Answers 2022 | All Weeks Assessment Answers [πŸ’―Correct Answer]

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About Bayesian Statistics: Time Series Analysis Course

This course is designed for working data scientists and aspiring statisticians alike.
This is the fourth and final course in a sequence of four that will introduce the fundamentals of Bayesian statistics.
It expands upon the topics covered in the course Bayesian Statistics: From Concept to Data Analysis, Techniques and Models, as well as Mixture models.

Analysis of time series focuses on modeling the dependencies that exist between the components of a sequence of variables that are time-related.
In order for you to do well in this class, you should have some background knowledge in calculus-based probability, the principles of maximum likelihood estimation, and Bayesian inference.

You will learn how to construct models that are able to describe temporal dependencies, as well as how to perform Bayesian inference and forecasting for the models that you have constructed.

Using the open-source and freely available software R, you will put what you’ve learned into practise using sample databases.

Your instructor, Raquel Prado, will guide you from the fundamental ideas behind modeling temporally dependent data to the implementation of particular classes of models.

Course Apply Link – Bayesian Statistics: Time Series Analysis

Bayesian Statistics: Time Series Analysis Quiz Answers

Practice Quiz : Objectives of the course

Q1. In this course will focus on models that assume that (mark all the options that apply):

  • The observations are realizations from spatial processes, where the random variables are spatially related
  • The observations are realizations from time series processes, where the random variables are temporally related
  • The observations are realizations from independent random variables

Q2. In this course we will focus on the following topics

  • Some classes of models for non-stationary time series
  • Models for univariate time series
  • Models for multivariate time series
  • Some classes of models for stationary time series

Q3. Some of the goals of time series analysis that we will illustrate in this course include:

  • Online monitoring
  • Analysis and inference
  • Forecasting
  • Clustering

Q4. In this course we will study models and methods for

  • Equally spaced time series processes
  • Discrete time processes
  • Unequally spaced time series processes
  • Continuous time processes

Q5. In this course you will learn about

  • Nonparametric methods of estimation for time series analysis
  • Normal dynamic linear models for non-stationary univariate time series
  • Bayesian inference and forecasting for some classes of time series models
  • Spatio-temporal models
  • Non-linear dynamic models for non-stationary time series
  • Autoregressive processes

Quiz: Stationarity, the ACF and the PACF

Q1. Ytβ€‹βˆ’Ytβˆ’1​=etβ€‹βˆ’0.8etβˆ’1​

How is this process written using backshift operator notation ({B}B) ?

  • (1βˆ’B)Yt​=(1βˆ’0.8B)et​
  • None of the above
  • BYt​=(1βˆ’0.8B)et​
  • B(Ytβ€‹βˆ’Ytβˆ’1​)=0.8Bet​

Q2. Which of the following plots is the most likely to correspond to a realization of a stationary time series process?

  • .
f39coV4hT 6 XKFeIT qA 8a89a85f6c5946f5b229a485081720f1 ts1
  • .
FHWMXVMUS m1jF1TFPv5oQ 9bb4afea3b0b42df9153c867541a62f1 ts3
  • .

Q3. If \{Y_t\}{Yt​} is a strongly stationary time series process with finite first and second moments, the following statements are true:

  • {Yt​} is also weakly or second order stationary
  • {Yt​} is a Gaussian process
  • The variance of Yt​, Var(Y_t),Var(Yt​),changes over time
  • The expected value of Yt​, E(Y_t),E(Yt​),does not depend on t.t.

Q4. If \{Y_t\}{Yt​} is weakly or second order stationary with finite first and second moments, the following statements are true:

  • If \{Y_t\}{Yt​} is also a Gaussian process then \{Y_t\}{Yt​} is strongly stationary
  • {Yt​} is also strongly stationary
  • None of the above

Q5. W​hich of the following moving averages can be used to remove a period d=8d=8 from a time series?

  • 1​/8ytβˆ’4​+41​(ytβˆ’3​+ytβˆ’2​+ytβˆ’1​+yt​+yt+1​+yt+2​+yt+3​)+81​yt+4​
  • 1/8βˆ‘j=βˆ’88​ytβˆ’k
  • 1/2​(ytβˆ’4​+ytβˆ’3​+ytβˆ’2​+ytβˆ’1​+yt​+yt+1​+yt+2​+yt+3​+yt+4​)
  • 1/8​(ytβˆ’4​+ytβˆ’3​+ytβˆ’2​+ytβˆ’1​+yt​+yt+1​+yt+2​+yt+3​+yt+4​)

Q6. Which of the following moving averages can be used to remove a period d=3d=3 from a time series?

  • 1/2​(ytβˆ’1​+yt​+yt+1​)
  • 1/3​(ytβˆ’1​+yt​+yt+1​)
  • None of the above

Quiz: The AR(1) definitions and properties

Q1. Which of the following plots corresponds to the PACF for an AR(1) with \phi = 0.8Ο•=0.8?

  • None of above is correct
  • .
O4eC8tQYS96HgvLUGBve w d7ee14b56b5e4dec839aadd83ee180f1 pacf1
  • .
R8PlNid9QYqD5TYnffGKlQ 98c9a17d5f7a4345a6212fb1485b56f1 pacf3
  • .
PJJ Ln9MR4 Sfy5 TLePrg 03642c7436f448a0864346020759bdf1 pacf2

Q2.

  1. Which of the following AR(1) processes are stable and therefore stationary?
  • Yt​=0.9Ytβˆ’1​+Ο΅t​,Ο΅tβ€‹βˆΌi.i.d.N(0,v)
  • Yt​=Ytβˆ’1​+Ο΅t​,Ο΅tβ€‹βˆΌi.i.d.N(0,v)
  • Yt​=βˆ’2Ytβˆ’1​+Ο΅t​,Ο΅tβ€‹βˆΌi.i.d.N(0,v)
  • Yt​=βˆ’0.8Ytβˆ’1​+Ο΅t​,Ο΅tβ€‹βˆΌi.i.d.N(0,v)

Q3. Which of the statements below are true?

  • The ACF coefficients of an AR(1) with AR coefficient \phi \in (-1,1)Ο•βˆˆ(βˆ’1,1) and \phi \neq 0ϕ​=0 are zero after lag 1
  • The PACF coefficients of an AR(1) with AR coefficient \phi \in (-1,1)Ο•βˆˆ(βˆ’1,1) and \phi \neq 0ϕ​=0 are zero after lag 1
  • The ACF of an AR(1) with coefficient \phi=0.5Ο•=0.5 decays exponentially in an oscillatory manner
  • The ACF of an AR(1) with AR coefficient \phi=0.8Ο•=0.8 decays exponentially

Q4. Which of the following corresponds to the autocovariance function at lag h=2,h=2, \gamma(2)Ξ³(2), of the autoregressive process Yt​=0.7Ytβˆ’1​+Ο΅t​,Ο΅tβ€‹βˆΌi.i.d.N(0,v), with v=2.v=2.

  • 3.9216
  • 0.490.49
  • 1.9216

Q5. What is the PACF coefficient at lag 1 for the AR(1) process

yt​=βˆ’0.7ytβˆ’1​+Ο΅t​ with \epsilon_t \stackrel{iid}\sim N(0,1)Ο΅tβ€‹βˆΌiidN(0,1)?

  • 0.70.7
  • -0.7βˆ’0.7
  • \approx 1.96β‰ˆ1.96
  • 00

Q6. What is the autovariance function at lag 1, \gamma(1)Ξ³(1) of the AR(1) process

yt​=0.6ytβˆ’1​+Ο΅t​ with \epsilon_t \stackrel{i.i.d.}{\sim} N(0,v)Ο΅tβ€‹βˆΌi.i.d.N(0,v) ? with variance v=2v=2.

  • 1.5625
  • 1.875
  • 1
  • 0.6

Q7. Consider an AR(1) process y_t = -0.5 y_{t-1} + \epsilon_t,yt​=βˆ’0.5ytβˆ’1​+Ο΅t​, with \epsilon_t \stackrel{i.i.d.}{\sim} N(0,1)Ο΅tβ€‹βˆΌi.i.d.N(0,1). Which of the following statements are true?

  • The autocovariance process of this function decays exponentially as a function of the lag hh and it is always negative
  • The autocovariance process of this function decays exponentially as a function of the lag hh and it is always positive
  • The PACF coefficient at lag 1 \phi(1,1)Ο•(1,1) is equal to -0.5βˆ’0.5
  • The PACF coefficients for lags greater than 1 are zero
  • The PACF coefficient at lag 1 \phi(1,1)Ο•(1,1) is equal to 0.50.5
  • The autocovariance process of this function decays exponentially as a function of the lag hh oscillating between negative and positive values

Week 02 : Properties of AR processes

Q1. Consider the following AR(2)AR(2) process,

Y_t = 0.5Y_{t-1} + 0.24Y_{t-2} + \epsilon_t, \quad \epsilon_t \sim \mathcal{N}(0, v).Yt​=0.5Ytβˆ’1​+0.24Ytβˆ’2​+Ο΅t​,Ο΅tβ€‹βˆΌN(0,v).

Give the value of one of the reciprocal roots of this process.

Q2. Assume the reciprocal roots of an AR(2)AR(2) characteristic polynomial are 0.70.7 and -0.2.βˆ’0.2.

Which is the corresponding form of the autocorrelation function \rho(h)ρ(h) of this process?

  • ρ(h)=(a+bh)0.3h,h>0, where $a$ and $b$ are some constants.
  • ρ(h)=a(0.7)h+b(βˆ’0.3)h,h>0, where aa and bb are some constants.
  • ρ(h)=(a+bh)0.7h,h>0, where aa and bb are some constants.
  • ρ(h)=(a+bh)(0.3h+0.7h),h>0, where $a$ and $b$ are some constants.

Q3. Assume that an AR(2) process has a pair of complex reciprocal roots with modulus r = 0.95r=0.95 and period \lambda = 7.1.Ξ»=7.1.

Which following options corresponds to the correct form of its autocorrelation function, \rho(h)ρ(h) ?

  • ρ(h)=a(0.95)hcos(7.1h+b), where aa and bb are some constants.
  • ρ(h)=a(0.95)hcos(2Ο€h/7.1+b), where aa and bb are some constants.
  • ρ(h)=a0.95h,h>0, where aa and bb are some constants.
  • ρ(h)=(a+bh)0.95h, where aa and bb are some constants.

Q4. Given the following AR(2)AR(2) process,

Y_t = 0.5Y_{t-1} + 0.36Y_{t-2} + \epsilon_t, \quad \epsilon_t \sim \mathcal{N}(0, v).Yt​=0.5Ytβˆ’1​+0.36Ytβˆ’2​+Ο΅t​,Ο΅tβ€‹βˆΌN(0,v).

The h=3h=3 steps-ahead forecast function f_t(3)ft​(3) has the following form:

  • ft​(3)=c1t​(1.1)3+c2t​(βˆ’2.5)3 for c_{1t}c1t​ and c_{2t}c2t​ constants.
  • ft​(3)=(0.9)3(c1t​+c2t​3) for c_{1t}c1t​ and c_{2t}c2t​ constants
  • ft​(3)=c1t​(3)0.9+c2t​(3)βˆ’0.4 for c_{1t}c1t​ and c_{2t}c2t​ constants.
  • ft​(3)=c1t​(0.9)3+c2t​(βˆ’0.4)3 for c_{1t}c1t​ and c_{2t}c2t​ constants.

Week 03: Practice Quiz The Normal Dynamic Linear Model

Q1. Which of the models below is a Dynamic Normal Linear Model?

  • Observation equation: y_t = a\theta^2_t + \epsilon_t, \quad \epsilon_t \sim \mathcal{N}(0, v), yt​=aΞΈt2​+Ο΅t​,Ο΅tβ€‹βˆΌN(0,v),
  • System equation: \theta_t = b\theta_{t-1} + c \frac{\theta_{t-1}}{1+ \theta^2_{t-1}} + \omega_t, \quad \omega_t \sim \mathcal{N}(0, w). ΞΈt​=bΞΈtβˆ’1​+c1+ΞΈtβˆ’12​θtβˆ’1​​+Ο‰t​,Ο‰tβ€‹βˆΌN(0,w).
  • Observation equation: y_t = \mu_t + \epsilon_t, \quad \epsilon_t \sim \mathcal{N}(0, v),yt​=ΞΌt​+Ο΅t​,Ο΅tβ€‹βˆΌN(0,v),
  • System equation: \mu_t = \mu_{t-1} + \omega_t, \quad \omega_t \sim \mathcal{N}(0, w).ΞΌt​=ΞΌtβˆ’1​+Ο‰t​,Ο‰tβ€‹βˆΌN(0,w).
  • Observation equation: y_t = \theta_t + \epsilon_t, \quad \epsilon_t \sim \mathcal{N}(0, v), yt​=ΞΈt​+Ο΅t​,Ο΅tβ€‹βˆΌN(0,v),
  • System equation: \theta_t = b\theta_{t-1} + c \frac{\theta_{t-1}}{1+ \theta^2_{t-1}} + \omega_t, \quad \omega_t \sim \mathcal{N}(0, w). ΞΈt​=bΞΈtβˆ’1​+c1+ΞΈtβˆ’12​θtβˆ’1​​+Ο‰t​,Ο‰tβ€‹βˆΌN(0,w).

Q2. Consider the Normal Dynamic Linear Model \mathcal{M}: \left\{\bm{F}_t, \bm{G}_t, \cdot, \cdot\right\}, M:{Ft​,Gt​,β‹…,β‹…}, for t = 1, \dots, T.t=1,…,T. Let’s assume \bm{F}_tFt​ is K \times 1KΓ—1 vector. What is the dimension of \bm{G}_t?Gt​?

  • T \times 1TΓ—1
  • T \times TTΓ—T
  • K \times KKΓ—K
  • K \times 1KΓ—1

Q3. Consider the third order polynomial Normal Dynamic Linear Model \mathcal{M}: \{\bm{F}, \bm{G}, \cdot, \cdot\}, M:{F,G,β‹…,β‹…}, where \bm{F} = (1 \quad 0 \quad 0)’F=(100)β€² and \bm{G} = \bm{J}_3(1),G=J3​(1), where \bm{J}J is Jordan block given by

J_3(1) = \left(

100110011

\right) J3​(1)=βŽβŽœβŽ›β€‹100​110​011β€‹βŽ βŽŸβŽžβ€‹

Given the posterior mean E (\bm{\theta}_t | D_t) = (m_t, b_t, g_t)’,E(ΞΈtβ€‹βˆ£Dt​)=(mt​,bt​,gt​)β€², which of the following options is the one corresponding to the forecast function f_t(h) \quad (h \geq 0)ft​(h)(hβ‰₯0) of the model?

  • ft​(h)=mt​+hbt​+h(hβˆ’1)gt​/2
  • ​ft​(h)=mt​+hbt​
  • ft​(h)=mt​+hbt​+h(h+1)gt​
  • ft​(h)=mt​+hbt​+h2gt​

Q4. Consider a forecast function of the following form:

ft​(h)=at,0​+at,1​xt+h​+at,3​+at,4​h, for h \geq 0.hβ‰₯0.

Which is the possible corresponding DLM \left\{ \bm{F}, \bm{G}, \cdot, \cdot \right\}?{F,G,β‹…,β‹…}?

  • F=(1xt​10)β€²

\bm{G} = \bm{I}_4,G=I4​,

where \bm{I}_4I4​ is the identity matrix of dimension 4 \times 44Γ—4

  • F=(1010)β€²

\bm{G} = \textrm{block diag} [\bm{I}_2, \bm{J}_2(1)],G=block diag[I2​,J2​(1)],

where \bm{I}_2I2​ is the identity matrix of dimension 2 \times 22Γ—2, and \bm{J}_2(1)J2​(1) is the Jordan block matrix given by

J_2(1) = \left(

1011 \right).J2​(1)=(10​11​).

  • F=(1xt​10)β€²

\bm{G} = \textrm{block diag} [\bm{I}_2, \bm{J}_2(1)],G=block diag[I2​,J2​(1)],

where \bm{I}_2I2​ is the identity matrix, and \bm{J}_2(1)J2​(1) is the Jordan block matrix given by

J_2(1) = \left(

1011 \right).J2​(1)=(10​11​).

  • None is correct.

Quiz : NDLM, Part I: Review

Q1. Assume we have following DLM representation:

\mathcal{M}: \qquad \left\{

(10)

,

(βˆ’0.7501βˆ’0.75)

, \cdot, \cdot \right\} M:{(10​),(βˆ’0.750​1βˆ’0.75​),β‹…,β‹…} and

E(\bm{\theta}_t | \mathcal{D}_t) = (1, 0.5)’E(ΞΈtβ€‹βˆ£Dt​)=(1,0.5)β€². What is the best description of the forecast function, f_t(h)ft​(h) for h \geq 0hβ‰₯0 of this model?

  • .
wsprqaDhSw Ka6mg4bsPiw d5de1d4d669e4ef1b62fb568ea6a5bf1 q1b
  • .
P2boaRUYRL m6GkVGMS Ag b17b544f335d44fba2fd2b7a838690f1 q1c
  • .
GkzhTtzmSDqM4U7c5ig6yw 2585c89e39f546d9af7abd7b341cabf1 q1d
  • .
o1A8k75TSfqQPJO U7n6TQ c31c28bfa5134f34baf6ed3e964767f1 q1a

Q2. Which of the options below correspond the DLMs \{\bm{F}, \bm{G}, \cdot, \cdot \}{F,G,β‹…,β‹…} with forecast function

f_t(h) = k_{t1}\lambda^h_1 – k_{t2}\lambda_2^h, ft​(h)=kt1​λ1hβ€‹βˆ’kt2​λ2h​,

where \lambda_1\lambda_2 \neq 0Ξ»1​λ2​​=0 and \lambda_1 \neq \lambda_2Ξ»1​​=Ξ»2​?

  • F=(1,0)β€²,

\bm{G} =

(Ξ»111Ξ»2).G=(Ξ»1​1​1Ξ»2​​).

  • F=(1,βˆ’1)β€²,

\bm{G} =

(Ξ»100Ξ»2).G=(Ξ»1​0​0Ξ»2​​).

  • F=(1,βˆ’1)β€²,

\bm{G} =

(Ξ»111Ξ»2).G=(Ξ»1​1​1Ξ»2​​).

  • F=(1,0)β€²,

\bm{G} =

(Ξ»100Ξ»2).G=(Ξ»1​0​0Ξ»2​​).

Q3. Consider a DLM with a forecast function of the form f_t(h) = k_{t,1} \lambda^h + k_{t,2} x_{t+h}.ft​(h)=kt,1​λh+kt,2​xt+h​. Which of the following representations corresponds to this DLM?

  • {Ft​,Gt​,β‹…,β‹…} with

\bm{F}_t = (1,x_t)’Ft​=(1,xt​)β€² and \bm{G}_t= \left(

Ξ»001 \right).Gt​=(Ξ»0​01​).

  • {Ft​,Gt​,β‹…,β‹…} with

\bm{F}_t = (1,0)’Ft​=(1,0)β€² and \bm{G}_t= \left(

Ξ»00xt \right).Gt​=(Ξ»0​0xt​​).

  • {Ft​,Gt​,β‹…,β‹…} with

\bm{F}_t = (1,x_t)’Ft​=(1,xt​)β€² and \bm{G}_t= \left(

1001 \right).Gt​=(10​01​).

Q4. Which of the following statements are true?

  • In order to obtain the filtering equations one must first obtain the smoothing equations
  • In a DLM of the form \{ \bm{F}_t, \bm{G}_t, v_t, \bm{W}_t\}{Ft​,Gt​,vt​,Wt​} with \bm{F}_t, \bm{G}_t, v_t, \bm{W}_tFt​,Gt​,vt​,Wt​ known for all t,t, the distribution of (\mathbf{\theta}_t| \mathcal{D}_t)(ΞΈtβ€‹βˆ£Dt​) is normal if the distribution of (\mathbf{\theta}_0 | \mathcal{D}_0)(ΞΈ0β€‹βˆ£D0​) is normal
  • The filtering equations allow us to obtain the moments of the distribution of (\mathbf{\theta}_t| \mathcal{D}_T)(ΞΈtβ€‹βˆ£DT​) for t=1:Tt=1:T and T >t.T>t.
  • The smoothing equations allow us to obtain the moments of the distribution of (\mathbf{\theta}_t| \mathcal{D}_T)(ΞΈtβ€‹βˆ£DT​) for t=1:Tt=1:T and T >t.T>t.
  • In a DLM of the form \{ \bm{F}_t, \bm{G}_t, v_t, \bm{W}_t\}{Ft​,Gt​,vt​,Wt​} with \bm{F}_t, \bm{G}_t, v_t, \bm{W}_tFt​,Gt​,vt​,Wt​ known for all t,t, the distribution of (\mathbf{\theta}_t| \mathcal{D}_t)(ΞΈtβ€‹βˆ£Dt​) is not normal even if the distribution of (\mathbf{\theta}_0 | \mathcal{D}_0)(ΞΈ0β€‹βˆ£D0​) is normal
  • In order to obtain the DLM smoothing equations one must first obtain the filtering equations

Week 04 : Quiz Seasonal Models and Superposition

Q1. Consider a full seasonal Fourier DLM with fundamental period p=3.p=3. Which of the choices below corresponds to the specification of \bm{F}F and \bm{G}G for such model?

  • F=(10)β€²

\bm{G} = \left(

βˆ’12βˆ’3√23√2βˆ’12 \right) G=(βˆ’21β€‹βˆ’23​​​23β€‹β€‹βˆ’21​​)

  • F=(10)β€²

\bm{G} = \left(

1011 \right) G=(10​11​)

  • F=(101)β€²

\bm{G} = \left(

0βˆ’1010000βˆ’1

\right) G=βŽβŽœβŽ›β€‹0βˆ’10​100​00βˆ’1β€‹βŽ βŽŸβŽžβ€‹

  • F=(10)β€²

\bm{G} = \left(

12βˆ’3√23√212 \right) G=(21β€‹βˆ’23​​​23​​21​​)

Q2. Assume monthly data have an annual cycle and so the fundamental period is p=12.p=12. Further assume that we want to fit a model with a linear trend and seasonal component to this dataset. For the seasonal component, assume we only consider the fourth harmonic, i.e., we only consider the Fourier component for the frequency \omega= 2\pi 4/12= 2 \pi/3.Ο‰=2Ο€4/12=2Ο€/3. What is the forecast function f_t(h), h \geq 0,ft​(h),hβ‰₯0, for a DLM with this linear trend and a seasonal component that considers only the fourth harmonic?

  • ft​(h)=at,0​+at,1​h
  • ft​(h)=at,0​+at,1​h+at,3​cos(32Ο€h​)+at,4​sin(32Ο€h​)
  • ft​(h)=at,1​cos(32Ο€h​)+at,2​sin(32Ο€h​)
  • ft​(h)=at,0​+at,1​h+at,3​cos(32Ο€h​)+at,4​sin(32Ο€h​)+at,5​(βˆ’1)h

Q4. A DLM \{ \bm{F}_t, \bm{G}_t, \cdot, \cdot \}{Ft​,Gt​,β‹…,β‹…} has the following forecast function:

f_t(h) = a_{t,1} x_{t+h} + (-1)^{h} a_{t,2}.ft​(h)=at,1​xt+h​+(βˆ’1)hat,2​.

What are the corresponding \bm{F}_tFt​ and \bm{G}_tGt​ matrices?

  • Ft​=(1,xt​)β€² and \bm{G}_t =\left(

1001 \right)Gt​=(10​01​)

  • Ft​=(1,xt​)β€² and \bm{G}_t =\left(

100βˆ’1 \right)Gt​=(10​0βˆ’1​)

  • Ft​=(1,0)β€² and \bm{G}_t =\left(

10xtβˆ’1 \right)Gt​=(10​xtβ€‹βˆ’1​)

Quiz : NDLM, Part II

Q1. Assume that we want a model with the following $2$ components:

  • Linear trend: \{\bm{F}_1, \bm{G}_1, \cdot, \cdot\}{F1​,G1​,β‹…,β‹…} with \bm{F}_1 = (1,0,0)’,F1​=(1,0,0)β€², \bm{G}_1 = \bm{J}_3(1) G1​=J3​(1) and state vector \bm{\theta}_{1t} = (1, -0.5, 0.1)’.ΞΈ1t​=(1,βˆ’0.5,0.1)β€².
  • Seasonal component: \{\bm{F}_2, \bm{G}_2, \cdot, \cdot\}{F2​,G2​,β‹…,β‹…} with \bm{F}_2 = (1, 0)’F2​=(1,0)β€², \bm{G}_2 = \bm{J}_2(\lambda, \omega)G2​=J2​(Ξ»,Ο‰), where \omega = \frac{\pi}{2}Ο‰=2π​ and \lambda = 0.9Ξ»=0.9 and state vector \bm{\theta}_{2t} = (1, 0.45)’.ΞΈ2t​=(1,0.45)β€².

Which graph is the description of forecast function f_t(h)ft​(h) with h \geq 0?hβ‰₯0?

  • .
HW82iWHaSlmvNolh2jpZeg f677681fc948403090f7ee6eec720bf1 wrong 3
  • .
myM6qWcVRpajOqlnFfaW2Q f326a46501b645179e3b8e3816a72ef1 wrong 1
  • .
7 gzBdK1RDq4MwXStcQ6ow 1703789779d64599a7dbd20de14fc0f1 wrong 2
  • .
U5DME34SSGiQzBN EqhoBw 67730175c5324b91886eb4c458daddf1 correct

Q2. Which is the right Fourier DLM model \{\bm{F}, \bm{G}, \cdot, \cdot \}{F,G,β‹…,β‹…} with period p = 6?p=6?

1 point

  • F=(1,0,1,0,0)β€²

\bm{G} =

βŽ›βŽβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœ12βˆ’3√20003√21200000βˆ’12βˆ’3√20003√2βˆ’1200000βˆ’1⎞⎠⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟G=βŽβŽœβŽœβŽœβŽœβŽœβŽœβŽ›β€‹21β€‹βˆ’23​​000​23​​21​000​00βˆ’21β€‹βˆ’23​​0​0023β€‹β€‹βˆ’21​0​0000βˆ’1β€‹βŽ βŽŸβŽŸβŽŸβŽŸβŽŸβŽŸβŽžβ€‹

  • F=(1,0,1,0,1)β€²

\bm{G} =

βŽ›βŽβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœ12βˆ’3√20003√21200000βˆ’12βˆ’3√20003√2βˆ’1200000βˆ’1⎞⎠⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟G=βŽβŽœβŽœβŽœβŽœβŽœβŽœβŽ›β€‹21β€‹βˆ’23​​000​23​​21​000​00βˆ’21β€‹βˆ’23​​0​0023β€‹β€‹βˆ’21​0​0000βˆ’1β€‹βŽ βŽŸβŽŸβŽŸβŽŸβŽŸβŽŸβŽžβ€‹

  • F=(1,0,1,0,0)β€²

\bm{G} =

βŽ›βŽβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœ3√21200012βˆ’3√2000003√2βˆ’12000βˆ’12βˆ’3√200000βˆ’1⎞⎠⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟G=βŽβŽœβŽœβŽœβŽœβŽœβŽœβŽ›β€‹23​​21​000​21β€‹βˆ’23​​000​0023β€‹β€‹βˆ’21​0​00βˆ’21β€‹βˆ’23​​0​0000βˆ’1β€‹βŽ βŽŸβŽŸβŽŸβŽŸβŽŸβŽŸβŽžβ€‹

  • F=(1,0,1,0,1)β€²

\bm{G} =

βŽ›βŽβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœ3√21200012βˆ’3√2000003√2βˆ’12000βˆ’12βˆ’3√200000βˆ’1⎞⎠⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟G=βŽβŽœβŽœβŽœβŽœβŽœβŽœβŽ›β€‹23​​21​000​21β€‹βˆ’23​​000​0023β€‹β€‹βˆ’21​0​00βˆ’21β€‹βˆ’23​​0​0000βˆ’1β€‹βŽ βŽŸβŽŸβŽŸβŽŸβŽŸβŽŸβŽžβ€‹

Q3. Consider a full seasonal Fourier DLM with fundamental period p=10.p=10. What is the dimension of the state vector \bm{\theta}_tΞΈt​ at each time tt?

  • None of the above
  • 10
  • 9
  • 11

Conclusion

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