- About The Coursera
- About Bayesian Statistics: Time Series Analysis Course
- Bayesian Statistics: Time Series Analysis Quiz Answers
- Practice Quiz : Objectives of the course
- Quiz: Stationarity, the ACF and the PACF
- Quiz: The AR(1) definitions and properties
- Week 02 : Properties of AR processes
- Week 03: Practice Quiz The Normal Dynamic Linear Model
- Quiz : NDLM, Part I: Review
- Week 04 : Quiz Seasonal Models and Superposition
- Quiz : NDLM, Part II
- Conclusion

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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. YtββYtβ1β=etββ0.8etβ1β

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

- (1βB)Ytβ=(1β0.8B)etβ
- None of the above
- BYtβ=(1β0.8B)etβ
- B(YtββYtβ1β)=0.8Betβ

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

- .

- .

- .

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

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

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

- If \{Y_t\}{
*Yt*β} is also a Gaussian process then \{Y_t\}{*Yt*β} is strongly stationary - {
*Yt*β} is also strongly stationary - None of the above

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

- 1β/8
*yt*β4β+41β(*yt*β3β+*yt*β2β+*yt*β1β+*yt*β+*yt*+1β+*yt*+2β+*yt*+3β)+81β*yt*+4β - 1/8β
*j*=β88β*yt*β*k* - 1/2β(
*yt*β4β+*yt*β3β+*yt*β2β+*yt*β1β+*yt*β+*yt*+1β+*yt*+2β+*yt*+3β+*yt*+4β) - 1/8β(
*yt*β4β+*yt*β3β+*yt*β2β+*yt*β1β+*yt*β+*yt*+1β+*yt*+2β+*yt*+3β+*yt*+4β)

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

- 1/2β(
*yt*β1β+*yt*β+*yt*+1β) - 1/3β(
*yt*β1β+*yt*β+*yt*+1β) - 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 = 0.8*Ο*=0.8?

- None of above is correct
- .

- .

- .

Q2.

- Which of the following AR(1) processes are stable and therefore stationary?

- Ytβ=0.9Ytβ1β+Ο΅tβ,Ο΅tββΌi.i.d.N(0,v)
- Ytβ=Ytβ1β+Ο΅tβ,Ο΅tββΌi.i.d.N(0,v)
- Ytβ=β2Ytβ1β+Ο΅tβ,Ο΅tββΌi.i.d.N(0,v)
- Ytβ=β0.8Ytβ1β+Ο΅tβ,Ο΅tββΌi.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*β(β1,1) and \phi \neq 0*Ο*ξ β=0 are zero after lag 1 - The PACF coefficients of an AR(1) with AR coefficient \phi \in (-1,1)
*Ο*β(β1,1) and \phi \neq 0*Ο*ξ β=0 are zero after lag 1 - The ACF of an AR(1) with coefficient \phi=0.5
*Ο*=0.5 decays exponentially in an oscillatory manner - The ACF of an AR(1) with AR coefficient \phi=0.8
*Ο*=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*β=0.7*Yt*β1β+*Ο΅t*β,*Ο΅t*ββΌ*i*.*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

*y**t*β=β0.7*y**t*β1β+*Ο΅**t*β with \epsilon_t \stackrel{iid}\sim N(0,1)*Ο΅**t*ββΌ*i**i**d**N*(0,1)?

- 0.70.7
- -0.7β0.7
- \approx 1.96β1.96
- 00

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

*y**t*β=0.6*y**t*β1β+*Ο΅**t*β with \epsilon_t \stackrel{i.i.d.}{\sim} N(0,v)*Ο΅**t*ββΌ*i*.*i*.*d*.*N*(0,*v*) ? with variance v=2*v*=2.

- 1.5625
- 1.875
- 1
- 0.6

Q7. Consider an AR(1) process y_t = -0.5 y_{t-1} + \epsilon_t,*yt*β=β0.5*yt*β1β+*Ο΅t*β, with \epsilon_t \stackrel{i.i.d.}{\sim} N(0,1)*Ο΅t*ββΌ*i*.*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 h
*h*and it is always negative - The autocovariance process of this function decays exponentially as a function of the lag h
*h*and it is always positive - The PACF coefficient at lag 1 \phi(1,1)
*Ο*(1,1) is equal to -0.5β0.5 - 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 h
*h*oscillating between negative and positive values

**Week 02 : Properties of AR processes**

Q1. Consider the following AR(2)*A**R*(2) process,

Y_t = 0.5Y_{t-1} + 0.24Y_{t-2} + \epsilon_t, \quad \epsilon_t \sim \mathcal{N}(0, v).*Y**t*β=0.5*Y**t*β1β+0.24*Y**t*β2β+*Ο΅**t*β,*Ο΅**t*ββΌN(0,*v*).

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

Q2. Assume the reciprocal roots of an AR(2)*A**R*(2) characteristic polynomial are 0.70.7 and -0.2.β0.2.

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

*Ο*(*h*)=(*a*+*bh*)0.3*h*,*h*>0, where $a$ and $b$ are some constants.**Ο(h)=a(0.7)h+b(β0.3)h,h>0, where aa and bb are some constants.***Ο*(*h*)=(*a*+*bh*)0.7*h*,*h*>0, where a*a*and b*b*are some constants.*Ο*(*h*)=(*a*+*bh*)(0.3*h*+0.7*h*),*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 = 0.95*r*=0.95 and period \lambda = 7.1.*Ξ»*=7.1.

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

*Ο*(*h*)=*a*(0.95)*h*cos(7.1*h*+*b*), where a*a*and b*b*are some constants.**Ο(h)=a(0.95)hcos(2Οh/7.1+b), where aa and bb are some constants.***Ο*(*h*)=*a*0.95*h*,*h*>0, where a*a*and b*b*are some constants.*Ο*(*h*)=(*a*+*bh*)0.95*h*, where a*a*and b*b*are some constants.

Q4. Given the following AR(2)*A**R*(2) process,

Y_t = 0.5Y_{t-1} + 0.36Y_{t-2} + \epsilon_t, \quad \epsilon_t \sim \mathcal{N}(0, v).*Y**t*β=0.5*Y**t*β1β+0.36*Y**t*β2β+*Ο΅**t*β,*Ο΅**t*ββΌN(0,*v*).

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

*ft*β(3)=*c*1*t*β(1.1)3+*c*2*t*β(β2.5)3 for c_{1t}*c*1*t*β and c_{2t}*c*2*t*β constants.*ft*β(3)=(0.9)3(*c*1*t*β+*c*2*t*β3) for c_{1t}*c*1*t*β and c_{2t}*c*2*t*β constants**ftβ(3)=c1tβ(3)0.9+c2tβ(3)β0.4 for c_{1t}c1tβ and c_{2t}c2tβ constants.***ft*β(3)=*c*1*t*β(0.9)3+*c*2*t*β(β0.4)3 for c_{1t}*c*1*t*β and c_{2t}*c*2*t*β 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*β=*aΞΈt*2β+*Ο΅t*β,*Ο΅t*ββΌN(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*β=*bΞΈt*β1β+*c*1+*ΞΈt*β12β*ΞΈt*β1ββ+*Οt*β,*Οt*ββΌN(0,*w*). - Observation equation: y_t = \mu_t + \epsilon_t, \quad \epsilon_t \sim \mathcal{N}(0, v),
*yt*β=*ΞΌt*β+*Ο΅t*β,*Ο΅t*ββΌN(0,*v*), - System equation: \mu_t = \mu_{t-1} + \omega_t, \quad \omega_t \sim \mathcal{N}(0, w).
*ΞΌt*β=*ΞΌt*β1β+*Οt*β,*Οt*ββΌN(0,*w*). - Observation equation: y_t = \theta_t + \epsilon_t, \quad \epsilon_t \sim \mathcal{N}(0, v),
*yt*β=*ΞΈt*β+*Ο΅t*β,*Ο΅t*ββΌN(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*β=*bΞΈt*β1β+*c*1+*ΞΈt*β12β*ΞΈt*β1ββ+*Οt*β,*Οt*ββΌN(0,*w*).

Q2. Consider the Normal Dynamic Linear Model \mathcal{M}: \left\{\bm{F}_t, \bm{G}_t, \cdot, \cdot\right\}, M:{*F**t*β,*G**t*β,β
,β
}, for t = 1, \dots, T.*t*=1,β¦,*T*. Letβs assume \bm{F}_t*F**t*β is K \times 1*K*Γ1 vector. What is the dimension of \bm{G}_t?*G**t*β?

- T \times 1
*T*Γ1 - T \times T
*T*Γ*T* - K \times K
*K*Γ*K* - K \times 1
*K*Γ1

Q3. Consider the third order polynomial Normal Dynamic Linear Model \mathcal{M}: \{\bm{F}, \bm{G}, \cdot, \cdot\}, M:{** F**,

**,β ,β }, where \bm{F} = (1 \quad 0 \quad 0)β**

*G***=(100)β² and \bm{G} = \bm{J}_3(1),**

*F***=**

*G***3β(1), where \bm{J}**

*J***is Jordan block given by**

*J*J_3(1) = \left(

100110011

\right) *J*3β(1)=ββββ100β110β011ββ βββ

Given the posterior mean E (\bm{\theta}_t | D_t) = (m_t, b_t, g_t)β,*E*(*ΞΈ**t*ββ£*Dt*β)=(*mt*β,*bt*β,*gt*β)β², which of the following options is the one corresponding to the forecast function f_t(h) \quad (h \geq 0)*ft*β(*h*)(*h*β₯0) of the model?

*ft*β(*h*)=*mt*β+*hbt*β+*h*(*h*β1)*gt*β/2- β
*ft*β(*h*)=*mt*β+*hbt*β *ft*β(*h*)=*mt*β+*hbt*β+*h*(*h*+1)*gt*β*ft*β(*h*)=*mt*β+*hbt*β+*h*2*gt*β

Q4. Consider a forecast function of the following form:

*f**t*β(*h*)=*a**t*,0β+*a**t*,1β*x**t*+*h*β+*a**t*,3β+*a**t*,4β*h*, for h \geq 0.*h*β₯0.

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

**,β ,β }?**

*G*=(1*F**xt*β10)β²

\bm{G} = \bm{I}_4,** G**=

**4β,**

*I*where \bm{I}_4** I**4β is the identity matrix of dimension 4 \times 44Γ4

=(1010)β²*F*

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

**2β,**

*I***2β(1)],**

*J*where \bm{I}_2** I**2β is the identity matrix of dimension 2 \times 22Γ2, and \bm{J}_2(1)

**2β(1) is the Jordan block matrix given by**

*J*J_2(1) = \left(

1011 \right).*J*2β(1)=(10β11β).

=(1*F**xt*β10)β²

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

**2β,**

*I***2β(1)],**

*J*where \bm{I}_2** I**2β is the identity matrix, and \bm{J}_2(1)

**2β(1) is the Jordan block matrix given by**

*J*J_2(1) = \left(

1011 \right).*J*2β(1)=(10β11β).

- None is correct.

**Quiz : NDLM, Part I: Review**

Q1. Assume we have following DLM representation:

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

(10)

,

(β0.7501β0.75)

, \cdot, \cdot \right\} M:{(10β),(β0.750β1β0.75β),β ,β } and

E(\bm{\theta}_t | \mathcal{D}_t) = (1, 0.5)β*E*(*ΞΈ**t*ββ£D*t*β)=(1,0.5)β². What is the best description of the forecast function, f_t(h)*f**t*β(*h*) for h \geq 0*h*β₯0 of this model?

- .

- .

- .

- .

Q2. Which of the options below correspond the DLMs \{\bm{F}, \bm{G}, \cdot, \cdot \}{** F**,

**,β ,β } with forecast function**

*G*f_t(h) = k_{t1}\lambda^h_1 β k_{t2}\lambda_2^h, *f**t*β(*h*)=*k**t*1β*Ξ»*1*h*ββ*k**t*2β*Ξ»*2*h*β,

where \lambda_1\lambda_2 \neq 0*Ξ»*1β*Ξ»*2βξ β=0 and \lambda_1 \neq \lambda_2*Ξ»*1βξ β=*Ξ»*2β?

=(1,0)β²,*F*

\bm{G} =

(*Ξ»*111*Ξ»*2).** G**=(

*Ξ»*1β1β1

*Ξ»*2ββ).

=(1,β1)β²,*F*

\bm{G} =

(*Ξ»*100*Ξ»*2).** G**=(

*Ξ»*1β0β0

*Ξ»*2ββ).

=(1,β1)β²,*F*

\bm{G} =

(*Ξ»*111*Ξ»*2).** G**=(

*Ξ»*1β1β1

*Ξ»*2ββ).

=(1,0)β²,*F*

\bm{G} =

(*Ξ»*100*Ξ»*2).** G**=(

*Ξ»*1β0β0

*Ξ»*2ββ).

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*β(*h*)=*kt*,1β*Ξ»h*+*kt*,2β*xt*+*h*β. Which of the following representations corresponds to this DLM?

- {
*F**t*β,*G**t*β,β ,β } with

\bm{F}_t = (1,x_t)β*F**t*β=(1,*x**t*β)β² and \bm{G}_t= \left(

*Ξ»*001 \right).*G**t*β=(*Ξ»*0β01β).

- {
*F**t*β,*G**t*β,β ,β } with

\bm{F}_t = (1,0)β*F**t*β=(1,0)β² and \bm{G}_t= \left(

*Ξ»*00*xt* \right).*G**t*β=(*Ξ»*0β0*xt*ββ).

- {
*F**t*β,*G**t*β,β ,β } with

\bm{F}_t = (1,x_t)β*F**t*β=(1,*x**t*β)β² and \bm{G}_t= \left(

1001 \right).*G**t*β=(10β01β).

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\}{
*F**t*β,*G**t*β,*vt*β,*W**t*β} with \bm{F}_t, \bm{G}_t, v_t, \bm{W}_t*F**t*β,*G**t*β,*vt*β,*W**t*β known for all t,*t*, the distribution of (\mathbf{\theta}_t| \mathcal{D}_t)(*ΞΈt*ββ£D*t*β) is normal if the distribution of (\mathbf{\theta}_0 | \mathcal{D}_0)(*ΞΈ*0ββ£D0β) is normal - The filtering equations allow us to obtain the moments of the distribution of (\mathbf{\theta}_t| \mathcal{D}_T)(
*ΞΈt*ββ£D*T*β) for t=1:T*t*=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)(
*ΞΈt*ββ£D*T*β) for t=1:T*t*=1:*T*and T >t.*T*>*t*. - In a DLM of the form \{ \bm{F}_t, \bm{G}_t, v_t, \bm{W}_t\}{
*F**t*β,*G**t*β,*vt*β,*W**t*β} with \bm{F}_t, \bm{G}_t, v_t, \bm{W}_t*F**t*β,*G**t*β,*vt*β,*W**t*β known for all t,*t*, the distribution of (\mathbf{\theta}_t| \mathcal{D}_t)(*ΞΈt*ββ£D*t*β) is not normal even if the distribution of (\mathbf{\theta}_0 | \mathcal{D}_0)(*ΞΈ*0ββ£D0β) 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}

**for such model?**

*G*=(10)β²*F*

\bm{G} = \left(

β12β3β23β2β12 \right) ** G**=(β21ββ23βββ23βββ21ββ)

=(10)β²*F*

\bm{G} = \left(

1011 \right) ** G**=(10β11β)

=(101)β²*F*

\bm{G} = \left(

0β1010000β1

\right) ** G**=ββββ0β10β100β00β1ββ βββ

=(10)β²*F*

\bm{G} = \left(

12β3β23β212 \right) ** G**=(21ββ23βββ23ββ21ββ)

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*=2*Ο*4/12=2*Ο*/3. What is the forecast function f_t(h), h \geq 0,*ft*β(*h*),*h*β₯0, for a DLM with this linear trend and a seasonal component that considers only the fourth harmonic?

*ft*β(*h*)=*at*,0β+*at*,1β*h**ft*β(*h*)=*at*,0β+*at*,1β*h*+*at*,3βcos(32*Οh*β)+*at*,4βsin(32*Οh*β)*ft*β(*h*)=*at*,1βcos(32*Οh*β)+*at*,2βsin(32*Οh*β)*ft*β(*h*)=*at*,0β+*at*,1β*h*+*at*,3βcos(32*Οh*β)+*at*,4βsin(32*Οh*β)+*at*,5β(β1)*h*

Q4. A DLM \{ \bm{F}_t, \bm{G}_t, \cdot, \cdot \}{*F**t*β,*G**t*β,β
,β
} has the following forecast function:

f_t(h) = a_{t,1} x_{t+h} + (-1)^{h} a_{t,2}.*f**t*β(*h*)=*a**t*,1β*x**t*+*h*β+(β1)*h**a**t*,2β.

What are the corresponding \bm{F}_t*F**t*β and \bm{G}_t*G**t*β matrices?

*F**t*β=(1,*xt*β)β² and \bm{G}_t =\left(

1001 \right)*G**t*β=(10β01β)

*F**t*β=(1,*xt*β)β² and \bm{G}_t =\left(

100β1 \right)*G**t*β=(10β0β1β)

*F**t*β=(1,0)β² and \bm{G}_t =\left(

10*xt*β1 \right)*G**t*β=(10β*xt*ββ1β)

**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\}{
1β,*F*1β,β ,β } with \bm{F}_1 = (1,0,0)β,*G*1β=(1,0,0)β², \bm{G}_1 = \bm{J}_3(1)*F*1β=*G*3β(1) and state vector \bm{\theta}_{1t} = (1, -0.5, 0.1)β.*J*1*ΞΈ**t*β=(1,β0.5,0.1)β². - Seasonal component: \{\bm{F}_2, \bm{G}_2, \cdot, \cdot\}{
2β,*F*2β,β ,β } with \bm{F}_2 = (1, 0)β*G*2β=(1,0)β², \bm{G}_2 = \bm{J}_2(\lambda, \omega)*F*2β=*G*2β(*J**Ξ»*,*Ο*), where \omega = \frac{\pi}{2}*Ο*=2*Ο*β and \lambda = 0.9*Ξ»*=0.9 and state vector \bm{\theta}_{2t} = (1, 0.45)β.2*ΞΈ**t*β=(1,0.45)β².

Which graph is the description of forecast function f_t(h)*ft*β(*h*) with h \geq 0?*h*β₯0?

- .

- .

- .

- .

Q2. Which is the right Fourier DLM model \{\bm{F}, \bm{G}, \cdot, \cdot \}{** F**,

**,β ,β } with period p = 6?**

*G**p*=6?

**1 point**

=(1,0,1,0,0)β²*F*

\bm{G} =

ββββββββββββββββ12β3β20003β21200000β12β3β20003β2β1200000β1ββ ββββββββββββββ** G**=βββββββββ21ββ23ββ000β23ββ21β000β00β21ββ23ββ0β0023βββ21β0β0000β1ββ ββββββββ

=(1,0,1,0,1)β²*F*

\bm{G} =

ββββββββββββββββ12β3β20003β21200000β12β3β20003β2β1200000β1ββ ββββββββββββββ** G**=βββββββββ21ββ23ββ000β23ββ21β000β00β21ββ23ββ0β0023βββ21β0β0000β1ββ ββββββββ

=(1,0,1,0,0)β²*F*

\bm{G} =

ββββββββββββββββ3β21200012β3β2000003β2β12000β12β3β200000β1ββ ββββββββββββββ** G**=βββββββββ23ββ21β000β21ββ23ββ000β0023βββ21β0β00β21ββ23ββ0β0000β1ββ ββββββββ

=(1,0,1,0,1)β²*F*

\bm{G} =

ββββββββββββββββ3β21200012β3β2000003β2β12000β12β3β200000β1ββ ββββββββββββββ** G**=βββββββββ23ββ21β000β21ββ23ββ000β0023βββ21β0β00β21ββ23ββ0β0000β1ββ ββββββββ

Q3. Consider a full seasonal Fourier DLM with fundamental period p=10.*p*=10. What is the dimension of the state vector \bm{\theta}_t*ΞΈ**t*β at each time t*t*?

- None of the above
- 10
- 9
- 11

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