# Bayesian Statistics: Time Series Analysis Coursera Quiz Answers

## All Weeks Bayesian Statistics: Time Series Analysis Coursera 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. Yt​−Yt−1​=et​−0.8et−1​

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

• (1−B)Yt​=(1−0.8B)et​
• None of the above
• BYt​=(1−0.8B)et​
• B(Yt​−Yt−1​)=0.8Bet​

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​} 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​/8yt−4​+41​(yt−3​+yt−2​+yt−1​+yt​+yt+1​+yt+2​+yt+3​)+81​yt+4​
• 1/8∑j=−88​ytk
• 1/2​(yt−4​+yt−3​+yt−2​+yt−1​+yt​+yt+1​+yt+2​+yt+3​+yt+4​)
• 1/8​(yt−4​+yt−3​+yt−2​+yt−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​(yt−1​+yt​+yt+1​)
• 1/3​(yt−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 = 0.8ϕ=0.8?

• None of above is correct
• .
• .
• .

Q2.

1. Which of the following AR(1) processes are stable and therefore stationary?
• Yt​=0.9Yt−1​+ϵt​,ϵt​∼i.i.d.N(0,v)
• Yt​=Yt−1​+ϵt​,ϵt​∼i.i.d.N(0,v)
• Yt​=−2Yt−1​+ϵt​,ϵt​∼i.i.d.N(0,v)
• Yt​=−0.8Yt−1​+ϵt​,ϵt​∼i.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)ϕ∈(−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)ϕ∈(−1,1) and \phi \neq 0ϕ​=0 are zero after lag 1
• The ACF of an AR(1) with coefficient \phi=0.5ϕ=0.5 decays exponentially in an oscillatory manner
• The ACF of an AR(1) with AR coefficient \phi=0.8ϕ=0.8 decays exponentially

Q4. Which of the following corresponds to the autocovariance function at lag h=2,h=2, \gamma(2)γ(2), of the autoregressive process Yt​=0.7Yt−1​+ϵt​,ϵt​∼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

yt​=−0.7yt−1​+ϵt​ with \epsilon_t \stackrel{iid}\sim N(0,1)ϵt​∼iidN(0,1)?

• 0.70.7
• -0.7−0.7
• \approx 1.96≈1.96
• 00

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

yt​=0.6yt−1​+ϵt​ with \epsilon_t \stackrel{i.i.d.}{\sim} N(0,v)ϵt​∼i.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​=−0.5yt−1​+ϵt​, with \epsilon_t \stackrel{i.i.d.}{\sim} N(0,1)ϵt​∼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 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)ϕ(1,1) is equal to -0.5−0.5
• The PACF coefficients for lags greater than 1 are zero
• The PACF coefficient at lag 1 \phi(1,1)ϕ(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​=0.5Yt−1​+0.24Yt−2​+ϵt​,ϵt​∼N(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.−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(−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.λ=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(2πh/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​=0.5Yt−1​+0.36Yt−2​+ϵt​,ϵt​∼N(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​(−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)−0.4 for c_{1t}c1t​ and c_{2t}c2t​ constants.
• ft​(3)=c1t​(0.9)3+c2t​(−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​=aθt2​+ϵt​,ϵt​∼N(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). θt​=bθt−1​+c1+θt−12​θt−1​​+ωt​,ωt​∼N(0,w).
• Observation equation: y_t = \mu_t + \epsilon_t, \quad \epsilon_t \sim \mathcal{N}(0, v),yt​=μt​+ϵt​,ϵt​∼N(0,v),
• System equation: \mu_t = \mu_{t-1} + \omega_t, \quad \omega_t \sim \mathcal{N}(0, w).μt​=μt−1​+ωt​,ωt​∼N(0,w).
• Observation equation: y_t = \theta_t + \epsilon_t, \quad \epsilon_t \sim \mathcal{N}(0, v), yt​=θt​+ϵt​,ϵt​∼N(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). θt​=bθt−1​+c1+θt−12​θt−1​​+ωt​,ωt​∼N(0,w).

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

• T \times 1T×1
• T \times TT×T
• K \times KK×K
• K \times 1K×1

Q3. Consider the third order polynomial Normal Dynamic Linear Model \mathcal{M}: \{\bm{F}, \bm{G}, \cdot, \cdot\}, M:{F,G,⋅,⋅}, where \bm{F} = (1 \quad 0 \quad 0)’F=(100)′ 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)=⎝⎜⎛​100​110​011​⎠⎟⎞​

Given the posterior mean E (\bm{\theta}_t | D_t) = (m_t, b_t, g_t)’,E(θt​∣Dt​)=(mt​,bt​,gt​)′, which of the following options is the one corresponding to the forecast function f_t(h) \quad (h \geq 0)ft​(h)(h≥0) of the model?

• ft​(h)=mt​+hbt​+h(h−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.h≥0.

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

• F=(1xt​10)′

\bm{G} = \bm{I}_4,G=I4​,

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

• F=(1010)′

\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 22×2, 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)′

\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:

(10)

,

(−0.7501−0.75)

, \cdot, \cdot \right\} M:{(10​),(−0.750​1−0.75​),⋅,⋅} and

E(\bm{\theta}_t | \mathcal{D}_t) = (1, 0.5)’E(θt​∣Dt​)=(1,0.5)′. What is the best description of the forecast function, f_t(h)ft​(h) for h \geq 0h≥0 of this model?

• .
• .
• .
• .

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

f_t(h) = k_{t1}\lambda^h_1 – k_{t2}\lambda_2^h, ft​(h)=kt1​λ1h​−kt2​λ2h​,

where \lambda_1\lambda_2 \neq 0λ1​λ2​​=0 and \lambda_1 \neq \lambda_2λ1​​=λ2​?

• F=(1,0)′,

\bm{G} =

(λ111λ2).G=(λ1​1​1λ2​​).

• F=(1,−1)′,

\bm{G} =

(λ100λ2).G=(λ1​0​0λ2​​).

• F=(1,−1)′,

\bm{G} =

(λ111λ2).G=(λ1​1​1λ2​​).

• F=(1,0)′,

\bm{G} =

(λ100λ2).G=(λ1​0​0λ2​​).

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​,⋅,⋅} with

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

λ001 \right).Gt​=(λ0​01​).

• {Ft​,Gt​,⋅,⋅} with

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

λ00xt \right).Gt​=(λ0​0xt​​).

• {Ft​,Gt​,⋅,⋅} with

\bm{F}_t = (1,x_t)’Ft​=(1,xt​)′ 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)(θt​∣Dt​) is normal if the distribution of (\mathbf{\theta}_0 | \mathcal{D}_0)(θ0​∣D0​) is normal
• The filtering equations allow us to obtain the moments of the distribution of (\mathbf{\theta}_t| \mathcal{D}_T)(θt​∣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)(θt​∣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)(θt​∣Dt​) is not normal even if the distribution of (\mathbf{\theta}_0 | \mathcal{D}_0)(θ0​∣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)′

\bm{G} = \left(

−12−3√23√2−12 \right) G=(−21​−23​​​23​​−21​​)

• F=(10)′

\bm{G} = \left(

1011 \right) G=(10​11​)

• F=(101)′

\bm{G} = \left(

0−1010000−1

\right) G=⎝⎜⎛​0−10​100​00−1​⎠⎟⎞​

• F=(10)′

\bm{G} = \left(

12−3√23√212 \right) G=(21​−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.ω=2π4/12=2π/3. What is the forecast function f_t(h), h \geq 0,ft​(h),h≥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(32πh​)+at,4​sin(32πh​)
• ft​(h)=at,1​cos(32πh​)+at,2​sin(32πh​)
• ft​(h)=at,0​+at,1​h+at,3​cos(32πh​)+at,4​sin(32πh​)+at,5​(−1)h

Q4. A DLM \{ \bm{F}_t, \bm{G}_t, \cdot, \cdot \}{Ft​,Gt​,⋅,⋅} 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​+(−1)hat,2​.

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

• Ft​=(1,xt​)′ and \bm{G}_t =\left(

1001 \right)Gt​=(10​01​)

• Ft​=(1,xt​)′ and \bm{G}_t =\left(

100−1 \right)Gt​=(10​0−1​)

• Ft​=(1,0)′ and \bm{G}_t =\left(

10xt−1 \right)Gt​=(10​xt​−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​,⋅,⋅} with \bm{F}_1 = (1,0,0)’,F1​=(1,0,0)′, \bm{G}_1 = \bm{J}_3(1) G1​=J3​(1) and state vector \bm{\theta}_{1t} = (1, -0.5, 0.1)’.θ1t​=(1,−0.5,0.1)′.
• Seasonal component: \{\bm{F}_2, \bm{G}_2, \cdot, \cdot\}{F2​,G2​,⋅,⋅} with \bm{F}_2 = (1, 0)’F2​=(1,0)′, \bm{G}_2 = \bm{J}_2(\lambda, \omega)G2​=J2​(λ,ω), where \omega = \frac{\pi}{2}ω=2π​ and \lambda = 0.9λ=0.9 and state vector \bm{\theta}_{2t} = (1, 0.45)’.θ2t​=(1,0.45)′.

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

• .
• .
• .
• .

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

1 point

• F=(1,0,1,0,0)′

\bm{G} =

⎛⎝⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜12−3√20003√21200000−12−3√20003√2−1200000−1⎞⎠⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟G=⎝⎜⎜⎜⎜⎜⎜⎛​21​−23​​000​23​​21​000​00−21​−23​​0​0023​​−21​0​0000−1​⎠⎟⎟⎟⎟⎟⎟⎞​

• F=(1,0,1,0,1)′

\bm{G} =

⎛⎝⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜12−3√20003√21200000−12−3√20003√2−1200000−1⎞⎠⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟G=⎝⎜⎜⎜⎜⎜⎜⎛​21​−23​​000​23​​21​000​00−21​−23​​0​0023​​−21​0​0000−1​⎠⎟⎟⎟⎟⎟⎟⎞​

• F=(1,0,1,0,0)′

\bm{G} =

⎛⎝⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜3√21200012−3√2000003√2−12000−12−3√200000−1⎞⎠⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟G=⎝⎜⎜⎜⎜⎜⎜⎛​23​​21​000​21​−23​​000​0023​​−21​0​00−21​−23​​0​0000−1​⎠⎟⎟⎟⎟⎟⎟⎞​

• F=(1,0,1,0,1)′

\bm{G} =

⎛⎝⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜3√21200012−3√2000003√2−12000−12−3√200000−1⎞⎠⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟G=⎝⎜⎜⎜⎜⎜⎜⎛​23​​21​000​21​−23​​000​0023​​−21​0​00−21​−23​​0​0000−1​⎠⎟⎟⎟⎟⎟⎟⎞​

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

• None of the above
• 10
• 9
• 11
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We are Team Networking Funda, a group of passionate authors and networking enthusiasts committed to sharing our expertise and experiences in the world of networking and team building. With backgrounds in [Your Background or Expertise], we bring a diverse range of perspectives and insights to help you navigate the challenges and opportunities of professional networking and teamwork.

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