Please check the question I upload below. For this assignment you do not need to do the question 6.

stat451assignment08.pdf

time_series_assignment.pdf

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Assignment 8 for Statistics 451, Spring 2019

1. Briefly explain why the family of Box-Cox transformations is preferred over the simpler-looking

family of power transformations.

2. The sample ACF and PACF’s are not the same as the “true” ACF and PACF’s.

(a) What inputs are needed to compute the sample ACF and PACF’s?

(b) What inputs are needed to compute the “true” ACF and PACF’s?

(c) How would you explain the difference between the sample ACF and PACF’s and the

“true” ACF and PACF’s to someone who has not has exposure to time series analysis

(be brief)?

3. Briefly explain:

(a) When and why the principle of parsimony is important in time series modeling.

(b) When the principle of parsimony is not important in time series modeling.

4. The filter representation of an ARMA model is

Zt =

θq (B)

at = ψ(B)at .

φp (B)

(1)

where the finite-length polynomials θq (B) and φp (B) were defined on page. The impulseresponse function (also known as the ψ weights) is given by the coefficients in the (potentially

infinite) ψ(B) polynomial. Use the RTseries commands

print(show.impulse.response(denominator=0.50,type=”arma”,

y.axis=TRUE, lag.max=7))

print(show.impulse.response(numerator= c(-0.5, -0.25, -0.125, -0.0625,

-0.03125, -0.015625), type=”arma”, y.axis=TRUE, lag.max=7))

print(show.impulse.response(numerator=0.40, type=”arma”,

y.axis=TRUE, lag.max=7))

to obtain the first seven coefficients of the impulse response functions for the specified AR(1),

MA(6), and MA(1) models:

Zt =

1

at = 0.50Zt−1 + at

(1 − 0.50B)

Zt =

(1 + 0.5B + 0.25B2 + 0.125B3 + 0.0625B4 + 0.03125B5 + 0.015625B6 )

at

1

= 0.5at−1 + 0.25at−2 + 0.125at−3 + 0.0625at−4 + 0.03125at−5 + 0.015625at−6 + at

Zt =

(1 − 0.40B)

at = −0.40at−1 + at

1

Notice the similarity of the impulse response functions for the AR(1) and the MA(6) models.

(a) What does the similarity of the impulse response functions for the specific AR(1) and

the MA(6) models imply?

1

(b) Find an AR(6) model that has an impulse response function that is similar to the specific

MA(1) model above (i.e., the MA(1) model with θ1 = 0.4). Check your answer using the

show.impulse.response function, as above.

(c) What is the important practical implication of having different models that have (approximately) the same impulse response functions?

(d) Explain when the ψ(B) polynomial will have a finite number of non-zero coefficients.

5. In time series modeling, both the ACF of the residuals and the Ljung-Box statistic are useful

diagnostics. Explain briefly how the Ljung-Box statistic complements the plot of the ACF of

the residuals.

6. Use the iden function to get the sample ACF function for simulated series #11 (the RTseries

object name is simsta11.tsd). Given the numerical values for the sample ACF from the

computer output, compute φb11 and φb22 for simulated series #11. Compare these with the

numerical values for the sample PACF from the computer output.

Note that you need to use a command like

iden(simsta11.tsd, print.table=TRUE)

in order to request the table of numerical values.

2

Assignment 8 for Statistics 451, Spring 2019

1. Briefly explain why the family of Box-Cox transformations is preferred over the simpler-looking

family of power transformations.

2. The sample ACF and PACF’s are not the same as the “true” ACF and PACF’s.

(a) What inputs are needed to compute the sample ACF and PACF’s?

(b) What inputs are needed to compute the “true” ACF and PACF’s?

(c) How would you explain the difference between the sample ACF and PACF’s and the

“true” ACF and PACF’s to someone who has not has exposure to time series analysis

(be brief)?

3. Briefly explain:

(a) When and why the principle of parsimony is important in time series modeling.

(b) When the principle of parsimony is not important in time series modeling.

4. The filter representation of an ARMA model is

Zt =

θq (B)

at = ψ(B)at .

φp (B)

(1)

where the finite-length polynomials θq (B) and φp (B) were defined on page. The impulseresponse function (also known as the ψ weights) is given by the coefficients in the (potentially

infinite) ψ(B) polynomial. Use the RTseries commands

print(show.impulse.response(denominator=0.50,type=”arma”,

y.axis=TRUE, lag.max=7))

print(show.impulse.response(numerator= c(-0.5, -0.25, -0.125, -0.0625,

-0.03125, -0.015625), type=”arma”, y.axis=TRUE, lag.max=7))

print(show.impulse.response(numerator=0.40, type=”arma”,

y.axis=TRUE, lag.max=7))

to obtain the first seven coefficients of the impulse response functions for the specified AR(1),

MA(6), and MA(1) models:

Zt =

1

at = 0.50Zt−1 + at

(1 − 0.50B)

Zt =

(1 + 0.5B + 0.25B2 + 0.125B3 + 0.0625B4 + 0.03125B5 + 0.015625B6 )

at

1

= 0.5at−1 + 0.25at−2 + 0.125at−3 + 0.0625at−4 + 0.03125at−5 + 0.015625at−6 + at

Zt =

(1 − 0.40B)

at = −0.40at−1 + at

1

Notice the similarity of the impulse response functions for the AR(1) and the MA(6) models.

(a) What does the similarity of the impulse response functions for the specific AR(1) and

the MA(6) models imply?

1

(b) Find an AR(6) model that has an impulse response function that is similar to the specific

MA(1) model above (i.e., the MA(1) model with θ1 = 0.4). Check your answer using the

show.impulse.response function, as above.

(c) What is the important practical implication of having different models that have (approximately) the same impulse response functions?

(d) Explain when the ψ(B) polynomial will have a finite number of non-zero coefficients.

5. In time series modeling, both the ACF of the residuals and the Ljung-Box statistic are useful

diagnostics. Explain briefly how the Ljung-Box statistic complements the plot of the ACF of

the residuals.

6. Use the iden function to get the sample ACF function for simulated series #11 (the RTseries

object name is simsta11.tsd). Given the numerical values for the sample ACF from the

computer output, compute φb11 and φb22 for simulated series #11. Compare these with the

numerical values for the sample PACF from the computer output.

Note that you need to use a command like

iden(simsta11.tsd, print.table=TRUE)

in order to request the table of numerical values.

2

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