Some questions about CMP CS2 CH13 to CH16

[ykai]

1.CMP CS2-CH13-question 13.9-(ii)
Why e_(n-2)=0 proved thar process is not Markov?

2.CMP CS2-CH13-question 13.12-(i)
Why E(X) can prove whether process is stationary or not?

3.Is the formula of PACF_2 for MA(p),AR(q),ARMA(p,q),ARIMA(p,d,q) ,p&q>2,must be (acf_2-acf^2_1)/(1-acf^2_1)?
Is the formula of PACF_1 for MA(p),AR(q),ARMA(p,q),ARIMA(p,d,q) ,p&q>2,must be acf_1?

4.CMP CS2-CH14-question 14.1 & CS2 Assignment X4-4.11-(i)
Why acf_0 could be a sign for stationary process of ARIMA?
Is this because the variance is small and the values vary less?

Why choose d=0 in CS2 Assignment X4-4.11-(i) by acf_k?

5.CMP CS2-CH14-question 14.2-(ii)
Is "inspection of the SACF and SPACF" because of "asymptotic result"?

6.CMP CS2-CH14-question 14.3-(i)
"If there is any conflict between the two criteria then we should use the principle of parsimony in choosing the value for d."
What is "the two criteria"?
What is "the principle of parsimony"?

7.CMP CS2-CH14-question 14.3-(ii)
Does it is because of acf of stationary prcoess decrease as lag become larger?

8.CMP CS2-CH14-question 14.4-(iv)&CS2 Assignment X4-4.8-(iii)-(a)
Why epsilon_120 of CMP CS2-CH14-question 14.4-(iv)=0?
Why epsilon_81 of CS2 Assignment X4-4.8-(iii)-(a) =0?
Why epsilon_82 CS2 Assignment X4-4.8-(iii)-(a) =0?

9.CMP CS2-CH14-question 14.5-(iv)
Is "Inspection of the values of the sample autocorrelation function based on their 95% confidence intervals"&"Inspection of the values of the sample partial autocorrelation function based on their 95% confidence intervals" both because of "asymptotic result"?

asymptotic result:acf&pacf approximate to N(0,1/n) for large n

10.CMP CS2-CH14-question 14.7-(iv)
Will acf and pacf for all MA,AR,ARIMA,ARMA never be 0?

11.CMP CS2-CH16-question 16.7-(ii)
How does it become chi-square test in line 2 of proof to line 3 of proof?

12.CS2 Assignment X4-4.5-(i)
How to recognize d of stationary?
I don't know how to chach its stationary.
Both acf(r) and pacf not have cut off.
There is no sign to chech it.

13.CS2 Assignment X4-4.6-(b)&(c)
Shouldn't MA,ARMA,ARIMA always be stationary?
Might ARMA be non-stationary?

I want to check something about AR,MA,ARMA,ARIMA. It confused me for a long time.
Is stationary AR = MA(infinity)?
Is invertible MA =AR(infinity)?
Is acf_k and actocov_k always equal alpha^(k-1)*acf_1 and alpha^(k-1)*actocov_1 for 4 process?
Does decay of acf and pacf of ARMA is exponentially?
Are acf of AR and MA both Yule-Walker equation?

 

[ykai]

I have found answer for question 3&13.

 

[Andrew Martin]

Hello

1. For it to be Markov, this needs to be true regardless of the value of e_(n-2). So, showing it works only when it is 0 means it can't be Markov.

2. For a process to be stationary (weakly stationary when we are considering time series), we require that the mean is constant and that the covariance depends only on the lag. Showing the mean is not constant is therefore sufficient to show it can't be stationary.

4. Not sure what you mean by acf_0? I recommending rereading Section 3.2 for this bit.

5. The result contributes to the inspection yes.

6. For the two criteria see Section 3.2 as above. The principle of parsimony here relates to choosing as simple a model as possible that appears to reasonably fit the situation.

7. Seeing spikes at lags 12, 24, 36 and so on in monthly data is indicative of an annual trend.

8. As time 120 is in the future, we use the expected value of white noise (0) for the prediction. Similarly for X4.8.

Hope this helps!

Andy

 

[ykai]

Thank you for your reply!
I have full understood question 1~9.

For last 5 question,I have found answer for them except for "Is invertible MA =AR(infinity)?".
For "Is invertible MA =AR(infinity)?",
Can the following sentence be used as a basis for this?
CH13-page39-"The invertibility condition ensures that the white noise process e can be written in terms of the X process."
If not, I want to ask if not invertible MA= AR(infinity) ,too?

 

[ykai]

I have found answer of question 11.

 

[ykai]

For question 13-c,
I want to ask if ARIMA could be not invertible?
I saw it's maganitide pf roots of MA is not > 1.

 

[Andrew Martin]

Hello

An invertible MA can be written as an AR of infinite order. A stationary AR can be written as an MA of infinite order.

An ARMA process is invertible if the roots of the characteristic equation for the MA terms are greater than 1 in magnitude. It is stationary if the roots of the characteristic equation for the AR terms are greater than 1 in magnitude. It is possible to have an ARMA process that is not invertible, not stationary or both.

Hope this helps!

Andy

 

[ykai]

Thank you for your reply!
All of question I have fully understood except for question 10-"Will acf and pacf for all MA,AR,ARIMA,ARMA never be 0?".

From CS2 CMP-CH13-page11,"The ACF of a purely indeterministic process satisfies acf_k ->0 as k ->infinity.","We do not expect two values of a (purely indeterministic) time series to be correlated if they are a long way apart.",so I think it said that acf_k and pacf_k of all process will be 0 for large k, or it is just "close to 0" not "equal to 0"?

Here I want to confirm whether this applies to all processes, because CS2 CMP-CH13-page11 seems to conflict with the solution 14.7-(iv).

 

[Andrew Martin]

Hello

The relevant results for the theoretical ACF and PACF are as follows (assuming we are considering a stationary series):

For an AR(p) process, the PACF cuts off after lag p (ie phi_k is exactly 0 for k > p) and the ACF decays to 0.
For an MA(q) process, the ACF cuts off after lag q (ie rho_k is exactly 0 for k > q) and the PACF decays to 0.
For an ARMA(p,q), both the ACF and PACF decay to 0.

Hope this helps!

Andy

 

[ykai]

Thank you for your reply!
I have fully understood!