Is Weak Stationarity of Y_k Achievable in AR(1) Process?

In summary, the conversation discusses an AR(1) process with a recursive sequence and Poisson random variables, and the question of whether the resulting process is weakly stationary. The key factors to consider are the mean and variance of the process over time, and whether they remain constant or change. The conversation provides some hints and resources for further exploration of this question.
  • #1
fireb
11
0
Consider AR(1) process \(X_t=bX_{t-1}+e_t\)
where \(e_t\) with mean of 0 and variance of \(\sigma^2\)
and |b| <1
Let \( a_k \) be a recursive sequence with \( a_1 \) =1 and \( a_{k+1} = a_k + P_k +1\) for \( k = 1, 2 ,...,\) where \(P_k \) is Poisson iid r.v with mean = 1
also, assume \(P_t\) and \(X_t\) are independent.
Is \(Y_k\)= \(X_{a_k}\) for k =1,2,... weakly stationary?There arent any similar problems in the my textbook and i have no clue how to begin
Im not looking for a straight answer, just something to point me in the right direction.
Thanks in advance
 
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  • #2
sam said:
Consider AR(1) process \(X_t=bX_{t-1}+e_t\)
where \(e_t\) with mean of 0 and variance of \(\sigma^2\)
and |b| <1
Let \( a_k \) be a recursive sequence with \( a_1 \) =1 and \( a_{k+1} = a_k + P_k +1\) for \( k = 1, 2 ,...,\) where \(P_k \) is Poisson iid r.v with mean = 1
also, assume \(P_t\) and \(X_t\) are independent.
Is \(Y_k\)= \(X_{a_k}\) for k =1,2,... weakly stationary?There arent any similar problems in the my textbook and i have no clue how to begin
Im not looking for a straight answer, just something to point me in the right direction.
Thanks in advance

Hi

I have some experience with AR but don't claim to be an expert or provide you with a correct solution. I can't post links yet...

According to a basic definition of weakly stationary (google wikipedia and weakly stationary) we need to see whether the mean and variance of the AR model changes over time. If it does, then the process is not weakly stationary. From your definition above the variance and mean of \(e_t\) is not modified and does not change over time. Whilst I cannot state or be confident that this is the answer, my hint would be that this points to weakly stationary.

However according to section 4.1 (google weakly stationary definition and see the result from ohio state) you need to show the expectation of \(X_t\) is finite and does not depend on t (which I don't think it does but this is where the problem may lie)

Hope this helps, again this is not the correct solution just some hints.
Cheers
 

1. What is weak stationarity?

Weak stationarity is a statistical property of a time series process where the mean, variance, and autocovariance do not change over time. This means that the process has a constant average level, a stable spread, and a consistent relationship between observations at different points in time.

2. How is weak stationarity different from strong stationarity?

Strong stationarity requires all statistical properties of the time series process to remain constant over time, while weak stationarity only requires the first and second moments (mean and variance) to be constant. This means that weak stationarity is a weaker assumption than strong stationarity, but it is still useful for analyzing time series data.

3. What is an AR(1) process?

An AR(1) process is a type of autoregressive model where the current value of a time series is a linear combination of the previous value and a random error term. The "AR" stands for "autoregressive" and the "1" indicates that the process only depends on the previous value.

4. Is weak stationarity achievable in an AR(1) process?

Yes, weak stationarity is achievable in an AR(1) process if the process is stationary. This means that the process has a constant mean and autocovariance that does not depend on time. However, if the AR(1) process is non-stationary, weak stationarity cannot be achieved.

5. How is weak stationarity tested in an AR(1) process?

There are several statistical tests that can be used to determine if an AR(1) process is weakly stationary. These include the Augmented Dickey-Fuller test, the Kwiatkowski-Phillips-Schmidt-Shin test, and the Phillips-Perron test. These tests compare the observed data to a null hypothesis of non-stationarity and provide a p-value that indicates the likelihood of the data being non-stationary.

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