Turning points test

[Molly]

Hi all,

is the turning points test hypothesis always:
H0: the residuals are from a white noise process H 1 : the residuals are not from a white noise process?

i dont really understand why this is the hypothesis? why not : the resiudals fit the data? what does this null hypothesis actually mean please?

Thanks,
Molly

 

[Andrew Martin]

Hi Molly

Say we're fitting an AR(2) model to some data, ie we're saying that we think that the data can be modeled as follows:

Xt = a1 Xt-1 + a2 Xt-2 + et

where {et} is a white noise process.

All ARMA models assume the {et} follow a white noise process and it is this assumption that we check when doing the diagnostics. We do this by calculating the residuals, which are the fitted values of the {et} for a particular model. For example, say we estimate a1 and a2 to be 0.3 and 0.1, then:

e^hat(t) = Xt - 0.3Xt-1 - 0.1Xt-2

We then check whether the e^hat(t) (the residuals) appear to be a white noise process. If not, then this suggests that we haven't got the structure of the model correct.

Hope this helps!

Andy

 

[Molly]

Hi Molly

Say we're fitting an AR(2) model to some data, ie we're saying that we think that the data can be modeled as follows:

Xt = a1 Xt-1 + a2 Xt-2 + et

where {et} is a white noise process.

All ARMA models assume the {et} follow a white noise process and it is this assumption that we check when doing the diagnostics. We do this by calculating the residuals, which are the fitted values of the {et} for a particular model. For example, say we estimate a1 and a2 to be 0.3 and 0.1, then:

e^hat(t) = Xt - 0.3Xt-1 - 0.1Xt-2

We then check whether the e^hat(t) (the residuals) appear to be a white noise process. If not, then this suggests that we haven't got the structure of the model correct.

Hope this helps!

Andy

Hi Andy,

that really does help so much thank you!

So we use the turning point test to check whether the residuals are white noise.
We can use the portmaneu test to do this too right? looking at April 2021, Question 9, although our actual model is ARMA(2,0) we have
m-(p+q)=3-(0+0) because we are testing for white noise so ARIMA(0,0).

So will this always be the case? should we always omit to ar part for this question? or is this test completely different, is this test for whether the process as a whole is white noise, where the turning points test is concerned only with whether the residuals are white noise?

Thanks so much

 

[Andrew Martin]

Hi Molly

That's right, the portmanteau test is also used to investigate whether the residuals appear to form a sample from a white noise process. When we do a portmanteau test we consider the magnitude of the sample autocorrelations. We have to choose how many autocorrelations (lags) to consider and this factors into the degrees of freedom for the relevant chi-squared distribution.

The degrees of freedom are m - (p+q) as you say, where m is the number of lags and p and q are the orders of the fitted model. So, if we have ARMA(2,0) then p + q = 2 and if m were 3 we would have 3 - (2) = 1. If we had fitted an ARMA(1,1) then p + q = 2 again etc.

If we are testing whether a sample set of data is white noise without first fitting an ARMA(p,q) model to it. Then we are essentially fitting an ARMA(0,0) to the data and in this case p + q = 0. However, when considering the residuals of a fitted ARMA(p,q) it is these values of p and q that feed into the dof.

Hope this helps!

Andy