[R] estimating an ARIMA model with constraints

[Laurent Duvernet]

Sorry if the notation is unclear. You got it right:

P(x)  = (1 - a_1*x - a_2*x^2) * (1 - b_1*x^23 - b_2*x^24) * (1 - c_1*x^168).

The a_i's, b_i's and c_i's are the coefs of the polynom P.

And there is also an MA part, which is "Q(B) epsilon(t)". Here epsilon(t) is 
the error process, and Q is another polynom of the same type as P (it could 
be different, that does not change the problem).

Q(x)  = (1 - alpha_1*x - alpha_2*x^2) * (1 - beta_1*x^23 - beta_2*x^24) * (1 
- gamma_1*x^168).

I can write "X(t) = ...", but I'm not sure it would be a lot clearer...

X(t) =  a_1*X(t-1) + a_2*X(t-2) + b_1*X(t-23) + (b_2 + a_1*b_1)*X(t-24) + 
(a_1*b_2 + a_2*b_1)*X(t-25) + a2*b2 X(t-26) + .... the terms around X(t-168) 
+ ... the MA part.

I hope everything is clear now.



>From: "Leeds, Mark (IED)" <Mark.Leeds at morganstanley.com>
>To: "Laurent Duvernet" <montcroix at hotmail.fr>, <r-help at stat.math.ethz.ch>
>Subject: RE: [R]  estimating an ARIMA model with constraints
>Date: Tue, 13 Mar 2007 10:56:47 -0400
>
>
>are the carats in your notation meant to be time subscripts ?
>
>also, I think I know what a and b are meant to be ( the coefficients of
>the polynomaisl corresponding
>To the ar part of the model but correct me if I'm wrong ) but is there
>an ma piece to it also ?
>And I don't see an error term ?
>
>
>I think you need to be clearer on your notation and write out the full
>model in terms of X(t) = whatever because then more people will reply.
>
>
>-----Original Message-----
>From: r-help-bounces at stat.math.ethz.ch
>[mailto:r-help-bounces at stat.math.ethz.ch] On Behalf Of Laurent Duvernet
>Sent: Tuesday, March 13, 2007 10:36 AM
>To: r-help at stat.math.ethz.ch
>Subject: [R] estimating an ARIMA model with constraints
>
>Hi,
>
>I am trying to estimate an ARIMA model in the case where I have some
>specific knowledge about the coefficients that should be included in the
>model. Take a classical ARIMA (or even ARMA) model:
>
>P(B) X(t) = Q(B) epsilon(t),
>
>where X(t) is the data, epsilon is a white noise, B is the backward
>operator and P and Q are some polynoms. Additionally, assume that you
>know in advance how P and Q look like. Typically, P could be something
>like this:
>
>P(x) = (1 - a(1)*x - a(2)*x^2) * (1 - b(1)*x^23 - b(2)*x^24) * (1 -
>c(1)*x^168)
>
>(That is in the case of hourly data, with lags 23 and 24 corresponding
>to the day, and lag 168 for the week.) How do you estimate this kind of
>model with R? The arima() and arima0() functions in the stats package do
>not allow this kind of constraints on the polynoms. I've searched in the
>packages dedicated to time series analysis, but I have not found a
>solution. Has anyone an idea?
>
>Thanks in advance!
>
>Laurent Duvernet
>EDF R&D
>
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