We know that the general ARIMA equation is given by:
Yt^ = μ + Φ1Yt-1 + …+ ΦpYt-p - θ1et-1 - … - θqet-q
The following data is given:
Period(s): 1 2 3 4 5 6 7
Yt: 25 30 29 31 35 34 -
Yt-1: * 25 30 29 31 35 34
d1=Yt-Yt-1 * 5 -1 2 4 -1 -
Using simple linear regression, we get μ = 0.8955 and Φ1 = 1.522
How to manually calculate the forecasted value of Yt^ for the 7th period using non-seasonal ARIMA (0,1,0) and non-seasonal ARIMA (1,1,2)?
[So far I know, the values of coefficients of the error terms or θ’s are estimated by non-linear optimization methods (hill-climbing) as the error may not be independent variable. But how the non-linear optimization method is manually performed in this case in order to find out the values of the error coefficients?]
Can anybody please help me?
First note that your equation holds for ARMA, not ARIMA models. For ARIMA models there are at least two ways to introduce a constant mu. Your first problem is trivial if mu is omitted since the e_t's are identical to the d_t's. If mu is present, the simplest is to take the average of the d_t's and subtract it from the d_t's and you are done. The other cases are more complex. There are several estimation methods but all make use of nonlinear optimization. The function to optimize can be a sum of squares or a (Gaussian) likelihood function, or an approximation of the latter. For a series length less than 50, say, I would strongly recommend maximum likelihood. Except perhaps for an ARIMA(0,1,1), computing the likelihood is difficult to show. I have a spreadsheet where I did it but it should be cleaned. Tell me if you are interested. Tell me first what is your objective: research or teaching. The main reference is still Box and Jenkins (1970), see Box et al. (2015) for an updated version which includes the likelihood approach. By the way my 1984 algorithm FLIKAM published in JRSS Ser. C Appl. Statist. is still available. Note also that there are also least squares methods in several steps like the method of Hannan and Rissanen.
Thanks Dr. Guy Mélard for your kind and valuable answer. My interest is actually research. However, I just want to know how this is manually performed. If you kindly send me the spreadsheet (if possible), I shall be highly obliged to you. I have more than 50 data points, almost 120 data points. So, in this case what kind of estimation method you suggest for nonlinear optimization? Also, I would like to know how different types of software handle this; do all software use same estimation method for a specific number (or limits) of data points?
1. I will try to locate the spreadsheet on my hard disc. Tomorrow probably.
2. For 120 data points, it depends largely on the value of the parameters but if the ARIMA model is not seasonal, a nonlinear least squares approach can be enough.
3. You should distinguish the optimization algorithm and the algorithm for computing the objective function. My focus here is on the latter. I have an old paper on the former too and I suppose it is on ResearchGate. I should try to find it. Tomorrow also.
4. Unfortunately my observations (a paper in preparation) is that there are small differences between different software packages. Generally 3 to 6 decimal correct digits can be obtained.
Thank you sir, thank you so much Dr. Mélard for your valuable suggestions. My email id is: [email protected]
I am now Honorary Visiting Professor, Indian Statistical Institute, Baranagar, Kolkata, India since 2014
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