# ARMA - coefficient interpretation

I would like to interpret my ARMA model. I would like to tell about all those numbers as much as possible. I tried to study it but there as still some issues I am not sure about. I will start by trying to explain the thing I hope I understand correctly (if not, please help me out) and then ask you guys how to interpret the rest. The data are inter-bank interest rates.

• The first is AIC and BIC. This should tell me how good the model is. I can use these numbers to compare with other models used on these data. For example I can try to do ARMA (5, 5) and see if this less complex but better interpretable model is better. The lower the AIC and BIC, the better. Am I correct?

• Coef of constant is just the constant of the function that tries to describe the data I model. Am I correct?

• ar.L1.PRIBOR_1m this is the autoregresive of the model. It means that if the value is changed by 1 in previous period, then the current value is changed by 1,9136. Am I correct?

• ar.L2.PRIBOR_1m this means that if the value in two periods back is changed by 1, then current value is changed by -1,4880. And so on with ar.L3.PRIBOR_1m etc. Am I Correct?

• the P>|z| is the p-value of the significance of the coef. If I decide that for me is 5 % enough then all the coefs where P>|z| is lover then 0,05 are OK. Am I correct?

I do not know how to interpret: - std err

• moving average part of the model. Can those coefs by somehow interpreted as autoregresive part?

• What does [95,0% Coef. Int.] mean?

• How can I write down this model as a function? Something like Y=....

• is the second part of the table (real, imaginary,...) worth describing?

Thank you very much for your help.

Here Graeme Walsh explains how to better look at the fit than just the coefficients: ARIMA model interpretation

Here David J Harris describes differences in the model selection parameters AIC, BIC, etc. AIC,BIC,CIC,DIC,EIC,FIC,GIC,HIC,IIC --- Can I use them interchangeably?

Here an earlier question regarding ARMA models How do I choose which parameters to estimate in an ARMA model in python statsmodel?

Your question: *I would like to tell about all those numbers as much as possible. *

1) The first part is descriptive (like name and selected model which is straightforward) + some measures like the AIC, BIC, HQIC, which are measures that mix the likelihood with the number of parameters and data points. Various texts explain how they relate with selecting an ARMA model. One example is:

from this course http://halweb.uc3m.es/esp/Personal/personas/amalonso/esp/tsa.htm

page 50 of the 9th lecture explains that AIC tends to overfit whereas BIC is consistent because it penalizes axtra parameters more, yet AIC is better to model a process that is potentially of infinite order (I am not sure what that means but imagine an order that grows with the sample size)

2) The second part are the values of the model parameters plus their estimates.

Standard error relates to an estimate of the error of the predicted value (difference of the predicted value with the underlying "true" model value). This is a frequentists concept. If the experiment would (potentially) been done many times (with the same distribution of the residuals), how would the error be distributed (in simple words: how strong is the influence of the residual error terms on my estimate of the parameters, say if i do another test with similar distribution of the error terms then how much different parameters may I find just because of a different set of residual error terms?). The error is called 'standard error' because one expresses the estimate of the error in terms of the standard deviation, or the estimate of the second moment. This error may possibly calculated by some exact formula or less exact formula, one can also estimate it computationally it by simply creating a very large number of random data.

z This seems to be the standardized value of the coefficient values. It is $z=coef/std.error$ Just another way to express it and not really new information. $P(\vert z \vert)$ seems to me to be a t-test.

95% conf interval Is also no new information and seems to be the coefficient value +/- a certain number of the standard error. The percentage refers to the probabilit/frequency that the procedure would include the correct value in this interval.

The typical statistical interpretation stuff applies to these three concepts. But note the comment from Graeme Walsh in the linked reference that it is not so good to look at all these parameters on an individual basis.

3) The third part, with the complex numbers, relates to the roots of the characteristic equation of the model.

This is explained for instance here: https://en.wikipedia.org/wiki/Autoregressive%E2%80%93moving-average_model#Specification_in_terms_of_lag_operator

Using the characteristic equation several theoretic results can be expressed. See also in the answer by Graeme Walsh ARIMA model interpretation

In your case you see for instance that the modulus is always >1 and therefore the process is stationary.

• I was actually looking for whether I should use the 95% confidence interval. Any links explaining that with more detail, especially on the presence of square brackets [] saying 0.025 and 0.975 (in Python statsmodels)? – Milind R Jan 7 '18 at 16:22

You can use the pi weights to help you interpret/decipher/explain the model. The pi weights are obtained by dividing the ma polynomial by the ar polynomial . Presented in this way (i.e. as a pure ar) the model's parameters are simply a weighted average of the past . Bye the way your model is in my experience way and I mean "way" over-parameterized as a result of a poor model i.e. a list-based selection strategy rather than an iterative self-checking multi-stage approach. Unusual values need to be dealt with by incorporating pulse indicators and not torturing ar and ma coefficients to explain the anomalies. Often-time you can also have a non-constant error variance which can be dealt with GLS or a suitable power transformation. Finally there maybe break-points in time where model parameters change. Trying to form one set of coefficients to heterogeneous data is ill-advised.