Suppose an ARIMA(p,d,q) model fits the data very poorly. What are some ways to improve the fit? Is there any way to do it without guessing and checking?
Edit. I am using auto.arima() to search for an ARIMA model. But it is a poor fit.
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In general ARIMA modeling means assuming Gaussian error terms. What you should do depends on what you tried and why it didn't fit. So without further clarification it is hard to give a specific answer. But is is possible that no ARIMA model will work even when adding in seasonal differencing asnd possible interventions. This could be due to outliers or the fatc that the residuals are not gaussian or the residual variance is not constant. Various alternatives are the ARCH and GARCH models ARIMAX models when the residuals are the problem because of "volatility, nonconstant variance or the need for exogenous variables. These are examples of generalizations outside the realm of the ARIMA models that I know of that can sometimes help.
Here is an interesting paper discussiong the GARCH(1,1) model for handling volatility. http://www-stat.wharton.upenn.edu/~steele/Courses/434/434Context/GARCH/HansenLunde01.pdf
If ARIMA is not fitting the data well, then ARIMA might be a bad model. There is no surety that ARIMA will give good results for any dataset. A simple example should be datasets which are usually modeled by GARCH (for eg Volatility)
In Econometrics, theory should precede modeling. If you can elaborate on the data and the process, then i would be easier to answer the vague question.
Regarding ARIMA(p,d,q), check for "d" stationarity of data, if stationary then move to estimate p and q, else difference and check for stationarity. Once stationarity is established then move to estimate p and q.
Rule of Thumb says that p and q should be small. The sparser the number of variables, the more robust the model is "expected" to be.
The auto.arima AIC procedure might work ok if there were no pulses, level shifts, seasonal pulses, local time trends, transient(changes) in parameters at particular points in time, transient (changes) in error variance at particular points in time. If you have any of these circumstances you may be at risk. While standard acf and pacf identification schemes are also subject to these "requirements" , modern procedures using robust EACF and other aggressively analytical procedures should be investigated. Your post is similar to many others reflecting on poor automatic identification leading to poor estimation results e.g. How to fit a model for a time series that contains outliers and What type of time series model would be good? and Auto.arima vs autobox do they differ? for recent activity on this important subject. You might want to post your data ,your potentially deficient model identification results and your potentially deficient estimation results and see if there are any takers to help you to come up with a potentially quality result.