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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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    $\begingroup$ This is awfully broad. Can you say more about your situation & what you've done? What is your current model fitting strategy? Do you typically just guess at $p$, $d$, & $q$? How have you checked & found your model to fit poorly? Note that the time-series topic & loosely similar questions are very common on this site, have you searched through the site and found anything helpful? $\endgroup$ – gung - Reinstate Monica Jul 20 '12 at 20:32
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    $\begingroup$ It might help to post a plot of your data as well. $\endgroup$ – jbowman Jul 20 '12 at 21:45
  • $\begingroup$ Based on your edit I assume that the automated procedure did not provide a good fit. Assuming it goes through the standard Box-Jenkins methodology of identification, estimation, diagnostic checking and possible iteration, what would make you think you could find a better model in the ARIMA family. I think I should defer to @TrishStat for expert advice on this but I think you need to understand why the model gives a poor fit and look for alternative model forms that might handle the deficiency as i suggested in my answer. $\endgroup$ – Michael R. Chernick Jul 21 '12 at 6:02
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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

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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.

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  • $\begingroup$ To extend the rule of thumb: d should probably be small too as one ought not over-difference the data. $\endgroup$ – Graeme Walsh May 23 '13 at 23:27
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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.

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