# Does ARIMA require normally distributed errors or normally distributed input data?

I have two questions related to time series forecasting with ARIMA:

1. Does ARIMA require normally distributed errors or normally distributed input data ?
2. Are there any assumptions on input time series data for ARIMA model and exogenous variables for ARIMAX model ?
• It depends; you can estimate parameters without assuming normality, but inference such as hypothesis tests and confidence intervals are based on that assumption. Dec 11 '13 at 23:08
• 1.Errors 2.Equal length I need at least 30 characters to post so I have these words here. Dec 12 '13 at 21:35
• thanks so much. I'm not sure then why people transform input data when there is really no assumption on the data, rather the assumption is on errors. Any insights appreciated as textbooks don't provide these type of information. Dec 13 '13 at 1:14
• @forecaster In order to identify an ARIMA model via an AIC criteria or to formally test the significance of estimated parameters via a T test or an F test the residuals should not exhibit non-randomness. THe Gaussian assumptions have all to do with the model's error process and nothing to do with the distribution of the observed series. For sure the originally observed series is functionally related to the error process via the model BUT there are no parametric distributional requirements for the observed series. My comments are all supportive of Glen B's reflections. Dec 16 '13 at 14:23
• @IrishStat, what is your take on having normally distributed errors is equivalent to having normally distributed observations for any linear time series model from Rob J. Hyndman's answer? Jun 2 '19 at 11:12

First, having normally distributed errors is equivalent to having normally distributed observations for any linear time series model.

Second, it is not necessary to assume normality of errors. Often, maximum likelihood is used to estimate the parameters of the model, and then a Gaussian likelihood is used, but it gives good results even with non-normal data. Where normality of errors is often assumed is in using the AIC for order selection, and in computing prediction intervals.

There are several specifications of ARIMA models with exogenous variables, and more than one such specification has been called an ARIMAX model, so it is not possible to precisely answer your second question without you specifying the model more accurately. For discussion of some of the models, see http://robjhyndman.com/hyndsight/arimax/

• Could you give any explanation why a Gaussian likelihood gives good results even with non-normal data? Or provide a reference? I understand how that works in a multiple regression but intuitively thought that ARIMA models may be more sensitive to deviations from normality (when normality is used in MLE). Mar 24 '15 at 17:09
• MLE with a Gaussian likelihood is asymptotically equivalent to least squares estimation. Minimizing the sum of squared errors should work ok for almost all distributions. Mar 24 '15 at 23:27
• I'm curious if it would for extreme values of sequences. For example, could I use ARIMA to predict the maximum price of a stock over 10 minute intervals? (Assuming the stock movement is not a totally random walk) May 18 '15 at 0:15

No -- the paper Maximum Likelihood Estimates of Non-Gaussian ARMA Models studies ARMA models with Student-t and Laplace distributed errors. Here is the abstract:

"We consider an approximate maximum likelihood algorithm for estimating parameters of possibly non-causal and non-invertible autoregressive moving average processes driven by independent identically distributed non-Gaussian noise. The normalized approximate maximum likelihood estimate has a global maximum which is consistent and ecient. The estimates and their associated asymptotic covariance matrix are calculated with a subroutine implemented in FORTRAN 77."

To answer your first question and to quote from [1],

Mathematics tell us that linear predictor can be optimal only when the process is Gaussian. When the process is non-Gaussian, a better predictor may be given by a non-linear dynamic model (Masani and Wiener, 1959).

Why statisticians, both young and old, keep themselves silent about it I do not know.

References:

[1] Ozaki, T. & Iino, M. An innovation approach to non-Gaussian time series analysis Journal of Applied Probability, 2001, 38, 78-92

• And yes, strictly speaking, ARIMA models are linear because their parameters are linear. Jun 3 '19 at 15:12