Solved – Fitting an ARFIMA Model with Covariates in R

arfimaarimartime series

In R's arima() function, one can specify a list of covariates while estimating the AR and MA coefficients using the xreg argument. For instance:

arima(lh, c(2,0,1), xreg= 1:length(lh))

returns a model with ARMA(2,0) disturbance and the linear effect of time series 1:length(lh):

Call:
arima(x = lh, order = c(2, 0, 1), xreg = 1:length(lh))

Coefficients:
         ar1      ar2      ma1  intercept  1:length(lh)
      1.3957  -0.6453  -1.0000     2.1072        0.0105
s.e.  0.1065   0.1082   0.0589     0.0585        0.0023

sigma^2 estimated as 0.1411:  log likelihood = -22.85,  aic = 57.7

Is there any equivalent method in R when fitting an ARFIMA (Autoregressive fractionally integrated moving average) model? I know it is possible to run a multiple regression on the residuals of an ARFIMA model, but this is different from estimating them together and would like to learn if anyone has a better suggestion.

Best Answer

Note that feeding the xreg parameter in arima() or auto.arima() does not fit an ARIMAX model, but a regression on xreg with ARIMA errors. See Rob Hyndman's "The ARIMAX model muddle" blog post.

This of course suggests how one could do a similar thing with ARFIMA: regress your observations on your covariates using lm() or similar, then fit an ARFIMA model to the residuals, e.g., using the arfima package. Alternatively, simply use the arfima() function in the package and use the xreg parameter.

However, if you really want to fit an ARFIMAX model, then I don't know of any R package that will help. (Same for ARIMAX, though I have never explicitly searched for this, since regression with ARIMA errors makes a lot more sense to me.)

Related Solutions

Solved – time series forecasting using auto.arima and exponential smoothing

Seasonality is probably not very strong. Different algorithms will give different results, unless seasonality is glaringly obvious.
The best measure is always to compare forecast accuracy on a holdout set: hold back the last $n$ observations, fit your models to all other observations, forecast into the last $n$ time periods with both models, then compare forecast accuracy using your error measure of choice (see 5 below).
Yes, this is a common complaint. I don't think there is an easy way to get the in-sample fit. But you can get the residuals: auto.arima(WWWusage)$residuals. Best to look into the code of auto.arima() to see whether you need to add or subtract them from the original series to get the fit. I'd say you have to subtract ("actuals=model+residuals"), but better check.
I recommend a good forecasting textbook. This is a very good start. Otherwise, read through the help pages.
The appropriate error measure will depend on your personal loss function. Is your pain symmetric, and will it increase more strongly with larger errors? Then use MSE. Is your pain proportional to absolute errors? Then use MAE. Best to look at multiple error measures.

One tip: averaging forecasts will usually improve accuracy. Consider taking the average of your two models' forecasts per future time bucket.
auto.arima() apparently fits no drift, even if you allow it.

Solved – Interpretation of TSA::arimax output model is presented in R

This is an old post, but since there is no accepted answer, I still want to provide an explanation for future readers. You have the following R output:

Coefficients:
         ar1      ar2      ma1  intercept     xreg
      1.4825  -0.6613  -0.8516  52745.107  -1.0132
s.e.  0.0295   0.0294   0.0064     40.828   0.0012

sigma^2 estimated as 0.08929:  log likelihood = -105.98,  aic = 221.97

(1) y = intercept + xreg * x series name + n_t(error term)

(2) n_t(error term) = ar1 * n(t-1) + ar2 * n(t-2) + white noise epsilon + ma1 * epsilon(t-1)

(3) white noise epsilon ~ NID(0,0.08929)

To explain it further:

(1) This formula focuses on xreg coefficient. The R outputs gives you a coefficient for xreg -- the effect of the exogenous variable on Y at it's current value.

(2) This formula focuses on unpacking the error term left unexplained by formula (1). Formula 1 and 2 together is also called regression with ARIMA error (ARMA error in this case). This error term itself is an ARIMA (2,0,1) process, according to your output. This is what ar1, ar2, ma2 in the R output corresponds to.

(3) This formula is describing what is left after formula 1 and 2. sigma^2 describes the variance of the white noise series -- epsilon in most Econometric textbooks.

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Solved – time series forecasting using auto.arima and exponential smoothing

Solved – Interpretation of TSA::arimax output model is presented in R

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