Solved – R auto.arima() with non-stationary covariates

arimapredictorrstationarity

I want to fit an ARMA model with covariates to a non-stationary time series.
I have daily measurements for water flow for 4 stations (S1-S4) and the time series is not stationary, so I will have to take a first difference.

I want to find an ARMA model that takes as covariates the other 3 stations.

What I have done:
a) m1 = auto.arima(S1, d=1, max.p, max.q, xreg=data.frame(S2,S3,S4))
b) m2 = auto.arima(diff(S1), d=0, max.p, max.q, xreg=data.frame(diff(S2),diff(S3),diff(S4)))

My question: which model is correct, considering the series are non-stationary?

I will appreciate any suggestion.

Thank you,
Patricia

Best Answer

They are algebraically equivalent models, although the estimated coefficients will be slightly different due to the way the models are estimated.

Assuming you are actually interested in S1 rather than its differences, then you should use m1.

Related Solutions

Solved – Arima time series forecast (auto.arima) with multiple exogeneous variables in R

If your external regressors are causal for $y$, but not the other way around and do not cause each other, then ARIMA is definitely appropriate. VAR makes sense if your different time series all depend on each other.

For auto.arima() to work with external regressors, collect your regressors into a matrix X, which you feed into the xreg parameter of auto.arima(). (Of course, X must have the same number of rows as the time series y you are modeling.)

For forecasting, you will need the future values of your regressors, which you then again feed into the xreg parameter of forecast.

The help pages are ?auto.arima and ?forecast.Arima (note the capital A - this is not a typo. Don't ask me...).

Solved – How to make time series data with changing variance stationary

I have time series data that reflects users activity on some platform, and it has clear daily seasonal pattern. I understand that in order to fit a model such as ARMA to this data I should first detrend it and remove its seasonal component, which is commonly performed using differencing.

Differencing is appropriate when the data has a stochastic trend (is integrated, has a unit root). It is not appropriate when the data is merely seasonal or has a deterministic trend (e.g. linear trend). By differencing in absence of a stochastic trend you will introduce a superfluous integrated MA(1) component.

Should the differencing eliminate the changing variance? I don't see why. If not, are there other methods to deal with data with such behavior?

No, differencing will not turn a time-varying variance to constant variance. But you could specify a model for the time-varying variance extra to the model for the time-varying mean (see my longer answer is this thread). An ARMA(p,q)-GARCH(s,r) model with exogenous regressors in the conditional variance equation (extra to those in the conditional mean equation) is such an example. It would look something like \begin{aligned} x_t &\sim D(\mu_t,\sigma_t^2); \\ \mu_t &= \varphi_1 \mu_{t-1} + \dotsc + \varphi_p \mu_{t-p} + (\varphi_1 + \theta_1) \varepsilon_{t-1} + \dotsc + (\varphi_m + \theta_m) \varepsilon_{t-m} \\ &+ \text{seasonal dummies or Fourier terms}; \\ \sigma_t^2 &= \omega + \alpha_1 \varepsilon_{t-1}^2 + \dotsc + \alpha_s \varepsilon_{t-s}^2 + \beta_1 \sigma_{t-1}^2 + \dotsc + \beta_r \sigma_{t-r}^2 \\ &+ \text{seasonal dummies or Fourier terms}. \\ \end{aligned} It might be that you do not need the regular GARCH terms (lagged $\varepsilon_t^2$ and lagged $\sigma_t^2$), then the conditional variance equation would collapse to $$ \sigma_t^2 = \omega + \text{seasonal dummies or Fourier terms}. $$ I do not know how to implement this directly, but there is a workaround: specify an ARMA(p,q)-GARCH(1,1) model with exogenous regressors $$ \sigma_t^2 = \omega + \alpha_1 \varepsilon_{t-1}^2 + \beta_1 \sigma_{t-1}^2 + \text{seasonal dummies or Fourier terms} $$ while fixing $\alpha_1=0$ and $\beta_1=1$. You can do this, for example, in "rugarch" package in R with functions ugarchspec and ugarchfit:

library(forecast)
library(rugarch)
p=2; q=2 # arbitrary choice, just for this example
x=rnorm(1000); x=ts(x,freq=12,start=c(1960,1)) # generated data just for this example
fourierterms=fourier(x,K=6)
spec=ugarchspec(variance.model=list(external.regressors=cbind(fourierterms)), mean.model=list(armaOrder=c(p,q), external.regressors=cbind(fourierterms)), 
fixed.pars=list(alpha1=0.0,beta1=1.0))
fit=ugarchfit(spec=spec,data=x)

Best Answer

Related Solutions

Solved – Arima time series forecast (auto.arima) with multiple exogeneous variables in R

Solved – How to make time series data with changing variance stationary

Related Question