Solved – Should I make a time series stationary before passing it as an input for ARIMA model

arimatime series

I'm working through this tutorial and this guy run SARIMAX model for a time series with both seasonal and trend components:

# create SARIMAX model with previously determined lags
# sar_m = sarimax.SARIMAX(ts15_train.values,
                    trend='n', 
                    order=(2,1,1), 
                    seasonal_order=(2, 1, 1, 24), 
                    simple_differencing=False).fit()

It seems wrong to me. Am I right here? I read some tutorials here and there and I believe we should eliminate both seasonal and trend components first to make a time series stationary (by performing some transforming operations like ts = log(ts) etc), then predict (e.g. by ARIMA model the next K values) and then bring back our seasonal and trend components (e.g. add a running mean, pow(2, x)).

Best Answer

No, what you are suggesting is almost entirely incorrect.

For forecasting, "eliminating" trend and seasonal components in the training period puts you in the awkward position of "bringing back" the trend and seasonal components in the forecast period. If your model doesn't detrend/seasonally adjust by defining dynamics for those components (which is the case with many such methods), you have no forecast available for those components; you have to make some kind of ad hoc decision as to what it should be (e.g. copy the last year's seasonal component), or fit another, separate model to those components.

Seasonal ARIMA models explicitly include a type of stochastic trend (if the differencing order is at least 1) and seasonality, which means that they are perfectly capable of accounting for those, including in the forecast period, and there is no reason to remove them first. In particular, ARIMA models are not assumed to be stationary (they are assumed to have a particular type of non-stationarity, though, so for example will not handle error variance that increases with the level).

At the end of your post, you seem to be suggesting having a deterministic trend ("pow(2,x)"?), which should also be done jointly with estimating the ARIMA model (an ARIMAX or regression with ARIMA errors model), not as a separate step. This is because the parts of a model are rarely independent: your estimate of the trend parameters depends on your estimate for the ARIMA parameters.

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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)

Solved – ACF indicates non-stationarity but but time series plot looks stationary

is a useful model for the most recent 78 values. There is a significant change in parameters over time (approx period 111) which is clearly visual from the time series plot. This conclusion was based on the CHOW test for constant parameters. Seasonal differencing (order 12) is needed without any power transform. Note the need for a quarterly auto-regressive effect often overlooked in the rush to just using 12 period effects suggesting tw0 seasonal effects. Finally a single anomaly is identified at the penultimate point in time (period 187)

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

Solved – ACF indicates non-stationarity but but time series plot looks stationary

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