# Contradictory Sources on Seasonality being a nonstationarity

I have been trying to figure out whether seasonality means nonstationarity, and the answers from many (often reliable) sources seem to be contradicting. (lets define stationarity as weakly stationarity )

Source 1:Forecasting: Principles and Practice (2nd ed)Rob J Hyndman and George Athanasopoulos

“A stationary time series is one whose properties do not depend on the time at which the series is observed. Thus, time series with trends, or with seasonality, are not stationary — the trend and seasonality will affect the value of the time series at different times. On the other hand, a white noise series is stationary — it does not matter when you observe it, it should look much the same at any point in time. Some cases can be confusing — a time series with cyclic behaviour (but with no trend or seasonality) is stationary. This is because the cycles are not of a fixed length, so before we observe the series we cannot be sure where the peaks and troughs of the cycles will be.”

Source 2: Forecast evaluation for data scientists: common pitfalls and best practices:

They list the following causes of nonstationarity:

• Seasonality • Trends (Deterministic, e.g., Linear/Exponential) • Stochastic Trends / Unit Roots • Heteroscedasticity
• Structural Breaks (sudden changes, often with level shifts)

Source 3: Does a seasonal time series imply a stationary or a non stationary time series The answers to these questions mostly say seasonality means nonstationarity

Source 4/5:

Here Dr. Huber says

a series contains "seasonality patterns," then a fortiori it is not stationary. The underlying concept of stationarity is that statistical properties of the series do not change over time, while the underlying concept of seasonality is that those properties do change, but in a periodic way.

Source 6:

I had job interviewer(who was also a professor) who asked me “You have a temperature series with seasonality. Is it stationary?”

When I said it was nonstationary due to seasonality, he claimed the answer was “seasonal series can be stationary”

Source 7:

A professor who taught my TSA class also told me That seasonal ARMA is stationary; Like $$-0.8x_{t-2}+\epsilon_t$$ (where $$\epsilon_t$$ is white noise) is just an AR(2) thats stationary.

Source 7

Mixed seasonal ARMA is a stationary process When we combine seasonal and non-seasonal operators we obtain a model $$\Phi\left(B^h\right) \phi(B) X_t=\Theta\left(B^h\right) \theta(B) > Z_t,$$ which is called mixed seasonal ARMA and it is denoted by $$> \operatorname{ARMA}(p, q) \times(P, Q)_h .$$

Source 8 ”Seasonality usually causes the series to be nonstationary because the average values at some particular times within the seasonal span (months, for example) may be different than the average values at other times. For instance, our sales of cooling fans will always be higher in the summer months.”

## **So my questions are: Are these differences due one side being mathematically wrong, or is this something disputed in the time series/DSP/forecasting communities? Is it because there might be more nuance with different types of seasonalities?

I looked at a couple more sources that go more deep into seasonality, and found the following

This source seems to mention 3 definitions of seasonality, and mentions stationary seasonality vs nonstationary seasonality, but I wasn't able to discern what exactly they are talking about regarding those 2.

This article introduces Lagrange multiplier tests of the null hypothesis of no unit roots at seasonal frequencies against the alternative of a unit root at either a single seasonal frequency or a set of seasonal frequencies. Application of the tests to three sets of seasonal variables shows that in most cases seasonality is nonstationary

Many reasonable time series models of seasonality are conceivable. One approach is to model seasonality as deterministic, as did Barsky and Miron (1989), or as periodic with unchanged periodicity, as done by Hansen and Sargent (1993). A second approach is to model seasonality as the sum of a deterministic process and a stationary stochastic process (Canova 1992). A third approach is to model seasonal patterns as nonstationary by allowing for (or imposing) seasonal unit roots (Box and Jenkins 1976)”

Source “Seasonality in Regression Paperback – September 23, 2014 by Svend Hylleberg (Author), Karl Shell (Editor)

In ch2, this source mentions 3 definitions for seasonality.

In the preceding sections we have touched on essentially three definitions of seasonality. The first definition is that the seasonal component of a time series is a component that is definitely periodic in character with period n. Such a component can be described by one of the equivalent models in (2.1), (2.2), or (2.6). However, in view of what economic time series actually look like (see the following examples), it is fair to say that the application of such a definition will render most seasonality problems unsolved. Even if many economic time series contain a deterministic seasonal component, they almost always contain components that are less regular but that nevertheless may be called seasonal and are modeled as such.

• I beg to differ: what exactly are you quoting? I don't believe I ever made a claim that you attribute to me. In the thread you attempt to reference and in a comment, I stated "If a series contains "seasonality patterns," then a fortiori it is not stationary." Could that be any less clear??
– whuber
Commented Apr 14 at 14:12
• @whuber I summarized the discussion regarding Seasonal ARMA in that link instead of pulling quotes; but your comment was "Your modified process is an example of an $\mathrm{AR}(12)$ model. Assuming the $\varepsilon_t$ are iid with finite variance, then, it must be stationary." Commented Apr 14 at 19:57
• That was pulled out of a longer context. Is it possible you haven't seen all the preceding comments because there are so many of them?
– whuber
Commented Apr 15 at 3:07
• Ok I caught my mistake, they added an additional term to their process, making it $y_t=\phi_1 y_{t-1}+\phi_2 y_{t-12}+\epsilon_t$ Commented Apr 16 at 19:27

This may be neither rigorous enough nor detailed enough to be an answer you are looking for, but let me offer a brief take on the matter. Seasonal time series can be either stationary or nonstationary.

• Autoregressive processes that do not contain a unit root (like seasonal ARMA) are stationary. The unconditional expected value and variance are constant over time.
• Deterministic patterns (like the ones modelled by seasonal dummy variables and Fourier terms) and unit-root type patterns (like a seasonally integrated version of SARIMA model) are nonstationary. The unconditional expected value of the former varies with time. The unconditional variance of the latter grows with time.

A way to distinguish between the two: can you predict the seasonal pattern from historical data that is infinitely old and does your forecast interval expand over time?

• For SARMA, you cannot. The forecast tends to a constant scalar (the unconditional mean) irrespective of which season you are in, and the forecast interval tends to a constant vector.
• For the other examples above, you either can predict the mean so that your forecast is different for different seasons (deterministic seasonality) or the forecast interval is expanding indefinitely (unit root seasonality).

I do not think the cases I have mentioned cover all the possibilities; you may formulate a seasonal model that does not fall into one of the categories above. However, I think they cover the most used models / modelling assumptions.

• Re the first bullet: there must be some strong implicit assumptions there. In particular, it doesn't even make sense unless by "stochastic pattern" you mean an AR process.
– whuber
Commented Apr 14 at 14:14
• @whuber, fair enough. Standard ARMA was what came to my mind first, but I thought there could be other, nonlinear variations of the same thing. I do not have an entirely clear picture in my head. Let me know if you think I should rather delete this answer. Commented Apr 14 at 19:31