You have identified your process as involving:

✓ no change(rather than change).

This implies that you expect all aspects of the process—such as the mean, variance, and dynamics—to remain constant over time.

Models for processes with no change:

ARMA model →

The family of autoregressive moving-average models forms a well-known category of stationary models within the time series literature. Typical of stationary models is that their mean, variance, auto-correlations and all other aspects that characterize a process remain stable over time.

AR model →

The class of autoregressive (AR) models is a subset of the broad family of autoregressive moving-average (ARMA) models. It is characterized by autoregression, meaning the observed variable is regressed on itself at earlier occasions. All ARMA models are stationary, meaning that their mean, variance, auto-correlations and all other aspects that characterize a process remain stable over time, despite the temporal fluctuations of the process itself.

MA model →

The class of moving-average (MA) models is a subset of the broad family of autoregressive moving-average (ARMA) models. It is characterized by the fact that the observed variable is presented as a weighted average of the current perturbation and one or more past perturbations that were given to the system. All ARMA models are stationary, meaning that their mean, variance, auto-correlations and all other aspects that characterize a process remain stable over time, despite the temporal fluctuations of the process itself.

ARMAX model →

The class of autoregressive moving-average models with exogenous inputs (ARMAX) can be used to model short-term dynamics in combination with one or more predictor variables. When the latter are stationary, then the ARMAX process will also be stationary; this means there is variability and dynamics in this variability, but there are no changes over time.

ARMA model →

DYnamic regression is another way of combining the autoregressive moving-average (ARMA) model with exogenous predictors. It separates these two components, and can be described as a regression model with ARMA residuals. When the predictors that are included are stationary, then the dynamic regression model is also stationary.

You may also want to read more about:

Estimation with state-space model →

Many of the models that can be used to capture processes that contain no change, can be formulated in state-space format, which then allows for their estimation using the Kalman filter. This is the case for the ARMA model. For the models that contain exogenous variables, you need a state-space model that allows for the inclusion of exogenous variables in the measurement equation and/or in the transition equation. To specify the ARMAX model, you include the exogenous predictors in the transition equation; for dynamic regression you include them in the measurement equation, to separate them from the dynamics of the model.

Stationarity →

Stationarity is an important concept in the time series literature. It implies that all the features of a process are stable over time, even though the process itself varies over time.

Stationarity tests →

There are various tests that can be used to determine whether your univariate \(N=1\) time series is stationary. While this can provide valuable insights, it is important to keep in mind that these tests always focus on a particular kind of non-stationarity; hence, you cannot use them to rule out all possible ways in which your process may be non-stationary.

Stationarity and change →

It may be easy to confuse the concept of stationarity with the idea that there is no change in your process. But change and non-stationarity are no synonyms. There are various models—like the threshold-autoregressive model or the Markov-switching autoregressive model—which are stationary, even though they are characterized by change in the form of recurrent switches between distinct regimes. The key issue here is that it is not possible to use time to predict when the switches will occur. As a result, these regime-switching processes fall within the category of stationary models.

Stationary models showing a trend →

Autoregressive models are by definition stationary, but in practice, they may generate trend-like behavior. This occurs when the system is moved far away from its long-run equilibrium, which is described as displacement of the system. The autoregression in the process then propagates this displacement over time, resulting in a gradual trajectory back towards the long-run mean that characterizes the process.

Absence of trends →

To be sure that your process is not characterized by any change, you may also investigate whether there is evidence that there is a trend. There are two broad classes of trends: deterministic versus stochastic trends. These show up as different temporal patterns and can therefore be detected in different ways.