You have identified your process as involving:

✓ change(rather than no change), which is
✓ smooth(rather than sudden), and
✓ reversible(rather than lasting).

This implies that you expect certain aspects of the process—such as the mean, variance, and/or dynamics—to transform gradually over time, and that these changes wax and wane over time.

Models that capture smooth reversible change:

Deterministic trends →

Characteristic of a deterministic trend is that it can be expressed as a direct function of time \(t\), and that it concerns some long-term change. Examples of deterministic trends that result in smooth reversible change are higher-order polynomials based on time \(t\), or a sine wave.

ARMAX model with a smoothly and reversibly changing exogenous input →

The class of autoregressive moving-average models with exogenous inputs (ARMAX) can be used to model short-term dynamics in combination with a deterministic pattern over time. Examples of allowing for a reversible pattern over time consist of including higher-order polynomials based on time \(t\), or a sine wave. The interpretation of the regression parameters of the exogenous inputs in an ARMAX model is considered challenging, as these inputs enter recursively into the model when there are autoregressive terms.

ARMAX model with autoregression and an input that changes back and forth →

When an ARMAX model includes a dummy variable that changes back and forth (e.g., to indicate whether it was a week or weekend day, or whether one is at work or not), the underlying changes are sudden. However, these sudden changes may translate into smooth changes in the observed outcome if the ARMAX process is characterized by autoregression.

Interrupted time series model with a press intervention that is switched on and off →

Interrupted time series modeling is a technique by which you can study the effect of an intervention on an ongoing process. It requires that you know when the intervention was administered. If you have an intervention that starts at some point, and after some time is terminated again, this can be described as a press intervention that was switched on and off. While the inclusion of such a dummy results in sudden rather than smooth changes in the underlying process, the transfer function of the model can translate this into a gradual onset and a gradual decay of the intervention effect for the outcome variable. In this way, the interrupted time series model can be used to represent smooth reversible change in the outcome based on sudden reversible change in the input. In contrast to the ARMAX model, the dynamics of the intervention effect are independent of the dynamics of the ongoing process in the interrupted time series model.

Dynamic regression with a smoothly and reversibly changing exogenous input →

In dynamic regression, you can model smooth reversible change by adding higher polynomials of time \(t\) or a sine wave as a predictor of the outcome, while the residuals are modeled as an ARMA process. The advantage of separating the predictors from the dynamics in the model—as is done in dynamic regression—is that the parameters of the predictors can be easily interpreted; this is in contrast to those in the ARMAX model.

Random walk →

A random walk process is characterized by a stochastic trend, which may change direction at any time and is inherently unpredictable. A random walk may also include drift, which results in the combination of a stochastic trend (which can go in any direction at any time), and a deterministic trend (which is either positive or negative when it stems from drift).

ARIMA model →

The general family of autoregressive integrated moving average (ARIMA) models includes unit root processes; these result in a stochastic trend over time. As with the random walk with drift—which can be thought of as a special case of the ARIMA model—the unit root processes with a constant result in a combination of a stochastic and a deterministic trend.

TAR model →

Threshold autoregressive (TAR) models are characterized by abrupt changes in the process features—including the variance and dynamics—when a specific variable passes a certain threshold. Hence, in terms of the underlying parameters, the model is more appropriate to capture sudden reversible change than smooth reversible change. However, because there is autoregression in the model, the switches may not manifest as abruptly in the outcome variable, and it may take the process some time to reach the new equilibrium after a switch. You can try this out with the interactive tool that is provided in the article about the TAR model. To be able to use this model, you should have observations of the threshold variable.

MS-AR model →

The Markov-switching autoregressive (MS-AR) model is based on having a latent Markov model (also known as a hidden Markov model), which is combined with an autoregressive model at the observed level. Although the underlying Markov model implies there are sudden rather than smooth changes in the model parameters, due to the autoregression in the observed part the switches may not manifest as abruptly in the outcome variable, as it may take the process some time to reach the new equilibrium after a switch. You can try this out with the interactive tool that is provided in the article about the MS-AR model. For this model, you do not need to have information on when the switches occurred (in contrast to the TAR model).

TV-AR models →

The time-varying autoregressive (TV-AR) model allows for any kind of change in any kind of process feature. Hecne, this also includes smooth reversible changes in the mean, the variance and/or the dynamics of the process.

You may also want to read more about:

Different trends →

There is an important distinction in time series modeling between deterministic and stochastic trends; both were mentioned as potential modeling candidates above. A stochastic trend can look like a deterministic trend, especially in the short-run. You may want to read more about their differences and why these are important to take into account when modeling a process.

Regime swtiching combined with autoregression →

Whether a regime switch–which is by definition sudden–translates into a gradual change in the outcome, depends on whether there is autoregression in the process. But additionally, it also depends on whether the autoregressive terms involve the outcome itself at previous occasions, or that these are based on the deviations of the observations from their regime-specific mean. In the latter case, the switch in the observed outcome is sudden rather than smooth. This is a nuance that may be easily overlooked.

Estimation with state-space model →

Some of the models that can be used to capture smooth reversible change, can be formulated in state-space format, which then allows for their estimation using the Kalman filter. 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 the transition equation. To specify the ARMAX model and interrupted time series 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. The former leads to smooth changes when there is autoregression; the latter would require you to specify the smoothness of the transitions explicitly yourself using the inputs.

AR processes close to stationarity bounds →

Autoregressive (AR) models are stationary and are thus not characterized by change. However, there are scenarios in which an AR model may seem to represent a smooth reversible trend. For instance, when you have an AR(1) process with an autoregressive parameter close to 1, the process may wander away from its long-term mean in either a positive or negative direction for a longer period of time before reverting back. Hence, while this is a process that is characterized by variability in the absence of change, it may seem to contain smooth reversible change. You can try this out yourself with the interactive tools included in the article about AR processes.