You have identified your process as involving:

✓ change(rather than no change), which is
✓ sudden(rather than smooth), 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 abruptly over time, and that these changes wax and wane over time.

Models for sudden reversible change

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 concerned with sudden 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. TAR models require you to have observations of the threshold variable; typically, this is not a dummy variable, but it is converted to a one or more dummies based on the threshold value(s). The threshold may be a prior version of the outcome variable \(y_t\); in that case the model is referred to as a self-exciting TAR (SETAR) model.

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. The underlying Markov model implies there are sudden rather than smooth changes in the model parameters. However, due to the autoregression in the observed part of the model, the switches may translate into less abrupt changes in the outcome variable, because 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; this is an important difference between the MS-AR and the TAR model.

ARMAX model with a dummy variable that changes back and forth →

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 \(X\). When the \(X\) variable changes repeatedly between 0 and 1, this results in abrupt changes in the intercept of the process. However, these abrupt changes only translate into abrupt changes in the observed variable, when there is no autoregression in the ARMAX model; when there are autoregressive terms, this will smoothen the transitions of \(y_t\) somewhat, making them less abrupt than those that govern the process.

Dynamic regression with a dummy variable that changes back and forth →

With dynamic regression, you can use various exogenous variables, while the residuals are modeled as an ARMA process. When you have a predictor variable that changes repeatedly between 0 and 1, this results in abrupt changes in the intercept of the process. Because the dynamic part of the model is separated from the part where the exogenous inputs are added, the latter only have direct effects on the outcome; as a result, changes in the exogenous inputs translate directly into changes in the output. Hence, sudden changes in the input variable result in sudden changes in the output variable; this makes dynamic regression different from the ARMAX model. Like the ARMAX model, dynamic regression requires the exogenous input to be an observed variable.

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; this is then represented with a dummy variable. If you have an intervention that was absent first, then is administered for some time after which it is stopped, the inclusion of a dummy representing this temporal pattern will allow you to study a sudden change in intercept and a change back after treatment is terminated. In the interrupted time series model, the effect of this sudden onset and sudden termination is accounted for by a tranfer function, which may contains its own dynamics, separate from the dynamics of the ongoing proces (as opposed to the ARMAX model, which uses the same dynamics for the exogenous inputs as for the ongoing process). The dynamics of the transfer function determine whether the sudden onset and termination translate into sudden switches in the observed outcome \(y_t\), or in more gradual onset and decay.

You may also want to read more about

Change point model →

Characteristic of change point models (also known as piece-wise regression, structural break models, or turnning point models) is that the process is a function of time \(t\) (e.g., a linear trend), and that the function suddenly changes (e.g., the intercept or slope of the linear trend). Hence, there may be a combination of smooth change and sudden change; but it is also possible that there is only sudden change. The timing of the change (i.e., the change point location) may be known or unknown. Although the change point model is not typical to consider for reversible change, it is able to account for this: In that case, you have at least two change points, resulting in three phases, where the third phase is characterized by the same features (e.g., mean, variance) as the first phase.

AR model with negative autoregression →

Autoregressive (AR) models are not characterized by change, but they may generate patterns that seem to suggest sudden reversible changes. For instance, an AR(1) process with a negative autoregressive parameter alternates almost constantly between postive and negativ deveations from the long-run equilibrium. Moreover, an AR(2) process can resulting in an oscillating pattern depending on the specficic combination of the two autoregressive parameters. You can try this out yourself with the interactive tools included in the article about AR processes.

TV-AR model →

The time-varying autoregressive (TV-AR) model allows for changes in all kind of model features; such changes may reverse or not. In contrast to the TAR model and MS-AR model, the TV-AR model is not characterized by different regimes; sudden changes will be the result of sudden changes in the parameters, whereas these can also change more slowly. The trigger of the sudden changes is not observed in TV-AR models.

Estimation with state-space model →

Some of the models that can be used to capture sudden reversible change, can be formulated in state-space format, which then allows for their estimation using the Kalman filter. For te models that contain exogenous variables, you need a state-space model that allows for the inclusion of exogenous variables in the measurement equation and transition equations. 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 latter translates into sudden changes in the observed process, whereas the former may result in smoother changes in the observed process.