You have identified your process as involving:

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

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 persist over time.

Models for sudden lasting change:

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, and there may be one or more change points.

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 \(X\). When the \(X\) variable changes from one constant to another (for instance from 0 to 1), this is sometimes regerred to as a press intervention. Including it in your ARMAX model results in an lasting change in the mean of the process. When there are autoregressive terms in the ARMAX model, this will smoothen the transition somewhat, resulting in a smooth rather than a sudden change in the outcome variable \(y_t\). Hence, the ARMAX model can be used to represent a smooth lasting change in the outcome based on a sudden lasting change in the input.

Dynamic regression →

With dynamic regression, you can use various exogenous variables that capture sudden changes in the intercept and/or slope of a deteministic trajectory over time, very similar to how this is done in cange point modeling. The major difference is that in dynamic regression the residualsare modeled as an ARMA process. If you have an intervention that was absent first, and once it was administered it is continued, this is referred to as a press intervention (also known as an AB design). Including this in your dynamic regression results in an instant change in the mean. You can also combine such sudden changes with smooth changes, as in the change point model. The parameters of the trend in dynamic regression can be easily interpreted, because the trend is separated from the dynamic part in this model.

Interrupted time series model with a press intervention →

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 was absent first, and once it was administered it is continued, this is referred to as a press intervention (also known as an AB design). It implies a sudden change in the intercept in the interrupted time series model. This may translate into a more sudden onset of the intervention effect in the outcome variable \(y_t\), due to the dynamics of the transfer function. These dynamics are separated from the dynamics of the ongoing process (as opposed to the ARMAX model, which uses the same dynamics for the exogenous inputs as for the ongoing process). As the ARMAX model, the interrupted time series model can be used to represent a smooth lasting change in the outcome based on a sudden lasting change in the input.

You may also want to read more about

Deterministic trends →

The sudden change may be combined with smooth change prior to and after the change; these can be accounted for with for instance a determinstic trend, which can be expressed as a direct function of time \(t\). A simple version of this is a linear trend over time, but there are many alternatives that allow for more flexible trajectories. To have lasting change, the trend cannot change direction, although it may plateau at some point in time.

Estimation with state-space model →

To estimate models that account for sudden lasting change, you may consider to formulate them in state-space format, which then allows for model 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.

MS-AR model with an absorbing state →

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 switches between different regimes, each of which is an autoregressive model with its own model parameters. When there is an absorbing state in the underlying Markov model, there is a lasting change; when there is little/no autoregression (in the absorbing state), the change is sudden.

TV-AR model →

Time-varying autoregressive (TV-AR) models allow for changes in the mean of a process, but also in the variance and/or the dynamics. Such changes may occur suddenly, although they may also occur more slowly. Moreover, the changes can be lasting, but may also be reversible. The trigger of the changes is not included in the TV-AR model.