You have identified your process as involving:

✓ change(rather than no change), which is
✓ smooth(rather than sudden), 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 gradually over time, and that these changes persist over time.

Models that capture smooth lasting change

Deterministic trends →

Characteristic of a determinstic trend is that it can be expressed as a direct function of time \(t\), and that it concerns some long-term change. A simple version of this is a linear trend over time, but there are many alternatives that allow for more flexible trajectories. To model lasting change, the trend you specify cannot change direction, although it may plateau at some point in time.

Random walk (with drift) →

A random walk process is charactersized by a stochastic trend. While this may look like a deterministic trend in the short run, in long run it is inherently unpredictable what direction it will develop in. A random walk may also include drift, which results in the combination of a stochastic and a deterministic trend; the latter is in a particular direction (depending on whether there is positive or negative drift). However, the overall trend may still deviate from this, depending on the size of the drift in comparison to the innovation variance of the random walk.

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, the unit root processes with a constant result in a combination of a stochastic and a deterministic trend. The ARMA part allows for further short-term dynamics in the process.

ARMAX model with a smoothly increasing or decreasing exogenous input →

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 exogenous inputs \(X\). By including time (or functions of time) as the exogenous inputs, you can explicitly model a smooth lasting change, much in the same way as you do when modeling a deterministic trend.

Dynamic regression with a smoothly increasing or decreasing exogenous input →

In dynamic regression, you can model smooth lasting change using functions of time \(t\) as the exogenous variables akin to modeling a deterministic trend; the added value of dynamic regression is that the residuals are modeled as an ARMA process. The parameters of the trend are more easily interpreted than those from the ARMAX model, because the part with the exogenous inputs is separated from the dynamic part of the model.

TV-AR model →

Whereas the other models mentioned here allow for changes in the mean of a process only, the time-varying autoregressive (TV-AR) model allows for changes in other aspects of the process as well; this includes change in the variance and/or the dynamics of the process. When using the TV-AR model, you do not specify the change process itself; as a result, all kinds of changes can occur, inlcuding smooth—but also more sudden—changes, and lasting—but also reversible—changes.

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 may look like a deterministic trend, especially in the shortrun. You may want to read more about their differences and why these are important to take into account when modeling a process.

Estimation with state-space model →

Many of the models that can be used to capture smooth lasting change, can be formulated in state-space format, which then allows for their estimation using the Kalman filter. This is the case for the random walk and the ARIMA model. 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.

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 lasting trend: For instance, when you have an AR(1) process with an autoregressive parameter close to 1 and you have observed this process for a shorter period of time, it may not be clear that this process is reverting to the same mean over time. A similar issue may arise for higher-order AR models, when the parameters are close to the bounds of stationarity. You can try this out yourself with the interactive tools included for an AR(1) and an AR(2) model in the article about the AR model.

ARMAX model with a press intervention →

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 exogenous inputs \(X\). When the exogenous input is a dummy variable that represents a press intervention (i.e., and intervention that once it is started, is continued), this results in a sudden rather than a smooth change in the intercept. However, when there is autoregression in the ARMAX model, this sudden change is translated in a more gradual onset of the intervention effect in the outcome variable \(y_t\). In this way, the ARMAX model can be used to represent a smooth lasting change in the outcome based on a sudden lasting change in the input.

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; this is represetned with a dummy variable. Hence, the underlying change is sudden rather than smooth. However, this sudden change is translated by the transfer function into a change in the outcome variable \(y_t\); depending on the dynamics of the transfer function, this may result in a gradual onset of the intervention effect in such a scenario. In this way, dynamic regression can be used to represent a smooth lasting change in the outcome based on a sudden lasting change in the input.The dynamics of this onset are independent of the dynamics of the ongoing process, as opposed to such dynamics in the ARMAX model.