## Time Series Regression Models

Time series regression models attempt to explain the current response using the response history (autoregressive dynamics) and the transfer of dynamics from relevant predictors (or otherwise). Theoretical frameworks for potential relationships among variables often permit different representations of the system.

Use time series regression models to analyze time series data, which are measurements that you take at successive time points. For example, use time series regression modeling to:

• Examine the linear effects of the current and past unemployment rates and past inflation rates on the current inflation rate.

• Forecast GDP growth rates by using an ARIMA model and include the CPI growth rate as a predictor.

• Determine how a unit increase in rainfall, amount of fertilizer, and labor affect crop yield.

You can start a time series analysis by building a design matrix (Xt), which can include current and past observations of predictors. You can also complement the regression component with an autoregressive (AR) component to account for the possibility of response (yt) dynamics. For example, include past measurements of inflation rate in the regression component to explain the current inflation rate. AR terms account for dynamics unexplained by the regression component, which is necessarily underspecified in econometric applications. Also, the AR terms absorb residual autocorrelations, simplify innovation models, and generally improve forecast performance. Then, apply ordinary least squares (OLS) to the multiple linear regression (MLR) model:

`${y}_{t}={X}_{t}\beta +{u}_{t}.$`

If a residual analysis suggests classical linear model assumption departures such as that heteroscedasticity or autocorrelation (i.e., nonspherical errors), then:

• You can estimate robust HAC (heteroscedasticity and autocorrelation consistent) standard errors (for details, see `hac`).

• If you know the innovation covariance matrix (at least up to a scaling factor), then you can apply generalized least squares (GLS). Given that the innovation covariance matrix is correct, GLS effectively reduces the problem to a linear regression where the residuals have covariance I.

• If you do not know the structure of the innovation covariance matrix, but know the nature of the heteroscedasticity and autocorrelation, then you can apply feasible generalized least squares (FGLS). FGLS applies GLS iteratively, but uses the estimated residual covariance matrix. FGLS estimators are efficient under certain conditions. For details, see [1], Chapter 11.

There are time series models that model the dynamics more explicitly than MLR models. These models can account for AR and predictor effects as with MLR models, but have the added benefits of:

• Accounting for moving average (MA) effects. Include MA terms to reduce the number of AR lags, effectively reducing the number of observation required to initialize the model.

• Easily modeling seasonal effects. In order to model seasonal effects with an MLR model, you have to build an indicator design matrix.

• Modeling nonseasonal and seasonal integration for unit root nonstationary processes.

These models also differ from MLR in that they rely on distribution assumptions (i.e., they use maximum likelihood for estimation). Popular types of time series regression models include:

• Autoregressive integrated moving average with exogenous predictors (ARIMAX). This is an ARIMA model that linearly includes predictors (exogenous or otherwise). For details, see `arima` or ARIMAX(p,D,q) Model.

• Regression model with ARIMA time series errors. This is an MLR model where the unconditional disturbance process (ut) is an ARIMA time series. In other words, you explicitly model ut as a linear time series. For details, see `regARIMA`.

• Distributed lag model (DLM). This is an MLR model that includes the effects of predictors that persist over time. In other words, the regression component contains coefficients for contemporaneous and lagged values of predictors. Econometrics Toolbox™ does not contain functions that model DLMs explicitly, but you can use `regARIMA` or `fitlm` with an appropriately constructed predictor (design) matrix to analyze a DLM.

• Transfer function (autoregressive distributed lag) model. This model extends the distributed lag framework in that it includes autoregressive terms (lagged responses). Econometrics Toolbox does not contain functions that model DLMs explicitly, but you can use the `arima` functionality with an appropriately constructed predictor matrix to analyze an autoregressive DLM.

The choice you make on which model to use depends on your goals for the analysis, and the properties of the data.

## References

[1] Greene, W. H. Econometric Analysis. 6th ed. Englewood Cliffs, NJ: Prentice Hall, 2008.