## Describe relationships and make predictions from time series data

Time series regression is a statistical method for predicting a future response based on the response history (known as autoregressive dynamics) and the transfer of dynamics from relevant predictors. Time series regression can help you understand and predict the behavior of dynamic systems from experimental or observational data. Time series regression is commonly used for modeling and forecasting of economic, financial, and biological systems.

You can start a time series analysis by building a design matrix ($$X_t$$), which can include current and past observations of predictors ordered by time (t). Then, apply ordinary least squares (OLS) to the multiple linear regression (MLR) model

$y_t=X_t\beta+u_t$

to get an estimate of a linear relationship of the response ($$y_t$$) to the design matrix. $$\beta$$ represents the linear parameter estimates to be computed and ($$u_t$$) represents the innovation terms. The residual terms can be extended in the MLR model to include heteroscedasticity or autocorrelation effects.

Other models that capture dynamics more explicitly include:

• Autoregressive integrated moving average with exogenous predictors (ARIMAX)
• Regression models with ARIMA time series errors
• Distributed lag models

The choice of model depends on your goals for the analysis and the properties of the data. See Econometrics Toolbox™ for more details.

See also: cointegration, GARCH models, DSGE, equity trading, predictive modeling