Note: This page has been translated by MathWorks. Click here to see

To view all translated materials including this page, select Country from the country navigator on the bottom of this page.

To view all translated materials including this page, select Country from the country navigator on the bottom of this page.

For a random variable *y _{t}*, the

For a *static* conditional mean model, the conditioning set
of variables is measured contemporaneously with the dependent variable
*y _{t}*. An example of a static
conditional mean model is the ordinary linear regression model. Given $${x}_{t},$$ a row vector of exogenous covariates measured at time

$$E({y}_{t}|{x}_{t})={x}_{t}\beta $$

(that is, the conditioning set is $${\Omega}_{t}={x}_{t}$$).

In time series econometrics, there is often interest in the dynamic behavior of a
variable over time. A *dynamic* conditional mean model
specifies the expected value of *y _{t}* as a
function of historical information. Let

Past observations,

*y*_{1},*y*_{2},...,*y*_{t–1}Vectors of past exogenous variables, $${x}_{1},{x}_{2},\dots ,{x}_{t-1}$$

Past innovations, $${\epsilon}_{1},{\epsilon}_{2},\dots ,{\epsilon}_{t-1}$$

By definition, a covariance stationary stochastic process has an unconditional
mean that is constant with respect to time. That is, if
*y _{t}* is a stationary stochastic
process, then $$E({y}_{t})=\mu $$ for all times

The constant mean assumption of stationarity does not preclude the possibility of
a dynamic conditional expectation process. The serial autocorrelation between lagged
observations exhibited by many time series suggests the expected value of
*y _{t}* depends on historical
information. By Wold’s decomposition [2], you can write the conditional mean of any stationary process

$$E({y}_{t}|{H}_{t-1})=\mu +{\displaystyle \sum _{i=1}^{\infty}{\psi}_{i}{\epsilon}_{t-i},}$$ | (1) |

Any model of the general linear form given by Equation 1 is a valid specification for the dynamic behavior of a stationary stochastic process. Special cases of stationary stochastic processes are the autoregressive (AR) model, moving average (MA) model, and the autoregressive moving average (ARMA) model.

[1] Box, G. E. P., G. M. Jenkins, and G. C. Reinsel.
*Time Series Analysis: Forecasting and Control*. 3rd ed.
Englewood Cliffs, NJ: Prentice Hall, 1994.

[2] Wold, H. *A Study in the Analysis of Stationary
Time Series*. Uppsala, Sweden: Almqvist & Wiksell,
1938.

- Econometric Modeler App Overview
- Specify Conditional Mean Models
- AR Model Specifications
- MA Model Specifications
- ARMA Model Specifications
- ARIMA Model Specifications
- Multiplicative ARIMA Model Specifications