## Conditional Mean Models

### Unconditional vs. Conditional Mean

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

*unconditional mean*is simply the expected value, $$E\left({y}_{t}\right).$$ In contrast, the

*conditional mean*of

*y*is the expected value of

_{t}*y*given a conditioning set of variables,

_{t}*Ω*. A

_{t}*conditional mean model*specifies a functional form for $$E\left({y}_{t}|{\Omega}_{t}\right).$$.

### Static vs. Dynamic Conditional Mean Models

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

*t*, and

*β*, a column vector of coefficients, the conditional mean of

*y*is expressed as the linear combination

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

(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

*H*

_{t–1}denote the history of the process available at time

*t*. A dynamic conditional mean model specifies the evolution of the conditional mean, $$E\left({y}_{t}|{H}_{t-1}\right).$$ Examples of historical information are:

Past observations,

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

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

### Conditional Mean Models for Stationary Processes

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

*t*.

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

*y*as

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

## References

[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.

## See Also

### Apps

### Objects

## Related Examples

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