## Conditional Mean Models

### Unconditional vs. Conditional Mean

For a random variable yt, the unconditional mean is simply the expected value, $E\left({y}_{t}\right).$ In contrast, the conditional mean of yt is the expected value of yt given a conditioning set of variables, Ωt. A 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 yt. 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 yt is expressed as the linear combination

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

(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 yt as a function of historical information. Let Ht–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, y1, y2,...,yt–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 yt is a stationary stochastic process, then $E\left({y}_{t}\right)=\mu$ 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 yt depends on historical information. By Wold’s decomposition [2], you can write the conditional mean of any stationary process yt as

 $E\left({y}_{t}|{H}_{t-1}\right)=\mu +\sum _{i=1}^{\infty }{\psi }_{i}{\epsilon }_{t-i},$ (1)
where $\left\{{\epsilon }_{t-i}\right\}$ are past observations of an uncorrelated innovation process with mean zero, and the coefficients ${\psi }_{i}$ are absolutely summable. $E\left({y}_{t}\right)=\mu$ is the constant unconditional mean of the stationary process.

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.