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## 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 , 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.

 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.

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

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