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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_t is the expected value of y_t given a conditioning set of variables, Ω_t. 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 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_t is expressed as the linear combination

(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:

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_t as

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.