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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 $$E\left({y}_{t}|{\Omega}_{t}\right).$$.
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
$$E({y}_{t}|{x}_{t})={x}_{t}^{\prime}\beta $$
(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, $${\epsilon}_{1},{\epsilon}_{2},\dots ,{\epsilon}_{t-1}$$
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 [1], you can write the conditional mean of any stationary process y_{t} as
$$E({y}_{t}|{H}_{t-1})=\mu +{\displaystyle \sum _{i=1}^{\infty}{\psi}_{i}{\epsilon}_{t-i},}$$ | (5-1) |
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. $$E({y}_{t})=\mu $$ is the constant unconditional mean of the stationary process.
Any model of the general linear form given by Equation 5-1 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] Wold, H. A Study in the Analysis of Stationary Time Series. Uppsala, Sweden: Almqvist & Wiksell, 1938.