## 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*_{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

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

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

### 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 [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.

## References

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

## See Also

`arima`

## Related Examples

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