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Unconditional vs. Conditional Mean |

For a random variable *y _{t}*,
the

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
a row vector of
exogenous covariates measured at time

(that is, the conditioning set is ).

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

Past observations,

*y*_{1},*y*_{2},...,*y*_{t–1}Vectors of past exogenous variables,

Past innovations,

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
for
all times

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

(5-1) |

where are past observations of an uncorrelated innovation process with mean zero, and the coefficients are absolutely summable. 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.

- Specify Conditional Mean Models Using arima
- AR Model Specifications
- MA Model Specifications
- ARMA Model Specifications
- ARIMA Model Specifications
- Multiplicative ARIMA Model Specifications

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