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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 $${x}_{t},$$ a row vector of
exogenous covariates measured at time

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

(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

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

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

$$E({y}_{t}|{H}_{t-1})=\mu +{\displaystyle \sum _{i=1}^{\infty}{\psi}_{i}{\epsilon}_{t-i},}$$ | (6-1) |

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 6-1 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] 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.

[2] 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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