Conditional Mean Models

Unconditional vs. Conditional Mean

For a random variable yt, the unconditional mean is simply the expected value, E(yt). In contrast, the conditional mean of yt is the expected value of yt given a conditioning set of variables, Ωt. A conditional mean model specifies a functional form for E(yt|Ωt)..

Static vs. Dynamic Conditional Mean Models

For a static conditional mean model, the conditioning set of variables is measured contemporaneously with the dependent variable yt. An example of a static conditional mean model is the ordinary linear regression model. Given xt, a row vector of exogenous covariates measured at time t, and β, a column vector of coefficients, the conditional mean of yt is expressed as the linear combination

E(yt|xt)=xtβ

(that is, the conditioning set is Ωt=xt).

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 yt as a function of historical information. Let Ht–1 denote the history of the process available at time t. A dynamic conditional mean model specifies the evolution of the conditional mean, E(yt|Ht1). Examples of historical information are:

  • Past observations, y1, y2,...,yt–1

  • Vectors of past exogenous variables, x1,x2,,xt1

  • Past innovations, ε1,ε2,,εt1

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 yt is a stationary stochastic process, then E(yt)=μ 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 yt depends on historical information. By Wold's decomposition [1], you can write the conditional mean of any stationary process yt as

E(yt|Ht1)=μ+i=1ψiεti,(5-1)

where {εti} are past observations of an uncorrelated innovation process with mean zero, and the coefficients ψi are absolutely summable. E(yt)=μ 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.

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