Weak exogeneity

In the context of linear time series regression, weak exogeneity is an identifying assumption which requires that the structural error term has a zero conditional expectation given the present and past values of the regressors. It is used to determine whether statistical inference about parameters of interest can be validly drawn from a conditional probability model alone, without needing to analyze the marginal distribution of the explanatory variables. While strict exogeneity is often implausible in macroeconomic and financial data due to feedback effects, weak exogeneity is the standard identifying assumption employed in these fields.

The concept was formalized by Jean-François Richard (1980) and further analyzed by Robert F. Engle, David F. Hendry, and Richard (1983) in Econometrica.

Definition

The variable $z_{t}$ is weakly exogenous for a set of specific parameters of interest, denoted as $\psi =(\phi _{1},\phi _{2})$, if the marginal density of $z_{t}$ contains no useful information for estimating $\psi$, that is

• the parameters of interest $\psi$ must depend only on the parameters of the conditional model ($\phi _{1}$) and not on the parameters of the marginal model ($\phi _{2}$)
• the parameters $\phi _{1}$ and $\phi _{2}$ must be variation-free. This means that the permissible range of values for $\phi _{1}$ does not depend on the values taken by $\phi _{2}$

In a linear regression framework defined by: $z_{t}=x_{t}^{\top }\beta +\epsilon _{t}$ where $z_{t}$ is the outcome variable, $x_{t}$ are the regressors (potentially containing past values of $z_{t}$), and $\epsilon _{t}$ is the structural error term, this implies that the errors are orthogonal to current and past regressors. This can be expressed by the following moment condition:

$\mathbb {E} [\epsilon _{t}\mid x_{t},x_{t-1},\dots ]=0$

This condition allows for the errors to be correlated with future realizations of the regressors, accommodating feedback mechanisms where an outcome variable in one period influences regressor values in future periods.

This is in contrast to strict exogeneity, a more restrictive assumption which requires that the error term has a zero conditional expectation conditional on the complete set of regressors, including past, present, and future values, that is $\mathbb {E} [\epsilon _{t}\mid x_{0},x_{1},...,x_{T}]=0$ where $T$ is the size of the sample.

Equivalently, weak exogeneity requires regressors and lagged response variables to be predetermined, that is determined prior to the current period.

Examples

A common example of a weakly exogenous variable is consumption in models with credit constraints and rational expectations. Here, consumption is predetermined but not strictly exogenous. An unpredictable negative income shock will be uncorrelated with past (and potentially current) consumption, but will surely be correlated with future consumption—the individual will be forced to adjust their future consumption to accommodate their poorer state, inducing correlation. If the shock affects current consumption, predeterminedness (defined now as lags only) provides potential instruments—lagged values of the variable.

The presence of predetermined variables is a motivating factor in the Arellano–Bond estimator.

See also

• Endogeneity (econometrics)
• Strict exogeneity