The endogeneity problem is particularly relevant in the context of
time series analysis of
causal processes. It is common for some factors within a causal system to be dependent for their value in period
t on the values of other factors in the causal system in period
t − 1. Suppose that the level of pest infestation is independent of all other factors within a given period, but is influenced by the level of rainfall and fertilizer in the preceding period. In this instance it would be correct to say that infestation is
exogenous within the period, but
endogenous over time.
Strict exogeneity Let the model be
y =
f(
x,
z) +
u. If the variable
x is sequential exogenous for parameter \alpha, and
y does not cause
x in
the Granger sense, then the variable
x is strongly/strictly exogenous for the parameter \alpha.
Weak exogeneity 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. 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. 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 ==