Endogeneity (econometrics)

In econometrics, endogeneity broadly refers to situations in which an explanatory variable is correlated with the error term.[1]

In simplest terms, endogeneity means that a factor or cause one uses to explain something as an outcome is also being influenced by that same thing. For example, education can affect income, but income can also affect how much education someone gets. When this happens, one's analysis might wrongly estimate cause and effect. The thing one thinks is causing change is also being influenced by the outcome, making the results unreliable.

The concept originates from simultaneous equations models, in which one distinguishes variables whose values are determined within the economic model (endogenous) from those that are predetermined (exogenous).[2]

Ignoring simultaneity in estimation leads to biased and inconsistent estimators, as it violates the exogeneity condition of the Gauss–Markov theorem. This issue is often overlooked in non-experimental research, which limits the validity of causal inference and the ability to draw reliable policy recommendations.[3]

Common solutions to address endogeneity include the use of instrumental variable techniques, which provide consistent estimators by introducing variables that are correlated with the endogenous explanatory variable but uncorrelated with the error term.

Besides simultaneity, correlation between explanatory variables and the error term can arise when an unobserved or omitted variable is confounding both independent and dependent variables, or when independent variables are measured with error.[4]

Definition of exogeneity

In a stochastic model, the notion of the usual exogeneity, sequential exogeneity, strong/strict exogeneity can be defined. Exogeneity is articulated in such a way that a variable or variables is exogenous for parameter \alpha. Even if a variable is exogenous for parameter \alpha, it might be endogenous for parameter \beta.

When the explanatory variables are not stochastic, then they are strong exogenous for all the parameters.

If the independent variable is correlated with the error term in a regression model then the estimate of the regression coefficient in an ordinary least squares (OLS) regression is biased; however if the correlation is not contemporaneous, then the coefficient estimate may still be consistent. There are many methods of correcting the bias, including instrumental variable regression and Heckman selection correction.

Static models

The following are some common sources of endogeneity.

Omitted variable

In this case, the endogeneity comes from an uncontrolled confounding variable, a variable that is correlated with both the independent variable in the model and with the error term. (Equivalently, the omitted variable affects the independent variable and separately affects the dependent variable.)

Assume that the "true" model to be estimated is but z_i is omitted from the regression model (perhaps because there is no way to measure it directly). Then the model that is actually estimated is where (thus, the z_i term has been absorbed into the error term).

If the correlation of x and z is not 0 and z separately affects y (meaning ), then x is correlated with the error term \varepsilon.

Here, x is not exogenous for \alpha and \beta, since, given x, the distribution of y depends not only on \alpha and \beta, but also on z and \gamma.

Measurement error

Suppose that a perfect measure of an independent variable is impossible. That is, instead of observing x^{*}_{i}, what is actually observed is where \nu_i is the measurement error or "noise". In this case, a model given by can be written in terms of observables and error terms as Since both x_i and u_i depend on \nu_i, they are correlated, so the OLS estimation of \beta will be biased downward.

Dynamic models

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.[5] 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.

External links

by Mark Thoma
Seth Godin's simple views on endogeneity

References

Jeffrey M. Wooldridge. Introductory Econometrics: A Modern Approach South-Western, 2009^
Jan Kmenta. Elements of Econometrics MacMillan, 1986^
John Antonakis. On making causal claims: A review and recommendations The Leadership Quarterly, December 2010^