endogeneity (econometrics)

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 $\backslash alpha$. Even if a variable is exogenous for parameter $\backslash alpha$, it might be endogenous for parameter $\backslash beta$.

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.

The following are some common sources of endogeneity.

{{further|Omitted-variable bias}}

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

: $y\_i\; =\; \backslash alpha\; +\; \backslash beta\; x\_i\; +\; \backslash gamma\; z\_i\; +\; u\_i$

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

: $y\_i\; =\; \backslash alpha\; +\; \backslash beta\; x\_i\; +\; \backslash varepsilon\_i$

where $\backslash varepsilon\_i=\backslash gamma\; z\_i\; +\; u\_i$ (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 $\backslash gamma\; \backslash neq\; 0$), then $x$ is correlated with the error term $\backslash varepsilon$.

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

Suppose that a perfect measure of an independent variable is impossible. That is, instead of observing $x^\{*\}\_\{i\}$, what is actually observed is $x\_i=x^\{*\}\_\{i\}+\; \backslash nu\_i$ where $\backslash nu\_i$ is the measurement error or "noise". In this case, a model given by

: $y\_i\; =\; \backslash alpha+\backslash beta\; x^\{*\}\_i\; +\; \backslash varepsilon\_i$

can be written in terms of observables and error terms as

: $$

\begin{align}

y_i & = \alpha+\beta(x_i-\nu_i) + \varepsilon_i \\[3pt]

y_i & = \alpha+\beta x_i +(\varepsilon_i - \beta\nu_i) \\[3pt]

y_i & = \alpha+\beta x_i +u_i \quad (\text{where } u_i=\varepsilon_i - \beta\nu_i)

\end{align}

Since both $x\_i$ and $u\_i$ depend on $\backslash nu\_i$, they are correlated, so the OLS estimation of $\backslash beta$ will be biased downward.

Measurement error in the dependent variable, $y\_i$, does not cause endogeneity, though it does increase the variance of the error term.

Suppose that two variables are codetermined, with each affecting the other according to the following "structural" equations:

:$y\_i\; =\; \backslash beta\_1\; x\_i\; +\; \backslash gamma\_1\; z\_i\; +\; u\_i$

:$z\_i\; =\; \backslash beta\_2\; x\_i\; +\; \backslash gamma\_2\; y\_i\; +\; v\_i$

Estimating either equation by itself results in endogeneity. In the case of the first structural equation, $E(z\_i\; u\_i)\; \backslash neq\; 0$. Solving for $z\_i$ while assuming that $1-\backslash gamma\_1\; \backslash gamma\_2\; \backslash neq\; 0$ results in

:$z\_i\; =\; \backslash frac\{\backslash beta\_2\; +\; \backslash gamma\_2\; \backslash beta\_1\}\{1-\backslash gamma\_1\; \backslash gamma\_2\}x\_i+\backslash frac\{1\}\{1-\backslash gamma\_1\; \backslash gamma\_2\}v\_i+\backslash frac\{\backslash gamma\_2\}\{1-\backslash gamma\_1\; \backslash gamma\_2\}u\_i$.

Assuming that $x\_i$ and $v\_i$ are uncorrelated with $u\_i$,

:$\backslash operatorname\; E(z\_i\; u\_i)\; =\; \backslash frac\{\backslash gamma\_2\}\{1-\backslash gamma\_1\; \backslash gamma\_2\}\backslash operatorname\; E(u\_i\; u\_i)\; \backslash neq\; 0$.

Therefore, attempts at estimating either structural equation will be hampered by endogeneity.

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.

Let the model be y = f(x, z) + u. If the variable x is sequential exogenous for parameter $\backslash alpha$, and y does not cause x in the Granger sense, then the variable x is strongly/strictly exogenous for the parameter $\backslash alpha$.

Generally speaking, simultaneity occurs in the dynamic model just like in the example of static simultaneity above.

• Virtuous circle and vicious circle
• Heterogeneity
• Dependent and independent variables

{{notelist}}

{{Reflist}}

• {{cite book |first=William H. |last=Greene |title=Econometric Analysis |location=Upper Saddle River |publisher=Pearson |edition=Sixth |year=2012 |isbn=978-0-13-513740-6 }}
• {{cite book |first=Peter |last=Kennedy |title=A Guide to Econometrics |edition=Sixth |location=Malden |publisher=Blackwell |year=2008 |page=139 |isbn=978-1-4051-8257-7 }}
• {{cite book |first=Jan |last=Kmenta |authorlink=Jan Kmenta |title=Elements of Econometrics |location=New York |publisher=MacMillan |edition=Second |year=1986 |isbn=0-02-365070-2 |pages=[https://archive.org/details/elementsofeconom0003kmen/page/651 651–733] |url=https://archive.org/details/elementsofeconom0003kmen/page/651 }}

• {{YouTube|dLuTjoYmfXs|Endogeneity: An inconvenient truth. Podcast with Prof. John Antonakis}}
• {{YouTube|id=WlOtUA8Rqw8&list=PLD15D38DC7AA3B737&index=14#t=7m42s|title=Lecture on Simultaneity Bias}} by Mark Thoma
• [http://sethgodin.typepad.com/seths_blog/2017/05/what-about-endogeneity.html Seth Godin's simple views on endogeneity]

Category:Causality

Category:Estimation theory

Category:Econometric models

Category:Corporate finance