From Wikipedia, the free encyclopedia
In , omitted-variable bias (OVB) occurs when a model created incorrectly leaves out one or more important factors. The "bias" is created when the model compensates for the missing factor by over- or underestimating the effect of one of the other factors.
More specifically, OVB is the
that appears in the estimates of
in a , when the assumed
is incorrect in that it omits an independent variable that is correlated with both the dependent variable and one or more included independent variables.
Two conditions must hold true for omitted-variable bias to exist in :
the omitted variable must be a
of the dependent variable (i.e., its true regression coefficient is not zero); and
the omitted variable must be correlated with an independent variable specified in the regression (i.e., cov(z,x), is not equal to zero).
Suppose the true cause-and-effect relationship is given by
{\displaystyle y=a+bx+cz+u}
with parameters a, b, c, dependent variable y, independent variables x and z, and error term u. We wish to know the effect of x itself upon y (that is, we wish to obtain an estimate of b). But suppose that we omit z from the regression, and suppose the relation between x and z is given by
{\displaystyle z=d+fx+e}
with parameters d, f and error term e. Substituting the second equation into the first gives
{\displaystyle y=(a+cd)+(b+cf)x+(u+ce).}
If a regression of y is conducted upon x only, this last equation is what is estimated, and the regression coefficient on x is actually an estimate of (b+cf ), giving not simply an estimate of the desired direct effect of x upon y (which is b), but rather of its sum with the indirect effect (the effect f of x on z times the effect c of z on y). Thus by omitting the variable z from the regression, we have estimated the
of y with respect to x rather than its
with respect to x. These differ if both c and f are non-zero.
As an example, consider a
of the form
{\displaystyle y_{i}=x_{i}\beta +z_{i}\delta +u_{i},\qquad i=1,\dots ,n}
xi is a 1 × p row vector of values of p
observed at time i or for the i th
β is a p × 1 column vector of unobservable parameters (the response coefficients of the dependent variable to each of the p independent variables in xi)
zi is a scalar and is the value of another independent variable that is observed at time i or for the i th
δ is a scalar and is an unobservable parameter (the response coefficient of the dependent variable to zi)
ui is the unobservable
occurring at time i or for the i th it is an unobserved realization of a
having  0 (conditionally on xi and zi);
yi is the observation of the
at time i or for the i th study participant.
We collect the observations of all variables subscripted i = 1, ..., n, and stack them one below another, to obtain the
Y, Z, and U:
{\displaystyle X=\left[{\begin{array}{c}x_{1}\\\vdots \\x_{n}\end{array}}\right]\in \mathbb {R} ^{n\times p},}
{\displaystyle Y=\left[{\begin{array}{c}y_{1}\\\vdots \\y_{n}\end{array}}\right],\quad Z=\left[{\begin{array}{c}z_{1}\\\vdots \\z_{n}\end{array}}\right],\quad U=\left[{\begin{array}{c}u_{1}\\\vdots \\u_{n}\end{array}}\right]\in \mathbb {R} ^{n\times 1}.}
If the independent variable z is omitted from the regression, then the estimated values of the response parameters of the other independent variables will be given by, by the usual
calculation,
{\displaystyle {\hat {\beta }}=(X'X)^{-1}X'Y\,}
(where the "prime" notation means the
of a matrix and the -1 superscript is ).
Substituting for Y based on the assumed linear model,
{\displaystyle {\begin{aligned}{\hat {\beta }}&=(X'X)^{-1}X'(X\beta +Z\delta +U)\\&=(X'X)^{-1}X'X\beta +(X'X)^{-1}X'Z\delta +(X'X)^{-1}X'U\\&=\beta +(X'X)^{-1}X'Z\delta +(X'X)^{-1}X'U.\end{aligned}}}
On taking expectations, the contribution of th this follows from the assumption that U is uncorrelated with the regressors X. On simplifying the remaining terms:
{\displaystyle {\begin{aligned}E[{\hat {\beta }}|X]&=\beta +(X'X)^{-1}X'Z\delta \\&=\beta +{\text{bias}}.\end{aligned}}}
The second term after the equal sign is the omitted-variable bias in this case, which is non-zero if the omitted variable z is correlated with any of the included variables in the matrix X (that is, if X'Z does not equal a vector of zeroes). Note that the bias is equal to the weighted portion of zi which is "explained" by xi.
states that regression models which fulfill the classical linear regression model assumptions provide the best, linear and unbiased estimators. With respect to , the relevant assumption of the classical linear regression model is that the error term is uncorrelated with the regressors.
The presence of omitted-variable bias violates this particular assumption. The violation causes the OLS estimator to be biased and . The direction of the bias depends on the estimators as well as the
between the regressors and the omitted variables. A positive covariance of the omitted variable with both a regressor and the dependent variable will lead the OLS estimate of the included regressor's coefficient to be greater than the true value of that coefficient. This effect can be seen by taking the expectation of the parameter, as shown in the previous section.
B Howland (2005). . Introductory Econometrics: Using Monte Carlo Simulation with Microsoft Excel. Cambridge University Press.
Clarke, Kevin A. (2005). "The Phantom Menace: Omitted Variable Bias in Econometric Research". Conflict Management and Peace Science. 22: 341–352. :.
Greene, W. H. (1993). Econometric Analysis (2nd ed.). Macmillan. pp. 245–246.
Wooldridge, Jeffrey M. (2009). "Omitted Variable Bias: The Simple Case". Introductory Econometrics: A Modern Approach. Mason, OH: Cengage Learning. pp. 89–93.  .
: Hidden categories:统计学总结_百度文库
两大类热门资源免费畅读
续费一年阅读会员，立省24元！
统计学总结
上传于||暂无简介
阅读已结束，如果下载本文需要使用1下载券
想免费下载本文？
定制HR最喜欢的简历
下载文档到电脑，查找使用更方便
还剩8页未读，继续阅读
定制HR最喜欢的简历
你可能喜欢第七章设定误差与数据问题(计量)_百度文库
两大类热门资源免费畅读
续费一年阅读会员，立省24元！
第七章设定误差与数据问题(计量)
上传于||暂无简介
阅读已结束，如果下载本文需要使用1下载券
想免费下载本文？
定制HR最喜欢的简历
下载文档到电脑，查找使用更方便
还剩3页未读，继续阅读
定制HR最喜欢的简历
你可能喜欢统计学中的相关与回归分析_图文_百度文库
两大类热门资源免费畅读
续费一年阅读会员，立省24元！
评价文档：
统计学中的相关与回归分析
上传于||文档简介
&&图​文​并​茂
大小：2.79MB
登录百度文库，专享文档复制特权，财富值每天免费拿！
你可能喜欢多元统计学_百度文库
两大类热门资源免费畅读
续费一年阅读会员，立省24元！
多元统计学
上传于||暂无简介
阅读已结束，如果下载本文需要使用2下载券
想免费下载本文？
定制HR最喜欢的简历
下载文档到电脑，查找使用更方便
还剩6页未读，继续阅读
定制HR最喜欢的简历
你可能喜欢