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{{More footnotes|date=July 2010}}
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In [[statistics]], '''omitted-variable bias''' ('''OVB''') occurs when a model is created which incorrectly leaves out one or more important causal 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 [[estimator bias|bias]] that appears in the estimates of [[parameter]]s in a [[regression analysis]], when the assumed [[specification (regression)|specification]] is incorrect in that it omits an independent variable that is correlated with both the dependent variable and one or more included independent variables.
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==Omitted-variable bias in linear regression==
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Two conditions must hold true for omitted-variable bias to exist in [[linear regression]]:
* the omitted variable must be a [[determinant]] of the dependent variable (i.e., its true regression coefficient is not zero); and
* the omitted variable must be correlated with one or more of the included [[independent variable]]s (i.e. the covariance of the omitted variable and the independent variable, cov(z,x), is not equal to zero).
{{confusing|date=December 2012}}
As an example, consider a [[linear model]] of the form
:<math>y_i = x_i \beta + z_i \delta + u_i,\qquad i = 1,\dots,n</math>


where
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* ''x''<sub>''i''</sub> is a 1&nbsp;×&nbsp;''p'' row vector of values of ''p'' [[independent variable]]s observed at time ''i'' or for the ''i'' <sup>th</sup> study participant;
 
* ''β'' is a ''p''&nbsp;×&nbsp;1 column vector of unobservable parameters (the response coefficients of the dependent variable to each of the ''p'' independent variables in ''x''<sub>''i''</sub>) to be estimated;
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* ''z''<sub>''i''</sub> is a scalar and is the value of another independent variable that is observed at time ''i'' or for the ''i'' <sup>th</sup> study participant;
 
* ''δ'' is a scalar and is an unobservable parameter (the response coefficient of the dependent variable to ''z''<sub>''i''</sub>) to be estimated;
  <li>[http://www.histoirepassion.eu/spip.php?article1895/ http://www.histoirepassion.eu/spip.php?article1895/]</li>
* ''u''<sub>''i''</sub> is the unobservable [[errors and residuals in statistics|error term]] occurring at time ''i'' or for the ''i'' <sup>th</sup> study participant; it is an unobserved realization of a [[random variable]] having [[expected value]]&nbsp;0 (conditionally on ''x''<sub>''i''</sub> and ''z''<sub>''i''</sub>);
 
* ''y''<sub>''i''</sub> is the observation of the [[dependent variable]] at time ''i'' or for the ''i'' <sup>th</sup> study participant.
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</ul>


We collect the observations of all variables subscripted ''i'' = 1, ..., ''n'', and stack them one below another, to obtain the [[matrix (mathematics)|matrix]] ''X'' and the [[vector (mathematics)|vectors]] ''Y'', ''Z'', and ''U'':
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:<math> X = \left[ \begin{array}{c} x_1 \\  \vdots \\ x_n \end{array} \right] \in \mathbb{R}^{n\times p},</math>


and
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: <math> 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}.</math>
 
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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 [[least squares]] calculation,
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:<math>\hat{\beta} = (X'X)^{-1}X'Y\,</math>


(where the "prime" notation means the [[transpose]] of a matrix and the -1 superscript is [[matrix inversion]]).
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Substituting for ''Y'' based on the assumed linear model,
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:<math>
 
\begin{align}
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\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 \\
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& =\beta + (X'X)^{-1}X'Z\delta + (X'X)^{-1}X'U.
 
\end{align}
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</math>
 
 
</ul>
On taking expectations, the contribution of the final term is zero; this follows from the assumption that ''U'' has zero expectation. On simplifying the remaining terms:
:<math>
\begin{align}
E[ \hat{\beta} | X ] & = \beta + (X'X)^{-1}X'Z\delta \\
& = \beta + \text{bias}.
\end{align}
</math>
 
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 ''z''<sub>''i''</sub> which is "explained" by ''x''<sub>''i''</sub>.
 
==Effects on ordinary least square==
The [[Gauss–Markov theorem]] states that regression models which fulfill the classical linear regression model assumptions provide the best, linear and unbiased estimators. With respect to [[ordinary least squares]], 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 [[Consistency_(statistics)|inconsistent]]. The direction of the bias depends on the estimators as well as the [[covariance]] 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.
 
==See also==
*[[Confounding variable]]
 
==References==
* {{cite book |last=Barreto |last2=Howland |chapter=Omitted Variable Bias |title=Introductory Econometrics: Using Monte Carlo Simulation with Microsoft Excel |publisher=Cambridge University Press |date=2005 |chapterurl=http://www3.wabash.edu/econometrics/EconometricsBook/chap18.htm}}
* {{cite journal |last=Clarke |first=Kevin A. |year=2005 |title=The Phantom Menace: Omitted Variable Bias in Econometric Research |journal=Conflict Management and Peace Science |volume=22 |issue= |pages=341–352 |doi=10.1080/07388940500339183 }}
* {{cite book |last=Greene |first=W. H. |title=Econometric Analysis |edition=2nd |publisher=Macmillan |date=1993 |pages=245–246}}
* {{cite book |last=Wooldridge |first=Jeffrey M. |year=2009 |chapter=Omitted Variable Bias: The Simple Case |pages=89–93 |title=Introductory Econometrics: A Modern Approach |location=Mason, OH |publisher=Cengage Learning |isbn=9780324660548 }}
 
{{Biases}}
 
[[Category:Regression analysis]]
[[Category:Bias]]

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When Wells Guthrie launched Copain winery in 1999, he knew steps to make wines that sell: His relatively bold, highoctane pinot noirs and syrahs earned him stellar ratings and a loyal following. But they lacked that old World refinement and finesse found in the Burgundies and Rh that Guthrie loved, so he made a decision to take his wines in another direction, picking grapes earlier and keeping alcohol levels in check. Some weren't pleased ("We lost customers and Vibram Kso Trek some distributors," Guthrie says), but he preferred these wines, and individuals have come around to his perspective..

Start Your Lawn Care Business Starting your personal lawn care business can be a "cash cow" if done the right way. When you were younger, maybe you cut grass for the neighbors and some relatives for a small fee. It was probably a nice little side income for you personally with little work.

One or more areas of a food that provides b12, such as milk, fortified soy milk, egg yolk, or fermented soy like miso or tempeh.Note that these amounts differ somewhat from the amounts recommended recommended through the USDA for pregnant women. Help Center Press Center E mail us Site MapMembership Newsletters Newsletters Register Manage Email Subscriptions Update Account Legal Privacy Policy Terms of UseBaby Products Britax Car SeatsThis Web page provides information of a general nature and it is designed for educational purposes only. For those who have any concerns about your own health or even the health of your child, it is best to consult with a physician or other healthcare professional.

White or pale colors, such as the beige most apartments are painted in, make objects recede, but dark or bright colors draw objects closer, which provides rooms a larger feel, the site says. However, Ms. Heydlauff, says lighter colors are the way to go when trying to make a room appear larger.. Louis Vuitton Women Shoes Australia

Everything will be as it always has been on Mothers Day, except our conversations about our youngsters. All of our children are boys and one topic often discussed is how to manage their fighting, their knocks and bumps and the developmental stages of growing up. The problem of fighting also exists with my very own son, but on a very different scale it is his battle to hold objects, to utter sounds, to weight bare and at times to simply keep living..