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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 i which is "explained" by x i.
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.
Omitted-variable bias is the bias that appears in estimates of parameters in regression analysis when the assumed specification omits an independent variable that should be in the model. Analysis methods
Although it is intended to mitigate the effects of extraneous factors and selection bias, depending on how the treatment group is chosen, this method may still be subject to certain biases (e.g., mean regression, reverse causality and omitted variable bias).
A variable omitted from the model may have a relationship with both the dependent variable and one or more of the independent variables (causing omitted-variable bias). [3] An irrelevant variable may be included in the model (although this does not create bias, it involves overfitting and so can lead to poor predictive performance). The ...
Bias is a distinct concept from consistency: consistent estimators converge in probability to the true value of the parameter, but may be biased or unbiased (see bias versus consistency for more). All else being equal, an unbiased estimator is preferable to a biased estimator, although in practice, biased estimators (with generally small bias ...
These are contrasted with confounders which are "good controls" and need to be included to remove omitted variable bias. [ 1 ] [ 2 ] [ 3 ] This issue arises when a bad control is an outcome variable (or similar to) in a causal model and thus adjusting for it would eliminate part of the desired causal path.
[citation needed] It has also been called unconfoundedness, selection on the observables, or no omitted variable bias. [1] This idea is part of the Rubin Causal Inference Model, developed by Donald Rubin in collaboration with Paul Rosenbaum in the early 1970s. The exact definition differs between their articles in that period.