Is there any ways to argue that the omitted variable problem is alleviated after adding a new variable to the model? Right now I'm basically just saying that adding this new variable significantly improves model fit (based on LR test). What else (such as some specific tests? ) can I do? Thanks!
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10
Take a look at
Oster, E. (2017). Unobservable Selection and Coefficient Stability: Theory and Evidence. Journal of Business & Economic Statistics, 37(2), 187–204. https://doi.org/10.1080/07350015.2016.1227711 (pdf)
She develops a formal bounding argument for omitted variable bias under the proportional selection relationship between observables and unobservables.
Stata code can be found here and here.
You can do something similar in Python using DoubleBML.
This is likely an incomplete list of implementations.
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1$\begingroup$ Interesting paper (+1). I'd appreciate, if you'd answer a couple of clarifying questions, as follows. What "proportional selection" in your answer refers to? Is this a general statistical approach or it is bound by certain assumptions or conditions? Is there a related functionality within
Recosystem? $\endgroup$ Commented Apr 13, 2015 at 5:48 -
3$\begingroup$ @AleksandrBlekh Consider a model $$ y=\alpha + \beta \cdot x + \Sigma_{j=1}^{C} \gamma_j \cdot x_j +\Sigma_{k=1}^{U} \eta_k \cdot z_k + \varepsilon = \alpha + \beta \cdot x + C + U + \varepsilon,$$ where $y$ is the outcome, $x$ is the variable of interest, $x_j$ are the controls and $z_k$ are the unobserved variables. The proportionality of selection relationship is defined as $$\delta = \frac{\frac{Cov(U,t)}{Var(U)}}{\frac{Cov(C,t)}{Var(C)}}.$$ Essentially, $\delta$ is the ratio of univariate regression coefficients of $x$ on $U$ and $x$ on $C$. $\endgroup$– dimitriyCommented Apr 13, 2015 at 5:52
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2$\begingroup$ A $\delta$ of one indicates that the observed and unobserved variables have an equally important effect on the coefficient of interest. A $\delta$ greater than one indicates that the unobserved variables are relatively more important. The other ingredient for the bias is the hypothetical $R^2$ from that regression if you had access to the $z$s. I am not aware of an
Rimplementation. $\endgroup$– dimitriyCommented Apr 13, 2015 at 5:53 -
$\begingroup$ Thank you very much for the comments. The subject is much clearer to me now - as I understand it's all about having a metric for assessing the relative importance of observed vs. unobserved factors. But why this distinction? I initially thought that the omitted variable bias term can (and should?) be applied to factors irrespective to their belonging to observed or latent classes, hence my mention of general approach. $\endgroup$ Commented Apr 13, 2015 at 6:29
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$\begingroup$ @AleksandrBlekh I am not sure what you mean. Once you make assumptions about $\delta$ and the hypothetical $R^2_{MAX},$ this offers a way to get at the magnitude of the OVB from how the $R^2$ and the coefficients change as you add observed covariates. $\endgroup$– dimitriyCommented Apr 13, 2015 at 17:39