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Thread: Can covariates be random variables? (logit)

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    Can covariates be random variables? (logit)




    Hi Everyone.

    I have been discussing a project at work. My colleague said that he is going to estimate a logit model and include as a covariates few random variables.

    He has pooled (mean) data for some (part of the dataset) of the covariates and he wants to generate random variables around these averages and assign these numbers to microdata. I do not get why he needs to do so. But the main question is - is it correct to include such random variables as independent variables in logit regression?

    By the OLS assumptions the eror term is uncorrelated with the covariates. Is it also the case for logit regression?

    Thanks.

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    Re: Can covariates be random variables? (logit)

    Sounds like they may be similating a component of data?

    Logit model typically means logistic regression, which doesn't have an error term or assumption.
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    Re: Can covariates be random variables? (logit)

    Also, in any model the covariates can be random... Simulating everything in life at random is totally fine. This is how our whole world came to being, conditional on what we know and what we will ever learn.

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    Re: Can covariates be random variables? (logit)

    But the main question is - is it correct to include such random variables as independent variables in logit regression?
    I have never seen this commented on. I don't understand what you would gain from this and adding meaningless variables to a model is rarely a good thing.
    "Very few theories have been abandoned because they were found to be invalid on the basis of empirical evidence...." Spanos, 1995

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    Re: Can covariates be random variables? (logit)

    If he is doing what we are assuming he is doing, we would have to hope the mean and dispersion parameters are a good representation of the variable, which may benefit from a larger sample size. So, that the variable approximates a normal distribution per se. So if the variable was skewed, a simulation may have issues if you did not address the actual distribution. Also, doing this may also require them to control for the variable's relationship with the dependent variable as well as other covariates if it is not completely independent.


    noetsi has a point in asking why is he/she doing this. Perhaps to attempt to adjust for a covariate and include it in the model, so the estimates of the real variable of interest is in a multidimensional model? Perhaps they are crazy or looking for a good time!


    So do you have any more info for us harel?
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    Re: Can covariates be random variables? (logit)


    By the OLS assumptions the error term is uncorrelated with the covariates. Is it also the case for logit regression?
    The answer is yes if by error term you mean the latent error term. In OLS the error is the dependent variable substracted the conditional mean y - E[y\lvert x] := \epsilon in the logit model there is no assumption made about y - E[y\lvert x] = y - Pr(y=1\lvert x) what is assumed is that Pr(y=1\lvert x) = L(x) where L() is the logit formula. However one way to arrive at this formula is to use the latent variable formulation:
    y = I[y^* > 0] where y^* is the latent variable. Then you make the assumption that y^* = \theta x + u
    and assume that u is distributed logistically. It then follows that Pr(y=1\lvert x) = Pr(y^* > 0 \lvert x) = Pr(\theta x + u>0 \lvert x) = Pr( u > - \theta x  \lvert x) .... then using the symmetry of the logistic dist. and assuming we have standardized the variance of the error we have the logit. So the latent error term is assumed to be conditionally independent and logistically distributed.

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