Solved – LOGIT Regression with multiple fixed effects – STATA

clogitclustered-standard-errorsfixed-effects-modelgeneralized linear modellogistic

For my thesis I am using as dependent variable the fraction of cash as part of the total price offered by the bidder. So, it's a fractional response that lies between [0,1]. I am not sure which regression should I use in Stata.

Also, my sample comprises 500 acquisitions in Europe announced in the period 2002-2016 from companies in different sectors (some companies have multiple acquisitions). The professor told me I should "control for year and industry (Fama French 12 – ffinds) fixed effects and adjust heteroskedasticity-robust standard errors for bidder clustering".

Here is what I have tried to do:

fracreg logit Y X1 X2 i.Year i.ffinds, vce(cluster ID)
glm Y X1 X2 i.Year i.ffinds, family(binomial) link(logit) robust nolog , but I cannot cluster for ID
I have tried to use xtlogit, but I am not able to apply it to multiple fixed effects

Which one is the correct approach? Also, using i.Year and i.ffinds I have too many dummies in the output. Is there a way to suppress them (like the option absorb used with reg)?

Best Answer

The first example is exactly how I would have done it.

Modeling proportions is what fracreg is for (although it's not the only way, with beta regression being the obvious alternative).
You're controlling for year and industry.
You're adjusting the standard errors in the way he requested.

The second example, even if you could get it to work right (offhand, I'm surprised you can't use a cluster VCE here), would give you the same answer as the first. That's how fractional logistic regression used to be done in Stata, using glm with certain options.

I strongly suspect the third example wouldn't work even if you could get the specification right; I don't know for sure, but I've never seen any research on estimating fixed-effect fractional logit models, let alone research that suggests you can just call the likelihood a quasi-likelihood and charge ahead.

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Solved – Presenting marginal effects of logit with fixed effects

In R the effects package can easily help with interpreting such coefficients by producing the appropriate graphs. From CRAN:

effects: Effect Displays for Linear, Generalized Linear, and Other Models

Graphical and tabular effect displays, e.g., of interactions, for various statistical models with linear predictors.

The package comes with two papers by John Fox describing the use of the package (along with examples) and the underlying implementation. The package takes good care of dealing with the subtle details and implementation difficulties for the supported models.

As an example:

require(car)
require(effects)
data(Cowles)
cowles.mod <- glm(volunteer ~ sex + neuroticism*extraversion,
                  data=Cowles, family=binomial)
Anova(cowles.mod)
plot(effect("neuroticism*extraversion", cowles.mod))
plot(effect("sex", cowles.mod))  ##for dummy, but should be similar for factors

Which will yield:

> Anova(cowles.mod)
Analysis of Deviance Table (Type II tests)

Response: volunteer
                         LR Chisq Df Pr(>Chisq)    
sex                        4.9184  1   0.026572 *  
neuroticism                0.3139  1   0.575316    
extraversion              22.1372  1  2.538e-06 ***
neuroticism:extraversion   8.6213  1   0.003323 ** 
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

And for the graph, which displays the predicted values of Y on the vertical axis, and the effect of X ("neuroticism") on Y given various levels of Z ("extraversion"):

enter image description here

Here's the effect plot for the dummy (but it should work similarly for factors):

enter image description here

Lastly, there seems to be some confusion surrounding partial effects (holding other predictors constant) vs marginal effects (ignoring other predictors). In my understanding the effects package is concerned with displaying "partial effects".

Solved – Can you have a conditional logit without fixed effects or a simple logit with conditional probabilities

After a week of intense learning we are now able to answer our own question. We post the answer here because it might be of some help to other people and because experts could still cross-validate it.

We came to the following conclusion. When the number of alternatives is large, estimating a conditional logit model is approximately equivalent to estimate an unconditional logit model with a dummy variable for each group and a constant and then conditioning the predicted odds for each alternative by the sum of the predicted odds for all the alternatives available to a given decision maker (i.e. all the alternatives within a group). The estimated coefficients and the predicted probabilities (after the transformation) would be approximately the same in the two cases. In fact, the coefficients of the conditional logit model refers to the linear prediction assuming that there is no fixed effect. In other words the coefficients measure the increase in the linear prediction (i.e. the log of the odds) for a unit increase in the independent variable. This is consistent with what explained in the Stata documentation of xtlogit and clogit. In particular, at page 15 of the latter, it is stated:

If Ti is large for all groups, the bias of the unconditional fixed-effects estimator is not a concern, and we can confidently use logit with an indicator variable for each group.

By conditioning to the sum of predicted odds within group, the conditional logit effectively removes any possible fixed effect because whatever is constant within group would not matter after dividing each predicted odds by the sum within group. This is so because whatever is constant within group would cancel out. Therefore, if you do not have any fixed effect, estimating an unconditional logit model with a constant and an indicator variable for each group or one with a constant but without the indicator variable, should return approximately the same coefficients and the same predicted odds. Then, you can just divide the predicted odds by the sum of the predicted odds within group. This would return the predicted probability that a decision maker chooses a given alternative. These probabilities would, by construction, sum to 1 within group.

This is what we think we learned, if this is wrong please correct us.

Best Answer

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Solved – Presenting marginal effects of logit with fixed effects

Solved – Can you have a conditional logit without fixed effects or a simple logit with conditional probabilities

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