Is it correct that the marginal effects of fractional respnse model is greater than 1?

[Eason Xu]

I am using command fracreg probit in stata to deal with my data. My dependent variable is the the ratio of newly established firms in each year, and my core independent variable is also a ratio. And my data is a short panel. Now I use command "margins, dydx()" to calculate the marginal effects, but the result shows the marginal effect ia larger than 1, around 1.5. Is this correct? How should I interpret this result?
Can someone help me with this or recommand some references？ Thank you!

[Clyde Schechter]

The marginal effect of what is larger than 1. If the variable whose marginal effect you are dealing with has a narrow range, (which it well might since you say it is a ratio) it may be that a unit change in that ratio simply is beyond the range of possibility. Remember that a marginal effect is interpreted as the rate of change in the outcome per unit increase in the regressor. If the regressor cannot change by that much, it is by no means problematic that the marginal effect exceeds 1.

Also, what is your -margins- command predicting, the conditional mean of the outcome, or xb (the linear prediction). The potential range for xb is from negative infinity to positive infinity; it is the probit part of the model that compresses that to the [0,1] interval. In Stata version 18, the conditional mean is the default prediction of -margins- following -fracreg-. But in earlier versions that may or may not have been so. See -help fracreg_postestimation##margins- to see what the default predictor is. If it is xb, then there is no reason to be concerned about a marginal effect > 1 in any situation. If your prediction is, indeed, the conditional mean, then consider the preceding paragraph.

[Eason Xu]

The marginal effect of what is larger than 1. If the variable whose marginal effect you are dealing with has a narrow range, (which it well might since you say it is a ratio) it may be that a unit change in that ratio simply is beyond the range of possibility. Remember that a marginal effect is interpreted as the rate of change in the outcome per unit increase in the regressor. If the regressor cannot change by that much, it is by no means problematic that the marginal effect exceeds 1.

Also, what is your -margins- command predicting, the conditional mean of the outcome, or xb (the linear prediction). The potential range for xb is from negative infinity to positive infinity; it is the probit part of the model that compresses that to the [0,1] interval. In Stata version 18, the conditional mean is the default prediction of -margins- following -fracreg-. But in earlier versions that may or may not have been so. See -help fracreg_postestimation##margins- to see what the default predictor is. If it is xb, then there is no reason to be concerned about a marginal effect > 1 in any situation. If your prediction is, indeed, the conditional mean, then consider the preceding paragraph.

Thank you for your reply! I have checked my stata,and find that the default predictor is conditional mean. And I have another confusion. If I increase the ratio by 100 times, like turning 0.05 into 5. And the marginal effect will turn into 0.015. Can I use this method to calculate the marginal effects? I still find hard to interpret the result greater than 1.

[Clyde Schechter]

Yes, it is standard in linear models that if you scale the predictor so that it takes on larger values, the coefficient takes on smaller values by the same proportion. And vice versa That's simple math. The important corollary from that is that regression coefficients must always be interpreted in light of the scale of all the variables.

Thinking of a concrete example, if the predictor variable were an amount of money measured in, say dollars, and if the outcome is something measured in very large numbers, then the coefficient will be very large as well. If you rescale the money variable to be measured in billions of dollars, and leave the outcome variable unchanged, then the coefficient will become smaller by a factor of a billion, and will probably be a relatively small number. Nothing in substances changes as a result of this. Neither the original coefficient nor the changed one can be understood without the unit of measure of the money variable as context.

The same is true for your ratio variable. If you increase it's magnitude by rescaling as you have done, then the coefficient (and hence the marginal effect which is a monotone increasing function of the coefficient) decreases by the same factor. So the original marginal effect, which made you feel uncomfortable, is perfectly sensible when presented with the context of the original variable's scale. If you prefer to present these same results using a scaled up ratio variable and a correspondingly scaled down marginal effect, that is probably OK. In fact, if your original ratio variable is scaled so that a unit change in its value is simply beyond the range of possibilities, or is at most very rare and strange, then it probably was a poor choice of measurement units to calculate it on that scale in the first place, and your rescaling would be an improvement. Not because a marginal effect > 1 is a problem--it's not! But because a variable whose range does not allow for unit changes is a poor candidate for calculating a marginal effect. Who cares what the consequences of a unit change in X are if X can never change by a unit!

On the other hand, remember, we do not do research for our own amusement: we do research for some audience of people who have an interest in the results. If the ratio is some well-known standard variable in your field and it is always calculated in certain units of measurement, then changing it may perplex your audience. At best they will be mildly confused before figuring out what you did. At worst they will wonder if you are pulling some scam to manipulate the analysis into providing some result you prefer. (That would not actually be the case since, substantively nothing changes with the rescaling. But those who aren't following the math might not realize that might worry that you are conning them.) So you have to really consider what, if any, expectations your audience has about this.

Finally, I really do hope you can gain some comfort with the possibility of marginal effects > 1 in a bounded model like probit. Suppose there is a road in your neighborhood that is only one km long. Even though the distance you travel on it can never exceed 1 km, there is nothing at all that prevents you from driving or biking down that road at 30 km/hr, even though 30 is much larger than 1. The implication, if you do that, is that the trip will only take you substantially less than an hour when carried out at that speed. Marginal effects are like speeds converting travel time (the independent variable) into distance traveled (the outcome variable.)

[Eason Xu]

Yes, it is standard in linear models that if you scale the predictor so that it takes on larger values, the coefficient takes on smaller values by the same proportion. And vice versa That's simple math. The important corollary from that is that regression coefficients must always be interpreted in light of the scale of all the variables.

Thinking of a concrete example, if the predictor variable were an amount of money measured in, say dollars, and if the outcome is something measured in very large numbers, then the coefficient will be very large as well. If you rescale the money variable to be measured in billions of dollars, and leave the outcome variable unchanged, then the coefficient will become smaller by a factor of a billion, and will probably be a relatively small number. Nothing in substances changes as a result of this. Neither the original coefficient nor the changed one can be understood without the unit of measure of the money variable as context.

The same is true for your ratio variable. If you increase it's magnitude by rescaling as you have done, then the coefficient (and hence the marginal effect which is a monotone increasing function of the coefficient) decreases by the same factor. So the original marginal effect, which made you feel uncomfortable, is perfectly sensible when presented with the context of the original variable's scale. If you prefer to present these same results using a scaled up ratio variable and a correspondingly scaled down marginal effect, that is probably OK. In fact, if your original ratio variable is scaled so that a unit change in its value is simply beyond the range of possibilities, or is at most very rare and strange, then it probably was a poor choice of measurement units to calculate it on that scale in the first place, and your rescaling would be an improvement. Not because a marginal effect > 1 is a problem--it's not! But because a variable whose range does not allow for unit changes is a poor candidate for calculating a marginal effect. Who cares what the consequences of a unit change in X are if X can never change by a unit!

On the other hand, remember, we do not do research for our own amusement: we do research for some audience of people who have an interest in the results. If the ratio is some well-known standard variable in your field and it is always calculated in certain units of measurement, then changing it may perplex your audience. At best they will be mildly confused before figuring out what you did. At worst they will wonder if you are pulling some scam to manipulate the analysis into providing some result you prefer. (That would not actually be the case since, substantively nothing changes with the rescaling. But those who aren't following the math might not realize that might worry that you are conning them.) So you have to really consider what, if any, expectations your audience has about this.

Finally, I really do hope you can gain some comfort with the possibility of marginal effects > 1 in a bounded model like probit. Suppose there is a road in your neighborhood that is only one km long. Even though the distance you travel on it can never exceed 1 km, there is nothing at all that prevents you from driving or biking down that road at 30 km/hr, even though 30 is much larger than 1. The implication, if you do that, is that the trip will only take you substantially less than an hour when carried out at that speed. Marginal effects are like speeds converting travel time (the independent variable) into distance traveled (the outcome variable.)

Thank you so much for this reply! It's really really really helpful! I have a deeper understanding for the marginal effect. Best regards！