Interpretation of Scaled AMEs after Fractional Response Model

[Alina Faruk]

Dear All,

I have am running a regression where both the dependent and independent variables are rates in %.

After some reading, I opted for the fractional response model using the -fracreg- command with probit option. For this purpose, I rescaled the variables to [0,1] interval by dividing them by 100.

After running the regression, I ran the -margins, dydx(*)- command for average marginal effects. But I am having trouble with interpretation now that my % variables are rescaled to proportions.

For instance, if the AME of x1 is 0.002, does that mean:

• a 1% increase in x1 will lead to a 2 percentage points rise in y?
• a 1% increase in x1 will lead to a 0.002 percentage points rise in y?
• a 1% increase in x1 will lead to a 2% rise in y?
• a 1% increase in x1 will lead to a 0.002% rise in y?
• a 0.01% increase in x1 will lead to a 2 percentage points rise in y?
• and so on,,,

Your help is much appreciated. Thank you..

[Maarten Buis]

The dydx() option results in additive effects, so in your case percentage points increase and not percentage. To move from proportions to percentage you multiply by 100, so 0.002*100 = 0.2 percentage points.

[Alina Faruk]

The dydx() option results in additive effects, so in your case percentage points increase and not percentage. To move from proportions to percentage you multiply by 100, so 0.002*100 = 0.2 percentage points.

Thank you so much. But since my independent variables also underwent the same scaling, should I say a 1% rise in x1 or a 0.01% rise in x1 leads to a 0.2 percentage point increase in y?

[Maarten Buis]

The easiest solution is to not rescale the explanatory/independent/right-hand-side/x-variable to a scale you don't like. Regardless, the effect is always in terms of a unit change in x. The rescaling changes the unit. For a percentage a unit is the entire theoretical range: the theoretical minimum is 0 and the theoretical maximum is 1. That is usually not that helpful. With percentages (not rescaled) the unit is 1 percentage point. That is sometimes fine, sometimes a bit too fine grained. In the later case, I usually change the unit to 10 percentage points (divide a percentage variable by 10 or multiply a proportions variable by 10).

And again, marginal effects is all additive not multiplicative, so you do *not* get the effect in terms of percentages but in term of percentage points.

[Alina Faruk]

The easiest solution is to not rescale the explanatory/independent/right-hand-side/x-variable to a scale you don't like. Regardless, the effect is always in terms of a unit change in x. The rescaling changes the unit. For a percentage a unit is the entire theoretical range: the theoretical minimum is 0 and the theoretical maximum is 1. That is usually not that helpful. With percentages (not rescaled) the unit is 1 percentage point. That is sometimes fine, sometimes a bit too fine grained. In the later case, I usually change the unit to 10 percentage points (divide a percentage variable by 10 or multiply a proportions variable by 10).

And again, marginal effects is all additive not multiplicative, so you do *not* get the effect in terms of percentages but in term of percentage points.

Thank you for your response! I will keep the independent variables on their original scale. I understand that the effect is in percentage points, but if my independent variable is a percentage, saying a 1% rise in x1 leads to 0.2 percentage point change in y, implies 1% of the percentage variable (x1) right? Basically a percentage change of a percentage? I am so sorry if this is not clear.

[Maarten Buis]

Saying a 1% rise means that you multiply whatever you start with by 1.01 to get a 1% increase. That is not what you are doing when you do marginal effects. Remember, marginal effects are all additive. So a unit increase means you add 1 (percentage point, foot, cow, whatever your unit is) to wherever you start. So if your explanatory/independent/right-hand-side/x-variable is measured in percentages and you compute a marginal effect, then the unit is a percentage point not a percentage.

[Alina Faruk]

Saying a 1% rise means that you multiply whatever you start with by 1.01 to get a 1% increase. That is not what you are doing when you do marginal effects. Remember, marginal effects are all additive. So a unit increase means you add 1 (percentage point, foot, cow, whatever your unit is) to wherever you start. So if your explanatory/independent/right-hand-side/x-variable is measured in percentages and you compute a marginal effect, then the unit is a percentage point not a percentage.

Thank you so much for the clarification!