# How can I interpret a regression model with a variable that appears as an inverse?

I have an equation where $$Employ = \beta_1 + \beta_2 * \frac{1}{GDP} + e$$ Then how can I explain the relationship between Employ and GDP? For example when GDP increase a certain unit or percentage, measure the change in Employ?

Is there any other way to interpret this type of regression model?

@AdamO: Thanks. I just want to know if there existed a way to explain the relationship between Y and X in the regression model : $$Y=\beta_1 +\beta_2\frac{1}{X}+\epsilon$$. (such as something similar to the interpretation of level-level model (or log-log one): $$Y=\beta_1 +\beta_2X+\epsilon$$: if X increases by 1 unit then Y rise by $$\beta_2$$ unit

@all: thank you!

• Employ and GPD are absolute values or relative variations? – Giorgio Spedicato Nov 21 '13 at 17:14
• Straightforwardly, for each unit change in the reciprocal of GDP, employment changes by $\beta_2$ units (the units are whatever you're measuring GDP & employment in). – Scortchi Nov 21 '13 at 17:19

## 2 Answers

Literally, you can interpret $\beta_2$ as a difference in employment associated with a unit difference in the inverse of GDP. However, I can't help but think "employ" is something like "employment rates" or "employment period" which means you're really trying to model a some rate or time in which the variance of the outcome depends upon the mean. Why make a big deal out of this? Because your model forces large GDP countries to largely determine $\beta_1$ and small GDP countries to determine $\beta_2$. What's the point when a simple t-test will infer differences between those groups?

Alternately, if you correctly account for a mean variance relationship, you can estimate a special type of regression model in the traditional sense, that is where each observation is iteratively reweighted to provide the correct contribution to estimation of the model parameters. Accounting for that relationship is achieved with a GLM and an inverse transform not of the predictor but of the outcome.

For instance, if I were modeling employment rates, I would first ask "are these data cross sectional?" and "can I measure the denominator?" Provided I could do that, I could consider a Poisson model with # of employed as the outcome and # employable as the offset adjusting for GDP linearly. I would then estimate a relative rate of employment associated with a unit difference in GDP, it's a multiplicative effect and can approximate the inverse curve very well, yielding predicted rates that are very consistent (and also constrained to be positive). I would go a few steps further with this model, but the conceptual parameter is, to me, useful.

Whenever possible, one option that can be useful is to visualize the data.

Consider the following models:

Model 1: $\text{Employ} = 1.5 -\frac{1}{\text{GDP}} + \epsilon$

Model 2: $\text{Employ} = 1.5 +\frac{1}{\text{GDP}} + \epsilon$

Now consider this data:

GDP = {1,2,3,4,5,6,7,8,9,10}. Looking to the scatterplot first thing to say is that Employ and GDP do not have linear relationship.

Model 1: Employ increases at decreasing rates as GDP increases.
Model 2: Employ decreases at decreasing rates as GDP increases.

• Careful with those causal interpretations! – AdamO Nov 21 '13 at 19:00
• Well, in a simulated setting as you proposed, it's okay to interpret the model parameters as either causal or observational, but in the context of the OP's problem description, there's no way such data were experimentally obtained. I would interpret $\beta_2$ as an associated difference in employ for a unit difference in the inverse of GDP. – AdamO Nov 21 '13 at 19:08