How to interpret a regression that includes GDP, GDP per capita, and population

While GDP per capita = GDP / population and are obviously related, these three measurements are not perfectly collinear and can be included in OLS just fine.

However, when interpreting the coefficient of GDP per capita, how does it make sense to say that we "keep GDP and population fixed in order to find the effect caused by GDP per capita"?

• Did you see researchers including all three in the models? While I could agree than including two of them is Ok, all three would be an issue. I'd strike that model down. Apr 14 '14 at 21:23
• Yes, but I'm asking what exactly the issue is? Algebraically it's not a problem because the three of them together, while having a deterministic relationship, do not have a perfect linear relationship, so the OLS still runs. Apr 14 '14 at 21:27
• I think you clearly stated the problem yourself. Your OLS results will not make any sense. Apr 14 '14 at 21:30

They are not perfectly collinear, but they are collinear to a problematic degree:

set.seed(123)
gdp <- exp(rnorm(100))
pop <- rnorm(100, 10000, 2000)
gdppercap <- gdp/pop
dv <- rnorm(100) #For collin, dv doesn't matter
library(perturb)
m1 <- lm(dv~gdp+pop+gdppercap)
summary(m1)
colldiag(m1)

The largest condition index is 21, which isn't outrageous but isn't good. And all the IVs have variance decomposition proportions over 0.75, which is also not good.

So, you should leave one of the IVs out, both for the logic of the model and for colinearity.

• Except that in the real model they do turn out to be significant still. But I accept the logical argument. Apr 15 '14 at 0:00
• Significance is often found with highly colinear variables. The problem is that a small change in the input data can make a large change in the model. Apr 15 '14 at 0:31
• Could you elaborate or point me to some resources to read? From what I'm taught I thought that high collinearity means that the $X'X$ matrix is really sparse, leading to large standard error? And why does a small change in data lead to a large change in the model? Apr 15 '14 at 2:57
• If you want all the details on collinearity, read Belsley. The small change large change is a consequence; and it is also an alternative diagnostic (see the perturb package in R) Apr 15 '14 at 9:28

There is not necessarily a problem in terms of getting estimates for each of the components. There is however a problem of interpreting them, as has been outlined above already.

Having said that, usually people include the logarithms of these variables in a regression. This may be more error prone. Since ln(GDP/POP)=ln(GDP)-ln(POP), a specification like

$y = \alpha + \beta_1 \ln(gdp) + \beta_2 \ln(gdp/pop) + \beta_3 \ln(pop) + \epsilon$

is equivalent to

$y = \alpha + \beta_1 \ln(gdp) + \beta_2 \ln(gdp) - \beta_2 \ln(pop) + \beta_3 \ln(pop) + \epsilon$

and hence

$y = \alpha + (\beta_1 + \beta_2) \ln(gdp) + (\beta_3- \beta_2) \ln(pop) + \epsilon$

$\ \ = \alpha + \hat\beta_1 \ln(gdp) + \hat\beta_2 \ln(pop) + \epsilon$

Here you see that only two parameters are identified $(\hat\beta_1, \hat\beta_2)$. Above, you attempt to estimate three parameters, though. This is a problem!

• Nothing is said in the question about which countries and/or periods are being included, but if the range of population or GDP is at all large, it is difficult to see that including GDP and population as they come makes sense, as any regression will just be dominated by its outliers. Hence on substantive grounds alone log transformation is indicated. That makes the point in this answer especially cogent. Apr 15 '14 at 0:11

Having all 3 in your model will introduce multicollinearity, which will prevent you from solving the system of equations necessary to produce the estimate coefficients of the least squares regression.

• I don't think so. The relationship between the three are deterministic but nonlinear. There is no way to obtain a variable as a linear combination of the other two Apr 14 '14 at 22:35

Regressing on GDP or log GDP is usually problematic because they are not stationary. In econometrics we usually regress on differences of log GDP. I hope you know the spurious regression concept.

If you insist on having levels, i.e. GDP, GDP/POP and POP, then the way to interpret your problem is the same as with the model y ~ x1 * x2 in R notation, where x1=GDP/POP and x2 = POP.

• Population isn't often constant either. Apr 15 '14 at 0:21
• @NickCox, GDP per capita may not be stationary either, if it's not deflated (real). Apr 15 '14 at 2:07
• We agree on all here. (I did study economics in high school and do read the newspaper!) Apr 15 '14 at 9:23
• Good point, but I think that the OP wants to use them in a cross-sectional setting. If not, stationarity is an obvious issue. Apr 16 '14 at 14:27
• @coffeinjunky, I didn't think this was cross sectional study, good point. Apr 17 '14 at 1:57