Why does my graph (residuals vs fitted values) look like this?

Question

My initial regression was # of newly constructed dwellings on avg per-capita personal income, unemployment %, GDP, # of current housing units, and population; all on the US county-level for 2022. I log-transformed all the variables except unemployment %. I then removed # of current housing units and population from the regression due to high multicollinearity with GDP. But there is an endogeneity issue with the income variable which is simultaneity (higher per-capita income may increase construction rates but higher construction rates may in turn create opportunities that increases income). So my plan turned to running an IV regression with educational attainment (measured by % of bachelor degree holding adults). I ran the regression and created this plot only to find it weirdly shaped. I then created many more graphs for random regressions and they all look like this. The only exception is when I regress construction rates on education and then plot the residuals against education itself.

My Questions:

Why do all my plots look like this (and not evenly scattered)? Is it because of further endogeneity issues between GDP, unemployment, income, etc?

Are there any other problems with the way I've gone about creating my model/running the regression?

Answer 1

Why do all my plots look like this (and not evenly scattered)? Is it because of further endogeneity issues between GDP, unemployment, income, etc

There will likely bee some endogeneity between your dependent variable and independent variables. For example, higher dwelling density might lead to more GDP growth by enabling more people to move in, and its well known that higher population density is related to (at least short run) economic growth.

However, endogeneity generally cannot be spot from the residual plot (see this great explanation) so I would not attribute that pattern to endogeneity. Generally, there are no good pure statistical approaches to detecting endogeneity, you need to use theory and reasoning when it comes to endogeneity.

What you can discover from residual plot are things like heteroskedasticity, autocorrelation or model misspecification (e.g. trying to fit linear model to non-linear relationship).

• Heteroskedasticity would show in not even spread of residuals (but this does not seem to be your problem).

• Autocorrelation shows in low residuals following other low residuals and high residuals other high residuals. This is typically problem in time series, but often students are not taught that spatial autocorrelation also exists. In your case I think based on the visuals there is autocorrelation present (although I always recommend people to run tests instead of eyeballing it).

• You can spot misspecification by seeing some relationship in your residual plot. This is clearly a problem in your case. In your case it does look like you are missing some important regressor.

I am also bit puzzled by the 'line' formed by residuals under the block of other residuals. It could signal that you have some truncated or possibly censored data. Its something I would recommend looking into.