Today, I was playing around with a small dataset and performed a simple OLS regression which I

expectedto fail due to perfect multicollinearity. However, it didn’t. This implies that my understanding of multicollinearity is wrong.My question is:

Wheream I wrong?

I think that I can show that one of my variables is a linear combination of the others. This should lead to a regressor matrix that has no full rank and thus the coefficients should not be identified.

I generated a small reproducible dataset

(code below):`exporter importer flow dist intraUS 1 Canada Canada 996.8677 6.367287 0 2 Florida Canada 995.8219 9.190562 0 3 Texas Canada 1001.6475 4.359063 0 4 Mexico Canada 1002.4371 7.476649 0 5 Canada Florida 1002.8789 5.389223 0 6 Florida Florida 1007.5589 6.779686 1 7 Texas Florida 996.8938 1.570600 1 8 Mexico Florida 1005.6247 5.910133 0 9 Canada Texas 999.9190 7.887672 0 10 Florida Texas 1004.1061 7.187803 1 11 Texas Texas 1004.5949 7.564273 1 12 Mexico Texas 1000.3728 2.021297 0 13 Canada Mexico 1003.0991 5.887743 0 14 Florida Mexico 999.2210 3.058495 0 15 Texas Mexico 997.6092 6.835883 0 16 Mexico Mexico 1006.7934 5.794425 0`

Each time exporter and importer are US states, the dummy

`intraUS`

is`1`

.Now I perform a regression of (trade)

`flow`

s on`exporter`

and`importer`

dummies,`dist`

ance and the`intraUS`

dummy. Feeding R with the following formula`lm(flow ~ dist + exporter + importer + intraUS, data = dat)`

returns estimates for all coefficients, no missing values or warnings about singularity:`(Intercept) dist exporterFlorida exporterTexas exporterMexico importerFlorida importerTexas importerMexico intraUS1 995.1033157 0.5744661 -1.2340338 -1.8792073 3.7375783 3.0361727 1.3256032 3.3225512 4.2429599`

This puzzles me, because the regressor matrix clearly indicates that

`intraUS`

is a linear combination of`exporterFlorida`

,`importerFlorida`

,`exporterTexas`

and`importerTexas`

:`> mmat <- data.frame(model.matrix(lm(flow ~ dist + exporter + importer + intraUS, data = dat))) X.Intercept. dist exporterFlorida exporterTexas exporterMexico importerFlorida importerTexas importerMexico intraUS1 1 1 6.367287 0 0 0 0 0 0 0 2 1 9.190562 1 0 0 0 0 0 0 3 1 4.359063 0 1 0 0 0 0 0 4 1 7.476649 0 0 1 0 0 0 0 5 1 5.389223 0 0 0 1 0 0 0 6 1 6.779686 1 0 0 1 0 0 1 7 1 1.570600 0 1 0 1 0 0 1 8 1 5.910133 0 0 1 1 0 0 0 9 1 7.887672 0 0 0 0 1 0 0 10 1 7.187803 1 0 0 0 1 0 1 11 1 7.564273 0 1 0 0 1 0 1 12 1 2.021297 0 0 1 0 1 0 0 13 1 5.887743 0 0 0 0 0 1 0 14 1 3.058495 1 0 0 0 0 1 0 15 1 6.835883 0 1 0 0 0 1 0 16 1 5.794425 0 0 1 0 0 1 0`

Calculating

`exporterFlorida * importerFlorida + exporterFlorida * importerTexas + exporterTexas * importerFlorida + exporterTexas * importerTexas`

gives – not surprisingly – exactly the values in`intraUS1`

.

So my question is, again: Why does this regressionnotfail, given that one variable is a linear combination of the others?

Below the complete code the reproduce the estimation:

`## Generate data #### set.seed(1) states <- c("Canada", "Florida", "Texas", "Mexico") dat <- expand.grid(states, states) colnames(dat) <- c("exporter", "importer") dat[, "flow"] <- NA dat[, "dist"] <- NA dat[, "intraUS"] <- 0 for (i in 1:nrow(dat)) { dat[i, c("flow", "dist")] <- c(rnorm(1, mean = 1000, sd = 5), rnorm(1, mean = 6, sd = 2)) if (dat[i, "exporter"] %in% states[2:3] && dat[i, "importer"] %in% states[2:3]) { dat[i, "intraUS"] <- 1 } } dat$intraUS <- factor(dat$intraUS) ## Run regression - works! #### summary(lm(flow ~ dist + exporter + importer + intraUS, data = dat)) ## Show that "intraUS1" is a linear combination of the dummies. #### mmat <- data.frame(model.matrix(lm(flow ~ dist + exporter + importer + intraUS, data = dat))) cbind(mmat, test = with(mmat, exporterFlorida * importerFlorida + exporterFlorida * importerTexas + exporterTexas * importerFlorida + exporterTexas * importerTexas ))[, c("intraUS1", "test")]`

**Answer**

`exporterFlorida * importerFlorida + exporterFlorida * importerTexas + exporterTexas * importerFlorida + exporterTexas * importerTexas`

This is *not* a linear combination of `exporterFlorida`

, `importerFlorida`

, `importerTexas`

and `exporterTexas`

. In a linear combination, the coefficients of the vectors must be *constants*. So something like

`2*importerFlorida + 3*importerTexas - exporterFlorida - 2*exporterTexas`

*is* a linear combination.

What you have could possibly be called a quadratic combination, but that’s stretching terminology into “I’m making stuff up” land.

**Attribution***Source : Link , Question Author : CL. , Answer Author : Matthew Drury*