# Why does this regression NOT fail due to perfect multicollinearity, although one variable is a linear combination of others?

Today, I was playing around with a small dataset and performed a simple OLS regression which I expected to fail due to perfect multicollinearity. However, it didn’t. This implies that my understanding of multicollinearity is wrong.

My question is: Where am 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
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 regression not fail, 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")]
``````

`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`