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When there is a perfect collinearity among more than two continuous variables, how do you deal with it and how are the regression results interpreted?

I have three independent variables which represent the percentage of different races within different cities, Say $x_1$ is the percentage of Hispanics, $x_2$ is the percentage of Blacks and $x_3$ is the percentage of Whites. Logically, $x_1 + x_2 + x_3 = 1$.

The dependent variable is collage attendance among the population in different cities. I have eliminated one of the independent variables ($x_1$) and estimated the following equation:

$\hat{y}=\beta_0 + \beta_1 X_2 + \beta_2 X_3 $

  1. Is this the best way to deal with this problem?

  2. How shall I interpret the coefficients of $\beta_1$ and $\beta_2$. If $\beta_1$ is equal to 3 for example, shall it be interpreted as: with 1 unit of increase in percentage of Blacks in the population of the city, the collage attendance increases by 3 units? Or shall I consider the omitted variable (X1: Hispanics percentage) as the base group and say that a unit of increase in Blacks percentage increases the collage attendance three times more than a unit of increase in Hispanics?

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The big problem is of course that you cannot interpret this as the effect of a change in one percentage while keeping the remainder constant: if the percentage in one group increases at least one of the remaining groups must decrease otherwise you would end up with more than 100%.

One solution that can help is to pick various racial compositions and compare the predicted outcomes of those. For example, you can take the racial compositions of two or more real cities and compare the predicted outcomes of those. Taking the compositions in real cities often helps the communication with the audience; It is easier to talk about comparing Berlin to Stuttgart (I am living in Germany), than comparing a city with $x_1\%$ Blacks, $y_1\%$ Hispanics and $z_1\%$ Whites with a city with $x_2\%$ Blacks, $y_2\%$ Hispanics and $z_2\%$ Whites.

You can also look at Chapter 12 of J. Aitchison (2003) The Statistical Analysis of Compositional Data The Blackburn Press.


On a different note: beware of the interpretation of such a regression using city level data. The results cannot tell you much about the behaviour of individuals. This is known as the ecological fallacy often attributed to:

W. S. Robinson (1950) "Ecological Correlations and the Behavior of Individuals" American Sociological Review, Vol. 15, No. 3, pp. 351-357.

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  • $\begingroup$ Thank you very much for your help; I have 100 cities in my dataset, so comparing each pair seems impossible. What if I only focus on one of the racial sets, say Blacks, then I will have an equation as: Y=beta0+beta1*X2; Can I then interpret beta1 as the increase in collage attendance by one unit of increase in black population percentage? $\endgroup$ Jun 11, 2014 at 19:38
  • $\begingroup$ I did not suggest you take all comparisons, I only suggest you take a few example cities to illustrate your results. $\endgroup$ Jun 12, 2014 at 7:54

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