Can a covariance matrix be recalculated for dummy variables?

Question

Say I have the following sample from a continuous variable $X$ and a categorical (dichotomous) variable $Y$:

  X Y
0.5 1
2.3 2
2.2 2
1.8 1

Moreover, the covariance matrix between $X$ and $Y$ is also known:

$$ \Sigma_{XY} = \begin{pmatrix} \sigma_{XX} & \sigma_{XY} \\ \sigma_{YX} & \sigma_{YY} \end{pmatrix} = \begin{pmatrix} 0.5 & 1.2 \\ 1.2 & 0.4 \end{pmatrix} $$

Now let's say we have dummy-coded $Y$, so our data looks like this:

  X Y1 Y2
0.5  1  0
2.3  0  1
2.2  0  1
1.8  1  0

My question is: how can we calculate the covariance matrix between $X$, $Y_1$ and $Y_2$?

$$ \Sigma_{XY_1Y_2} = \begin{pmatrix} \sigma_{XX} & \sigma_{XY_1} & \sigma_{XY_2} \\ \sigma_{Y_1X} & \sigma_{Y_1Y_1} & \sigma_{Y_1Y_2} \\ \sigma_{Y_2X} & \sigma_{Y_2Y_1} & \sigma_{Y_2Y_2} \end{pmatrix} = \begin{pmatrix} 0.5 & ? & ? \\ ? & ? & ? \\ ? & ? & ? \end{pmatrix} $$

In my actual problem, I can often bootstrap my way into getting a fair estimate for this matrix, but it is very important for me to find the analytical solution as well. For what it's worth, the proportion of each category in the population is also known.

Answer 1

We've got:

$$Y_1 = 1 - Y_2 \;\;(1)$$ $$Y = Y_2 + 1 \;\;(2)$$ $$Y = 2 - Y_1 \;\;(3)$$

So, using equation $(1)$: $$\sigma_{Y_1Y_1} = Var(Y_1) = 0 + (-1)^2 Var(Y_2) = \sigma_{Y_2Y_2}$$ $$\sigma_{Y_1Y_2} = \sigma_{Y_2Y_1} = Cov(Y_1, Y_2) = -Cov(Y_1, Y_1) = -\sigma_{Y_1Y_1}$$

... and using equations (2, 3):

$$Var(Y) = 0.4 = Var(Y_2) = Var(Y_1)$$

As for the other two $\sigma$s, assuming that the covariance $\sigma_{XY}$ was calculated using their numerical values :

$$\sigma_{XY} = Cov(X, Y) = E((X - E(X))(Y - E(Y))) = 1.2$$ $$= E((X-E(X))(Y_2 + 1 - 1 - E(Y_2))) = \sigma_{XY_2}$$

Similarly:

$$\sigma_{XY} = E((X-E(X))(2 - Y_1 - 2 + E(Y_1))) = -\sigma_{XY_1}$$

So the final answer is:

$$\Sigma_{XY} = \begin{pmatrix} 0.5 & -1.2 & 1.2 \\ -1.2 & 0.4 & -0.4 \\ 1.2 & -0.4 & 0.4 \end{pmatrix}$$

... and that should finish the answer if I've not made any errors!