Correlation Range with Four Variables
You have four random variables $a$, $b$, $c$, and $d$. Every pairwise correlation equals $\frac{1}{2}$, except for $\text{Corr}(a, d)$, which is unknown. Call it $\rho$.
What is the range of valid values for $\rho$? In other words, for which values of $\rho$ does a valid joint distribution exist?
Hints
- A set of correlations is realizable if and only if the corresponding correlation matrix is positive semi-definite. What does that require of its determinant and principal minors?
- Write out the $4 \times 4$ correlation matrix with $\rho$ in the $(1,4)$ and $(4,1)$ positions and $1/2$ everywhere else off-diagonal. The key constraint will come from $\det(R) \geq 0$.
- Expand the $4 \times 4$ determinant to get $\det(R) = \frac{1}{4} + \frac{\rho}{2} - \frac{3\rho^2}{4}$. Factor the resulting quadratic inequality.
Worked Solution
How to Think About It: The constraint is that the $4 \times 4$ correlation matrix must be positive semi-definite (PSD). That is the necessary and sufficient condition for a set of correlations to be jointly realizable. The matrix has ones on the diagonal, $\rho$ in the $(1,4)$ and $(4,1)$ entries, and $1/2$ everywhere else off-diagonal. We need to find the range of $\rho$ that keeps all eigenvalues non-negative, which is equivalent to requiring all leading principal minors (or equivalently the determinant and all sub-determinants) to be non-negative.
Quick Estimate: With three variables all at correlation $1/2$, the "average direction" picture is: each variable makes the same angle with the others. Adding a fourth variable $a$ that also correlates at $1/2$ with $b$ and $c$, the correlation with $d$ is constrained by geometry. The unconstrained range is $[-1, 1]$, but the existing network of $1/2$ correlations forces the lower bound upward. By symmetry the upper bound should still be $1$ (all four variables identical up to scaling). The lower bound will be somewhere around $-1/4$ to $-1/2$ -- let us compute.
Approach: Write out the full $4 \times 4$ correlation matrix and enforce $\det(R) \geq 0$ along with all principal minor conditions.
Formal Solution:
The correlation matrix is:
$$R = \begin{pmatrix} 1 & 1/2 & 1/2 & \rho \\ 1/2 & 1 & 1/2 & 1/2 \\ 1/2 & 1/2 & 1 & 1/2 \\ \rho & 1/2 & 1/2 & 1 \end{pmatrix}$$
Step 1 -- Check sub-matrices. All $1 \times 1$ minors are $1 > 0$. All $2 \times 2$ minors involving only the known correlations give $1 - 1/4 = 3/4 > 0$. The $2 \times 2$ minor for $(a, d)$ gives $1 - \rho^2 \geq 0$, so $\rho \in [-1, 1]$.
The $3 \times 3$ sub-matrix for $(b, c, d)$ has all off-diagonal entries $1/2$, with determinant:
$$\det = 1 - 3 \cdot \frac{1}{4} + 2 \cdot \frac{1}{8} = \frac{1}{2} > 0$$
The $3 \times 3$ sub-matrix for $(a, b, d)$ is:
$$M = \begin{pmatrix} 1 & 1/2 & \rho \\ 1/2 & 1 & 1/2 \\ \rho & 1/2 & 1 \end{pmatrix}$$
$$\det(M) = \frac{1}{2} + \frac{\rho}{2} - \rho^2$$
Setting $\det(M) \geq 0$: the roots of $\rho^2 - \rho/2 - 1/2 = 0$ are $\rho = 1$ and $\rho = -1/2$, giving $\rho \in [-1/2, 1]$. By the same calculation, the $(a, c, d)$ minor gives the same constraint.
Step 2 -- The $4 \times 4$ determinant. Expanding by cofactors along the first row:
$$\det(R) = \frac{1}{4} + \frac{\rho}{2} - \frac{3\rho^2}{4}$$
Setting $\det(R) \geq 0$ and multiplying through by $-4$:
$$3\rho^2 - 2\rho - 1 \leq 0$$
Factoring:
$$(3\rho + 1)(\rho - 1) \leq 0$$
This gives $\rho \in [-1/3, 1]$.
Step 3 -- Combine all constraints. The binding constraint comes from the $4 \times 4$ determinant: $\rho \in [-1/3, 1]$. The $3 \times 3$ constraint $\rho \geq -1/2$ is weaker and automatically satisfied.
Verification: At $\rho = -1/3$, we get $\det(R) = 1/4 - 1/6 - 1/12 = 0$, confirming the matrix is PSD (singular) at the boundary. At $\rho = 1$, all four variables are perfectly correlated, which is trivially valid.
Answer: The correlation between $a$ and $d$ must satisfy $\rho \in \left[-\dfrac{1}{3},\, 1\right]$.
Intuition
The core idea is that correlations are not free parameters you can set independently -- they are constrained by geometry. Each random variable can be represented as a unit vector in some inner product space, and the correlation between two variables is the cosine of the angle between their vectors. Once you fix the angles between most pairs of vectors, the remaining angles are constrained by the geometry of the space. In four dimensions with five of six pairwise angles fixed, the sixth has very little room to move.
This shows up constantly in portfolio construction and risk modeling. If you know how most of your assets correlate with each other, the remaining correlations are already partially determined. A risk manager who plugs in an "unreasonable" correlation into a matrix (one that violates the PSD condition) will get nonsensical results -- negative variances for some portfolios, for instance. The PSD check via determinants and eigenvalues is the standard tool for catching these errors. In practice, estimated correlation matrices from noisy data often fail to be PSD, which is why techniques like nearest-PSD-matrix projection (Higham's algorithm) or shrinkage estimators are essential in production systems.