Chapter 9 Why Correlation Fails
Chapters 7 and 8 developed descriptive statistics and distribution estimation using directional partial moments.
Those results showed that many classical statistical quantities arise from aggregations of directional deviations relative to benchmarks.
The next topic is dependence between variables.
Classical statistics measures dependence primarily through covariance and correlation.
These statistics summarize relationships between variables with a single number.
However, these measures possess fundamental limitations. They
- measure only linear association,
- aggregate directional information symmetrically,
- and can obscure nonlinear, asymmetric, or tail-specific relationships.
Directional statistics provides a deeper perspective.
Just as classical moments arise from aggregations of partial moments, covariance and correlation arise from aggregations of directional co-partial moments.
This chapter explains why correlation can fail and establishes the connection between covariance and directional partial-moment matrices.
9.1 Classical Dependence Measures
For two random variables \(X\) and \(Y\), the covariance is
\[ Cov(X,Y)=E[(X-\mu_X)(Y-\mu_Y)]. \]
Covariance measures the joint variation of the two variables relative to their means.
The Pearson correlation coefficient standardizes covariance:
\[ \rho(X,Y)= \frac{Cov(X,Y)}{\sigma_X\sigma_Y}. \]
The statistic lies in the interval
\[ -1 \le \rho(X,Y) \le 1. \]
Values near
- \(1\) indicate strong positive linear association,
- \(-1\) indicate strong negative linear association,
- \(0\) indicate no linear association.
The key limitation is that correlation measures only linear relationships.
9.2 Directional Co-Partial Moments
Directional statistics partitions the joint distribution relative to benchmark values \(t_X\) and \(t_Y\).
Four regions arise:
\[ X \le t_X,\; Y \le t_Y \]
\[ X \le t_X,\; Y > t_Y \]
\[ X > t_X,\; Y \le t_Y \]
\[ X > t_X,\; Y > t_Y \]
These regions correspond to combinations of directional deviations.
Benchmarks may be chosen in different ways depending on the application.
- External benchmarks, such as policy targets or required returns.
- Internal benchmarks, such as sample means.
The covariance decomposition developed in Section 9.3 uses the means:
\[ t_X = \mu_X, \quad t_Y = \mu_Y. \]
Define the positive-part operator
\[ (x)^+ = \max(x,0). \]
The directional co-partial moments of order \(r,s\) are:
9.3 Covariance from Co-Partial Moments
Covariance can be expressed directly in terms of directional co-partial moments.
Let the benchmarks equal the means:
\[ t_X=\mu_X, \quad t_Y=\mu_Y. \]
From Chapter 2,
\[ x = x^+ - (-x)^+. \]
Applying this to deviations:
\[ X-\mu_X = (X-\mu_X)^+ - (\mu_X-X)^+ \]
\[ Y-\mu_Y = (Y-\mu_Y)^+ - (\mu_Y-Y)^+ \]
Define
\[ A=(X-\mu_X)^+,\quad B=(\mu_X-X)^+ \]
\[ C=(Y-\mu_Y)^+,\quad D=(\mu_Y-Y)^+ \]
Then
\[ (X-\mu_X)(Y-\mu_Y)=(A-B)(C-D). \]
Expanding gives
\[ (A-B)(C-D)=AC + BD - AD - BC. \]
Substituting the definitions yields
\[ (X-\mu_X)(Y-\mu_Y) = (X-\mu_X)^+(Y-\mu_Y)^+ + (\mu_X-X)^+(\mu_Y-Y)^+ - (X-\mu_X)^+(\mu_Y-Y)^+ - (\mu_X-X)^+(Y-\mu_Y)^+. \]
Taking expectations gives
\[ Cov(X,Y) = CoUPM_{1,1}(X,Y) + CoLPM_{1,1}(X,Y) - DLPM_{1,1}(X,Y) - DUPM_{1,1}(X,Y). \]
Thus covariance is simply the signed aggregation of four directional co-partial moments.
This mirrors the earlier variance decomposition
\[ Var(X)=U_2(\mu;X)+L_2(\mu;X). \]
9.4 Covariance Matrices from Partial-Moment Matrices
For a system of \(n\) variables, directional co-partial moments form matrices.
Define
\[ CLPM_{ij}=CoLPM_{1,1}(X_i,X_j) \]
\[ CUPM_{ij}=CoUPM_{1,1}(X_i,X_j) \]
\[ DLPM_{ij}=DLPM_{1,1}(X_i,X_j) \]
\[ DUPM_{ij}=DUPM_{1,1}(X_i,X_j) \]
Each matrix captures directional co-movement across the variables.
The classical covariance matrix can be written as
\[ \Sigma = CLPM + CUPM - DLPM - DUPM. \]
The diagonal elements satisfy
\[ \Sigma_{ii}=Var(X_i). \]
This follows because when \(i=j\), the divergent partial moments \(DLPM\) and \(DUPM\) vanish, leaving only the lower and upper components.
The expression therefore reduces to the variance decomposition derived earlier:
\[ Var(X_i)=U_2(\mu_i;X_i)+L_2(\mu_i;X_i). \]
Like their univariate counterparts, these directional matrices can be estimated empirically from sample data using sample co-partial moments.
library(NNS)
set.seed(123)
x <- rnorm(100)
y <- rnorm(100)
cov.mtx <- PM.matrix(LPM_degree = 1, UPM_degree = 1, target = 'mean', variable = cbind(x, y), pop_adj = TRUE)
cov.mtx
## $cupm
## x y
## x 0.4299250 0.1033601
## y 0.1033601 0.5411626
##
## $dupm
## x y
## x 0.0000000 0.1469182
## y 0.1560924 0.0000000
##
## $dlpm
## x y
## x 0.0000000 0.1560924
## y 0.1469182 0.0000000
##
## $clpm
## x y
## x 0.4033078 0.1559295
## y 0.1559295 0.3939005
##
## $cov.matrix
## x y
## x 0.83323283 -0.04372107
## y -0.04372107 0.93506310
# Reassembled Covariance Matrix
cov.mtx$clpm + cov.mtx$cupm - cov.mtx$dlpm - cov.mtx$dupm
## x y
## x 0.83323283 -0.04372107
## y -0.04372107 0.93506310
# Standard Covariance Matrix
cov(cbind(x, y))
## x y
## x 0.83323283 -0.04372107
## y -0.04372107 0.935063109.5 Correlation as a Normalized Covariance
The correlation matrix is obtained by standardizing covariance:
\[ \rho_{ij}= \frac{\Sigma_{ij}} {\sqrt{\Sigma_{ii}\Sigma_{jj}}}. \]
Since covariance itself is derived from directional matrices, correlation represents a further aggregation.
The information hierarchy becomes
\[ (CLPM,CUPM,DLPM,DUPM) \rightarrow \Sigma \rightarrow \rho. \]
Directional matrices therefore preserve more structural information about dependence than correlation alone.
9.6 Nonlinear Dependence
Correlation measures linear association and therefore fails when relationships are nonlinear.
Consider
\[ Y=X^2 \]
with \(X\) symmetrically distributed around zero.
In this case
\[ Corr(X,Y)=0. \]
For example, if \(X\sim N(0,1)\),
\[ Corr(X,X^2)=0. \]
Despite zero correlation, the variables are perfectly dependent.
Directional co-partial moments reveal this structure.
With benchmark \(t_X=0\),
- \(CoUPM\) captures strong dependence when \(X>0\)
- \(DLPM\) captures the mirrored dependence when \(X<0\)
The directional matrices therefore expose strong dependence that the aggregated covariance cancels.
9.7 Asymmetric Dependence
Asymmetric dependence refers to dependence that differs between the upper and lower regions of the joint distribution.
Examples include
- financial assets that move together primarily during crashes
- economic variables responding differently to positive and negative shocks
- risk exposures concentrated in losses
Directional matrices isolate these effects directly.
For example,
\[ CLPM \]
captures joint downside deviations, while
\[ CUPM \]
captures joint upside co-movement.
If dependence is concentrated in one region, the directional matrices reveal it even when overall covariance appears modest.
9.8 Tail Dependence
Extreme events often drive the most consequential relationships.
Correlation averages dependence across the entire distribution and therefore may understate tail relationships.
Directional co-partial moments of higher order emphasize extreme deviations:
\[ CoLPM_{r,s}, \quad CoUPM_{r,s}. \]
Increasing \(r\) and \(s\) increases sensitivity to extreme observations.
This concept is closely related to tail dependence in copula theory, which Chapter 10 examines in detail.
9.9 Information Loss in Aggregation
The mapping from directional matrices to covariance is many-to-one.
Different directional dependence structures can produce identical covariance values.
Similarly, many covariance matrices produce identical correlation matrices after normalization.
Thus correlation discards substantial structural information about joint distributions.
Directional methods preserve this information by retaining contributions from each directional region separately.
9.10 Summary
This chapter examined the limitations of classical correlation and covariance.
Key observations include:
- Correlation measures only linear association.
- Covariance aggregates directional co-deviations across the joint distribution.
- Covariance itself arises from directional co-partial moments.
- The covariance matrix equals the aggregation of directional partial-moment matrices.
- Correlation is the normalized version of this aggregate.
Directional statistics therefore provides a richer representation of dependence structure.
The following chapter develops directional dependence measures built from directional co-partial moments.