6.3 Multicollinearity, Correlated Predictors & Aliasing
Key Takeaways
- Task B-2 names mitigating multicollinearity as an explicit PCPA skill, and the project rubric asks candidates to interpret multicollinearity and correlated variables.
- Moderate correlation among predictors is a GLM strength, not a defect: unlike a univariate analysis, the GLM apportions the effect and avoids double-counting.
- Severe correlation makes coefficients erratic and standard errors large, so small changes in the data swing the estimates wildly; such a model is called unstable.
- Multicollinearity is a linear combination of predictors predicting another, so it can hide from a pairwise correlation matrix; the variance inflation factor detects it, with VIF above 10 a common rule of thumb.
- Perfectly correlated predictors are aliased and the GLM has no unique solution; software usually drops one, but near-perfect correlation may slip through and produce nonsensical coefficients.
Correlation Is Usually the Point
Rating variables in property and casualty insurance are correlated by construction. Vehicle value correlates with driver age. Building age correlates with construction class. Territory correlates with almost everything.
That is precisely why GLMs replaced one-way analyses. A univariate territory indication double-counts whatever share of the territory signal actually belongs to vehicle type. A multivariate GLM apportions the response between correlated predictors and produces each variable's unique effect, holding the others constant.
So moderate correlation is not a problem to be removed. It is the reason to use the model. Understanding the correlation structure before fitting does two useful things: it prepares you for the fact that GLM relativities will differ from one-way indications, and it lets you explain why they differ — which is a rubric criterion.
When Correlation Turns Destructive
Problems begin when correlation between predictors is very high. The same information is entering the model twice, and the fitting procedure — forbidden from double-counting — must split the effect between the two variables. How to split it becomes a source of great uncertainty, with three visible symptoms:
- Erratic coefficients. Extremely large positive and negative values that offset each other. A pair of correlated predictors may show $+2.8$ and $-2.6$ where each alone would show $+0.15$.
- Inflated standard errors. The model tells you it is unsure, even though the pair jointly predicts well.
- Instability. Refit on a bootstrap sample or drop 5% of the rows and the coefficients swing.
A model with these symptoms is called unstable. Notice that predictive accuracy may look fine — it is the individual coefficients that are unreliable. For a rating plan, whose whole output is the individual coefficients, that is fatal.
Aliasing: Perfect Correlation
Where two predictors are perfectly correlated they are aliased, and the GLM has no unique solution: infinitely many coefficient pairs produce the identical fit. Most fitting software detects this and automatically drops one predictor, often reporting NA or a singularity warning.
Aliasing in insurance data is usually structural rather than accidental:
- A territory code and a ZIP-based group where each ZIP maps to exactly one territory.
- A vehicle symbol and a vehicle-value band derived from the symbol.
- A dummy variable set that includes every level plus an intercept — the classic dummy-variable trap.
- Two versions of a field created during a system conversion where one is a deterministic recode of the other.
The dangerous case is near-perfect correlation. Software may not catch it, the fitting procedure may fail to converge, and if it does converge the coefficients are nonsense. If a GLM fails to converge, correlation structure is the first thing to check.
Multicollinearity: The Hidden Case
Pairwise correlation is easy to find in a correlation matrix. Multicollinearity is the subtler problem where two or more predictors together strongly predict a third, even though no single pair is highly correlated. Because the effect is a linear combination rather than a pair, a correlation matrix will not reveal it.
The detection tool is the variance inflation factor. For predictor $j$, fit a linear model with $x_j$ as the target and every other predictor as inputs, obtain that model's $R_j^2$, and compute:
The VIF measures how much the squared standard error of predictor $j$ is inflated by collinearity with the other predictors. A reading of 4 means the variance of the coefficient is four times what it would be with orthogonal predictors, so the standard error is doubled.
| $R_j^2$ | VIF | Interpretation |
|---|---|---|
| 0.00 | 1.0 | No collinearity |
| 0.50 | 2.0 | Mild |
| 0.80 | 5.0 | Noticeable |
| 0.90 | 10.0 | Common rule-of-thumb threshold for "high" |
| 0.99 | 100.0 | Severe; coefficients unusable |
The rule of thumb is VIF > 10 is high, but it is a rule of thumb rather than a decision rule. Where large VIFs appear, the Monograph's guidance is to look deeper into the collinearity structure and make an informed decision, not to delete variables mechanically.
Remedies
| Remedy | What it does | Cost |
|---|---|---|
| Keep one of the correlated group | Simplest fix; removes duplicated information | Unique information in the dropped predictors is lost |
| Combine into a single engineered variable | E.g. one "vehicle risk" score in place of symbol and value | Needs a defensible construction |
| Group or re-band levels | Reduces near-duplication in categoricals | Loses granularity |
| Dimensionality reduction (PCA, factor analysis) | Produces uncorrelated components from a correlated group | Components are hard to explain in a rate filing |
| Shrinkage (ridge-type penalties) | Stabilises coefficients by accepting some bias | Coefficients are no longer unbiased; less transparent |
| Collect more data | Correlation that is a sampling artefact may dissolve | Not available inside a project window |
For the PCPA Project, the first three are usually the right answers: they are quick, explainable in a sentence, and directly support the rubric criterion that the candidate describes why a variable was or was not included.
[!WARNING] Do not diagnose multicollinearity from a correlation matrix alone. A variable can have a pairwise correlation below 0.3 with every other predictor and still have a VIF above 20, because it is nearly a linear combination of three of them. Run VIFs on the final candidate predictor set, not just a correlation heat map.
A GLM includes vehicle symbol and vehicle value, which correlate at 0.97. The fitted coefficients are +3.1 and -2.9 with very large standard errors. What has happened, and what is the right response?
A predictor's pairwise correlation with every other predictor is below 0.35, yet its variance inflation factor is 14. What does this indicate?
A GLM fails to converge after a territory code and a ZIP-group variable are both included, where each ZIP group maps to exactly one territory. What is the underlying problem?