Varyans Enflasyon Faktörü (VIF) ve Çoklu Doğrusal Bağlantı

What is multicollinearity?

Multicollinearity means that one of the explanatory variables in a regression can be explained linearly by the others. The extreme example is putting the same length into the model in both centimetres and metres: one column is an exact multiple of the other. In practice you meet softer versions far more often — income and spending, visitor count and session count, store size and headcount.

As collinearity rises, the model struggles to separate each variable's individual contribution to the outcome. Coefficient standard errors grow, confidence intervals widen, p-values rise, and the coefficients move around when the sample changes slightly — they can even flip sign.

What does VIF measure?

The logic of VIF is surprisingly simple. For each explanatory variable, you fit an auxiliary regression that predicts it from the other explanatory variables. Let the coefficient of determination of that auxiliary model be R²_j. Then:

What the formula says: the better variable j is explained by the others, the larger VIF gets. R² = 0 gives VIF = 1, meaning no overlap at all. R² = 0.9 gives VIF = 10.

The "variance inflation" in the name is literal: the variance of the coefficient estimate is VIF times what it would be with no collinearity. Because the standard error is the square root of the variance, the increase in the standard error is roughly √VIF. That is the number you will actually use.

Making the values concrete

Read the table from the "standard error factor" column. VIF = 10 means that coefficient's confidence interval is about three times wider than it would be without collinearity. A coefficient that comes out non-significant may be so not because it has no effect, but simply because you could not measure it.

Why isn't a correlation matrix enough?

The first method that comes to mind is to look at the correlation matrix, and that is useful — but incomplete. Pearson correlation measures only the pairwise relationship. A variable can show no strong correlation with any single other variable and still be explained almost perfectly by a combination of several.

The classic example: the model contains "total revenue", "product A revenue" and "product B revenue" together. Because total = A + B there is an exact linear dependence, yet the pairwise correlation between total and A might be an innocent-looking 0.7. The correlation matrix misses this; VIF does not.

On the other hand, VIF does not tell you everything either: not the direction of the relationship, not whether two variables are conceptually interchangeable, and not the model's predictive quality. Read it alongside the correlation table, scatter plots and domain knowledge.

Computing VIF in Python

statsmodels wants the design matrix and the index of the column to compute. If you use the formula interface, the intercept is part of that matrix; do not interpret the intercept's VIF — it only reflects the centring of the means.

import pandas as pd
import statsmodels.formula.api as smf
from statsmodels.stats.outliers_influence import variance_inflation_factor

model = smf.ols(
    "sales ~ adspend + price + visitors + web_sessions",
    data=df
).fit()

X = model.model.exog
names = model.model.exog_names

vif = pd.DataFrame({
    "variable": names,
    "VIF": [variance_inflation_factor(X, i)
            for i in range(X.shape[1])],
})

print(vif[vif["variable"] != "Intercept"].round(2))

Be careful with categorical variables entered as dummies: you get a separate VIF for each dummy column, and those are naturally linked to one another. A high VIF on a many-level categorical variable is not by itself a warning sign; a generalised VIF (GVIF) for the variable group is the more appropriate measure there.

Reading a table step by step

Suppose a sales model produces this VIF table:

Let us work web sessions out: with VIF = 9.1, the auxiliary R² is 1 − 1/9.1 = 0.89. The other explanatory variables explain roughly 89% of the variation in web sessions linearly. Its standard error is about √9.1 ≈ 3 times wider than it would be without collinearity.

The right conclusion is not "drop web sessions from the model". It is this: visitor count and web sessions may be measuring the same phenomenon two different ways, so interpreting those two coefficients separately is not reliable. The decision comes from domain knowledge, not from statistics — do these two metrics measure different things in business terms?

Thresholds are warnings, not rules

Thresholds of 5 and 10 appear often in the literature, but they are not statistically derived boundaries; they are rough signals drawn from practice. Applying one mechanically produces errors in both directions.

A low VIF does not show the model is correctly specified — omitted variables, the wrong functional form and endogeneity never show up in VIF at all. A high VIF does not make a model unusable either. The real question is: what am I going to use this coefficient for?

What to do when VIF is high

1. Look for a structural mistake first. Was the same variable added twice? Are a total and its components both in the model? Are all dummy levels present (the dummy-variable trap)? Errors like these inflate VIF artificially and are the easiest to fix.
2. Look at the domain meaning. Do the two variables measure the same phenomenon, or do they represent different mechanisms? "Visitor count" and "web sessions" are near-duplicates; "ad spend" and "store size" are not.
3. Make a reasoned choice. If one of two similar indicators stands out on measurement quality, data availability or fit with the business question, keep it and write the decision down.
4. Consider a summary measure or regularisation. Several similar indicators can be turned into a clearly defined index. If the goal is prediction, ridge regression shrinks correlated coefficients and buys stability — though coefficient interpretation then differs from OLS.
5. Report the effect. Show coefficients, standard errors and predictive performance before and after the change, side by side.

One more note: for interaction and polynomial terms, centring (subtracting the mean) can reduce artificial numerical collinearity. It does not remove a real overlap in information, only the part that comes from scale.

The limits of VIF

Checklist

• Is there a structural duplicate or a total-and-components clash in the design matrix?
• Was VIF computed for every explanatory column, with the intercept excluded?
• Were high values read in terms of auxiliary R² and √VIF?
• Were the correlation matrix and scatter plots examined too?
• Were coefficient standard errors and confidence intervals reviewed?
• Was any decision to drop a variable justified with domain knowledge?
• Is it clear whether the goal is prediction or inference?
• Was the before/after effect on the coefficients reported?

Conclusion

VIF is a diagnostic that shows how reliably a coefficient can be measured — not an exam to be passed. Rather than comparing the number to a threshold, translate it through √VIF into "how many times wider did this coefficient's confidence interval get?". That is when the number starts to mean something.

And do not lose sight of the real question: are you building this model to predict, or to interpret the effect of individual variables? The answer determines whether a high VIF is a serious problem or a footnote.