statsmodels.stats.multicollinearity.vif#

statsmodels.stats.multicollinearity.vif(data, *, is_moment=False, has_constant='raise')[source]#

Variance inflation factors of all variables

The variance inflation factor, VIF, of a variable is \(1 / (1 - R^2)\), where \(R^2\) is the coefficient of determination of the regression of the variable on all other variables and a constant. It measures by how much the variance of the estimate of the coefficient of the variable is inflated by the correlation with the other variables.

Parameters:

dataarray_like: The explanatory variables with observations in rows and variables in columns. The data should not include a constant, see has_constant. If is_moment is True, then this is a square covariance or correlation matrix of the variables.
is_momentbool, optional: If True, then data is a symmetric, positive semi-definite moment matrix of the variables instead of the observations. Only the correlations are used, so the matrix can be a covariance or a correlation matrix. The default is False.
has_constant{“raise”, “skip”}, optional: How to treat constant columns, which have zero variance. If “raise”, the default, then a ValueError is raised. If “skip”, then the constant columns are ignored and the corresponding elements of the result are NaN. The other variables are not affected because the regressions always include a constant.

Returns:

ndarray or Series: The variance inflation factors, one for each column of data. This is a pandas Series with the column labels as index if data is a pandas DataFrame. The VIF of a variable that is a linear combination of other variables is infinite.

Raises:

ValueError: If data contains missing or infinite values, if data has constant columns and has_constant is “raise”, or if is_moment is True and data is not a valid moment matrix.

See also

statsmodels.stats.outliers_influence.variance_inflation_factor: VIF of a single variable, estimated by OLS.
vif_ridge: VIF of the parameter estimates of ridge regression.
collinear_index: Columns that are perfectly collinear with preceding columns.

Notes

The VIF of all variables are the diagonal elements of the inverse of the correlation matrix of the variables. This needs one matrix decomposition instead of one regression for each variable, which is much faster than calling variance_inflation_factor for each column. Up to numerical precision, the results are the same as those of that function with its default standardize=True. The latter limits the VIF to \(10^{15}\) and warns if the design matrix is ill-conditioned, while this function returns inf for variables that are collinear with other variables.

Constants

The VIF requires a constant in the regressions. The variables are demeaned, which is equivalent to including a constant, and the constant column of a design matrix, for example res.model.exog, has to be removed or skipped. A set of dummy variables that sums to one is an implicit constant. If the design matrix does not include another constant, then it has to be dropped, because the dummy variables are perfectly collinear after demeaning and their VIF is infinite.

If is_moment is True and the moment matrix is not centered, for example the cross-product matrix x.T @ x, then the VIF are those of regressions without a constant.

Singular and nearly singular data

The computation uses the eigenvalues of the correlation matrix. The VIF is infinite for variables that are numerically perfectly collinear with other variables, i.e. if the correlation matrix is singular up to floating point precision. A variable that is not part of such a linear relationship has a finite VIF. VIFs above about \(10^{12}\) should be read as indicating that the variable is collinear because they are affected by floating point noise. Floating point precision of the data matters: a linear combination of single precision (float32) variables is collinear only up to the precision of the single precision data, and is not detected as perfectly collinear.

References

[1]

Belsley, D. A., E. Kuh, and R. E. Welsch. 1980. Regression Diagnostics: Identifying Influential Data and Sources of Collinearity. Wiley.

Examples

The VIF of the explanatory variables of the Longley data, which are highly collinear.

>>> import statsmodels.api as sm
>>> from statsmodels.stats.api import vif
>>> exog = sm.datasets.longley.load_pandas().exog
>>> vif(exog).round(1)
GNPDEFL     135.5
GNP        1788.5
UNEMP        33.6
ARMED         3.6
POP         399.2
YEAR        759.0
dtype: float64

The design matrix of a model that is estimated with a formula has a constant, which is skipped.

>>> import statsmodels.formula.api as smf
>>> mod = smf.ols("TOTEMP ~ GNPDEFL + GNP + UNEMP + ARMED + POP + YEAR",
...               data=sm.datasets.longley.load_pandas().data)
>>> vif(mod.data.orig_exog, has_constant="skip").round(1)
Intercept       NaN
GNPDEFL       135.5
GNP          1788.5
UNEMP          33.6
ARMED           3.6
POP           399.2
YEAR          759.0
dtype: float64