rvif (version 3.0)

multicollinearity: Decision Rule to Detect Troubling Multicollinearity

Description

Given a multiple linear regression model with n observations and k independent variables, the degree of near-multicollinearity affects its statistical analysis (with a level of significance of alpha%) if there is a variable i, with i = 1,...,k, that verifies that the null hypothesis is not rejected in the original model and is rejected in the orthogonal model of reference.

Usage

multicollinearity(y, x, alpha = 0.05)

Value

The function returns the value of the RVIF and the established thresholds, as well as indicating whether or not the individual significance analysis is affected by multicollinearity at the chosen significance level.

Arguments

y

A numerical vector representing the dependent variable of the model.

x

A numerical design matrix that should contain more than one regressor (intercept included in the first column).

alpha

Significance level (by default, 5%).

Author

Román Salmerón Gómez (University of Granada) and Catalina B. García García (University of Granada).

Maintainer: Román Salmerón Gómez ([e-mail])

Details

This function compares the individual inference of the original model with that of the orthonormal model taken as reference.

Thus, if the null hypothesis is rejected in the individual significance tests in the model where there are no linear relationships between the independent variables (orthonormal) and is not rejected in the original model, the reason for the non-rejection is due to the existing linear relationships between the independent variables (multicollinearity) in the original model.

The second model is obtained from the first model by performing a QR decomposition, which eliminates the initial linear relationships.

References

Salmerón, R., García, C.B. and García, J. (2025). A Redefined Variance Inflation Factor: overcoming the limitations of the Variance Inflation Factor. Computational Economics, 65, 337-363, doi: https://doi.org/10.1007/s10614-024-10575-8.

Overcoming the inconsistences of the variance inflation factor: a redefined VIF and a test to detect statistical troubling multicollinearity by Salmerón, R., García, C.B and García, J. (working paper, https://arxiv.org/pdf/2005.02245).

See Also

rvifs

Examples

### Example 1
	
  set.seed(2024)
  obs = 100
  cte = rep(1, obs)
  x2 = rnorm(obs, 5, 0.01)  # related to intercept: non essential
  x3 = rnorm(obs, 5, 10)
  x4 = x3 + rnorm(obs, 5, 0.5) # related to x3: essential
  x5 = rnorm(obs, -1, 3)
  x6 = rnorm(obs, 15, 0.5)
  y = 4 + 5*x2 - 9*x3 -2*x4 + 2*x5 + 7*x6 + rnorm(obs, 0, 2)
  x = cbind(cte, x2, x3, x4, x5, x6)
  multicollinearity(y, x)

### Example 2
### Effect of sample size
  
  obs = 25 # by decreasing the number of observations affected to x4 
  cte = rep(1, obs)
  x2 = rnorm(obs, 5, 0.01)  # related to intercept: non essential
  x3 = rnorm(obs, 5, 10)
  x4 = x3 + rnorm(obs, 5, 0.5) # related to x3: essential
  x5 = rnorm(obs, -1, 3)
  x6 = rnorm(obs, 15, 0.5)
  y = 4 + 5*x2 - 9*x3 -2*x4 + 2*x5 + 7*x6 + rnorm(obs, 0, 2)
  x = cbind(cte, x2, x3, x4, x5, x6)
  multicollinearity(y, x)

### Example 3
  
  y = 4 - 9*x3 - 2*x5 + rnorm(obs, 0, 2)
  x = cbind(cte, x3, x5) # independently generated
  multicollinearity(y, x)
  
### Example 4
### Detection of multicollinearity in Wissel data
  
  head(Wissel, n=5)
  y = Wissel[,2]
  x = Wissel[,3:6]
  multicollinearity(y, x)
  
### Example 5
### Detection of multicollinearity in euribor data
  
  head(euribor, n=5)
  y = euribor[,1]
  x = euribor[,2:5]
  multicollinearity(y, x)
  
### Example 6
### Detection of multicollinearity in Cobb-Douglas production function data

  head(CDpf, n=5)
  y = CDpf[,1]
  x = CDpf[,2:4]  
  multicollinearity(y, x)
  
### Example 7
### Detection of multicollinearity in number of employees of Spanish companies data
  
  head(employees, n=5)
  y = employees[,1]
  x = employees[,3:5]
  multicollinearity(y, x)
  
### Example 8
### Detection of multicollinearity in simple linear model simulated data
  
  head(SLM1, n=5)
  y = SLM1[,1]
  x = SLM1[,2:3]
  multicollinearity(y, x)

  head(SLM2, n=5)
  y = SLM2[,1]
  x = SLM2[,2:3]
  multicollinearity(y, x)
    
### Example 9
### Detection of multicollinearity in soil characteristics data

  head(soil, n=5)
  y = soil[,16]
  x = soil[,-16] 
  x = cbind(rep(1, length(y)), x) # the design matrix has to have the intercept in the first column
  multicollinearity(y, x)
  multicollinearity(y, x[,-3]) # eliminating the problematic variable (SumCation)
  
### Example 10
### The intercept must be in the first column of the design matrix
  
  set.seed(2025)
  obs = 100
  cte = rep(1, obs)
  x2 = sample(1:500, obs)
  x3 = sample(1:500, obs)
  x4 = rep(4, obs)
  x = cbind(cte, x2, x3, x4)
  u = rnorm(obs, 0, 2)
  y = 5 + 2*x2 - 3*x3 + 10*x4 + u
  multicollinearity(y, x)
  multicollinearity(y, x[,-4]) # the constant variable is removed