# Multicollinearity in Zero Inflated Negative Binomial Regression

I am trying to model counts govt, based on the counts lp,const,opp and another independent variable govtno. govt has many zeros, so I am using a zero-inflated negative binomial regression. The counts lp,const,opp also have many zeros. The pairwise correlations between these counts might indicate the presence of multicollinearity between predictors lp,const,opp:

       govt     const       lp       opp
A   1.0000000 0.2883734 0.4135134 0.3913364
B   0.2883734 1.0000000 0.4138627 0.5478605
C   0.4135134 0.4138627 1.0000000 0.5315744
D   0.3913364 0.5478605 0.5315744 1.0000000


1) How can I really check for multicollinearity in this model? I do not know how to calculate VIFs for zero inflated regression models.
2) How can I address this multicollinearity? My final goal is to test significance of the predictors, so solution(s) should allow for statistical significance testing.

Here is the summary output of the zero-inflated negative binomial regression:

>summary(m4 <- zeroinfl(govt ~ govtno + const + lp + opp, data = dat1b.w.nc, dist="negbin"))

Call:
zeroinfl(formula = govt ~ govtno + const + lp + opp, data = dat1b.w.nc, dist = "negbin")

Pearson residuals:
Min       1Q   Median       3Q      Max
-0.71953 -0.14796 -0.11066 -0.08794 15.45473

Count model coefficients (negbin with log link):
Estimate Std. Error z value Pr(>|z|)
(Intercept) -0.123334   0.272388  -0.453   0.6507
govtno      -0.013671   0.006024  -2.269   0.0232 *
const        0.028129   0.015127   1.860   0.0630 .
lp           0.024683   0.014829   1.665   0.0960 .
opp          0.155652   0.036760   4.234 2.29e-05 ***
Log(theta)  -0.639797   0.137549  -4.651 3.30e-06 ***

Zero-inflation model coefficients (binomial with logit link):
Estimate Std. Error z value Pr(>|z|)
(Intercept)  3.943508   0.351314  11.225  < 2e-16 ***
govtno      -0.027054   0.008617  -3.139  0.00169 **
const       -0.052898   0.057112  -0.926  0.35433
lp          -1.045437   0.187422  -5.578 2.43e-08 ***
opp         -1.881200   0.349475  -5.383 7.33e-08 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Theta = 0.5274
Number of iterations in BFGS optimization: 37
Log-likelihood: -1422 on 11 Df