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I’ve simulated some data consisting of one response variable (‘y’) and two collinear predictor variables (‘Amount’ and ‘MPS’), where collinearity arises from one of two causes: (1) Amount causes MPS, or (2) Amount and MPS are jointly affected by an unmeasured variable.

What I'm trying to do is figure out whether path analysis can discriminate between these two causes of collinearity. But I'm having trouble specifying a path model for collinearity scenario (2).

My question: Is it possible to specify a path model that implies that two exogenous variables are jointly influenced by an unmeasured variable?

I'm working in lavaan, but answers for how to do this conceptually would also be appreciated (if you aren't familiar with lavaan).

Here are my data, simulated in R:

# collinearity cause (1)
Amount <- rnorm(n=350, mean=0, sd=1)   
MPS <- rnorm(n=350, mean=0.76*Amount, sd=0.653) 
y <- rnorm(n=350, mean=0.367*Amount + 0.367*MPS, sd=0.72) 

# collinearity cause (2)
Lurking <- rnorm(n=350, mean=0, sd=1) 
Amount <- rnorm(n=350, mean=0.872*Lurking, sd=0.486)  
MPS <- rnorm(n=350, mean=0.872*Lurking, sd=0.486)  
y <- rnorm(n=350, mean=0.367*Amount + 0.367*MPS, sd=0.72)         

And this is my path model for (1), specified in lavaan:

model1 <- '
  #regressions
  y ~ Amount
  y ~ MPS
  MPS ~ Amount
  '

And this is a path model I tried for (2):

model2<- '
  #regressions
  y ~ Amount
  y ~ MPS
  #residual correlations
  MPS ~~ Amount
  '

so for path model (2) my approach was to specify a residual correlation between MPS and Amount. I'm uncertain if this is the correct approach. but even if it is, it doesn't work – to make it work I have to specify that exogenous variables are not fixed, and this uses up my degrees of freedom so I can't test the model.

If anyone has any suggestions for how I can do this – or if it is possible at all - I'd really appreciate it.

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