I have a reasonable understanding of why multicollinearity is a problem is regression models, along the lines of this excellent post.
To summarise my understanding, for a regression model of $y = \alpha + \beta_1x + \beta_2z$ (where $x$ and $z$ are correlated), beta coefficient estimates (as well as being unstable) are difficult to interpret, as a situation where you might increase $z$ without increasing $x$ is unlikely to occur, and not supported by the data.
I understand multicollinearity is less harmful to purely predictive as opposed to explanatory or descriptive models.
I'm interested in another interpretation:
If I decided to increase $z$, and let $x$ vary as it pleases in reaction, what would I see happen to $y$, accounting for the fact that $x$ is likely to move with $z$, and also have it's own effect?
In other words, accepting the causal interpretation that $x$ and $z$ both cause $y$, and are themselves correlated to some extent (.7 say), how would all three variables move if $z$ is (linearly) increased by some amount?
I've tried to model this sort of thing before, fitting $y = \alpha + \beta_1x + \beta_2z$ (model 1), and $x = \alpha + \beta_1z$ (model 2). Hypothetical increased $z$ values are produced, and resulting $x$ values are predicted with model 2. The hypothetical $x$ and $z$ values are used to predict $y$ using model 1. However this feels very unsatisfactory, complicated simulations are required to capture uncertainty (I used
arm). Additionally, my gut tells me that apart from being painfully inelegant, it's a bad idea for other reasons I can't put my finger on.
- Is such an 'observational'/conditional-when-I-feel-like-it interpretation possible?
- Does anyone know of a better method for this interpretation?
- Can anyone recommend a paper or
Rpackage along these lines?
- Is the above multi-model mess at-all valid?
I'm aware that a model along the lines of $y = \alpha + \beta_1z$ would yield a similar answer to the two-stage mess above, but would lose information in $x$.
I understand that these ideas are similar to structural equation modelling, but apart from having scant knowledge of SEM, I'm yet to find an
R package which allows flexibly extending these models with different link functions for proportional odds models, etc.