# Translating a Method of Moments/Instrumental Variable Estimator to Bayesian

Hello everyone!

I’m working on production function estimation, which seeks to interpret the betas in the following regression causally:

\ln Y_t = \beta_0 + \beta_1 \ln L_t + \beta_2 \ln K_t + \beta_3 \ln M_t + \omega_t + \eta_t

Where M_t is assumed to be correlated with observed input K_t and unobserved productivity process \omega_t: M_t = f(K_t, \omega_t), and \omega_t has some time series dynamics to it: \omega_t = g(\omega_{t-1}). \eta_t is i.i.d normal noise uncorrelated with the other variables.

The popular approach to this problem is do a two-stage procedure that involves first doing preliminary regressions, then taking those estimates and forming a method of moments estimator. I’m wondering if I can do better (especially with propagation of estimation error through the two stages, since estimated coefficients are used as fixed inputs in the second stage) by jointly estimating the probability of the whole system.

I’m thinking the probabilistic model I want to write down is something like:
\begin{align}\ln Y_t &\sim N(\mu_t, \sigma^2_Y) \\ \ \mu_t &= \beta_0 + \beta_1 \ln L_t + \beta_2 \ln K_t + \beta_3 \ln M_t + \omega_t \\ M_t &\sim N(\phi_t, \sigma^2_M) \\ \phi_t &= f(K_t, \omega_t) \\ \omega_t &\sim GRW(0, \sigma_\omega^2) \\ \beta_k &\sim N(0, \sigma^2_\beta) \\ \sigma^2_k &\sim \text{Gamma}(2, 10) \end{align}

Does this look right for an “instrumental variable” type model? Adding the endogenous variable as a separately likelihood? Would I then be able to directly interpret the parameters \beta_1, \beta_2, and \beta_3 causally? Or would I have to do something else with the \phi function?

Regarding the \phi function, in the original paper they assume the function is invertible, and so use it to estimate \omega_t = f(M_t, K_t), and use a 4th order polynomial for f. I’m curious if I can do better than that though? For example, is it possible to directly model M_t as a GP in data K_t and latent variable \omega_t? I’m a bit fuzzy on how to use GPs as priors – would I want to model the mean function or the covariance function in this case?

Summary:

1. Does anyone have experience with Bayesian IV? Does adding a 2nd likelihood function cut it, or do I need to do more work?
2. Can I use a GP prior in my model for the \phi function? How would that work?

Thanks!

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Interesting question!

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