Replicate Blundell and Bond (2000) results using R
3. How do we recover parameters from production function estimates (INCOMPLETE ANSWER - will be updated with how to do this in R once I have time to figure it out, or if somebody else knows...)
Blundell and Bond aren't estimating the parameters of a Cobb-Douglas production function. They're estimates the parameters of a &quot;dynamic (common factor) representation&quot; of a production function, specifically:
$(1) \ \ y_{it} = \beta_n n_{i,t} - \rho \beta_n n_{i,t-1} + \beta_k k_{i,t} - \rho \beta_k k_{i,t} + \rho y_{i,t-1} + (\gamma_t - \rho \gamma_{t-1}) + (\eta_i(1-\rho) + e_{it} + m_{it} - \rho m_{i,t-1})$
...which they get by quasi-differencing their production function and doing something weird with the $v_{it}$ term (not shown above). Changing the symbols, they re-write this as:
$(2) \ \ y_{it} = \pi_1n_{it} + \pi_2 n_{i,t-1} + \pi_3 k_{it} + \pi_4 k_{i,t-1} + \pi_5 y_{i,t-1} + \gamma^*_t + (\eta^*_i +w_{it})$
Where the * presumably indicates steady-state values, but again, I wish I had access to a teacher to explain precisely why they write that. Anyway, what their SYS-GMM estimator returns is a vector of parameter values $\boldsymbol{\pi} = (\pi_1,...,\pi_5)'$. We now have to use this to get the vector of parameters $\boldsymbol{\theta} = (\beta_n,\beta_k,\rho)$.
The only place I can find an explanation of how one does that is this paper applying SYS-GMM to Cuba. They refer to Woolridge 2002 (I don't have it), and assert that one can consider the problem as defined by the relationship $\boldsymbol{\pi} = h(\boldsymbol{\theta})$. We have $\boldsymbol{\pi}$, we need $\boldsymbol{\theta}$, and we will have to employ some function h() to make the link. It sounds like h() is taken to be an instance of a general class of minimum distance estimators, and we can (according to Woolridge 2002, apparently) solve the problem by solving:
$min (\boldsymbol{\theta}) = (\hat{\pi}-h(\theta))'\Omega(\hat{\pi}-h(\theta)) $
where $\Omega$ is some weighting function. And again, apparently $\Omega$ should be chosen as the variance of the matrix of parameters, obtained by &quot;the delta method&quot;.
...but now it gets confusing, because although it's easy to find info on the delta method online, the matrix of parameters includes the unknowns $\boldsymbol{\theta}$. I guess the way to tackle this is to multiply it all out and plug into optim, using some arbitrary choice on starting values (factor shares of output, if $ valued?). I will update if I get a chance to try that...
In going from equation (1) to (2) it is pretty obvious that:
$$h(\beta_n,\beta_k,\rho) = \begin{pmatrix} \beta_n \\ - \rho \beta_n \\ \beta_k \\ - \rho \beta_k \\ \rho \end{pmatrix}= \begin{pmatrix} \pi_1 \\ \pi_2 \\ \pi_3  \\ \pi_4 \\ \pi_5 \end{pmatrix},$$
and $$\Omega = var(\pi)$$