GMM Estimation of a parameter from Intratemporal Euler of capital constraints I'm trying to estimate $\lambda$ from this intratemporal Euler equation:

$\left[ \dfrac{C_t^{-\sigma}}{C_t^{*-\sigma}} \dfrac{P_t}{S_t P_t^*} \right]^{\lambda} \left[ \dfrac{\bar{P_t} Y_t - \Delta (FR_t)}{P_t C_t} \right]^{1-\lambda}  = 1 $

Log-linearizing this gives: (done in paper)

$\Rightarrow \lambda \left[ \sigma (c_t - c_t^*) - \ln (RER) \right] = (1-\lambda) \left[ y_t - c_t - \ln (P_t / \bar{P_t}) - (fr_t - fr_{t-1}) \right] $

Now, there are a bunch of assumptions and approximations taken into account, but at the end of the day, I will have data on all the macro variables, and my project is to estimate the values of $\lambda$ for different countries (FYI: $\lambda$ represents the presence of risk sharing).

I've heard that there exists GMM methods for such estimations, but Hayashi only has a small section of intertemporal Euler equation GMM work. I was wondering if you guys could advise me on how I can estimate for this $\lambda$ parameter, or perhaps some guideline as to where I can look. Just to note again, I will have all the variables' data. :)

Thanks!
To elaborate on what has been said in the comments already, using GMM based on Euler Equations generally involves uncertainty that motivates some sort of expected orthogonality between a moment equation and some instruments. Here is a common example of a "Consumption-Based Asset-Pricing Model" (see, for example, Campbell, 1993, 1996) posted by Dave Giles:



A representative agent chooses a consumption time-path to maximize expected discounted utility

$E\left[\sum_{i=0}^\infty\beta^i U(c_{t+i})|\Omega_t\right]$,

where $\Omega_t$ is the information set at time $t$, subject to the inter-temporal budget constraint,

$c_t + p_tq_t = r_tq_{t-1} + w_t$, for all $t$.

The optimal consumption path satisfies:

$p_tU'(c_t) = \beta E[r_{t+1}U'(c_{t+1})|\Omega_t]$, for all $t$

Which gives the Euler equation:

$E[\beta(r_{t+1}/p_t)[U'(c_{t+1}) / U'(c_t)]|\Omega_t] - 1 = 0$.

He then imposes the CRRA utility function $U(c_t) = c_t^{1-\gamma}/(1-\gamma)$, making the Euler equation become

$E[\beta(r_{t+1}/p_t)(c_{t+1}/c_t)^{-\gamma}|\Omega_t] - 1 = 0$.

This yields the moment equations:

$E[\{\beta(r_{t+1} / p_t)(c_{t+1} / c_t)^{-\gamma} - 1\}z_t] =
 E[E[\beta(r_{t+1} / p_t)(c_{t+1} / c_t)^{-\gamma} - 1|\Omega_t]z_t] = 0$

with $z_t$ a vector of instruments belonging to $\Omega_t$.



In this scenario, the uncertainty referred to is captured in the expectation operator over the information set $\Omega$. Notice that the interpretation of $\Omega$ motivates which instruments to use in the estimation.

Takeaway: if you want to use Euler equations to motivate GMM, you need some kind of uncertainty. Use uncertainty structure to decide instruments.



Sources

Campbell, J. Y., 1993. Intertemporal asset pricing without consumption data. American Economic Review, 83, 487-512.

Campbell, J. Y., 1996. Understanding risk and return. Journal of Political Economy, 104, 298-345.