GMM Estimation of a parameter from Intratemporal Euler of capital constraints

Question

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!

Answer 1

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:

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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 $\Omega_t$.

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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.

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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.

Answer 2

You do not need any fancy GMM here. You can consistently estimate the parameter of interest using Non-linear least squares.