<p>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 <a href="http://davegiles.blogspot.com/2013/05/estimating-euler-equation-using-gmm.html" rel="noreferrer">Dave Giles</a>:</p>

<hr>

<p>A representative agent chooses a consumption time-path to maximize expected discounted utility</p>

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

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

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

<p>The optimal consumption path satisfies:</p>

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

<p>Which gives the Euler equation:</p>

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

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

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

<p>This yields the moment equations:</p>

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

<p>with $z_t$ a vector of instruments belonging to $\Omega_t$.</p>

<hr>

<p>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.</p>

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

<hr>

<p>Sources</p>

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

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

CC BY-SA 3.0
2015-06-12T17:51:06.570Z
2015-06-12T17:51:06.570Z
philE
<p>You do not need any fancy GMM here. You can consistently estimate the parameter of interest using Non-linear least squares. </p>

CC BY-SA 3.0
2015-09-03T14:32:26.797Z
2015-09-03T14:32:26.797Z
ChinG
<p>I'm trying to estimate $\lambda$ from this intratemporal Euler equation:</p>

<p>$\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 $</p>

<p>Log-linearizing this gives: (done in paper)</p>

<p>$\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] $</p>

<p>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).</p>

<p>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. :)</p>

<p>Thanks!</p>

CC BY-SA 3.0
2015-06-10T21:07:11.533Z
GMM is based on sample analogues of theoretical moment conditions. Theoretical moment conditions are expected values. Where are the expected values in your equation?
Alecos Papadopoulos
CC BY-SA 3.0
2015-06-10T21:48:15.860Z
Yes to be able to estimate statistically you should have some model with uncertainty, or income fluctuation.
user157623
<macroeconomics><open-economy-macro><capital-structure>
GMM Estimation of a parameter from Intratemporal Euler of capital constraints
Nafis S.
# GMM Estimation of a parameter from Intratemporal Euler of capital constraints

(Asked by: username_0 on 2015-06-10)

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!

**Tags:** macroeconomics, open-economy-macro, capital-structure

> **Comments:**
> **username_1**: GMM is based on sample analogues of theoretical moment conditions. Theoretical moment conditions are expected values. Where are the expected values in your equation?  
> **username_2**: Yes to be able to estimate statistically you should have some model with uncertainty, or income fluctuation.  

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## 2 answers

### Answer by username_3 on 2015-06-12. Score: 5

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](http://davegiles.blogspot.com/2013/05/estimating-euler-equation-using-gmm.html):

---

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.


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### Answer by username_4 on 2015-09-03. Score: 0

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


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