Worked GMM For Asset-Pricing Euler Equations

Why Euler Equations Become GMM Moments

Asset-pricing theory links the return on an asset to how its payoff covaries with investors’ marginal utility. Under constant relative risk aversion (CRRA) preferences, a convenient specification of that link uses consumption growth and a gross asset return. The resulting Euler equation is an expectation restriction. GMM makes it possible to estimate preference parameters from several such restrictions, including restrictions formed by multiplying the Euler equation by valid instruments.

The worked examples use hypothetical data only. They illustrate the construction and interpretation of moments; they are not empirical findings about any market or investor population.

Definition: Let $G_{t+1}=C_{t+1}/C_t$ be gross consumption growth, $R_{j,t+1}$ the gross return on asset $j$, and $\beta$ the discount factor. Under CRRA utility with coefficient $\gamma$, define the pricing error $$ u_{j,t+1}(\theta)=\beta G_{t+1}^{-\gamma}R_{j,t+1}-1,\qquad \theta=(\beta,\gamma)'. $$ The population Euler equation is $E[u_{j,t+1}(\theta_0)]=0$ at the true parameter $\theta_0$, when the model and its expectation restriction are valid.

Here, $C_t$ is consumption at date $t$, and $R_{j,t+1}$ is a gross return: a value of 1.05 represents a 5% return over the period. If information $z_t$ is available at date $t$, the conditional Euler restriction implies unconditional moments

E\!\left[z_t u_{j,t+1}(\theta_0)\right]=0.

The instrument must be known at date $t$ and satisfy the relevant orthogonality restriction under the model. A constant instrument gives the unconditional Euler equation; additional instruments create additional moment conditions. These moments are not automatically valid merely because a variable is observed before the return: their validity is an assumption about the economic model and information set.

From Observations to an Overidentified System

Suppose there are $m$ moment conditions and $p$ unknown parameters. For a time series of length $T$, write the observation-level vector of moments as $g_t(\theta)$. For example, with one asset and instruments $z_t=(1,z_{2t},z_{3t})'$, the vector is $z_t u_{t+1}(\theta)$. Its sample mean is

\bar g_T(\theta)=\frac{1}{T}\sum_{t=1}^{T}g_t(\theta).

The system is overidentified when $m>p$: there are more moment restrictions than parameters. Unless the sample moments happen to equal zero at the same parameter value, estimation finds a parameter that makes the moments small according to a chosen weighting matrix. In two-step feasible GMM, a first-step estimate is used to estimate the long-run covariance matrix of the moments, and its inverse supplies the second-step weight. This tutorial uses that efficient-weight setup in the numerical example.

GMM criterion: For a positive-definite weighting matrix $W$, the GMM estimator minimizes $$ Q_T(\theta)=\bar g_T(\theta)'W\bar g_T(\theta). $$ With a consistent estimate $\widehat{\Omega}$ of the long-run covariance matrix of $\sqrt{T}\bar g_T(\theta_0)$, the efficient second-step choice is $W=\widehat{\Omega}^{-1}$. Inference also requires identification, suitable regularity conditions, and a consistent covariance estimate; the robust sandwich variance accounts for the sampling variability and dependence represented by $\widehat{\Omega}$.

Worked Example: Calculating Euler Contributions

Consider two hypothetical periods, one asset, $\beta=0.98$, $\gamma=2$, and a constant instrument $z_t=1$. In the first period let $G_{t+1}=1.00$ and $R_{t+1}=0.96886$; in the second let $G_{t+1}=1.00$ and $R_{t+1}=1.07913$. Calculate each observation’s moment contribution.

For the first period, $$ z_tu_{t+1}=1\left[0.98(1.00)^{-2}(0.96886)-1\right] =0.9494828-1=-0.0505172. $$ For the second period, $$ z_tu_{t+1}=1\left[0.98(1.00)^{-2}(1.07913)-1\right] =1.0575474-1=0.0575474. $$

The sample moment is the average: $$ \bar g_T(\theta)=\frac{-0.0505172+0.0575474}{2}=0.0035151. $$ The small positive sample moment says that, at these chosen parameter values, the average discounted payoff contribution exceeds zero by about 0.003515 per period in this two-observation illustration. It is a sample discrepancy, not proof that the population Euler equation is false.

Estimating a Discount Factor with Multiple Moments

A simple calculation makes the weighted estimation step transparent. Suppose $\gamma$ is fixed at 2, and use a constant instrument and one additional instrument $z_t$. The two sample moments can be written $\bar g_T(\beta)=\beta a-b$, where $a$ and $b$ collect averages formed from the observed data:

a=\frac{1}{T}\sum_{t=1}^{T} \begin{pmatrix} G_{t+1}^{-2}R_{t+1}\\ z_tG_{t+1}^{-2}R_{t+1} \end{pmatrix}, \qquad b=\frac{1}{T}\sum_{t=1}^{T} \begin{pmatrix} 1\\ z_t \end{pmatrix}.

Thus there are two moment conditions and one unknown parameter, so the model is overidentified. The example below supplies hypothetical sample summaries rather than listing all 200 observations.

Worked Example: Solving the Weighted GMM Problem

Suppose $T=200$ hypothetical periods produce $a=(1.02,1.00)'$ and $b=(0.99,0.98)'$. For this illustration, suppose the estimated long-run covariance matrix is $\widehat{\Omega}=0.04I_2$, where $I_2$ is the two-by-two identity matrix. Estimate $\beta$ using $W=\widehat{\Omega}^{-1}=25I_2$.

Since $Q_T(\beta)=(\beta a-b)'W(\beta a-b)$, differentiate with respect to $\beta$ and set the derivative to zero:

\frac{\partial Q_T(\beta)}{\partial\beta} =2a'W(\beta a-b)=0 \quad\Longrightarrow\quad \widehat\beta=\frac{a'Wb}{a'Wa}.

Because $W=25I_2$, the factor 25 cancels: $$ \widehat\beta =\frac{(1.02)(0.99)+(1.00)(0.98)} {(1.02)^2+(1.00)^2} =\frac{1.9898}{2.0404} =0.975201. $$ The estimated discount factor is about 0.9752. It is dimensionless and, in this model, represents the estimated factor applied to next period’s marginal utility. Its interpretation depends on the maintained CRRA Euler equation and on the validity of the moments used.

The fitted sample moments are $$ \bar g_T(\widehat\beta) =\widehat\beta a-b \approx \begin{pmatrix} 0.004705\\ -0.004799 \end{pmatrix}. $$ They need not both equal zero: the single parameter cannot generally set two sample moments to zero simultaneously.

Robust Uncertainty and the J Test

Let $D$ be the $m$-by-$p$ Jacobian of the population moment vector with respect to the parameters, evaluated at the true parameter. With efficient GMM weighting and regularity conditions, the asymptotic covariance matrix of $\sqrt{T}(\widehat\theta-\theta_0)$ is $(D'\Omega^{-1}D)^{-1}$. In practice, the robust sandwich estimate uses sample estimates of the Jacobian and long-run covariance matrix. In the scalar example above, the derivative of $\bar g_T(\beta)$ is $a$, so the corresponding estimated variance is $(a'\widehat\Omega^{-1}a)^{-1}/T$.

Worked Example: Robust Standard Error and Confidence Interval

Continue with the hypothetical summaries $a=(1.02,1.00)'$, $T=200$, and $\widehat{\Omega}=0.04I_2$. The estimated robust variance of $\widehat\beta$ is $$ \widehat{\operatorname{Var}}(\widehat\beta) =\frac{1}{T}\left(a'\widehat{\Omega}^{-1}a\right)^{-1} =\frac{1}{200\left[25(1.02^2+1.00^2)\right]} =\frac{1}{10202} \approx0.00009802. $$ Therefore, $SE(\widehat\beta)=\sqrt{0.00009802}\approx0.009901$. A large-sample 95% confidence interval is $$ 0.975201\pm1.96(0.009901) \approx[0.9558,\;0.9946]. $$

Under the stated model, instrument, covariance estimate, and large-sample approximation, this interval is a procedure that would cover the discount factor in 95% of repeated samples. It does not mean there is a 95% probability that the fixed parameter lies in this particular realized interval.

Overidentifying restrictions can be assessed using Hansen’s $J$ statistic. With efficient weighting, it is $T\bar g_T(\widehat\theta)'\widehat\Omega^{-1}\bar g_T(\widehat\theta)$. Under the null that all specified population moments are valid, and under the regularity conditions for the test, it has an asymptotic chi-square distribution with $m-p$ degrees of freedom. The degrees of freedom count the excess moments, not the number of assets or observations.

Worked Example: Computing and Interpreting the J Test

For the same estimate, the two fitted moments are approximately $0.004705$ and $-0.004799$, with $T=200$, $m=2$, and $p=1$. Therefore, $$ J =200 \begin{pmatrix}0.004705&-0.004799\end{pmatrix} (25I_2) \begin{pmatrix}0.004705\\-0.004799\end{pmatrix} \approx0.2258. $$ Under the null, the reference distribution is $\chi^2_1$, since $m-p=1$. The upper-tail p-value is approximately 0.635. At significance level $\alpha=0.05$, this result does not reject the joint moment restrictions.

This is not evidence that the model or instruments have been proved correct. The test may have limited power, and its interpretation depends on the maintained assumptions and consistent covariance estimation. A rejection would indicate that the specified moments are difficult to reconcile jointly with the data under the test’s assumptions; it would not by itself identify which moment condition is responsible.

Practical Checks and a Common Pitfall

Before reporting an asset-pricing GMM result, verify the timing and construction of every quantity. Consumption growth must be $C_{t+1}/C_t$, the return must be gross rather than a percentage entered as though it were gross, and an instrument dated $t$ must be available at the time specified by the economic argument. Also state which parameters are estimated and which are fixed. For instance, the numerical estimation example fixes $\gamma=2$; its estimate is not a joint estimate of risk aversion and the discount factor.

A common mistake is to treat a non-rejection of the J test as confirmation that every instrument is valid. The test assesses the joint compatibility of the moment restrictions under its assumptions; a high p-value does not establish their economic validity. Another mistake is to read a small sample moment as a zero population moment. The sample average is random, and its uncertainty is summarized through the moment covariance and the resulting inference, not by rounding the average to zero.

Key takeaway: An asset-pricing GMM analysis begins with a stated Euler equation and justified instruments, then forms sample moments, estimates parameters by weighted minimization, and quantifies uncertainty using a suitable long-run covariance estimate. The J test evaluates the joint overidentifying restrictions; it is a diagnostic, not a proof of the model.

Check Your Understanding

Answer each question using the definitions and calculations above.

Why does multiplying the Euler pricing error by an instrument dated $t$ yield a moment restriction only when the relevant orthogonality condition holds?
If there are four moment conditions and two estimated parameters, what are the asymptotic degrees of freedom for the J test under its null?
In the weighted estimation example, why do the two fitted sample moments not both equal zero at the estimated discount factor?
What does the robust standard error measure in the hypothetical discount-factor example, and what assumptions are needed for its interpretation?
Explain why a J-test p-value of approximately 0.635 does not prove that the asset-pricing model is correct.