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An Adversarial Approach to Structural Estimation
\defcitealias{dfj}{DFJ}
\gdef\@thefnmark\@footnotetext{We thank Mariacristina De Nardi and John Jones for sharing the data and codes for the empirical application and for very helpful discussion. We also thank Isaiah Andrews, Manuel Arellano, Stephane Bonhomme, Aureo De Paula, Costas Meghir, Chris Hansen, Koen Jochmans, Whitney Newey, Luigi Pistaferri, Bernard Salanie, Dennis Kristensen, Anna Mikusheva, Zhenling Jiang, Xintong Han, and Daniel Waldinger, as well as numerous participants in conferences and venues for helpful discussion. Elsie Hoffet, Yijun Liu, Ignacio Ciggliutti, and Marcela Barrios provided superb research assistance. We gratefully acknowledge the support of the NSF by means of the Grant SES\hyp1824304 and the Richard N.\ Rosett Faculty Fellowship and the Liew Family Faculty Fellowship at the University of Chicago Booth School of Business.}
Structural estimation is a useful tool to learn about the effects of policies that are yet to be implemented. Structural models are naturally articulated as parametric models and, as such, may be estimated using maximum likelihood (MLE). However, likelihood functions are sometimes too complex to evaluate or may not exist in closed form. This has spurred large literature on simulation\hypbased estimation methods.
A prominent example of such methods is the simulated method of moments (SMM) m1989. If we want identification and estimation of the parameters to rely on specific features, SMM is a natural tool as long as such features can be expressed as moments. At the same time, a naive strategy to stack many moments is known to yield poor finite sample properties altonji1996small. This tradeoff is especially pronounced in models with rich heterogeneity, where the number of moments may grow rapidly with the number of covariates. While this problem may be resolved if we can reduce the moments to a handful of informative ones, such a choice is often not obvious.
This paper proposes a new simulation\hypbased estimation method, which we call {\em adversarial estimation}. It is inspired by the {\em generative adversarial networks (GAN)}, a machine learning algorithm developed by goodfellow2014generative to generate realistic images. We adopt their adversarial framework to estimate the structural parameters that generate realistic economic data. The proposed estimator achieves efficiency under correct specification and the parametric rate under misspecification. Thus, our method is useful in applications where the likelihood is not computable but simulation is feasible and it can be a more efficient alternative to SMM.
The generative adversarial estimation framework is a minimax game between two components---the {\em discriminator} and the {\em generator}---over classification accuracy: \[ \min_{\{\text{\em generator}\}}\max_{\{\text{\em discriminator}\}}\text{\em classification accuracy}. \] The generator is an algorithm that produces the simulated data; its objective is to find a data\hypgenerating process that confuses the discriminator. The discriminator is a classification algorithm that distinguishes the observed data from the simulated data; it takes an observation as input and classifies whether it comes from observed data or simulated data; its objective is to maximize the accuracy of its classification.
In the original GAN, both the discriminator and the generator are given as neural networks (hence the name). In this paper, we take the generator to be the structural model we intend to estimate and the discriminator to be an arbitrary classification algorithm (while our primary choice is a neural network). For classification accuracy, we employ the cross\hypentropy loss, following goodfellow2014generative. \footnote{There are also other losses considered in the literature. In machine learning, they concern high\hypdimensional data such as images, sounds, and texts, and the Wasserstein distance has gained huge popularity for its ability to measure the distance of disjoint probability distributions. It is also used in economic applications imbensGANs.}
Interestingly, our framework casts a bridge between SMM and MLE. When we use a logistic discriminator, the resulting estimator is asymptotically equivalent to optimally\hypweighted SMM ((ref)). When we use the oracle discriminator, the resulting estimator is equivalent to MLE under the condition that the simulation sample size increases faster than the actual sample size. Of particular interest is the middle case, in which the oracle discriminator is not available but a sufficiently rich discriminator capable of approximating it is used. Under some conditions, the resulting estimator enjoys the desirable properties of both SMM and MLE: the user has the flexibility to choose moments if desired, a closed\hypform likelihood is not required, and the asymptotic efficiency is attained.
We illustrate the theoretical properties of our estimator in simulations using simple models. We show that the curvature of the classification accuracy is comparable to that of the log likelihood function for a suitable choice of discriminator. In addition, we show that the estimator can achieve the parametric rate under misspecification, and finally that compared to SMM, the adversarial estimator suffers less from the small sample bias. We also showcase the implementation of the method using a Roy Model with two occupations over two time periods.
Using the adversarial estimation framework, we revisit investigation of the elderly's saving motives in dfj. Understanding different channels of saving motives is vital in evaluating social insurances. We aim to disentangle three reasons to save: survival risk, medical expense risk, and bequest motive. The structural model is dynamic and agents face heterogeneous risk by gender, age, health status, and permanent income. We demonstrate the capacity of adversarial estimation with two specifications: first with the inputs representing similar identifying variation as dfj, and second the inputs augmented with gender and health.
The rest is organized as follows. (ref) defines the adversarial estimation framework. (ref) illustrates the estimator with simple examples. (ref) develops the asymptotic properties. (ref) applies the method to dfj.
The adversarial estimation has two main components: simulation and discrimination. The simulation component is the same as other simulation\hypbased estimation methods, such as SMM or indirect inference, but the discriminator component is new. The essence of the adversarial framework is to find a parameter value for which the corresponding simulated data is indistinguishable from the real data according to the discriminator. We now describe each component in turn.
Suppose we have data $\{X_i\}_{i=1}^n$ drawn i.i.d.\ from an unknown distribution $P_0$. Suppose we have a fully parametric model $\{P_\theta:\theta\in\Theta\}$ for which the likelihood is not tractable but simulation is feasible. \footnote{This is the case for many structural models in economics involving dynamic optimal decision making.} Our target is the parameter $\theta$ that best describes the distribution of the data $P_0$ through the model $P_\theta$.
We formalize the simulation process as follows: for a given $\theta$, and a given sample size $m$, we can obtain a sample of simulated observations, $\{X_{i,\theta}\}_{i=1}^m$, according to model $P_\theta$ by taking draws $\{\tilde{X}_i\}_{i=1}^m$ from a known distribution $\tilde{P}_0$ and applying a transformation $T_\theta$ to them, $X_{i,\theta}=T_\theta(\tilde{X}_i)$. \footnote{If $P_\theta$ is discrete, e.g., Bernoulli with parameter $\theta$, we can generate $\tilde{X}_i\sim U[0,1]$ and apply the inverse transform sampling, e.g., $X_{i,\theta}=\mathbbm{1}\{\tilde{X}_i\geq 1-\theta\}$. }
For illustration, take the example of a normal location model with known variance $1$ and unknown mean $\theta$, $P_\theta=N(\theta,1)$. When we generate a simulated observation $X_{i,\theta}$ from $P_\theta$, we can generate a standard normal observation $\tilde{X}_i\sim N(0,1)$ and convert it into $P_\theta$ through $X_{i,\theta}=\theta+\tilde{X}_i$.
We now turn to the discriminator. The discriminator is the novelty in the estimation framework and is the key component in the construction of the objective function for the adversarial estimator. For some $\theta$ and $x$, consider the problem of assessing whether $x$ is from $P_\theta$ or $P_0$. If $P_\theta$ is very different from $P_0$, it should be easy to distinguish realizations of $P_\theta$ from those of $P_0$. If they are close, it should be harder. The idea, therefore, is to pick a classification algorithm that takes a value $x$ and predicts which distribution it came from, and to search for the value of $\theta$ for which the algorithm can classify the least.
If we had access to the probability density functions corresponding to $P_0$ and $P_\theta$, it would be easy to assign the provenance of $x$ according to the likelihood of $x$ for each distribution. This suggests an estimation strategy based on the search of $\theta$ for which the probability that any draw $X_{i,\theta}$ is drawn from $P_0$ versus $P_\theta$ is $0.5$. Since we do not have access to the probability distributions, this strategy is infeasible. However, we can take advantage of the availability of samples $\{X_i\}_{i=1}^n$ and $\{X_{i,\theta}\}_{i=1}^m$ to estimate the extent to which, for a given $\theta$, these two distributions are different. In particular, we use the fitted predictions of a discrete choice model (called the discriminator), where the dependent variable is $1$ if the data is real and $0$ if it is simulated, and the explanatory variables are $X_i$ if the data is real, and $X_{i,\theta}$ if it is simulated. When $\theta$ is a poor candidate to describe the observed data, the predictions will be either close to $1$ or close to $0$. However, as $\theta$ becomes a better candidate to describe the real data, the distribution of the prediction will concentrate around $1/2$.
Formally, classification is defined as a function $D:\mathcal{X}\to[0,1]$ such that $D(x)$ represents the likelihood of $x$ being an actual observation; $D(x)=1$ means that $x$ is classified as “actual” with certainty; $D(x)=0$ that $x$ is classified as “simulated” with certainty. Denote by $\mathcal{D}_n$ the class of classification functions we consider. The dependence on $n$ allows us to use a richer classification algorithm as the sample size gets larger. The choice of $\mathcal{D}_n$ is an important one for the researcher as it impacts the properties of the estimator. While any class of binary choice models would work, certain choices will have attractive properties, as we discuss below.
The {\em adversarial estimator} is defined by the following minimax problem: \footnote{Minimization and maximization need not be solved exactly ((ref)).} \[ \hat{\theta}=\mathop{\arg\min}_{\theta\in\Theta}\operatorname*{max\vphantom{g}}_{\vphantom{\theta}D\in\mathcal{D}_n} \frac{1}{n}\sum_{i=1}^n\log D(X_i)+\frac{1}{m}\sum_{i=1}^m\log(1-D(X_{i,\theta})). \] Since $D$ is between $0$ and $1$, both $\log D$ and $\log(1-D)$ are nonpositive. If $\{X_i\}$ and $\{X_{i,\theta}\}$ are very different from each other, the discriminator may be able to find $D$ that assigns $1$ on the support of $\{X_i\}$ and $0$ on the support of $\{X_{i,\theta}\}$, in which case the inner maximization attains the value of zero. Meanwhile, regardless of the values of $\{X_i\}$ and $\{X_{i,\theta}\}$, the discriminator can always attain the classification accuracy of $2\log(1/2)$ by setting $D\equiv1/2$. \footnote{This is of course provided that a constant function $1/2$ is in $\mathcal{D}_n$, which is usually the case.} In general, therefore, the inner maximization will give a number between $2\log(1/2)$ and $0$, and the closer it is to $2\log(1/2)$, the less able the discriminator is to classify the observations.
When we let $n$ and $m$ grow, we obtain the population counterpart of the problem \[ \min_{\theta\in\Theta}\max_{\vphantom{\theta}D\in\mathcal{D}_n}\,\mathbb{E}_{X_i\sim P_0}[\log D(X_i)]+\mathbb{E}_{X_{i,\theta}\sim P_\theta}[\log(1-D(X_{i,\theta}))]. \] If there is no restriction on $\mathcal{D}_n$ (so any function $D:\mathcal{X}\to[0,1]$ is allowed), the optimum classifier for the population inner maximization is known to be \[ D_\theta(x)\coloneqq\frac{p_0(x)}{p_0(x)+p_\theta(x)}, \] where $p_0$ and $p_\theta$ are the densities of $P_0$ and $P_\theta$ with respect to some common dominating measure goodfellow2014generative. We call this $D_\theta$ the {\em oracle discriminator}. If the model is correctly specified, then $\theta_0$ is the unique solution to the outer minimization goodfellow2014generative. In the normal location model, if we assume $P_0=N(0,1)$, the oracle discriminator is given by \( D_\theta(x)=\Lambda(\frac{1}{2}\theta^2-\theta x) =\Lambda(-\theta(x-\frac{1}{2}\theta)) \). Since $\Lambda$ is a standard logistic cdf, $\Lambda(0)=1/2$, $\lim_{t\to\infty}\Lambda(t)\to 1$, and $\lim_{t\to-\infty}\Lambda(t)\to 0$. Therefore, if $\theta<0$, positive deviation of $x$ from $\theta/2$ is classified as more likely an actual observation, and negative deviation as less likely; if $\theta=0$, whatever value of $x$ has an equal chance of being actual.
The choice of $\mathcal{D}_n$ gives rise to a few special cases. First, if we use the oracle discriminator $D_\theta$ in lieu of maximization, the resulting estimator for $\theta$ becomes efficient under correct specification and $m\gg n$ kmp2021. In the normal location model, we see that as $m\to\infty$, the oracle estimator solves \[ \hat{\theta}=\mathop{\arg\min}_{\theta\in\Theta}\frac{1}{n}\sum_{i=1}^n\log\Lambda\Bigl(\frac{1}{2}\theta^2-\theta X_i\Bigr)+\mathbb{E}_\theta\Bigl[\log\Bigl(1-\Lambda\Bigl(\frac{1}{2}\theta^2-\theta X_{i,\theta}\Bigr)\Bigr)\Bigr]. \] The FOC combined with the first\hyporder Taylor expansion of $\Lambda$ around $0$ yields \[ 0=\frac{1}{n}\sum_{i=1}^n(\theta-X_i)\Bigl[1-\Lambda\Bigl(\frac{\theta^2}{2}-\theta X_i\Bigr)\Bigr]-\mathbb{E}_\theta\Bigl[(\theta-X_{i,\theta})\Lambda\Bigl(\frac{\theta^2}{2}-\theta X_{i,\theta}\Bigr)\Bigr] \approx\frac{1}{2n}\sum_{i=1}^n(\theta-X_i). \] Therefore, $\hat{\theta}$ is approximately the sample average, which is the MLE.
Second, if we use the logistic discriminator, the cross\hypentropy loss can be interpreted as the log likelihood of the logistic regression where the actual observations are labeled $1$ and the simulated $0$. \footnote{When $n\neq m$, the two sets of observations are weighted differently.} The resulting estimator for $\theta$ is then asymptotically equivalent to the optimally\hypweighted SMM with moments $\mathbb{E}[X_i]$ under $m\gtrsim n$ ((ref)). In practice, we may use a sieve of discriminators that can represent oracle $D_\theta$ asymptotically, e.g., the sieve of neural networks or the sieve of logistic discriminators with an increasing number of polynomials of $X$. In fact, we can regard $D_\theta$ as the nuisance parameter estimated in the inner maximization. (ref) presents conditions under which the estimation of $D_\theta$ via nonparametric estimation makes the adversarial estimator efficient.
The estimation algorithm is given in (ref). As is customary in simulation\hypbased methods, we use the same shocks $\{\tilde{X}_i\}_{i=1}^m$ to generate $\{X_{i,\theta}\}_{i=1}^m$ across different $\theta$. Note that since the optimal discriminator depends on $\theta$, we need to solve the inner maximization for each candidate value of $\theta$. When the transformation $T_\theta$ is differentiable in $\theta$, line (ref) can be performed by gradient descent.
The asymptotic distribution of the adversarial estimator depends on the choice of $\mathcal{D}_n$. If the discriminator is logistic, the asymptotic variance of the adversarial estimator coincides with SMM ((ref)). If $\mathcal{D}_n$ is a nonparametric discriminator, under some conditions, the asymptotic variance will be a function of the score and Hessian of the likelihood ((ref)). When the likelihood is intractable, estimating this asymptotic variance formula is not an easy task; we recommend using bootstrap in which we resample both $\{X_i\}_{i=1}^n$ and $\{\tilde{X}_i\}_{i=1}^m$ with replacement, even though it might be computationally costly.
We overview our estimator with simple examples to provide insights. For the case where our method is of practical interest, see (ref). The first example we consider is a logistic location model in which the mean is unknown and the variance is known. We illustrate three points using this example: (1) the adversarial estimator achieves parametric efficiency under correct specification; (2) the adversarial estimator is asymptotically normal under model misspecification; (3) the adversarial estimator is less sensitive to the curse of dimensionality compared to SMM. Next, we consider a Roy model with two occupations over two periods of time. This example illustrates the whole procedure of estimation and inference in a case when the likelihood is intractable.
We write $\mathbb{L}_\theta\coloneqq-\frac{1}{2n}\sum_{i=1}^n\log p_\theta(X_i)$ for minus half the log likelihood and $\mathbb{M}_\theta(D)\coloneqq\frac{1}{n}\sum_{i=1}^n\log D(X_i)+\frac{1}{m}\sum_{i=1}^m\log(1-D(X_{i,\theta}))$ for the sample objective function. Also, we let $\phi(x)\coloneqq\frac{1}{\sqrt{2\pi}}\exp(-x^2/2)$ be the standard normal pdf and $\Phi(x)\coloneqq\int_{-\infty}^x\phi(t)dt$ the standard normal cdf.
Suppose we have $n$ i.i.d.\ observations $X_1,\dots,X_n$ from the standard logistic distribution with pdf $p_0(x)=\Lambda(x)(1-\Lambda(x))$. Our structural model is the logistic distribution with unit scaling, i.e., $p_\theta(x)=\Lambda(x-\theta)(1-\Lambda(x-\theta))$. The oracle discriminator is given by \( D_\theta(x) =\Lambda(-\theta-2\log(1+e^{-x})+2\log(1+e^{-(x-\theta)})) \). The synthetic data is generated as $X_{i,\theta}=T_\theta(\tilde{X}_i)\coloneqq\theta+\tilde{X}_i$ where $\tilde{X}_i$ follows the standard logistic distribution. We set $n=m=300$ and run 500 replications.
To yield a discriminator capable of representing the oracle, we consider \( D_\lambda(x)=\Lambda(\lambda_0-2\log(1+e^{-x})+2\log(1+e^{-x+\lambda_1})) \) parameterized by $\lambda\in\mathbb{R}^2$. This class of discriminator is “correctly specified” in the sense that the oracle discriminator is given by $\lambda_\theta\coloneqq(-\theta,\theta)^\top$; thus, it allows us to obliterate the approximation error of the nonparametric estimator for $D_\theta$ and focus on the aspects conducive to efficiency. Nonetheless, we also present results with a nonparametric estimator, a shallow neural network, at the end of this section.
An intuition behind efficiency is that the curvature of $\mathbb{M}_\theta(\hat{D}_\theta)$ at $\theta_0$ is proportional to the Fisher information. (ref) illustrates this point. First, the curvature of $\mathbb{L}_\theta$ is a quarter of the Fisher information, and so is the curvature of the oracle loss $\mathbb{M}_\theta(D_\theta)$ ((ref)). Second, the estimated loss $\mathbb{M}_\theta(\hat{D}_\theta)$ traces $\mathbb{M}_\theta(D_\theta)$ very well. As a result, the curvature of $\mathbb{M}_\theta(\hat{D}_\theta)$ also becomes a quarter of the Fisher. This is somewhat surprising given that $\hat{D}_\theta$ is estimated separately for each $\theta$ ((ref), line 8); the plot of $\mathbb{M}_\theta(\hat{D}_\theta)$ could have been zigzag if maximization was noisy each time.
An important practice that effects “smooth” $\mathbb{M}_\theta(\hat{D}_\theta)$ is to use a deterministic algorithm for the inner maximization. Here, we use Matlab's fminsearch for maximization, which employs a deterministic algorithm. However, if some stochastic optimization is to be used, we advise that the random seed be reset to the same value each time maximization is carried out. For a logistic discriminator with differentiable $T_\theta$, (ref) shows that the estimated loss $\mathbb{M}_\theta(\hat{D}_\theta)$ will be smooth in $\theta$ if $\{\tilde{X}_i\}$ are fixed and the exact maximum is attained at the inner step for each $\theta$.
With the curvature of $\mathbb{M}_\theta(\hat{D}_\theta)$ matching $\mathbb{M}_\theta(D_\theta)$, the asymptotic variance of the adversarial estimator is $1+n/m$ times the inverse Fisher ((ref), (ref)). In this example, the theoretical asymptotic standard deviation of MLE is 1.73 while of the adversarial estimator is 2.45, which are closely reproduced in (ref).
Similar results hold when $m$ is increased (figures omitted); the curvatures of $\mathbb{M}_\theta(D_\theta)$ and $\mathbb{M}_\theta(\hat{D}_\theta)$ match closely with $\mathbb{L}_\theta$, and the adversarial estimator gets closer to MLE. For example, when $m=3{,}000$ (so $m=10 n$), the standard error of the adversarial estimator decreases to 2.00 (theoretically 1.94).
To see how a nonparametric discriminator fares, we also try a shallow neural network discriminator. The input is a one\hypdimensional observation $X$; there are three nodes in one hidden layer with a hyperbolic tangent activation function; the output is a sigmoid function. The neural network discriminator is trained for each $\theta$ using Matlab's train function, which is deterministic. (ref) shows that the estimated loss $\mathbb{M}_\theta(\hat{D}_\theta)$ still gives a good approximation to $\mathbb{M}_\theta(D_\theta)$. It is notable that as we increase $m$, the {\em level} of $\mathbb{M}_\theta(\hat{D}_\theta)$ becomes off from $\mathbb{M}_\theta(D_\theta)$, but the {\em curvature} is still correctly estimated ((ref)). If we adjust the level, it becomes clear the curvature matches that of the log likelihood ((ref)). According to our theory, the quality of the adversarial estimator hinges on the curvature of $\mathbb{M}_\theta(\hat{D}_\theta)$ but {\em not} on the level of $\mathbb{M}_\theta(\hat{D}_\theta)$ being close to $\mathbb{M}_\theta(D_\theta)$. Thus, the resulting estimator is very close to the oracle (figures omitted).
We also examine if bootstrap works on the adversarial estimator. The bootstrap consists of 500 replications with resampling both $\{X_i\}_{i=1}^n$ and $\{\tilde{X}_i\}_{i=1}^m$ with replacement but holding fixed the specification of the discriminator. The bootstrap standard error for the logistic discriminator is $2.29$ and for the neural network discriminator $2.52$, which are close to the theoretical limit $2.45$.
We now move to explore how the adversarial estimator behaves under misspecification. Suppose we misspecify the model to be a normal location family with unit variance, $p_\theta(x)=\frac{1}{\sqrt{2\pi}}\exp(-\frac{(x-\theta)^2}{2})$, while the true distribution is still the standard logistic distribution that has variance $\pi^2/3\approx 3.3$. The oracle discriminator is \( D_\theta(x) =\Lambda(\log\sqrt{2\pi}-x+\tfrac{1}{2}(x-\theta)^2-2\log(1+e^{-x})) \). Here, we use the correctly specified discriminator \( D_\lambda(x)=\Lambda(\lambda_0+\lambda_1 x+\lambda_2 x^2+\lambda_3\log(1+e^{-x})) \) parameterized by $\lambda\in\mathbb{R}^4$.
(ref) shows that the curvature of $\mathbb{L}_\theta$ is much steeper than $\mathbb{M}_\theta(D_\theta)$ due to misspecification (particularly to misspecification of variance). However, the estimated loss $\mathbb{M}_\theta(\hat{D}_\theta)$ still estimates the curvature of the oracle loss correctly. (ref) shows that the oracle adversarial estimator is approximately normal and comparable with quasi\hypMLE. A slight inflation of the variance is due to the fact that the adversarial estimator uses the synthetic data and gets affected by their randomness while quasi\hypMLE does not. (ref) shows that the adversarial estimator is very close to the oracle one. The theoretical asymptotic standard deviation of the adversarial estimator implied by (ref) is 2.27 while of quasi\hypMLE is 1.81. The observations for the increased synthetic sample size $m$ and for the neural network discriminator are analogous to (ref) and hence omitted for space.
Finally, we compare the adversarial estimator with SMM. As discussed, the adversarial estimator with a logistic discriminator is asymptotically equivalent to SMM. However, it is known that stacking up many moments yields poor finite\hypsample performance of SMM. To compare our estimator in this regard, the logistic location model is a particularly interesting one. Unlike the normal distribution, the sample average is not a sufficient statistic for the mean of a logistic distribution. Indeed, the collection of order statistics is known to be a minimal sufficient statistic. Technically speaking, therefore, the higher\hyporder moments $\mathbb{E}[X_i^2]$, $\mathbb{E}[X_i^3]$, $\dots$ do contribute in identifying the mean. In light of this, we carry out the following exercise.
For SMM, we consider matching (1) three moments $\mathbb{E}[X_i]$, $\mathbb{E}[X_i^2]$, $\mathbb{E}[X_i^3]$, (2) seven moments $\mathbb{E}[X_i],\dots,\mathbb{E}[X_i^7]$, and (3) eleven moments $\mathbb{E}[X_i],\dots,\mathbb{E}[X_i^{11}]$. Since the optimally\hypweighted SMM beats the unweighted SMM in all cases in our simulation, we only present the optimally\hypweighted SMM for comparison; the weights are estimated with the real data. For the adversarial estimator, we use the same set of moments as the inputs to the discriminator. In particular, the discriminator is the logistic classifier of the form \( D_\lambda(x)=\Lambda(\lambda_0+\lambda_1 x+\cdots+\lambda_d x^d) \) for $d=3,7,11$ parameterized by $\lambda\in\mathbb{R}^{1+d}$. In contrast to the one in (ref), this discriminator is “misspecified” but is good enough to yield a reasonable estimator for $\theta$. As discussed in (ref), the optimally\hypweighted SMM is asymptotically equivalent to the adversarial estimator with this choice of the discriminator. However, their finite\hypsample properties are subject to debate. For this exercise, we decrease the sample sizes to $n=m=200$ to emphasize the finite\hypsample performance.
(ref) shows the plots of the cross\hypentropy loss and the log likelihood for varying numbers of inputs. It is noteworthy that the curvature of the estimated loss $\mathbb{M}_\theta(\hat{D}_\theta)$ is very close to the oracle one up to seven moments. We see nonnegligible deviation of the curvature for eleven moments, but as we see below, it is still sharp enough to yield a much better estimator than SMM.
The first row of (ref) shows the histogram of the optimally\hypweighted SMM. The horizontal scales of the figures are adjusted to match the distribution of SMM; MLE is the same for all figures and serves as the reference point. We see that the precision of SMM deteriorates quickly as the number of moments increases. For eleven moments, the standard error is eight times as large as MLE. The second row of (ref) presents the adversarial estimator. Even for seven inputs, the adversarial estimator is as tight as MLE, and for eleven moments, it is still comparable (three\hyptimes larger standard error). This shows that the adversarial estimator is less sensitive to the number of moments compared to SMM. This can especially be an advantage when we do not know which moments to match.
We also note that since the moments are highly correlated, the estimation of the discriminator gives warnings of multicollinearity, but it does not impair the quality of the subsequent estimator $\hat{\theta}$. This is insightful for a more general neural network discriminator since neural network weights are not identified uniquely. This observation is in line with our theory that depends on the quality of the estimator $\hat{D}_\theta$ for $D_\theta$ but not on the quality of the estimator $\hat{\lambda}_\theta$ for $\lambda_\theta$.
The improvement of our method relative to SMM is analogous to the improvement of empirical likelihood to GMM i2002. SMM, like GMM, suffers from substantial bias when the number of moments is large; our method, like empirical likelihood, has better finite\hypsample and large\hypsample properties at the expense of computational cost. The idea of both comes from treating the nuisance component as a kind of a nonparametric maximum likelihood problem. Meanwhile, both SMM and GMM retain the advantage of simplicity to easily accommodate time series settings.
We consider the following model of comparative advantage, for which the likelihood is not available under some configurations of the parameter values. Suppose there are two sectors and two periods. In each period, an agent chooses the sector to work in to maximize her present and discounted future expected wages. The wage $w_{i1s}$ for agent $i$ in period 1 in sector $s$ is determined by $\log w_{i1s}=\mu_s+\varepsilon_{i1s}$, and the wage $w_{i2s}$ for agent $i$ in period 2 in sector $s$ by $\log w_{i2s}=\mu_s+\gamma_s\mathbbm{1}\{d_{i1}=s\}+\varepsilon_{i2s}$ where $d_{i1}$ is the sector choice of agent $i$ in period 1. The parameter $\mu_s$ represents the base wage in sector $s$ and $\gamma_s$ the returns to experience in sector $s$. The error terms are observable to the agent in respective periods (so she observes $\varepsilon_{i1\mathpalette\bigcdot@{.5}}$ in period 1 and $\varepsilon_{i2\mathpalette\bigcdot@{.5}}$ in period 2) and distribute as \[
=N\!\left(
,
\right)\!. \] Observable to us is the quartet $X_i=(\log w_{i1},d_{i1},\log w_{i2},d_{i2})$ of realized log wages and sector choices in both periods. They are functions of above variables by $w_{i1}=w_{i1d_{i1}}$, $d_{i1}=\mathop{\arg\max}_{s\in\{1,2\}}w_{i1s}+\beta\mathbb{E}[w_{i2}\mid d_{i1}=s]$, $w_{i2}=w_{i2d_{i2}}$, and $d_{i2}=\mathop{\arg\max}_{s\in\{1,2\}}w_{i2s}$ where $\beta$ is the discount factor. We fix $\beta=0.9$, so $\beta$ is not a free parameter.
As a first exercise, we show that the adversarial estimator has a computational advantage over MLE. To this end, we fix $\rho_t=0$ to have a tractable likelihood. Thus, the parameter of interest is $\theta=(\mu_1,\mu_2,\gamma_1,\gamma_2,\sigma_1,\sigma_2,\rho_s)$. The true value is $\theta_0=(1.8,2,0.5,0,1,1,0.5)$. We set the sample sizes at $n=m=300$.
Although the likelihood is available, the correct functional form of $D_\theta$ is not easy to derive. So, we skip the correctly specified discriminator and use the neural network discriminator for the feasible adversarial estimator. The neural network has one hidden layer with 10 nodes with a hyperbolic tangent activation function. The input is $X_i$ without transformation. The output layer uses a sigmoid function.
Note that if $w_{i11}+\beta\mathbb{E}[w_{i2}\mid d_{i1}=1]<\beta\mathbb{E}[w_{i2}\mid d_{i1}=2]$, there is no way that agent $i$ chooses sector 1 in period 1. Therefore, if we see a pair $(w_{i1},d_{i1})=(w_{i11},1)$ that satisfies this inequality for a particular $\theta$, this observation is not supported by $P_\theta$. This is indeed a common phenomenon. (ref) plots the loss and the log likelihood against each parameter, holding all other parameters to the truth. The range of the figures reflects the range of MLE and the adversarial estimator. In this “relevant” region, we see that $\mathbb{L}_\theta$ sometimes breaks off; this is because the discontinued part does not support the real data so $\mathbb{L}_\theta$ is infinity.
Aside from possible inefficiency, this is not a problem for MLE insofar as the likelihood maximizer can be found. However, there may be a trouble when the initial value of $\theta$ does not support the real data. In fact, if we do not pick the initial value carefully, Matlab's fminsearch wanders around the unsupported region and returns a meaningless value after the evaluation count hits the limit. Meanwhile, (ref) indicates that such a problem does not occur for the cross\hypentropy loss; indeed, $\mathbb{M}_\theta(D_\theta)$ extends a nice curve throughout the “unsupported” region. The key is in the robustness of the sample Jensen--Shannon divergence \[ \frac{1}{2}\mathbb{M}_\theta(D_\theta)=\frac{1}{2n}\sum_{i=1}^n\log\frac{p_0(X_i)}{p_0(X_i)+p_\theta(X_i)}+\frac{1}{2m}\sum_{i=1}^m\log\frac{p_\theta(X_{i,\theta})}{p_0(X_{i,\theta})+p_\theta(X_{i,\theta})}. \] When a single observation $X_i$ is not on the support of $p_\theta$, the corresponding fraction is $1$, which does not ruin the sum so we can still calculate a meaningful distance using remaining observations; hence the curve continues. Moreover, even if the entire observations $\{X_i\}$ go outside the support, the divergence still works as long as (some of) synthetic data are on the support of $p_0$ and the second sum is informative. It is only when both the entire real sample $\{X_i\}$ and the synthetic sample $\{X_{i,\theta}\}$ are outside the supports of $p_\theta$ and $p_0$ respectively that the Jensen--Shannon divergence gets fixated at $0$ and loses guidance on $\theta_0$. \footnote{If the supports of $p_0$ and $\{p_\theta\}$ are fully disjoint, the Jensen--Shannon projection $\theta_0$ is not defined.} This is the intuition why the adversarial estimator does not suffer from the support issue in the Roy model. We can also see this as a virtue of estimating the likelihood ratio as opposed to the raw likelihood.
This is not to say, however, that the cross\hypentropy loss works for any kind of disjointly supported distributions. When GAN is used to generate high\hypdimensional data like images, the cross\hypentropy loss is known to be very hard to train, partly because of the severe disjoint support problem. An alternative is the Wasserstein loss that puts a nice gradation on the distance between completely disjoint distributions.
(ref) also illustrates that, despite having discrete observables (sector choices), the objective functions are very smooth thanks to continuous observables (wages), so there is no need for smoothing in contrast to (ref). The resulting estimators are comparable with MLE just as in the previous examples ((ref)).
Now, we illustrate the whole procedure of estimation and inference using the Roy model with intractable likelihood. Let us consider the same model as (ref) without assuming $\rho_t=0$ , so the parameter of interest is $\theta=(\mu_1,\mu_2,\gamma_1,\gamma_2,\sigma_1,\sigma_2,\rho_t,\rho_s)$. The true values are the same as before. We first pre\hypestimate the model with a logistic discriminator and then estimate it with a neural network discriminator using the logistic estimator as the initial value. Since it is naturally speculated that identification comes from the moments of the log wages, we consider the logistic discriminator of the form $D_\lambda(\log w_1,d_1,\log w_2,d_2)=\Lambda(\lambda_0+\lambda_1\log w_1+\lambda_2 d_1+\lambda_3\log w_2+\lambda_4 d_2+\lambda_5(\log w_1)^2+\lambda_6(\log w_2)^2+\lambda_7\log w_1\log w_2)$.
As the curvature of the second logistic loss is quite sharp, we may in practice stop here and go with the logistic estimator. For illustration, we move on to the neural network discriminator with the same configuration as (ref). The loss is plotted as the blue line in (ref). The vertical blue dotted lines indicate the neural network estimator. \footnote{Note that the global minimizer is not the same as the local minimizers of the figures since the parameters are fixed at the logistic estimator.}
Next, we use bootstrap to compute the standard errors. We resample both the actual data $\{X_i\}_{i=1}^n$ and the simulation shocks $\{\tilde{X}_i\}_{i=1}^m$ with replacement, pre\hypestimate the model with the second logistic discriminator, and then estimate the model with the neural network discriminator. (ref) shows the bootstrap samples of the logistic estimator (red) and the neural network estimator (blue). Due to some degree of roughness of the neural network loss and difficulty in identifying the global minimum, the neural network sample is overall more dispersed than the logistic estimator. However, the neural network estimator is still comparable and sometimes produces even tighter estimates (for $\sigma_1$ and $\sigma_2$). Note that the neural network takes as inputs the raw quartet but not the higher\hyporder moments. So, the neural network with one hidden layer of 10 nodes “figures out” the correct moments to match and produces an estimator comparable with (and partly superior to) the logistic discriminator whose inputs were deliberately chosen.
(ref) presents the estimates and the standard errors (not multiplied by $\sqrt{n}$). Along with the adversarial estimator, we present the results of SMM. SMM matches the same seven moments as the inputs to the second logistic discriminator: $\mathbb{E}[\log w_{i1}]$, $\mathbb{E}[d_{i1}]$, $\mathbb{E}[\log w_{i2}]$, $\mathbb{E}[d_{i2}]$, $\mathbb{E}[(\log w_{i1})^2]$, $\mathbb{E}[(\log w_{i2})^2]$, and $\mathbb{E}[\log w_{i1}\log w_{i2}]$. The optimal weights are estimated with the actual data. We see that the adversarial estimator with the logistic discriminator is slightly more precise than SMM.
Not every aspect of our method is superior to alternatives. First, the theoretical results in this paper do not cover time series data. The Roy model has a dynamic choice of individuals, but we have i.i.d.\ observations of individuals. This is not to say that the adversarial framework cannot be extended thereto, but it would require a careful design of the discriminator to incorporate the structure of serial correlation.
Second, the adversarial estimator can be time\hypconsuming. A logistic discriminator is as fast as II, but a neural network discriminator can take a long time to train. In the logistic location model, both MLE and the adversarial estimator with a logistic discriminator take less than a second, while the adversarial estimator with a neural network discriminator takes about 30 seconds on a laptop without a GPU or parallelization. For this, we recommend pre\hypestimation with a logistic discriminator or other existing methods to start with a good initial value.
The third drawback is a possible roughness of the loss surface. As seen in (ref), a logistic discriminator tends to yield a very smooth objective function ((ref)) while a neural network discriminator may sometimes get bumpy and have spurious local minima ((ref)). Some degree of roughness can be smoothed with the choice of a training method or an increased number of iterations; additionally, we can estimate the discriminator several times and take their average and/or use an optimization method tailored for noisy functions. If the initial value is good enough, we may also employ grid search in the neighborhood to skip estimation of the gradient. At any rate, we recommend plotting the loss surface before outer minimization.
Fourth, being comparable with MLE, the asymptotic variance of the adversarial estimator depends on the score and Hessian ((ref)), which is not easy to compute given intractable likelihood. Therefore, we may resort to resampling methods like bootstrap to obtain a variance estimator, which can cost additional time.
This section derives the asymptotic properties of the adversarial estimator. For more general results, we refer the reader to our earlier version kmp2022.
Let $\tilde{X}_i\sim\tilde{P}_0$ be a common random shock used in simulation. The simulated observation $X_{i,\theta}\sim P_\theta$ is then constructed by transforming $\tilde{X}_i$ through a map, $X_{i,\theta}=T_\theta(\tilde{X}_i)$. For a function $f$, the sample averages of $f(X_i)$ and $f(X_{i,\theta})$ are denoted by $\mathbb{P}_0 f\coloneqq\frac{1}{n}\sum_{i=1}^n f(X_i)$ and $\mathbb{P}_\theta f\coloneqq\frac{1}{m}\sum_{i=1}^m f(X_{i,\theta})$. Their population counterparts are denoted as $P_0 f\coloneqq\int f(x)dP_0$ and $P_\theta f\coloneqq\int f(x)dP_\theta$. We denote the population objective function by $M_\theta(D)\coloneqq P_0\log D+P_\theta\log(1-D)$ as well as the previously defined sample objective function $\mathbb{M}_\theta(D)\coloneqq\mathbb{P}_0\log D+\mathbb{P}_\theta\log(1-D)$. We also define the distance on $\Theta$ by $h(\theta_1,\theta_2)\coloneqq\sqrt{\int(\sqrt{p_{\theta_1}}-\sqrt{p_{\theta_2}})^2}$.
Suppose that observables can be written as $X_i=(y_i,x_i)$ where $\theta$ affects only the conditional distribution of $y_i$ given $x_i$. Such $x_i$ is called the covariate. In the maximum likelihood literature, it is known that an efficient estimator is obtained by maximizing the conditional likelihood of $y_i$ given $x_i$, so the marginal distribution of $x_i$ can be left unspecified. The same observation holds true in the adversarial framework. Namely, the oracle discriminator $D_\theta$ does not depend on the marginal distribution of $x_i$, so the distributions $P_0$ and $P_\theta$ can be regarded as specifying only the conditional distribution of $y_i$ given $x_i$. In our theory, we save notational complexity by allowing this implicitly. One possible complication this might bring is the method to draw covariates for the simulated data. In (ref), we set $n=m$ and use the same sets of covariates in the actual data. Another possibility is to bootstrap the covariates.
The adversarial estimator is consistent if the estimated loss $\mathbb{M}_\theta(\hat{D}_\theta)$ converges uniformly to the oracle loss $\mathbb{M}_\theta(D_\theta)$ and $\hat{\theta}$ finds a global minimizer. As the maximized cross\hypentropy loss is effectively bounded between $2\log(1/2)$ and $0$, uniform convergence on $\Theta$ is not an unreasonable assumption.
This theorem does not assume that the generative model is parametric, so it also applies to possibly “nonparametric” generators.
To obtain a rate of convergence of the generator, we assume that the structural model is parametric.
To establish asymptotic results in terms of $n$, we next assume that the synthetic sample size $m$ grows as fast as $n$. It is allowed (but not required) that $m$ diverges faster than $n$, in which case $n/m$ converges to $0$.
The next assumption ensures that the estimation procedure finds a good minimum and that the derivative of the estimated loss converges to that of the oracle. The first property hinges on the estimation procedure employed, the tolerance level, etc. The second property is used in semiparametric $M$\hypestimation to obtain a regular estimator orthogonal to nuisance estimation ks1993. We revisit the plausibility of this condition in (ref).
The next assumption consists of three parts. The first is a stronger identification condition than in (ref) that ensures a quadratic curvature at $\theta_0$; this is implied by the positive definiteness of $\tilde{I}_{\theta_0}$ in (ref). The second assumes a degree of smoothness needed for $T_\theta$; this is trivial with $\tau_n\equiv 0$ if $n/m\to 0$ or (ref) holds; otherwise, if $T_\theta$ and $D_{\theta_0}$ are differentiable in $\theta$ and $x$ respectively and (ref) holds, there is a closed\hypform expression for $\tau_n$, which we derive in (ref). Third, we need that $P_0$ is “close enough” to $P_{\theta_0}$ in the sense that convergence of $\theta$ to $\theta_0$ takes place on the support of $P_0$; this is also trivial under (ref).
The second condition of (ref), which we call orthogonality, is essential in the rate of convergence for $\hat{\theta}$ in (ref). Even in the best scenario, we can only expect $\mathbb{M}_\theta(\hat{D}_\theta)-\mathbb{M}_\theta(D_\theta)=O_P(n^{-1})$, so the convergence of $\hat{D}_\theta$ alone does not grant orthogonality. The key to satisfying it is, therefore, some extent of the convergence of the {\em derivative} of $\hat{D}_\theta$ with respect to $\theta$ to that of $D_\theta$. Note that this is different from the derivative of $\hat{D}_\theta$ with respect to $x$, so it does not follow from the convergence of the derivative of a nonparametrically estimated function. Rather, it is the structure of the nested optimization that brings about orthogonality.
Take the logistic discriminator $D_\lambda(x)=\Lambda(x^\top\lambda)$ as considered in (ref). We can check that orthogonality holds if the following conditions are met. Let $\mathbb{E}_n f(X)\coloneqq\frac{1}{n}\sum_{i=1}^n f(X_i)$ and $\mathbb{E}_m f(X_\theta)\coloneqq\frac{1}{m}\sum_{i=1}^m f(X_{i,\theta})$ and denote the differentiation with respect to a row vector $\theta^\top$ by a dot, e.g., $\dot{\lambda}_\theta=\frac{\partial}{\partial\theta^\top}\lambda_\theta$.
For ease of notation, we assume that $\lambda$ and $\theta$ are one\hypdimensional; however, the argument below applies equally to the vector case. The FOC for $\hat{\lambda}_\theta$ yields \( \mathbb{E}_n[(1-\Lambda(X\hat{\lambda}_\theta))X]-\mathbb{E}_m[\Lambda(X_\theta\hat{\lambda}_\theta)X_\theta]=0 \). This holds for every $\theta$, so we may differentiate both sides by $\theta$, which can be solved for the derivative of $\hat{\lambda}_\theta$ with respect to $\theta$,
Note that $\lambda_\theta$ satisfies the population FOC, which leads to the population counterpart of the same expression, so $\dot{\hat{\lambda}}_\theta$ is consistent for $\dot{\lambda}_\theta$. Moreover, by the uniform convergence assumptions, we deduce $\sup_\theta\|\dot{\hat{\lambda}}_\theta-\dot{\lambda}_\theta\|=O_P(n^{-1/2})$. Thus, the derivative of the discriminator converges.
To derive orthogonality, we first Taylor\hypexpand it in $\lambda$ around $\hat{\lambda}_\theta$. In doing so, the first\hyporder term can be ignored thanks to Condition (ref). For arbitrary $\theta$,
Next, we expand it further in $\theta$ around $\hat{\theta}$.
This is $O_P(n^{-1})$ for fixed $\theta$, so we can take a shrinking neighborhood of $\theta$ around $\theta_0$ that contains $\hat{\theta}$ to make the supremum of this $o_P(n^{-1})$, yielding orthogonality. If the neighborhood shrinks only slightly slower than $n^{-1/2}$, then convergence of $\hat{\lambda}_\theta$ and $\dot{\hat{\lambda}}_\theta$ can be relaxed to as slow as $o_P(n^{-1/4})$ if possibly a few more degrees of differentiability and finite moments are granted. It is also straightforward to relax the exact FOC condition to allow for errors of negligible order and to allow for nonlinear but parametric logistic discriminators, such as small neural networks. An interesting conclusion of this is that the curvature of the estimated loss converges faster than the level, as observed throughout (ref).
For a general nonparametric discriminator, it is not trivial to obtain a similar low\hyplevel condition. (ref) develops conditions for $\hat{D}_\theta$ to converge faster than $n^{-1/4}$ (pointwise in $\theta$), which seems necessary but is not sufficient to derive orthogonality. \footnote{In a similar situation where the derivative of the nuisance parameter identifies $\theta$, ks1993 exploit the structure of a kernel density estimator to show the convergence of the derivative, whereby obtaining a corresponding orthogonality condition.} In (ref), the plots of $\mathbb{M}_\theta(\hat{D}_\theta)$ confirm orthogonality in examples with or without differentiability.
We may derive a closed\hypform expression for $\tau_n$ in (ref) when $T_\theta$ is differentiable in $\theta$ and $D_{\theta_0}$ in $x$. Suppose that $\mathcal{X}$ and $\tilde{\mathcal{X}}$ are Euclidean spaces; denote the differentiation with respect to an argument by a prime and with respect to a subscript by a dot, e.g., $f_\theta'(x)=\frac{\partial}{\partial x}f_\theta(x)$ and $\dot{f}_\theta(x)=\frac{\partial}{\partial\theta^\top}f_\theta(x)$. Observe that
Thus, we find \( \tau_n=\dot{T}_{\theta_0}^\top(D_{\theta_0}[p_{\theta_0}'/p_{\theta_0}-p_0'/p_0]\circ T_{\theta_0}) \).
Note that this assumption only matters for the misspecified case. (ref) verifies this assumption for the misspecified model in (ref) in the regions relevant for the asymptotic distribution. The red dashed lines plot the LHS of (ref), \( n(\tilde{\mathbb{P}}_0-\tilde{P}_0)(\log(1-D_{\theta_0})\circ T_{\theta+h/\sqrt{n}}-\log(1-D_{\theta_0})\circ T_{\theta_0}) \). For the normally\hypmisspecified logistic location model in (ref), we see that this line is already very linear and its slope corresponds to $\tau_n$.
To derive the asymptotic distribution of the adversarial estimator, we need the structural model to be differentiable as in maximum likelihood.
We further impose a mild smoothness condition on $T_\theta$.
Similarly to (ref), (ref) is trivial if $n/m\to 0$. Moreover, under the low\hyplevel conditions (and notation) in (ref),
Therefore, if $\tilde{\mathbb{P}}_0[\dot{T}_{\theta_0}^\top(D_{\theta_0}'\dot{\ell}_{\theta_0}+D_{\theta_0}\dot{\ell}_{\theta_0}')\circ T_{\theta_0}]$ is consistent for $\tilde{P}_0[\dot{T}_{\theta_0}^\top(D_{\theta_0}'\dot{\ell}_{\theta_0}+D_{\theta_0}\dot{\ell}_{\theta_0}')\circ T_{\theta_0}]$, (ref) holds even if $m\sim n$.
(ref) verifies (ref) for the normally\hypmisspecified model in (ref). The blue dotted lines plot the LHS of (ref), \( \sqrt{n}[(\mathbb{P}_{\theta_0+h/\sqrt{n}}-P_{\theta_0+h/\sqrt{n}})-(\mathbb{P}_{\theta_0}-P_{\theta_0})]D_{\theta_0}\dot{\ell}_{\theta_0} \).
A stronger efficiency result holds if the structural model is correctly specified.
Thus, if $n/m\to0$, the adversarial estimator attains parametric efficiency.
Our theory assumes that $\mathcal{D}$ is a sieve that eventually is capable of representing $D_\theta$. In finite samples, however, we do not know how well $\mathcal{D}$ approximates $D_\theta$. Therefore, it is interesting to see what happens when $\mathcal{D}$ is not a sieve but a fixed class of functions. Although the complete treatment of this case is beyond our scope, we examine what happens to the population problem as we enrich $\mathcal{D}$, e.g., by gradually adding nodes and layers to the neural network. \footnote{The case where $\mathcal{D}$ is fixed to be the class of logistic discriminators is analyzed in (ref).}
For simplicity, we maintain (ref) and assume that $\mathcal{D}$ contains a constant function $1/2$. Let $\tilde{D}_\theta$ be the population maximizer of $M_\theta(D)$ in $\mathcal{D}$. Since $M_\theta(D)-M_\theta(D_\theta)=-2d_\theta(D,D_\theta)^2+o(d_\theta(D,D_\theta)^2)$ by (ref), $\tilde{D}_\theta$ is equivalent to a minimizer of $d_\theta(D,D_\theta)^2$ in $\mathcal{D}$ up to $o(d_\theta(D,D_\theta)^2)$. Under (ref), $\tilde{D}_{\theta_0}=D_{\theta_0}\equiv1/2$ and $M_{\theta_0}(1/2)=M_\theta(1/2)$. By (ref),
Note that by (ref), \( d_\theta(D_{\theta_0},D_\theta)^2 =\tfrac{1}{2}\int\tfrac{p_0}{p_0+p_0}(\sqrt{p_0}-\sqrt{p_\theta})^2+\tfrac{1}{2}\int\tfrac{p_\theta}{p_\theta+p_\theta}(\sqrt{p_0}-\sqrt{p_\theta})^2+o(h(p_0,p_\theta)^2) =\tfrac{1}{2}h(p_0,p_\theta)^2+o(h(p_0,p_\theta)^2) \). Thus, we obtain \[ M_{\theta_0}(\tilde{D}_{\theta_0})-M_\theta(\tilde{D}_\theta)=-h(p_0,p_\theta)^2+2d_\theta(\tilde{D}_\theta,D_\theta)^2+o(h(p_0,p_\theta)^2). \] If $\mathcal{D}$ contains $D_\theta$, the second term is zero and the Hellinger curvature allows us to estimate $\theta$ efficiently; if $\mathcal{D}$ is a singleton set that contains only $1/2$, the first and second terms cancel and the objective function becomes completely flat, rendering estimation of $\theta$ impossible. Therefore, the second term represents the loss in efficiency due to the limited capacity of $\mathcal{D}$. For the regular logit case, we know that $\mathcal{D}$ is already rich enough that the curvature admits $\sqrt{n}$\hypestimation. Then, as we enrich $\mathcal{D}$, it becomes more capable of minimizing $d_\theta(\tilde{D}_\theta,D_\theta)^2$, getting closer to efficiency.
Using the adversarial framework, we examine the elderly's saving, following dfj (henceforth \citetalias{dfj}). The elderly save for various reasons---uncertainty on survival, bequest motive, or ever\hyprising medical expenses as they age. Different motives for saving yield different implications on policy evaluation such as Medicaid and Medicare. Hence, it is an important and active area of research.
The risk the elderly face is highly heterogeneous, depending on their gender, age, health status, and permanent income. This implies potentially large heterogeneity in the saving motive across individuals; not accounting for this can bias the estimates of utility. For example, the rich live several years more than the poor on average. Failure to reflect this difference can make the rich look thriftier than they are. On the other hand, existing estimation methods such as SMM may suffer from severe lack of precision when various heterogeneity is introduced. This motivates adversarial estimation as a more efficient alternative to SMM. Indeed, our adversarial estimates, using the same model and the same data as in \citetalias{dfj}, will see considerable gains in precision.
We focus on the behavior of single, retired individuals of age 70 and older. In each period, a surviving single retired agent receives utility $u(c)$ from consumption $c$ and, if they die in that period, additional utility $\phi(e)$ from leaving estate $e$, where \[ u(c)\coloneqq\frac{c^{1-\nu}}{1-\nu}, \qquad\quad \phi(e)\coloneqq\vartheta\frac{(e+k)^{1-\nu}}{1-\nu}, \] and $\nu$ is the relative risk aversion and $\vartheta$ and $k$ are the intensity and curvature of the bequest motive. Each individual is associated with gender $g$ and permanent income $I$, and carries six state variables: age $t$, asset $a_t$, nonasset income $y_t$, health status $h_t$, medical expense shock $\zeta_t$, and survival $s_t$. Health and survival are binary, where $h_t=1$ means they are healthy at age $t$, and $s_t=1$ they survive to the next period.
They face three channels of uncertainty: health, survival, and medical expenses. Heath and survival evolve as Markov chains. We denote \[ \pi_{H}(g,h_t,I,t)\coloneqq\Pr(h_{t+1}=1\mid g,h_t,I,t), \quad \pi_{S}(g,h_t,I,t)\coloneqq\Pr(s_{t+1}=1\mid g,h_t,I,t). \] The medical expenses they incur are given by \( \log m_{t}=m(g,h_t,I,t)+\sigma(g,h_t,I,t)\psi_{t} \), where $m$ and $\sigma$ are deterministic functions, $\psi_{t}=\zeta_{t}+\xi_{t}$, $\xi_{t}\sim N(0,\sigma_{\xi}^{2})$, $\zeta_{t}=\rho\zeta_{t-1}+\epsilon_{t}$, and $\epsilon_{t}\sim N(0,\sigma_{\epsilon}^{2})$. The nonasset income evolves deterministically as $y_t=y(g,I,t)$. The asset evolves as \( a_{t+1}=a_{t}+y_{n}(ra_{t}+y_{t},\tau)+b_{t}-m_{t}-c_t \), where $b_{t}\geq0$ is the {\em government transfer}, $r$ the {\em risk\hypfree pretax rate of return}, $y_{n}(\cdot,\tau)$ the {\em posttax income}, and $\tau$ the {\em tax structure}. The agent faces a borrowing constraint $a_t\geq0$ while social insurance guarantees minimum consumption $c_t\geq\underline{c}$; government transfer $b_t$ is positive only when both constraints cannot be satisfied without it.
The timing in each period is given as follows. Heath $h_t$ and medical expenses $m_t$ realize; then the individual chooses consumption $c_t$; then survival $s_t$ realizes; if $s_t=0$, they leave the remaining assets as bequest; if $s_t=1$, move on to the next period.
Denoting the {\em cash\hypon\hyphand} by $x_t\coloneqq c_t+a_{t+1}$, the agent's Bellman equation is \[ V_{t}(x,g,h,I,\zeta)=\max_{c,x'} \, u(c,h)+\beta[s\mathbb{E}_tV_{t+1}(x',g,h',I,\zeta')+(1-s)\phi(e)] \] subject to $x'=(x-c)+y_{n}(r(x-c)+y',\tau)+b'-m'$, $e=(x-c)-\max\{0,\tilde{\tau}(x-c-\tilde{x})\}$, and $x\geq c\geq\underline{c}$. The first constraint is the budget constraint; the second the bequest (taxed at rate $\tilde{\tau}$ with deduction $\tilde{x}$); the last the borrowing and consumption constraints.
We also look at two transformations: the {\em marginal propensity to consume at the moment of death} $\text{MPC}\coloneqq(1+r)/(1+r+[\beta\vartheta(1+r)]^{1/\nu})$ and the {\em implied asset floor} $\underline{a}\coloneqq k/[\beta\vartheta(1+r)]^{1/\nu}$ above which individuals get utility from bequeathing. \footnote{The {\em marginal propensity to bequeath (MPB)} is defined by $1-\text{MPC}$.}
We use the same data as \citetalias{dfj}, taken from {\em Assets and Health Dynamics Among the Oldest Old (AHEAD)}. The sample consists of non\hypinstitutionalized individuals of age 70 and older in 1994. It contains 8,222 individuals in 6,047 households (3,872 singles and 2,175 couples). The survey took place biyearly from 1994 to 2006. We focus on 3,259 single retired individuals, 592 of which are men and 2,667 women. \footnote{Single individuals are those who were neither married nor cohabiting at any point in the analysis.} Of those, 884 were alive in 2006. We drop the first survey in 1994 for reliability, following \citetalias{dfj}.
The survey collects information on age $t$, financial wealth $a_t$, nonasset income $y_t$, medical expenses $m_t$, and health status $h_t$. Financial wealth includes real estate, autos, several other liquid assets, retirement accounts, etc. Nonasset income includes social security benefits, veteran's benefits, and other benefits. Medical expenses are total out\hypof\hyppocket spending; the average yearly expenses are \$3,700 with standard deviation \$13,400. The permanent income is not observed, but we use as a proxy the ranking of individual average income over time. The health status is a binary variable indicating whether the individual perceives herself as healthy.
Following \citetalias{dfj}, we carry out estimation in two steps: (1) estimate $\pi_H$, $\pi_S$, $m$, $\sigma$, $\rho_m$, $\sigma_\xi$, $\sigma_\epsilon$ (in fact, we borrow numbers from \citetalias{dfj}), (2) estimate $\nu$, $\text{MPC}$, and $k$ using our adversarial approach. The parameters $r$, $\tau$, $\tilde{\tau}$, and $\tilde{x}$ are fixed as in the original paper, and $\beta = 0.971$. For $\underline{c}$, we fix it at \$4,500 to reflect annual social security payments. \footnote{In their preferred specification, \citetalias{dfj} estimate $\beta$ and $\underline{c}$ in addition to $\nu$, MPC, and $k$. Instead, we fix $\beta$ and $\underline{c}$ to reasonable values in the literature. Changing $\underline{c}$ mostly affects estimates of $\nu$.} After the second step, we can also recover $\vartheta$ and $\underline{a}$.
We consider two different sets of inputs to the discriminator. The first set consists of the log age of an individual in 1996, permanent income (the aforementioned proxy), the profile (full history) of asset holdings, and the profile of survival indicators, \footnote{All individuals are alive in 1996, so we drop $s_{t_{1996}}$.} \[ X_1\coloneqq(1,\log t_{1996},I,a_{t_{1996}},\dots,a_{t_{2006}},s_{t_{1998}},\dots,s_{t_{2006}})\in\mathbb{R}^{14}. \] This is intended to capture similar identifying variation as \citetalias{dfj}. \footnote{\citetalias{dfj} use median assets as moments. Unlike in (ref), median moments cannot be translated into logistic inputs that yield an asymptotically equivalent estimator.} The second set is augmented with gender and the profile of health status, \[ X_2\coloneqq(X_1,g,h_{t_{1996}},\dots,h_{t_{2006}})\in\mathbb{R}^{21}, \] aiming to capture more variation for the bequest motive as explained in (ref).
We use cross validation to configure the discriminator. We focus on feed\hypforward neural networks with sigmoid activation functions with at most two hidden layers. We fix $\theta$ at a preliminary estimate; split the actual data into sample 1 (80%) and sample 2 (20%); estimate $D$ with sample 1, varying the numbers of nodes and layers; evaluate their classification accuracy with sample 2; \footnote{We use the classification accuracy provided by Keras's ADAM based on thresholding.} pick the network configuration with the highest accuracy. The selected neural network discriminator consists of two hidden layers, the first with 20 nodes and the second 10 nodes.
We compare our estimates with SMM in \citetalias{dfj}. They use 150 moments of median assets of groups divided by the cohort and permanent income quintile in each year. The cohort is defined on a four\hypyear window; Cohort 1 are those who were 72--6 years old in 1996; Cohort 2 were 77--81; Cohort 3 were 82--6; Cohort 4 were 87--91; Cohort 5 were 92 and older. Details are in \citetalias{dfj}. We note that accounting for health and gender is infeasible in SMM since it yields too many moments.
(ref) gives the parameter estimates from \citetalias{dfj} and our adversarial method with specifications $X_1$ and $X_2$. Parenthesized numbers are the standard errors; we use honore2017poor to compute them for the adversarial estimates. The first row is the SMM estimates in \citetalias{dfj}. The second and third rows come from the adversarial estimation; the second uses $X_1$ (14 variables) and the third $X_2$ (21 variables).
A major difference between our estimates and \citetalias{dfj}'s is the curvature of the utility of bequests $k$. Ours is an order of magnitude smaller, which has an important implication: while \citetalias{dfj} conclude only the super rich would obtain utility from bequeathing, our estimate suggests bequeathing matters across the entire permanent income distribution. A related number is the implied asset floor $\underline{a}$. We obtain estimates of \$1,320 and \$4,243, which are on the lower side of the estimates known in the literature. However, they correspond respectively to the 22nd and 24th percentiles of the distribution of assets one period before deaths (see (ref)) in our sample. We interpret these numbers as providing a sensible fit of the data. In contrast, \citetalias{dfj}'s implied asset floor is \$36,215, which corresponds to the 40th percentile.
Overall, the intensity of the bequest motive is minor in \citetalias{dfj} and $X_1$ but non\hypnegligible in $X_2$. While $k$ is low for both $X_1$ and $X_2$, MPC is almost twice as large in $X_1$ compared to $X_2$. Consequently, individuals care about bequests less than their own consumption according to $X_1$. \citetalias{dfj} and adversarial also differ in risk aversion $\nu$. A large value of risk aversion rationalizes the observed saving patterns when the consumption floor $\underline{c}$ is fixed at \$4,500, a reasonable value in the literature. \footnote{\citetalias{dfj}'s risk aversion estimate increases from 3.84 to 6.04 in an alternative specification where $\underline{c}$ is fixed at \$5,000. However, according to their criterion, the fit of the model decreases substantially.}
In line with our theory, the adversarial estimation provides substantial gains in precision relative to \citetalias{dfj}. The decrease in standard errors reflects that the data is sufficiently powerful to conclude the importance of the bequest motive, especially when exploiting additional variation in gender and health.
The last column reports the cross\hypentropy loss for each method. To make a fair comparison, we take the estimates of each method and train the discriminator using $X_2$ as the input. The loss for adversarial $X_1$ does not improve over \citetalias{dfj} but does so substantially for adversarial $X_2$, which is consistent with our observation that gender and health provide useful identification for the bequest motive.