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Locally Robust Semiparametric Estimation
Many economic and causal parameters depend on nonparametric or high dimensional first steps. Examples include dynamic discrete choice, games, average consumer surplus, and treatment effects. This paper shows how to construct moment functions for GMM estimators that are locally robust, referred to henceforth as orthogonal, where moment conditions have a zero derivative with respect to first steps. We show that such moment functions can be constructed by adding to identifying moment functions the nonparametric influence function from the effect of the first step on identifying moments. This construction follows practice where identifying moment conditions are often derived first from economic or causal models and then orthogonal moment functions constructed. Also the nonparametric influence function is entirely determined by the identifying moment functions and first step which motivates automatic ways to estimate the nonparametric influence function. In constructing sample moment functions we also cross-fit, a form of sample splitting where the moment function for each observation is evaluated at first step estimators that only use other observations, which further reduces bias. A GMM estimator based on orthogonal moment functions with cross-fitting used in construction of sample moments is referred to here as a debiased GMM estimator.
Debiased GMM has several advantages over plug-in GMM where only the identifying moment functions are used. First, standard confidence intervals for plug-in GMM are invalid under local alternatives when there is model selection in the first step while confidence intervals with orthogonal moment functions remain valid. Thus GMM with orthogonal moments is preferred over plug-in GMM in the many applications with first step model selection. Second, with regularized first steps debiased GMM will be root-n consistent under conditions where plug-in GMM is not. Model selection and/or regularization is often an important feature of machine learning, which is useful for estimating econometric models with many regressors or state variables, making debiased GMM especially important with machine learning first steps. Third, orthogonal moment functions will be doubly robust when the orthogonal moment functions are affine in the first step. We give this double robustness characterization and use it to derive new classes of doubly robust estimators. Fourth, in some important settings debiased GMM is known to have faster remainder rates or smaller 2nd order mean square error than plug-in GMM. In addition regularity conditions for debiased GMM are general and simple relative to those for plug-in GMM. We show asymptotic normality for debiased GMM for any first step estimator where certain mean square consistency conditions hold and either one (under double robustness) or two (more generally) mean square rates hold. We also show that conditions for plug-in GMM do not share this generality and simplicity due to an additional remainder that is specific to the first step estimator and complicated.
Debiased GMM is computationally more complicated than plug-in GMM in requiring estimation of additional unknown functions on which the nonparametric influence function depends. Also plug-in GMM has been innovative and useful in a variety of settings, including Powell, Stock, and Stoker (1989), Hotz and Miller (1993), Newey (1994a), Shen (1997), Chen and Shen (1998), Ai and Chen (2003, 2007), and many others. The advantages of debiased GMM discussed here motivate its use as an alternative to plug-in GMM, especially for machine learning or any other first steps involving model selection and/or regularization.
Machine learning is useful for estimating economic and causal models where there are high dimensional covariates or state variables, e.g. as in Belloni et al. (2012), Belloni, Chernozhukov, and Wei (2013), Robins et al. (2013), Belloni, Chernozhukov, and Hansen (2014), Farrell (2015), Kandasamy et al. (2015), Belloni, Chernozhukov, and Kato (2015), Belloni et al. (2017), and Athey, Imbens, and Wager (2018). Machine learning methods that are useful for these purposes include Lasso, Dantzig, neural nets, boosting, and others. Orthogonal moment functions reduce model selection and/or regularization biases which are common for machine learning first steps. Cross-fitting for debiased GMM avoids the need for Donsker conditions, which do not hold for many machine learning first steps, and reduces own observation bias. The large sample theory given here imposes only mean square convergence properties which will hold for a variety of machine learning first steps. The advantages of debiased GMM make it preferred to plug-in GMM for many machine learning first steps.
Previous to this paper debiased GMM with machine learning first step was based on orthogonal moment functions constructed in various ways. Constructing orthogonal moment functions by adding the nonparametric influence function to identifying moments opens the way to debiased GMM for many objects of interest. We illustrate by constructing debiased GMM estimators for functionals of conditional quantiles and for parameters of dynamic discrete choice models. The dynamic discrete choice estimator is based on machine learners of conditional choice probabilities allowing for high dimensional state variables. The estimator incorporates a novel Lasso estimator of conditional value function differences where the Lasso left-hand side variable is a function of a machine learner of the conditional choice probability. The estimator and the results we give provide a prototype for using machine learning for dynamic structural models. The conditional quantile estimator allows for high dimensional regressors.
This paper gives automatic estimators of the additional functions on which the nonparametric influence function depends. The approach uses just the orthogonal moment functions and the first step to construct estimators of the additional functions. The conditional quantile estimator employs this automatic estimator. This approach generalizes the automatic method in Chernozhukov, Newey, and Singh (2018).
We show that the nonparametric influence function has a useful robustness property. The robustness property is that the expected value of the nonparametric influence function is zero when the additional functions are not equal to the truth but the first step is. Consequently, the estimator of these additional functions is not required to converge faster than $n^{-1/4}$. We also show orthogonality using a standard Gateaux derivative characterization of the nonparametric influence function and some regularity conditions.
Orthogonal estimators for functionals of a density constructed by adding the nonparametric influence function have previously been given by Hasminskii and Ibragimov (1978), Pfanzagl and Wefelmeyer (1981), and Bickel and Ritov (1988). Newey, Hsieh, and Robins (1998, 2004) suggested the construction of orthogonal moment functions from adding the nonnparametric influence function. This construction was considered in Chernozhukov et al. (2016), Chernozhukov et al. (2018), and Bravo, Escanciano, and van Keilegom (2020). This paper innovates by showing robustness of the nonparametric influence function to the additional unknown functions on which it depends, so that these additional functions need not be estimated at a $n^{-1/4}$ rate. Also, regularity conditions are given for any first step to have no first order effect on expected orthogonal moments, the precise orthogonality condition for used in the asymptotic theory. These results are obtained using a standard Gateaux derivative characterization of the influence function. None of the Theorems in this paper appear in previous work. The relationship of the orthogonalization results in this paper to previous literature is discussed more fully in Section 4.
The Robinson (1988) semiparametric regression and Ichimura (1993) index regression estimators have first order conditions that are orthogonal moment functions. The object of interest in those papers minimizes an objective function that is an expectation that is also minimized by the first step. The objects of interest we consider are much more general in including many economic and causal parameters that do not minimize the same expectation as the first step.
Doubly robust moment functions have been constructed by Robins, Rotnitzky, and Zhao (1994, 1995), Robins and Rotnitzky (1995), Scharfstein, Rotnitzky, and Robins (1999), Robins, Rotnitzky, and van der Laan (2000), Robins and Rotnitzky (2001), Graham (2011), and Firpo and Rothe (2019). This paper innovates by deriving large classes of new doubly robust moment functions, including affine functionals of nonparametric regressions, functions satisfying other conditional moment restrictions, and density estimators, and by characterizing double robustness, all based on adding the nonparametric influence function. We also give related, partial robustness results where original moment conditions are satisfied even when the first step is not equal to the truth.
Targeted maximum likelihood, Van der Laan and Rubin (2006), based on machine learners has been considered by Van der Laan and Rose (2011) and large sample theory given by Luedtke and Van Der Laan (2016), Toth and Van der Laan (2016), and Zheng et al. (2016). Here we directly target parameters of interest via GMM based on adding the nonparametric influence function, with automatic estimation of additional unknown parameters and general and simple regularity conditions for asymptotic inference.
Recent work on debiased machine learning by Chernozhukov et al. (2018), Chernozhukov, Newey, and Robins (2018), and Chernozhukov, Newey, and Singh (2018) is partly based on and is also generalized by this paper. The construction of orthogonal moments here was described in Chernozhukov et al. (2018), which cited this paper for that construction and contains no results from this paper. The asymptotic theory in this paper uses the orthogonal moment construction here to improve on the asymptotic theory Chernozhukov et al. (2018), as described in Section 8. The doubly robust moment conditions considered in Chernozhukov, Newey, and Robins (2018) and Chernozhukov, Newey, and Singh (2018) were derived in this paper and the asymptotic theory in those other papers uses theory given in this paper. The automatic machine learner of the additional unknown functions given here generalizes that in Chernozhukov, Newey, and Singh (2018). In addition Newey and Robins (2017) and Hirshberg and Wager (2018) are concerned with linear functions of a regression that are formulated here. Furthermore, Bonhomme and Weidner (2018) have shown the importance of orthogonal moment functions in specification analysis, Foster and Srygkanis (2019) in deriving rates of convergence for machine learners, Chernozhukov, Hausman, and Newey (2019) for demand analysis with endogenous total expenditure, Semenova (2019) for machine learning for partially identified models, Singh and Sun (2019) for machine learning of complier effects, and Chernozhukov, Semenova, and Newey (2019) for machine learning of weighted average value functions in dynamic structural models.
There are other sources of bias arising from nonlinearity of moment conditions in the first step. Cattaneo and Jansson (2018) and Cattaneo, Jansson, and Ma (2018) give useful bootstrap and jackknife methods that reduce nonlinearity bias. Newey and Robins (2017) show that one can also remove this bias by cross fitting in some settings. We use cross-fitting in this paper.
To summarize the contributions of this paper, we consider GMM estimation with nonparametric first steps, with orthogonal moment conditions constructed by adding the nonparametric influence function to identifying moment functions. We give novel such estimators of functionals of high dimensional conditional quantiles and of dynamic discrete choice parameters with high dimensional state variables, including a novel Lasso estimator of conditional value function differences. We show that that the nonparametric influence function is robust to additional unknown functions on which it depends, so that $n^{-1/4}$ consistency is not required there. We show orthogonality of the constructed moment functions using the standard Gateaux derivative characterization of the influence function, including regularity conditions. We give examples showing that plug-in GMM is severely biased by model selection and/or first step regularization whereas debiased GMM is not. We give a general approach to estimating additional unknown functions in the nonparametric influence function and use it to automatically debias estimators of functionals of high dimensional conditional location learners, including regression quantiles. We give a variety of new doubly robust moment equations and characterize double robustness. We give general and simple regularity conditions that improve on previous conditions and apply these for asymptotic inference on functionals of high dimensional regression quantiles and dynamic discrete choice parameters with high dimensional state variables.
Section 2 describes orthogonal moments and debiased GMM, and gives the conditional quantile estimator. Section 3 gives the dynamic discrete choice estimator and reports results of a Monte Carlo study. Section 4 shows orthogonality and the robustness of the nonparametric influence function. Section 5 compares the properties of debiased and plug-in GMM estimators. Section 6 gives automatic estimators of the additional functions. Section 7 gives novel classes of doubly robust moment functions and characterizes double robustness. Section 8 provides general and simple asymptotic theory for debiased GMM.
The subject of this paper is GMM estimators of parameters identified by moment functions that depend on a first step unknown function. In this Section we describe this type of estimator and give examples.
To describe such an estimator let $\theta$ denote a finite dimensional parameter vector of interest, $\gamma$ an unknown function, and $W$ a data observation. We assume that there is a vector $g(w,\gamma,\theta)$ of known functions of a possible realization $w$ of $W$ such that \[ E[g(W,\gamma_{0},\theta_{0})]=0, \] where $\theta_{0}$ and $\gamma_{0}$ are the true parameter vector and function. We will assume that the parameter is identified by these moments, i.e. that $\theta_{0}$ is the unique solution to $E[g(W,\gamma_{0},\theta)]=0$ over $\theta$ in some set $\Theta.$
The true function $\gamma_{0}$ is unknown so a first step estimator $\hat{\gamma}$ of $\gamma_{0}$ is used$.$ Let $W_{1},...,W_{n}$ be a sample of i.i.d. data observations. Estimated sample moment functions can be formed by plugging in $\hat{\gamma}$ into $g(W_{i},\gamma,\theta)$ and averaging over data observations to obtain $\sum_{i=1}^{n}g(W_{i},\hat{\gamma},\theta)/n.$ One could form a "plug-in" GMM estimator by minimizing a quadratic form in these estimated sample moments, but such an estimator will be highly biased by first step model selection and/or regularization as further detailed in Section 5. This bias can be reduced by using orthogonal moment functions$.$
Orthogonal moment functions are based on influence functions. To describe them we need to explain some additional concepts and notation. Let $F$ denote a possible CDF for a data observation $W$ and suppose that $\hat{\gamma}$ has a probability limit $\gamma(F)$ when $F$ is the true distribution of $W$. Here $\gamma(F)$ is the probability limit of $\hat{\gamma}$ under general misspecification, similar to Newey (1994a), so that $F$ is unrestricted except for regularity conditions such as existence of $\gamma(F)$ or the expectation of certain functions of the data. For example if $\hat{\gamma}(x)$ is a nonparametric estimator of $E[Y|X=x]$ then $\gamma(F)(x)=E_{F}[Y|X=x]$ is the conditional expectation function when $F$ is the true distribution of $W$, which is well defined under the regularity condition that $E_{F}[\left\vert Y\right\vert ]$ is finite.
Next, let $F_{0}$ denote the true distribution of $W,$ $H$ some alternative distribution, and $F_{\tau}=(1-\tau)F_{0}+\tau H$ for $\tau\in\lbrack0,1].$ We assume that $H$ is chosen so that $\gamma(F_{\tau})$ exists for $\tau$ small enough and possibly other regularity conditions are satisfied. We also make the key assumption that there exists a function $\phi(w,\gamma,\alpha,\theta)$ such that
Here $\alpha$ is an additional unknown function on which $\phi(w,\gamma _{0},\alpha_{0},\theta)$ depends and $d/d\tau$ is the derivative from the right (i.e. for nonnegative values of $\tau$) at $\tau=0.$ This equation is the well known characterization of the influence function $\phi(w,\gamma _{0},\alpha_{0},\theta)$ of $\mu(F)=E[g(W,\gamma(F),\theta)]$ as the Gateaux derivative of $\mu(F),$ as in Von Mises (1947), Hampel (1974), Huber (1981). The restriction that $\gamma(F_{\tau})$ exists allows $\phi(w,\gamma _{0},\alpha_{0},\theta)$ to be the influence function when $\gamma(F)$ is only well defined for certain types of distributions, such as when $\gamma(F)$ is a conditional expectation or density. The function $\phi(w,\gamma,\alpha ,\theta)$\ will generally exist when $E[g(W,\gamma(F),\theta)]$ has a finite semiparametric variance bound. Also $\phi(w,\gamma,\alpha,\theta)$ will generally be unique because we are not restricting $H$ except for regularity conditions. We will refer to $\phi(w,\gamma,\alpha,\theta)$ as the nonparametric influence function as it characterizes the local effect of the first step function $\gamma$ on the expected moment $\mu(F).$ The nonparametric influence function can be calculated from the derivative in equation ((ref)) or as described in Newey (1994a); see Ichimura and Newey (2017).
Orthogonal moment functions are constructed by adding the nonparametric influence function to the identifying moment functions to obtain \[ \psi(W,\gamma,\alpha,\theta)=g(W,\gamma,\theta)+\phi(W,\gamma,\alpha,\theta). \] Estimation of the unknown functions $\gamma$ and $\alpha$ will have no first order effect on the expected value of $\psi(W,\gamma,\alpha,\theta)$ and estimation of $\theta$ will not affect the expectation of $\phi(W,\gamma ,\alpha,\theta)$ as we will show. Debiased sample moments can then be constructed by evaluating at first step estimators of $\gamma,$ $\alpha,$ and $\theta$ and averaging over the data.
Constructing orthogonal moment functions is greatly facilitated by the wide variety of known $\phi(w,\gamma,\alpha,\theta).$ The form of $\phi (w,\gamma,\alpha,\theta)$ for density weighted average derivatives is in Powell, Stock, and Stoker (1989). For first step least squares projections (including conditional expectations), density weighted conditional means, and their derivatives $\phi(W,\gamma,\alpha,\theta)$ is given in Newey (1994a). Hahn (1998) and Hirano, Imbens, and Ridder (2003) used those results to obtain $\phi(W,\gamma,\alpha,\theta)$ for treatment effect estimators. Bajari, Hong, Krainer, and Nekipelov (2010) and Bajari, Chernozhukov, Hong, and Nekipelov (2009) derived $\phi(W,\gamma,\alpha,\theta)$ for some first steps used in structural estimation. Hahn and Ridder (2013, 2019) derived $\phi (W,\gamma,\alpha,\theta)$ for generated regressors that depend on first step conditional expectations. Ai and Chen (2007) and Ichimura and Newey (2017) derived $\phi(W,\gamma,\alpha,\theta)$ for first step estimators of functions satisfying conditional moment restrictions. Semenova (2018) derived $\phi(W,\gamma,\alpha,\theta)$ for support functions used in partial identification. Many of these derivations did not directly use equation ((ref)) but Ichimura and Newey (2017) show that various $\phi (W,\gamma,\alpha,\theta)$ solve equation ((ref)). These wide variety of known $\phi(W,\gamma,\alpha,\theta)$ can be used to construct orthogonal moment functions.
We also use cross-fitting, a form of sample splitting, in the construction of debiased sample moments; e.g. see Schick (1986) and Klaassen (1987). Partition the observation indices $(i=1,...,n)$ into $L$ groups $I_{\ell},$ $(\ell=1,...,L).$ Let $\hat{\gamma}_{\ell},$ $\hat{\alpha}_{\ell}$, and $\tilde{\theta}_{\ell}$ be estimators that are constructed using all observations not in $I_{\ell}.$ Debiased sample moments functions are
A debiased GMM estimator is
where $\hat{\Upsilon}$ is a positive semi-definite weighting matrix and $\Theta$ is the set of parameter values. A choice of $\hat{\Upsilon}$ that minimizes the asymptotic variance of $\hat{\theta}$ will be $\hat{\Upsilon }=\hat{\Psi}^{-1},$ for \[ \hat{\Psi}=\frac{1}{n}\sum_{\ell=1}^{L}\sum_{i\in I_{\ell}}\hat{\psi}_{i\ell }\hat{\psi}_{i\ell}^{\prime},\text{ }\hat{\psi}_{i\ell}=g(W_{i},\hat{\gamma }_{\ell},\tilde{\theta}_{\ell})+\phi(W_{i},\hat{\gamma}_{\ell},\hat{\alpha }_{\ell},\tilde{\theta}_{\ell}). \] There is no need to account for the presence of $\hat{\gamma}$ and $\hat{\alpha}$ in $\hat{\psi}_{i\ell}$ because the presence of $\phi (W_{i},\hat{\gamma}_{\ell},\hat{\alpha}_{\ell},\tilde{\theta}_{\ell})$ removes the first order effect on $\hat{\psi}(\theta)$ of $\hat{\gamma}_{\ell}$ and $\hat{\alpha}_{\ell}$ under conditions we give. An estimator $\hat{V}$ of the asymptotic variance of $\sqrt{n}(\hat{\theta}-\theta_{0})$ is
The cross-fitting used in this estimator, where $\hat{\psi}(\theta)$ is averaged over observations not used to form $\hat{\gamma}_{\ell}$ and $\hat{\alpha}_{\ell},$ eliminates bias due to averaging over observations that are used to construct the first step$.$ Eliminating such "own observation" bias helps remainders converge faster to zero, e.g. as in Newey and Robins (2017), and can be important in practice, e.g. as in Angrist and Krueger (1995). It also eliminates the need for Donsker conditions for $\hat{\gamma }_{\ell}$ and $\hat{\alpha}_{\ell},$ which is important for many machine learner first steps that are not known to satisfy such conditions, as discussed in Chernozhukov et al. (2018).
The debiased moments require a first step estimator $\hat{\alpha}_{\ell}$ of unknown functions that appear in $\phi(W,\gamma,\alpha,\theta).$ When the form of $\alpha_{0}$ is known one can plug-in an estimator $\hat{\alpha}_{\ell}.$ Also, in Section 6 we use the debiased moment functions to construct an automatic estimator $\hat{\alpha}_{\ell}$ that does not requiring knowing the form of $\alpha_{0}$. This automatic method generalizes that of Chernozhukov, Newey, and Singh (2018) beyond functionals of least squares projections.
The efficiency of debiased GMM is entirely determined by the choice of moment functions, first step, and weighting matrix. The matrix $\hat{\Psi}^{-1}$ is an optimal choice of weighting matrix as usual for GMM. The efficient choice of moment functions and first steps will depend on a model and as further discussed in Section 4. The presence of $\hat{\phi}$ in the orthogonal moment functions $\hat{\psi}(\theta)$ does not affect identification of $\theta$. The second, mean zero condition in equation ((ref)) holds for all possible distributions of $W$ so that $\hat{\phi}$ will converge in probability to zero. The sole purpose of including $\hat{\phi}$ is to remove the first-order effect of $\hat{\gamma}_{\ell}$ on the moment functions. It accomplishes this in a nonparametric way that does not depend on any model assumptions, as further discussed in Section 4.
The initial estimator $\tilde{\theta}_{\ell}$ can be based on the identifying moment conditions and constructed as \[ \tilde{\theta}_{\ell}=\arg\min_{\theta\in\Theta}\hat{g}_{\ell}(\theta )^{\prime}\hat{\Upsilon}_{\ell}\hat{g}_{\ell}(\theta),\text{ }\hat{g}_{\ell }(\theta)=\frac{1}{n-n_{\ell}}\sum_{\ell^{\prime}\neq\ell}\sum_{i\in I_{\ell^{\prime}}}g(W_{i},\tilde{\gamma}_{\ell^{\prime}},\theta). \] where $\hat{\Upsilon}_{\ell}$ uses only observations not in $I_{\ell},$ $n_{\ell}$ is the number of observations in $I_{\ell}$, and $\tilde{\gamma }_{\ell^{\prime}}$ uses observations not in $\ell$ and not in $\ell^{\prime}.$ One could iterate on the initial estimator $\tilde{\theta}_{\ell}$ in the debiased moments $\hat{\psi}(\theta)$ and/or $\hat{\Psi}$ by calculating $\hat{\theta}$ and/or $\hat{\Psi}$ a second time with $\tilde{\theta}_{\ell}$ being a debiased GMM estimator obtained from a prior iteration.
Example 1: An example that will be used to illustrate the theory has a data observations $W=(Y,X,Z)$ and $\theta_{0}=E[Z\gamma_{0}(X)]=E[\alpha _{0}(X)\gamma_{0}(X)]$ for $\gamma_{0}(X)=E[Y|X]$ and $\alpha_{0}(X)=E[Z|X].$ This example is of interest in its own right as the component of the expected conditional covariance $E[Cov(Z,Y|X)]=E[ZY]-\theta_{0}$ that depends on unknown functions, which covariance is useful for the analysis of covariance and for estimation of a partially linear model, Robinson (1988). We specify the identifying moment function with implied nonparametric influence function as \[ g(w,\gamma,\theta)=z\gamma(x)-\theta,\text{ }\phi(w,\gamma,\alpha )=\alpha(x)[y-\gamma(x)]. \] The debiased GMM estimator is \[ \hat{\theta}=\frac{1}{n}\sum_{\ell=1}^{L}\sum_{i\in I_{\ell}}\left\{ Z_{i}\hat{\gamma}_{\ell}(X_{i})+\hat{\alpha}_{\ell}(X_{i})[Y_{i}-\hat{\gamma }_{\ell}(X_{i})]\right\} =\frac{1}{n}\sum_{\ell=1}^{L}\sum_{i\in I_{\ell} }\left\{ Z_{i}\hat{\gamma}_{\ell}(X_{i})+\hat{\alpha}_{\ell}(X_{i})Y_{i} -\hat{\alpha}_{\ell}(X_{i})\hat{\gamma}_{\ell}(X_{i})\right\} . \]
The object of interest in this example is an expected linear functional of a conditional quantile function where \[ \theta_{0}=E[m(W,\gamma_{0})],\text{ }\gamma_{0}=\arg\min_{\gamma\in\Gamma }E[v(Y-\gamma(X))],\text{ }v(u)=[\lambda-1(u<0)]u. \] Here $Y$ is a dependent variable of interest and $m(w,\gamma)$ is a linear functional of $\gamma.$ An example is a weighted average derivative of $\gamma_{0}$ where $m(w,\gamma)=\int d(x)[\partial\gamma(x)/\partial x_{1}]dx.$ Here the identifying moment function is $g(w,\gamma,\theta )=m(w,\gamma)-\theta$. The nonparametric influence function is $\phi (w,\gamma,\alpha)=\alpha(x)v_{u}(y-\gamma(x)),$ where $v_{u}(u)=\lambda -1(u<0)$ denotes the derivative of $v(u)$ away from zero, as follows from Ai and Chen (2007, p. 40) and Ichimura and Newey (2017).
We can construct a debiased GMM estimator of $\theta_{0}$ from any learner of the conditional quantile $\gamma_{0}$ that converges sufficiently fast in mean square. Let $\hat{\gamma}_{\ell}(x)$ be a learner of $\gamma_{0}$, computed from observations not in $I_{\ell}$, and $\hat{\gamma}_{\ell,\ell^{\prime} }(x)$ be computed from observations not in $\ell$ or $\ell^{\prime}.$ Also let $K(u)$ be a bounded, univariate kernel, with $\int K(u)du=1$ and $\int K(u)udu=0,$ and $h$ a bandwidth. Let $b(x)$ be a $p\times1$ vector of functions of $x.$ A debiased GMM estimator is
The $\hat{\alpha}_{\ell}(x)$ is a special case of an automatic Lasso minimum distance learner given and motivated in Section 6.
This estimator depends on the regularization term $r_{\lambda}$ in the objective function for $\hat{\rho}_{\ell}$. This $r_{\lambda}$ should be chosen larger than the conventional Lasso regularization parameter because $\hat{Q}_{\ell}$ depends on the nonparametric estimator $\hat{\gamma} _{\ell,\ell^{\prime}}(X_{i})$, as further discussed in Section 8. The nested sample splitting used for $\hat{\gamma}_{\ell,\ell^{\prime}}(X_{i})$ requires that the first step learner be computed for $L^{2}$ subsamples. Use of $\hat{\gamma}_{\ell}(x)$ as a starting value for computation of each $\hat{\gamma}_{\ell,\ell^{\prime}}(X_{i})$ may aid in this computation. The nested sample splitting allows for a very general first step that need only have a mean square convergence rate. Here $\hat{\gamma}_{\ell}(X_{i})$ could be used in place of $\hat{\gamma}_{\ell,\ell^{\prime}}(X_{i})$ if$\ \sum _{i\notin I_{\ell}}\left\vert \hat{\gamma}_{\ell}(X_{i})-\gamma_{0}\left( X_{i}\right) \right\vert /(n-n_{\ell})$ converged to zero at some rate that is a power of $n.$
For dynamic discrete choice with high dimensional state variables we estimate structural parameters via learners of conditional choice probabilities. This approach replaces computation of expected value functions with nonparametric estimation as suggested in Hotz and Miller (1993). For simplicity we focus this example on binary choice, providing methods and results that will be available for the more complicated models employed more widely in practice. In particular we provide a Lasso estimator of conditional value function differences where the dependent variable is a function of estimated future choice probabilities. In Section 8 we provide convergence rate results for this estimator. We also give the nonparametric influence function for this kind of first step. This analysis provides a prototype for estimation of dynamic structural models with high dimensional state variables.
In dynamic binary choice individuals choose between two alternatives $j=1$ and $j=2$ to maximize the expected present discounted value of per period utility $U_{tj}=D_{j}(X_{t})^{\prime}\theta_{0}+\varepsilon_{tj},$ $(j=1,2;t=1,...,T),$ where $\varepsilon_{jt}$ is i.i.d. with known CDF, independent of the entire history $\{X_{s}\}_{s=1}^{\infty}$ of a state variable vector $X$, and $X_{t}$ is Markov of order $1$ and stationary. The parameter of interest is $\theta_{0}$. We develop an estimator that allows for high dimensional $X_{t}$.
We assume that choice $1$ is a renewal choice where the conditional distribution of $X_{t+1}$ given $X_{t}$ and choice $1$ does not depend on $X_{it}$. We also assume that $D_{1}(X_{t})=(-1,0^{\prime})^{\prime}$ and $D_{21}(X_{t})=0$ so that the first element in $\theta$ is a binary choice constant. Let $Y_{jt}$ equal a dummy variable equal to $1$ when choice $j$ is made and $\gamma_{10}(X_{t})=\Pr(Y_{2t}=1|X_{t})$ be the conditional choice probability of alternative 2. Also let $V(X_{t})$ denote the expected value function. As in Hotz and Miller (1993) there is a known function $H(p)$ such that for $\gamma_{20}(X_{t})=E[H(\gamma_{10}(X_{t+1}))|X_{t},Y_{2t}=1]$ and $\gamma_{30}=E[H(\gamma_{10}(X_{t+1}))|Y_{1t}=1],$
For example when $\varepsilon_{1t}$ and $\varepsilon_{2t}$ are independent Type I\ extreme value this equation is satisfied for $H(p)=.5227-\ln(1-p).$ Then for the CDF $\Lambda(a)$ of $\varepsilon_{t1}-\varepsilon_{t2},$ $D(X_{t})=D_{2}(X_{t})-D_{1}(X_{t})$, and $\delta$ the discount factor the conditional choice probability for $j=2$ is
We consider data of i.i.d. observations on individuals each followed for $T$ time periods, that also includes the $T+1$ observation $X_{T+1}$ of the state variables, where $W=(X_{1}^{\prime},Y_{21},...,X_{T}^{\prime},Y_{2T} ,X_{T+1}^{\prime})^{\prime}.$ An estimator of $\theta_{0}$ can be obtained by constructing first step estimators $\hat{\gamma}_{2}(x)$ and $\hat{\gamma} _{3}$, substituting these estimators for $\gamma_{2}$ and $\gamma_{3}$ in $a(x,\theta,\gamma_{2},\gamma_{3})$ in equation ((ref)), and then maximizing a binary choice log-likelihood as if $\hat{\gamma}_{2}(x)$ and $\hat{\gamma}_{30}$ were true. We specify as identifying moment functions the derivative of the pseudo log-likelihood associated with the binary choice probability in equation ((ref)) with respect to $\theta$,
Estimators of $\gamma_{10}(x)$, $\gamma_{20}(x)$, and $\gamma_{30}$ are needed as a first step $\hat{\gamma}_{\ell}$ for the identifying moment function. We will consider any $\hat{\gamma}_{1\ell}(x)$ that converges sufficiently quickly in mean square. For example $\hat{\gamma}_{1\ell}$ could be logit Lasso or a linear Lasso estimator with dependent variable $Y_{2t}$. We use Lasso to construct $\hat{\gamma}_{2\ell}(x)$ in order to control for estimation error that results from an estimated dependent variable. Let $\hat{\gamma}_{1\ell,\ell^{\prime}}(x)$ be an estimator of the conditional choice probability computed from observations not in $I_{\ell}$ or $I_{\ell^{\prime}}.$ Let $b(x)$ denote a $p\times1$ dictionary of functions of the state variables $x$. We form $\hat{\gamma}_{2\ell}(x)$ and $\hat{\gamma }_{3\ell}$
Here $\hat{\gamma}_{2\ell}(x)$ is Lasso with left hand side variable $H\left( \hat{\gamma}_{1\ell,\ell^{\prime}}(X_{i,t+1}\right) )$ and right-hand side variables $b(X_{it})Y_{2it}$ and $\hat{\gamma}_{3\ell}$ is a sample mean conditional on $Y_{1it}=1.$ Here $\hat{\gamma}_{2\ell}$, and $\hat{\gamma }_{3\ell}$ use all observations all observations with $i\notin I_{\ell}.$ The nested sample splitting in $\hat{\gamma}_{1\ell,\ell^{\prime}}(x)$ is useful in only requiring mean square convergence rates for conditional choice probabilities although it is somewhat complicated. When $L$ is moderate size (e.g. $L=5)$ there may be many $\hat{\gamma}_{1\ell,\ell^{\prime}}(x)$ to compute (e.g. 25), although an estimate for a particular $\ell$ and $\ell^{\prime}$ could provide a good starting value for other splits. As in Example 2 the nested cross-fit estimator $\hat{\gamma}_{1\ell,\ell^{\prime} }(x)$can be replaced by single sample splitting if $\hat{\gamma}_{\ell}(x)$ converged in a sample absolute value terms.
The three first steps $\hat{\gamma}_{1\ell}(x),$ $\hat{\gamma}_{2\ell}(x)$, and $\hat{\gamma}_{3\ell}$ result in a nonparametric influence function that is the sum of three terms, one term for each first step, with
The form of each $\phi_{j}(w,\gamma,\alpha,\theta)$ follows from Proposition 4 of Newey (1994a) by treating only $\gamma_{j}$ as unknown and holding the other first steps fixed at their true values, as in Newey (1994a, p. 1357). The true values $\alpha_{10},$ $\alpha_{20}$, and $\alpha_{30}$ are
\qquad\qquad
To construct $\hat{\alpha}_{2\ell},$ $\hat{\alpha}_{3\ell}$, and $\hat{\alpha }_{1\ell}$, obtain an initial estimator $\tilde{\theta}_{\ell}$ from binary choice pseudo maximum likelihood over $i\notin I_{\ell}$, $t\leq T$ with $\gamma_{2}$ and $\gamma_{3}$ replaced by $\hat{\gamma}_{2\ell}$ and $\hat{\gamma}_{3\ell}$ respectively in the choice probability formula. Also let $\hat{a}_{it}=a(X_{it},\hat{\theta}_{\ell},\hat{\gamma}_{2\ell} ,\hat{\gamma}_{3\ell})$ and construct $\hat{\alpha}_{2\ell}(X_{it} ,Y_{2it},\tilde{\theta}_{\ell})$ by substituting $\hat{a}_{it}$ for $a(x)$, $X_{it}$ for $x$, $Y_{2it}$ for $y_{2},$ and $\tilde{\theta}_{\ell}$ for $\theta_{0}$in the formula for $\alpha_{20}(x,y_{2},\theta_{0})$ in equation ((ref)). Next obtain $\hat{\alpha}_{3\ell}$ by replacing $\alpha_{20}(X_{t},Y_{2t},\theta_{0})$ by $\hat{\alpha}_{2}(X_{it} ,Y_{2it},\tilde{\theta}_{\ell})$ and population expectations by sample averages over $i\notin I_{\ell},t\leq T$ in the third line of equation ((ref)). Also, obtain $\hat{\alpha}_{1\ell}(x,\theta)$ by replacing $\alpha_{30}$ and $\gamma_{10}(x)$ by $\hat{\alpha}_{3\ell}$ and $\hat{\gamma }_{1\ell}(x)$ respectively in the first line of equation ((ref)) and by replacing the two conditional expectations by the predicted values from Lasso regressions over $i\notin I_{\ell},t\leq T$ with regressors $b(X_{i,t+1}),$ dependent variables equal to each element of $\hat{\alpha }_{2\ell}(X_{it},Y_{2it},\tilde{\theta}_{\ell})$ and $Y_{1it}$ respectively, and regularization factors $r_{2}$ and $r_{3}$ respectively, analogously to $\hat{\gamma}_{2\ell}(x)$. Finally substitute $\hat{\alpha}_{1\ell},$ $\hat{\alpha}_{2\ell},$ and $\hat{\alpha}_{3\ell}$ for $\alpha_{10} ,\ \alpha_{20}$, and $\alpha_{30}$ and $\hat{\gamma}_{1\ell}$, $\hat{\gamma }_{2\ell}$ and $\hat{\gamma}_{3\ell}$ for $\gamma_{1},$ $\gamma_{2}$, and $\gamma_{3}$ in the formulas for $\phi_{1}$, $\phi_{2}$, and $\phi_{3}$ and construct a debiased GMM estimator as in Section 2.
To explore the finite sample properties of this estimator we carried out a Monte Carlo study for a model similar to that of Rust (1987). The state variables consisted of a positive variable $x_{1}$ (mileage) and other variables $x_{2},...,x_{6}$ with transition
where $(X_{2t},...,X_{6t})$ is i.i.d. over $t$, $X_{2t},$ $X_{4t}$, and $X_{6t}$ are chi-squared with one degree of freedom and $X_{3t}$ and $X_{5t}$ are binary with $\Pr(X_{kt}=1)=1/2$, $k=3,5.$ We specified that $D(x)$ is two dimensional with $D_{1}(x)=(-1,0)^{\prime}$ and $D_{2}(x)=(0,\sqrt{x_{1} })^{\prime}$ and that $\varepsilon_{1t},$ $\varepsilon_{2t}$ are independent Type I\ extreme value, so that $\Lambda(a)=e^{a}/(1+e^{a})$ corresponds to binary logit.
To generate the data we solved the Bellman equation on a finite grid using the fact that the state space has a two dimensional structure in terms of $x_{1}$ and $\sum_{k=1}^{5}c_{k}x_{k+1},$ with linear interpolation between grid points. We did not enforce this index structure in estimation, so that the estimation treated the state space as dimension six. We carry out 500 Monte Carlo replications for $T=10$ and $n=100,$ $300,$ $1000$, and $10,000.$ We specified five fold cross fitting where $L=5.$ We consider three specifications of the vector $b(x)$ used by Lasso, consisting of a) the elements of $x$, b) those from a) and squares of elements of $x$; c) those from b) and all products of two elements of $x$. The conditional choice probability estimators $\hat{\gamma}_{1\ell,\ell^{\prime}}(x)$ and $\hat{\gamma}_{1\ell}(x)$ were logit Lasso truncated to be between $.0001$ and $.9999.$ We used the MATLAB Lasso and logit Lasso procedures for computation. The regularization value $r$ for Lasso was chosen by two fold regularization. Although we do not know whether the resulting $r$ satisfies the conditions in the asymptotic theory of Section 8, we uses this $r$ so that the estimator in the Monte Carlo would be based on an "off the shelf" machine learner of unknown functions.
The results are reported in Tables 1, 2, and 3. The PI labels the GMM estimator based only on identifying moment functions, DB the debiased GMM, Bias is the absolute value of bias, Med SE denotes the median of the estimated standard errors corresponding to equation ((ref)), SD denotes standard deviation, and Cvg denotes coverage probability of a nominal 95 percent confidence interval.
In all cases debiased GMM has much smaller bias than the plug-in estimator. For the richest dictionary $b(X)$ in Table 3 coverage probabilities are quite close to the nominal value though conservative. In contrast plug-in GMM has large bias and confidence interval coverage probabilities that are far from their nominal values in all cases. Remarkably, for larger sample sizes or smaller dimensional $b(x)$, debiased GMM is no more variable than plug-in GMM, and in several cases is less variable. Overall, the performance of the debiased GMM estimator in this example with an "off the shelf" machine learner suggests that debiased GMM for dynamic discrete choice and other structural models could be useful in practice.
The low variance of debiased GMM could result partly from the fact that the bias correction "partials out" the effect of $\hat{\gamma}_{\ell}$ in the identifying moments, with the effect of changing $\gamma$ in the identifying moment functions being approximately cancelled by the effect of $\hat{\gamma }_{\ell}$ in the nonparametric influence function estimator. In the next Section we explain this "partialling out" effect.
Neyman orthogonality refers to the unknown functions $\gamma$ and $\alpha$ having no first order effect on the moments \[ \bar{\psi}(\gamma,\alpha,\theta):=E[\psi(W,\gamma,\alpha,\theta)]\text{.} \] To show Neyman orthogonality let $\alpha(F)$ denote the probability limit of $\hat{\alpha}$ when $F$ is the CDF of $W,$ similarly to $\gamma(F).$ Because $\phi(w,\gamma,\alpha,\theta)$ is the nonparametric influence function it will satisfy the mean zero condition in equation ((ref)) identically in $F$, so that $0\equiv E_{F}[\phi(W,\gamma(F),\alpha(F),\theta)].$ Substituting $F_{\tau}$ for $F$ and differentiating this identity with respect to $\tau$ gives
where the first equality follows by the chain rule, the second equality follows from the influence function formula in equation ((ref) ), and the third from the definition of $\bar{\psi}(\gamma,\alpha,\theta).$ This equation shows that the functions $\gamma$ and $\alpha$ have no first order effect on $\bar{\psi}(\gamma,\alpha,\theta)$ along the path $(\gamma(F_{\tau}),\alpha(F_{\tau}))$. The second equality shows how the presence of $E[\phi(W,\gamma,\alpha,\theta)]$ "partials out" the effect of varying $\tau$ on $E[g(W,\gamma(F_{\tau}),\theta)]$. The zero mean property of the nonparametric influence function implies that the local effect of $\gamma$ on $E[g(W,\gamma,\theta_{0})]$ along the path $\gamma(F_{\tau})$ is cancelled, or "partialled out," by the effect of varying $\gamma$ and $\alpha$ on $E[\phi(W,\gamma,\alpha,\theta)]$ along the path $\gamma(F_{\tau})$ and $\alpha(F_{\tau}).$ The following result gives precise conditions for equation ((ref)):
Theorem 1: If i) equation ((ref)) is satisfied; ii) $\int\phi(w,\gamma(F_{\tau}),\alpha(F_{\tau}),\theta)F_{\tau}(dw)=0$ for all $\tau\in\lbrack0,\bar{\tau})$ with $\bar{\tau}>0,$ and iii) $\int\phi(w,\gamma(F_{\tau}),\alpha(F_{\tau}),\theta)F_{0} (dw)$ and $\int\phi(w,\gamma(F_{\tau}),\alpha(F_{\tau}),\theta )H(dw)$\textit{ are continuous at }$\tau=0$\textit{ then equation ((ref)) is satisfied.}
The proofs of this result and others are given in Appendix A.
Example 1: Here $g(w,\gamma,\theta)=z\gamma(x)-\theta$ and $\phi(w,\gamma,\alpha)=\alpha(x)[y-\gamma(x)]$ so that
When $\alpha(X)=\alpha_{0}(X)$ the presence of $\gamma$ in the identifying moment $E[g(W,\gamma,\theta)]$ is exactly cancelled, or partialled out, by the presence of $\gamma$ in $E[\phi(W,\gamma,\alpha_{0})]=E[\alpha_{0} (X)\{\gamma_{0}(X)-\gamma(X)\}].$
Equation ((ref)) is a total zero derivative condition for joint variation in $\left( \gamma,\alpha\right) $ along the path $(\gamma(F_{\tau }),\alpha(F_{\tau})).$ In many cases $\gamma(F)$ and $\alpha(F)$ are distinct objects so that it is possible to choose $F_{\tau}$ so that $\alpha(F_{\tau})$ varies with $\tau$ and $\gamma(F_{\tau})=\gamma_{0}$ remains equal to its true value. For example $\gamma(F)$ and $\alpha(F)$ may be determined by the distributions of different random variables and so be distinct objects. In such cases equation ((ref)) implies that $\phi(W,\gamma_{0} ,\alpha,\theta)$ has mean zero even when $\alpha\neq\alpha_{0}.$
Theorem 2: For any $\alpha$ if i) there is $F_{\alpha}$ such that $\alpha\left( F_{\alpha}\right) =\alpha $ and $\gamma(F_{\tau}^{\alpha})=\gamma_{0}$ for $F_{\tau }^{\alpha}=(1-\tau)F_{\alpha}+\tau F_{0}$ \textit{and all }$\tau\in \lbrack0,\bar{\tau}),$ $\bar{\tau}>0$; \textit{ii) }$d\int g(w,\gamma(F_{\tau }^{\alpha}),\theta)F_{\alpha}(dw)/d\tau=\int\phi(w,\gamma_{0},\alpha ,\theta)F_{0}(dw)$ \textit{then}
Noting that hypothesis ii) is just the characterization of the nonparametric influence function when the true distribution is $F_{\alpha}$, we see that the nonparametric influence function has zero expectation when the function $\alpha$ and/or the parameter $\theta$ are not equal to their true values $\alpha_{0}$ and $\theta_{0}$ and there is some distribution $F_{\alpha}$ such that $\alpha=\alpha(F_{\alpha})$. In all the examples of which we are aware equation ((ref)) is easy to confirm by inspection of $\phi(W,\gamma_{0},\alpha,\theta)$.
Example 1: Here $E[\phi(W,\gamma_{0},\alpha,\theta)]=E[\alpha (X)\{Y-\gamma_{0}(X)\}]=0$ for any $\alpha(X)$ by $\gamma_{0}(X)=E[Y|X]$ and iterated expectations.
Theorem 2 shows that equation ((ref)) is a general property of the nonparametric influence function and is not confined to a particular set of examples.
Neyman orthogonality with respect to only $\gamma(F_{\tau})$ follows from choosing $F_{\tau}$ so that $\alpha(F_{\tau})=\alpha_{0}$ when $\gamma(F)$ and $\alpha(F)$ are distinct objects. Then equation ((ref)) implies $\partial\bar{\psi}(\gamma(F_{\tau}),\alpha_{0},\theta_{0})/\partial\tau=0$. For asymptotic theory it is useful to have a zero derivative with respect to $\gamma$ that is stronger than this pathwise derivative condition. The following result shows that equation ((ref)) implies that $\bar{\psi }(\gamma,\alpha_{0},\theta_{0})$ has a zero Hadamard derivative with respect to $\gamma$ if the set of pathwise derivatives $d\gamma(F_{\tau})/d\tau$ is rich enough.
Theorem 3: If there is a norm $\left\Vert \gamma\right\Vert $, a linear set $\Gamma,$ and a set $\mathcal{H} $\ such that for all $H\in\mathcal{H}$; i) $\alpha(F_{\tau })=\alpha_{0}$\textit{ and equation ((ref)) is satisfied; ii) } $\bar{\psi}(\gamma,\alpha_{0},\theta_{0})$\textit{ is Hadamard differentiable at }$\gamma_{0}$\textit{ tangentially to }$\Gamma$\textit{ with derivative} $\bar{\psi}_{\gamma}(\delta,\alpha_{0},\theta_{0}),$\textit$\delta\in \Gamma$; \textit{iii) }$\gamma(F_{\tau})$\textit{ is Hadamard differentiable at }$\tau=0;$ \textit{iv) the closure of }$\{\partial\gamma(F_{\tau} )/\partial\tau:H\in$\textit$\mathcal{H}\}$\textit{ is }$\Gamma$\textit{ then }
Furthermore, if $\bar{\psi}(\gamma,\alpha_{0},\theta_{0})$ is twice continuously Frechet differentiable in a neighborhood of $\gamma_{0} $ then there is $C>0$ such that for $\left\Vert \gamma-\gamma_{0}\right\Vert $ small enough
Hadamard and Frechet differentiability are defined and discussed e.g. in van der Vaart (1998, 20.2). The first conclusion of this result is that the expected value $\bar{\psi}(\gamma,\alpha_{0},\theta_{0})$ of $\psi (w,\gamma,\alpha,\theta)$ moment function has zero Hadamard derivative with respect to $\gamma$. Theorems 2 and 3 combined show that adding the nonparametric influence function $\phi(w,\gamma,\alpha,\theta)$ to identifying moment functions $g(w,\gamma,\theta)$ makes $\psi(w,\gamma,\alpha ,\theta)=g(w,\gamma,\theta)+$ $\phi(w,\gamma,\alpha,\theta)$ Neyman orthogonal. Equation ((ref)) bounds the departure from zero of the expected moments. This bound is useful for formulating regularity conditions for root-n consistency of debiased GMM when $\bar{\psi}(\gamma,\alpha _{0},\theta_{0})$ is nonlinear in $\gamma,$ as we will explain in Section 8. When formulating regularity conditions for particular moment functions and first step estimators it may be simpler to directly confirm equation ((ref)). In many cases equation ((ref)) will be satisfied under specific regularity conditions when $\left\Vert \gamma-\gamma _{0}\right\Vert $ is a mean square norm, because $\bar{\psi}(\gamma ,\alpha,\theta)$ is an expected value. Equation ((ref)) with a mean square norm $\left\Vert \gamma-\gamma_{0}\right\Vert $ has wide applicability to machine learning first steps where many mean square convergence rates are available.
If $\gamma$ were taken to be the limit of a nonparametric estimator $\hat{\gamma}$ for fixed bandwidth, number of series terms, or regularization then we can think of $\left\Vert \gamma-\gamma_{0}\right\Vert $ as bias in $\gamma$ from nonparametric estimation. Frechet differentiability of $\bar{\psi}(\gamma,\alpha_{0},\theta_{0})$ in $\gamma$ and equation ((ref)) then imply that $\bar{\psi}(\gamma,\alpha_{0},\theta _{0})=o(\left\Vert \gamma-\gamma_{0}\right\Vert )$, so that the orthogonal moments $\bar{\psi}(\gamma,\alpha_{0},\theta_{0})$ shrink to zero faster than the nonparametric bias. Thus orthogonal moment functions have the small bias property considered in Newey, Hsieh, and Robins (1998, 2004). As usual for GMM the estimator $\hat{\theta}$ will inherit this property of the orthogonal moment functions.
Equation ((ref)) is similar to Theorem 2.2 of Robins et al. (2008) but different in applying directly to the standard influence function characterization in equation ((ref)) without specifying a score function (derivative of the log-likelihood of a model). Proceeding in this way allows us to show Neyman orthogonality of $\psi(w,\gamma,\alpha,\theta)$ using equation ((ref)). To the best of our knowledge equation ((ref)) has not appeared in this form previously. Also, Theorems 1-3 appear to be novel in specifying regularity conditions for Neyman orthogonality of moment functions obtained from adding the nonparametric influence function.
The construction of orthogonal moment functions we consider has antecedents in the literature on functional estimation, where the identifying moment conditions are $g(w,\gamma,\theta)=m(\gamma)-\theta$ for some explicit functional of $m(\gamma)$ of $\gamma.$ Here $\phi(w,\gamma,\alpha)$ is the influence function of $m(\gamma(F))$ and $\psi(w,\gamma,\alpha,\theta )=m(\gamma)-\theta+\phi(w,\gamma,\alpha)$. Examples of such moment functions when $\gamma$ is a density function were given by Hasminskii and Ibragimov (1978), Pfanzagl and Wefelmeyer (1981), and Bickel and Ritov (1988). Robins and Rotnitzky (1992) gave an orthogonal moment function with identifying moment function $m(w,\gamma)-\theta.$ Newey, Hsieh, and Robins (1998, 2004) suggested adding the nonparametric influence to identifying moment functions and showed that the remainder will be second order. Robins et al. (2008), and Van der Vaart (2014) showed that in general the remainder is second order for functional estimators. GMM with moment functions obtained by adding the nonparametric influence to identifying moment functions was considered in Chernozhukov et al. (2018) and Bravo, Escanciano, and van Keilegom (2020). The framework described in those papers originated in the joint research for this paper.
Robustness of the nonparametric influence function to the additional functions on which it depends, as in Theorem 2, is not shown in any of the work cited in the previous paragraph. The absence of a first order effect of $\gamma$ on $\bar{\psi}(\gamma,\alpha,\gamma)$ was also not shown, although that is implicit in the remainder analyses previously given. Theorems 2 and 3 provide key conditions for first step estimation to have no first order effect on the asymptotic variance of the debiased GMM estimator $\hat{\theta}$ as needed for the asymptotic theory in Section 8.
The orthogonal moment functions $\psi(w,\gamma,\alpha,\theta)$ could be constructed as an efficient influence function of a semiparametric model, as in Robins and Rotnitzky (1992) and many others since. The construction we give bypasses the semiparametric efficiency framework and is based on the simpler influence function characterization in equation ((ref)) and the limit $\gamma(F)$ for any semiparametric estimator as in Newey (1994a). This construction highlights the distinct roles of $g(w,\gamma,\theta)$ as identifying moments and $\phi(w,\gamma,\alpha,\theta)$ as a bias correction that does not affect identification and is entirely determined by $g(w,\gamma,\theta)$ and $\gamma(F)$. The distinct role of $\phi (w,\gamma,\alpha,\theta)$ leads directly to the robustness result of Theorem 2 and motivates its automatic estimation in Section 6. For these reasons we choose to construct orthogonal moments by adding the nonparametric influence function to identifying moments rather than finding a semiparametric efficient influence function.
The orthogonalization given here is nonparametric, i.e. is estimator based rather than model based, relying only on $g(w,\gamma,\theta)$ and the nonparametric limit $\gamma(F)$ of $\hat{\gamma}$ and not on the specification of any model. Consequently, the Neyman orthogonality shown in Theorems 1-3 does not depend on correct specification of any model. For any first step the orthogonal moments $\bar{\psi}(\gamma,\alpha,\theta)$ will have the properties given in the conclusions of Theorems 2 and 3 under the stated regularity conditions. The absence of any model specification from those regularity conditions demonstrates the nonparametric nature of these orthogonal moment conditions.
There are also model based approaches to orthogonalization that generalize the efficient influence function. For example Newey (1990), Belloni, Chernozhukov, and Kato (2015), and Belloni et al. (2017) showed that in a semiparametric model the residual from the projection of identifying moment functions on the tangent set is orthogonal for certain kinds of first steps. The approach here has the advantage that it does not depend on correct specification of a model.
In some cases the identifying moment functions $g(w,\gamma,\theta)$ may already be orthogonal, so that $\phi(w,\gamma,\alpha,\theta)=0$. An important class of orthogonal moment functions are those where $g(w,\gamma,\theta)$ is the derivative with respect to $\theta$ of an objective function where nonparametric parts have been concentrated out. That is, suppose that there is a function $q(w,\theta,\zeta)$ such that $g(w,\gamma,\theta)=\partial q(w,\theta,\zeta(\theta))/\partial\theta$ and $\zeta(\theta)=\arg\max_{\zeta }E[q(W,\theta,\zeta)]$, where $\gamma$ includes $\zeta(\theta)$ and possibly additional functions. Proposition 2 of Newey (1994a) and Lemma 2.5 of Chernozhukov et al. (2018) then imply that $g(w,\gamma,\theta)$ is orthogonal. This class of moment functions includes those of Robinson (1988), Ichimura (1993), and various partially linear regression models where $\zeta$ represents a conditional expectation. It also includes the efficient score for a semiparametric model when the nonparametric component estimates the maximum of the expected log likelihood; see Severini and Wong (1992), Newey (1994a, pp. 1358-1359), and van der Vaart (1998, pp. 391-396).
The nonparametric influence function $\phi(W,\gamma,\alpha,\theta)$ is unique because $H$ in equation ((ref)) is unrestricted except for regularity conditions. This uniqueness provides another way to understand orthogonality of $\psi(w,\gamma,\alpha,\theta)$. The influence function of $E_{F} [g(W,\gamma(F),\theta_{0})]$ at $F_{0}$ is $\psi(W,\gamma_{0},\alpha _{0},\theta_{0}),$ under the moment condition $E[g(W,\gamma_{0},\theta _{0})]=0.$ This result follows by differentiating $E_{F_{\tau}}[g(W,\gamma (F_{\tau}),\theta_{0})]$ with respect to $\tau$, applying the chain rule, and using ((ref)). The sample average of the debiased moment function $\hat{\psi}(\theta_{0})$ is a nonparametric estimator of $E_{F}[g(W,\gamma (F),\theta_{0})].$ If $\hat{\psi}(\theta_{0})$ is asymptotically equivalent to a sample average and locally regular, meaning that for $H$ and data are i.i.d. with CDF $F_{\tau_{n}}=(1-\tau_{n})F_{0}+\tau_{n}H$ and $\tau_{n}=O(1/\sqrt {n})$ the limiting distribution of $\hat{\psi}(\theta_{0})-E_{F_{\tau_{n}} }[g(W,\gamma(F_{\tau_{n}}),\theta_{0})]$ does not depend on $\tau_{n},$, then uniqueness of $\phi$ implies
as in Van der Vaart (1991), Newey (1994a), and Ichimura and Newey (2017). In other words, the only sample average of a function of the data that $\hat{\psi}(\theta_{0})$ can be asymptotically equivalent to, and also be locally regular, is $\sum_{i=1}^{n}\psi(W,\gamma_{0},\alpha_{0},\theta _{0})/n.$ Equation ((ref)) is precisely an asymptotic version of orthogonality where the sample average $\hat{\psi}(\theta_{0})$ is asymptotically equivalent to the sample average of the same function with estimators $\hat{\gamma}_{\ell},$ $\hat{\alpha}_{\ell},$ and $\tilde{\theta }_{\ell}$ replaced by their limits. In this way orthogonality is justified by uniqueness of $\phi(w,\gamma,\alpha,\theta)$.
Equation ((ref)) is the key orthogonality property for the asymptotic theory of Section 8, where primitive regularity conditions for equation ((ref)) are given. The primitive conditions are motivated by the analysis of this Section, including the conclusions of Theorems 2 and 3. Equation ((ref)) also helps in the comparison of debiased GMM with plug-in GMM, to which we now turn.
To highlight the role of orthogonal moment functions we compare the properties of debiased GMM with a corresponding cross-fit plug-in GMM estimator \[ \tilde{\theta}=\arg\min_{\theta\in\Theta}\hat{g}(\theta)^{\prime}\hat {\Upsilon}\hat{g}(\theta). \] Confidence intervals based on the plug-in GMM estimator are invalid with first step model selection and plug-in GMM is so biased that it is not root(n) consistent with a Lasso and other regularized first steps. Debiased GMM does not suffer from these problems. Plug-in GMM is simpler than debiased GMM in not requiring computation of $\hat{\alpha}_{\ell}$ and $\hat{\phi}$ but that reduced computational does not justify it when it has large biases due to model selection and/or regularization. In this Section we discuss these and other comparative properties of debiased and plug-in GMM.
To compare the properties of debiased and plug-in GMM it is helpful to compare the key asymptotic property of debiased GMM in equation ((ref)) with a corresponding key property of plug-in GMM,
This property is very important. When equation ((ref)) is not satisfied plug-in GMM can have invalid confidence intervals or not be root(n) consistent. As shown in Section 8, the corresponding condition in equation ((ref)) for debiased GMM will be satisfied under general and simple regularity conditions. In contrast, equation ((ref)) requires an additional condition that is specific to the first step and more complicated. Let
Theorem 4: If equation ((ref)) is satisfied for $\hat{\alpha}_{\ell}=\alpha_{0}$ and $\tilde{\theta}_{\ell} =\theta_{0}$ then equation ((ref)) is satisfied if and only if $\tilde{\phi}=o_{p}(n^{-1/2})$.
Equation ((ref)) for $\hat{\alpha}_{\ell}=\alpha_{0}$ and $\tilde{\theta}_{\ell}=\theta_{0}$ is even simpler and more general than ((ref)) for a debiased GMM estimator, making $\sqrt{n}\tilde{\phi }\overset{p}{\longrightarrow}0$ the key regularity condition for plug in GMM. This condition will fail with first step model selection or regularization.
Example 1: Here $\psi(W,\hat{\gamma}_{\ell},\alpha_{0},\theta _{0})-\psi(W,\gamma_{0},\alpha_{0},\theta_{0})=[Z-\alpha_{0}(X)][\hat{\gamma }_{\ell}(X)-\gamma_{0}(X)]$, so that equation ((ref)) will be satisfied under a weak, mean square consistency condition for $\gamma_{0},$ as in Assumption 1 of Section 8. Then equation ((ref)) will be satisfied if and only if
This condition is approximate sample orthogonality of the cross-fit residual $Y_{i}-\hat{\gamma}_{\ell}(X_{i})$ with $\alpha_{0}(X_{i}).$ This condition can fail to hold under model selection, where $\hat{\gamma}_{\ell}(X)$ only includes some variables on which $\alpha_{0}(X)$ depends, so that the residuals $Y_{i}-\hat{\gamma}_{\ell}(X_{i})$ are not approximately orthogonal to $\alpha_{0}(X_{i})$ in the sample. This condition can also fail to hold when $\hat{\gamma}_{\ell}(X_{i})$ is a regularized estimator where residuals are not approximately orthogonal to functions of $X_{i}$ on which $\alpha _{0}(X_{i})$ depends. Thus we see that the properties of the plug-in estimator will depend heavily on the nature of $\alpha_{0}(X_{i})$ even though no estimator of $\alpha_{0}(X_{i})$ is explicitly present.
If the first step $\hat{\gamma}$ incorporates model selection then plug-in GMM gives invalid asymptotic confidence intervals. This occurs because model selection that leads to a correct model with probability approaching must also choose the same model under local alternatives in a root-n neighborhood where that model is incorrect. This local misspecification has a first order effect on plug-in GMM, giving a limiting distribution with nonzero mean under local alternatives, so that the usual asymptotic confidence intervals, based on a zero mean limiting distribution, are invalid. Debiased GMM does not suffer from this problem because the bias from the incorrect first step is second-order. This feature of debiased GMM makes it preferred to plug-in GMM in the many applications where the first step incorporates model selection.
Example 1: Here plug-in GMM is $\tilde{\theta}=\sum_{\ell=1}^{L} \sum_{i\in I_{\ell}}Z_{i}\hat{\gamma}_{\ell}(X_{i})/n$. Suppose that $\gamma_{0}(x)=\breve{b}(x)^{\prime}\breve{\beta}$ for finite dimensional vectors $\breve{b}(x)$ and $\breve{\beta},$ where $\breve{\beta}$ has no zero components. Consider $\hat{\gamma}_{\ell}$ that incorporates model selection such that $\hat{\gamma}_{\ell}(x)$ is equal to the least squares regression of $Y_{i}$ on $\breve{b}(X_{i})$ for all $i\in I_{\ell}$ with probability approaching one. A variety of estimators have this property, including post Lasso or post Lasso with thresholding, when $\breve{b}(X_{i})$ are included among the Lasso regressors and are not too correlated with the other Lasso regressors. Here equation ((ref)) will not hold when $\alpha_{0}(X)$ is not a linear combination of $\breve{b}(X_{i})$, because $\alpha_{0}(X_{i})$ need not be approximately orthogonal in the sample to the residuals $Y_{i}-\hat{\gamma}_{\ell}(X_{i})$ in that case. The precise behavior of $\tilde{\theta}$ is given in the follow result.
Theorem 5: If i) $\gamma_{0}(x)=\breve{b}(x)^{\prime} \breve{\beta};$ ii) $\breve{G}=:E[\breve{b}(X)\breve{b}(X)^{\prime} ]$ is nonsingular; iii) $\alpha_{0}(X)$ and $\breve{b} (X)$ are bounded; and iv) with probability approaching one $\hat{\gamma}_{\ell}(x)$\textit{ is equal to ordinary least squares from regressing }$Y_{i}$\textit{ on }$\breve{b}(X_{i})$\textit{ over }$i\notin I_{\ell};$\textit{ then for }$\bar{\alpha}(x)=\breve{b}(x)^{\prime}\breve {G}^{-1}E[\breve{b}(X)\alpha_{0}(X)]$\textit{,} \[ \tilde{\theta}=\theta_{0}+\frac{1}{n}\sum_{i=1}^{n}\zeta(W_{i})+o_{p} (n^{-1/2})\text{, }\zeta(W)=Z\gamma_{0}(X)-\theta_{0}+\bar{\alpha} (X)[Y-\gamma_{0}(X)].\text{ } \] \textit{Also if iv) the distribution of }$Y$\textit{ conditional on } $(X,Z)$\textit{ has a pdf }$f_{0}(y|x,z)$\textit{ such that there is } $C>0$\textit{ with }$E[\int\{\sup_{\left\vert a\right\vert \leq C} \{[df_{0}(y+a|X,Z)/da]^{2}/f_{0}(y+a|X,Z)\}\}dy]<\infty,$\textit{ then for }$\bar{\sigma}^{2}=E[\{\alpha_{0}(X)-\bar{\alpha}(X)\}^{2}],$\textit{ any }$\mu$, \textit{and }$W_{1},...,W_{n}$ \textit{i.i.d. with CDF }$F_{n}$ \textit{having conditional pdf }$f_{0}(\tilde{y}-n^{-1/2}\mu\{\alpha _{0}(X)-\bar{\alpha}(X)\}|x,z)$ \textit{for }$Y_{i}$ given $(X_{i} ,Z_{i})=(x,z)$ \textit{and CDF }$F_{0}(x,z)$ \textit{for }$(X_{i},Z_{i})$ \textit{we have} \[ \sqrt{n}(\tilde{\theta}-\theta_{n})\overset{d}{\longrightarrow}N(\mu \bar{\sigma}^{2},V), \] \textit{where }$\theta_{n}=E_{F_{n}}[Z\cdot E_{F_{n}}[Y|X]]$\textit{ is the parameter of interest for }$F_{n}.$
The first conclusion shows that with model selection plug-in GMM is asymptotically equivalent to a sample average but the function being averaged is not $\psi(W,\gamma_{0},\alpha_{0},\theta_{0})$ if $\bar{\alpha} (X)\neq\alpha_{0}(X)$. As a result the second conclusion follows, with the limiting distribution of plug-in GMM having a nonzero mean under a local alternative when $\bar{\alpha}(X)\neq\alpha_{0}(X)$, so that the usual asymptotic confidence interval based on a zero mean limiting distribution is invalid. The first conclusion leads to the second conclusion because the nonparametric influence function is unique. The asymptotic equivalence of plug-in GMM to a sample average that is not $\psi(W,\gamma_{0},\alpha _{0},\theta_{0})$ implies that there are local alternatives where the limiting distribution has non zero mean. In this way the invalidity of standard asymptotic confidence intervals for plug-in GMM $\tilde{\theta}$ under model selection can be thought of as resulting from asymptotic equivalence of $\sqrt{n}(\tilde{\theta}-\theta_{0})$ to a sample average of a function that is not $\psi(W,\gamma_{0},\alpha_{0},\theta_{0})$.
The reason that model selection creates a problem for plug-in GMM is that $\psi(W,\gamma_{0},\alpha_{0},\theta_{0})$ depends on two functions, one being $\gamma_{0}(X)$ and the other $\alpha_{0}(X).$ Model selection for $\hat{\gamma}(x)$ gives a good estimator of $\gamma_{0}(X)$ but the variables selected for estimating $\gamma_{0}(X)$ may not include all the variables on which $\alpha_{0}(X)$ depends, leading to $\bar{\alpha}(X)\neq\alpha_{0}(X)$ and hence $\bar{\sigma}_{{}}^{2}>0.$ As mentioned previously in this Section $\sqrt{n}\tilde{\phi}\overset{p}{\longrightarrow}0$ fails in this case because $\alpha_{0}(X_{i})$ will not be approximately asymptotically orthogonal in the sample to the cross-fit least square residuals.
Plug-in GMM will be asymptotically equivalent to the sample average of $\psi(W,\gamma_{0},\alpha_{0},\theta_{0})$ when $\bar{\alpha}(X)=\alpha _{0}(X)$, i.e. when $\alpha_{0}(X)$ is a linear combination of the same variables on which $\gamma_{0}(X)$. However, there is generally no good reason to impose that condition a priori. The one case we are aware of where that condition does hold a priori is where $Z=Y$, so that $\alpha_{0}(X)=\gamma _{0}(X).$ In this case the parameter of interest is $\theta_{0}=E[\gamma _{0}(X)^{2}]=E[\{E[Y|X]\}^{2}]$ and plug-in asymptotic confidence intervals would be valid.
In general plug-in GMM estimators will have similar properties to this example. The nonparametric influence function $\phi(W,\gamma_{0},\alpha _{0},\theta_{0})$ depends on $\alpha_{0}$ in addition to the first step function $\gamma_{0}$, and a first step model selection will only fit $\gamma_{0}$ well and not $\alpha_{0}$. Consequently model selection for the first step will make asymptotic confidence intervals invalid for plug-in GMM. There may be some exceptions where this model selection problem does not hold, as discussed in Example 1, but these are few and far between.
In some cases the double selection procedure of Belloni, Chernozhukov, and Hansen (2014) also corrects for model selection. That procedure includes variables in the estimation of $\gamma_{0}$ if they are also important in estimation of $\alpha_{0}$. Such a double selection procedure is not available for many first step machine learners that are not constructed by selecting variables in a regression. Also, double selection can lead to much less parsimonious models than would be used by debiased GMM, which allows different variables to be selected for estimation of $\alpha_{0}$ than are selected for estimation of $\gamma_{0}$. It is beyond the scope of this paper to compare debiased GMM with double selection.
Another way to avoid the model selection problem for plug-in GMM is to limit selection to models that can approximate any unknown function. For example Newey (1994a) did this for series estimation by requiring that selection is made only among models that can approximate any function in large samples. Forcing a flexible approximation in this way is not very feasible in high dimensional settings and is not needed for debiased GMM, where model selection can be applied separately to the various first step estimators.
That model selection can make confidence intervals invalid was shown by Leeb and Potscher (2005, 2008) for least squares and Lasso. The model selection problem for plug-in GMM was pointed out by Belloni, Chernozhukov, and Kato (2015) and Chernozhukov, Hansen, and Spindler (2015) who suggested Neyman orthogonal moment functions as a solution to this problem. Theorem 4 gives precise asymptotic theory showing the asymptotic equivalence of plug-in GMM to a sample average of a function other than $\psi(W,\gamma_{0},\alpha_{0} ,\theta_{0}).$ This feature of plug-in GMM under model selection is reminiscent of the Hodges estimator, a well known example of an estimator that is not asymptotically equivalent to the sample average of the influence function of the mean.
Debiased GMM avoids the model selection problem because it has the small bias property discussed in Section 4 following Theorem 3. Model selection imparts a bias to the first step of size $1/\sqrt{n}$ under local alternatives. The small bias property for debiased GMM means that its bias vanishes faster than first step bias, i.e. faster than $1/\sqrt{n}$. Hence the asymptotic distribution of debiased GMM will have zero mean and asymptotic confidence intervals will be valid. Given the common feature of model selection in machine learning and other econometric methods the robustness of debiased GMM to first step model selection motivates its use in practice.
Many machine learners employ regularization to obtain estimators of functions that approximately balance bias and standard deviation. For nonparametric estimation or machine learning with large sets of predictors the standard deviation of the predictor will shrink slower than $1/\sqrt{n},$ and hence so will be bias. This bias may pass through to plug in GMM and result in $\tilde{\theta}$ not being root(n) consistent. This bias problem is clearly present for Lasso where penalization leads to bias for $\tilde{\theta}$ of size $\sqrt{\ln(p)/n}$. With bias of that size $\sqrt{n}$ times the bias will be of size $\sqrt{\ln(p)}$ which goes to infinity, so that that plug-in GMM is not root-n consistent, as we show in this subsection. Debiased GMM has the small bias property discussed in Section 2 and so will be root-n consistent under sufficient regularity conditions, with bias being second order (size $\ln(p)/n$ for Lasso) permitting debiased GMM to be root-n consistent (by $\sqrt{n}\ln(p)/n\longrightarrow0$ for Lasso).
Example 1: To illustrate the regularization bias problem for plug-in GMM we give its properties when $\gamma_{0}(X)$ is a linear combination of a finite number of functions and $\hat{\gamma}_{\ell}$ is Lasso; \[ \hat{\gamma}_{\ell}(x)=b(x)^{\prime}\hat{\beta}_{\ell},\text{ }\hat{\beta }_{\ell}=\arg\min_{\beta}\frac{1}{n-n_{\ell}}\sum_{i\notin I_{\ell}} [Y_{i}-b(X_{i})^{\prime}\beta]^{2}+r\sum_{j=1}^{p}\left\vert \beta _{j}\right\vert , \] where $b(x)$ is a $p\times1$ vector of functions.
Theorem 6: If i) there is an $s\times1$ subvector $\breve{b}(x)$ of $b(x)$ such that $\gamma_{0}(X)=\breve {b}(X)^{\prime}\breve{\beta},$ all elements of $\breve{\beta} $\textit{ are nonzero, }$\breve{G}=:E[\breve{b}(X)\breve{b}(X)^{\prime} ]$\textit{ is nonsingular; ii) }$\alpha_{0}(X)$\textit{ is bounded and }$c=E[\alpha_{0}(X)\breve{b}(X)^{\prime}]\breve{G}^{-1}\breve{e}\neq0$\textit{ for }$\breve{e}=(sgn(\breve{\beta}_{1}),...,sgn(\breve{\beta}_{s}))^{\prime} $\textit{; iii) with probability approaching one }$1(\hat{\beta}_{\ell j}=0)=1(\beta_{j}=0)$\textit{ for all }$j;$\textit{ iv) }$\sqrt{\ln (p)/n}=O(r)$\textit{ then }$\tilde{\theta}=\sum_{\ell=1}^{L}\sum_{i\in I_{\ell}}Z_{i}\hat{\gamma}_{\ell}(X_{i})/n$\textit{ satisfies} \[ \sqrt{n}\left\vert \tilde{\theta}-\theta_{0}\right\vert =\left\vert O_{p}(1)-c\sqrt{n}r\right\vert \overset{p}{\longrightarrow}\infty. \]
Because $\gamma_{0}(X)$ is a linear combination of a finite number of elements of $b(X)$ condition iii) is known to be satisfied when all coefficients from regressing each $b_{j}(X_{i})$ on $\breve{b}(X_{i})$ are small enough in absolute value; see Zhao and Yu (2006).
Here we see that the plug in estimator is root-n consistent for Lasso in Example 1 when $\gamma_{0}(X)$ is a linear combination of a finite number of elements of $b(X)$ and condition iii)\ is satisfied. In general plug-in GMM will not be root-n consistent with a Lasso first step, though it is beyond the scope of this paper to show this. More generally plug-in GMM will also have large bias for first step machine learners other than Lasso, e.g. as found for random forests in a Monte Carlo example in Chernozhukov et al. (2018).
The robustness of debiased GMM to model selection and its low bias relative to plug-in GMM are both reasons to prefer debiased GMM over plug-in GMM for an any application where there is first step model selection or where the regularization bias in the first step estimator passes through to the estimator of the parameter of interest. Both features are present for many machine learning first steps making debiased GMM especially useful there. We note that this preference is based on first order properties which dominate any second order comparison of debiased and plug-in GMM.
Doubly robust moment conditions are those that hold when either $\gamma =\gamma_{0}$ or $\alpha=\alpha_{0}$. Such moment conditions are of wide interest. We will show in Section 7 that orthogonal moment conditions are doubly robust if and only if $\bar{\psi}(\gamma,\alpha_{0},\theta_{0})$ is affine in $\gamma.$ Plug in GMM moment conditions have more limited robustness properties, being satisfied when $\gamma$ has a form related to $\alpha_{0}$ for some cases, as also discussed in Section 7.
When there is no first step model selection and regularization does not destroy root(n) consistency of plug-in GMM, debiased GMM has been shown to have better 2nd order properties than plug-in GMM $\ $in some cases. Newey, Hsieh, and Robins (1998, 2004) showed that debiased leave one out kernel estimators of density weighted averages have smaller asymptotic and small sample mean square error and than corresponding plug-ins, for a wide range of bandwidths. Newey and Robins (2017) gave weaker conditions for root(n) consistency of doubly robust estimators than for plug-in estimators. These comparisons seem specific to the type of estimator, so it may be too much to expect that debiased GMM\ always has better second order properties than plug-in GMM. The first order advantages of debiased GMM when the first step incorporates model selection is regularized give compelling reasons for its use there.
From Theorem 4 we see that the key difference between asymptotic theory for debiased and plug-in GMM is that for debiased GMM it is sufficient that equation ((ref)) holds while for plug-in GMM it is sufficient that ((ref)) holds with $\hat{\alpha}_{\ell}=\alpha_{0}$ and $\tilde{\theta}_{\ell}=\theta_{0}$ and that equation ((ref)) holds. Equation ((ref)) is a little more involved for debiased GMM because $\hat{\alpha}_{\ell}$ is estimated but sufficient conditions are general and simple in only involving mean square consistency and one or two mean square rate conditions, as described in Section 8. In contrast showing equation ((ref)) for plug-in GMM seems very specific to the nature of $\hat{\gamma}_{\ell}$ and quite complicated. Using cross-fitting equation ((ref)) will follow from \[ \sqrt{n}\int\phi(w,\hat{\gamma}_{\ell},\alpha_{0},\theta_{0})F_{0} (dw)\overset{}{\overset{p}{\longrightarrow}0,\text{ }(\ell=1,...,L)} \] but showing this also seems quite specific to the form of $\hat{\gamma}$ and complicated. In this way regularity conditions for plug-in GMM are less general and more complicated than for debiased GMM.
Debiased GMM requires the nonparametric influence function $\phi (w,\gamma,\alpha,\theta)$ and an estimator $\hat{\alpha}$ of the unknown function $\alpha_{0}$. As previously discussed $\phi(w,\gamma,\alpha,\theta)$ is readily available in many important settings. In this Section we give a general approach to constructing $\hat{\alpha}$ that is based only on the orthogonal moment functions, show how that can be used for debiased GMM estimation of objects that depend on location functions, and apply that to obtain the $\hat{\alpha}_{\ell}$ of Example 2.
The orthogonality of $\bar{\psi}(\gamma,\alpha,\theta)$ can be used to estimate $\alpha_{0}.$ Let $\Gamma$ be the linear set from Theorem 2 that is known to contain $\gamma_{0}$ and $\hat{\gamma}$ and let $\delta\in\Gamma.$ The zero Hadamard derivative in Theorem 3 implies a zero Gateaux derivative with respect to $\delta$ so that
This can be thought of as a population moment condition for $\alpha_{0}$. We can form a sample moment function corresponding to this population moment condition by replacing the expectation by a sample average, $\gamma_{0}$ by $\hat{\gamma}_{\ell}$, and $\theta_{0}$ by $\hat{\theta}_{\ell}$ to obtain
We can then replace $\alpha$ by a sieve (i.e. parametric approximation) and estimate the sieve parameters using these sample moments for a variety of choices of $\delta.$ We can also regularize to allow for a high dimensional specification for $\alpha.$ The sample moments in equation ((ref)) depend only on observations not in $I_{\ell}$ so that the resulting $\hat{\alpha}_{\ell}$ will also, as required for debiased GMM.
Example 1: Here $\alpha_{0}$ is an function of $X$ that has finite second moment and
This is a sample moment corresponding to population moment condition $E[\left\{ Z-\alpha_{0}(X)\right\} \delta(X)]=0,$ which holds by $\alpha _{0}(X)=E[Z|X].$ If $\alpha(X)$ was replaced by a linear combination $\rho^{\prime}b(x)$ of a dictionary $b(x)=(b_{1}(x),...,b_{p}(x))^{\prime}$ and $\delta(X)$ replaced by element $b_{j}(X)$ then the sample moment function \[ \hat{\psi}_{\gamma}(b_{j},\rho^{\prime}b)=\frac{1}{n-n_{\ell}}\sum_{i\notin I_{\ell}}\left[ Z_{i}-\rho^{\prime}b(X_{i})\right] b_{j}(X_{i}). \] The collection of sample moments $\hat{\psi}_{\gamma}(b_{j},\rho^{\prime}b),$ $(j=1,...,p)$ can be used to construct a Lasso or Dantzig estimator of $\alpha_{0}(X)=E[Z|X].$
Example 4: Consider an additional example where the parameter of interest is a linear function $\theta_{0}=E[m(W,\gamma_{0})]$ of an unknown function $\gamma_{0}$ with
$v(u)$ is a convex function, $\Gamma$ is a set of functions that is closed in mean square, and $\bar{\alpha}(X)$ is the Riesz representer for the linear functional $E[m(X,\gamma)]$. Here $\gamma_{0}(X)$ measures the conditional location of $Y$ given $X$. For example $\gamma_{0}(X)=E[Y|X]$ when $v(u)=u^{2}/2$ and $\gamma_{0}(X)$ is the $\lambda^{th}$ conditional quantile when $v(u)=[\lambda-1(u<0)]u.$ For identifying moment function $g(w,\gamma ,\theta)=m(w,\gamma)-\theta$ the nonparametric influence function is \[ \phi(w,\gamma,\alpha,\theta)=\alpha(x)v_{u}(Y-\gamma(X)),\text{ }\alpha _{0}(X)=\frac{\bar{\alpha}(X)}{\bar{v}_{ \operatorname{u} \operatorname{u} }(X)},\text{ }\bar{v}_{ \operatorname{u} \operatorname{u} }(X)=\frac{d}{d\tau}E[v_{u}(Y-\gamma_{0}(X)+\tau)|X]. \] This influence function appears in Ai and Chen (2007) for a series estimator of $\gamma_{0}$ and Ichimura and Newey (2017) show it satisfies equation ((ref)). Replacing $\alpha(X)$ by a linear combination $\rho^{\prime }b(X)$ of a dictionary $b(X)$ and choosing $\delta=b_{j}$ gives the sample moment
where we have assumed that $v_{u}(u)$ has derivative $v_{ \operatorname{u} \operatorname{u} }(u)$ and $e_{j}$ is the $j^{th}$ unit vector.
Lasso or Dantzig estimators of $\alpha_{0}(x)$ can be constructed from the collection of sample moments $\hat{\psi}_{\gamma}(b_{j},\rho^{\prime}b),$ $(j=1,...,p).$ Let $\hat{M}_{\ell}=(\hat{M}_{\ell1},...,\hat{M}_{\ell p})^{\prime}$. A Lasso estimator of $\alpha_{0}(x)$ is
This is a Lasso minimum distance estimator of $\alpha_{0}(x)$ that generalize that of Chernozhukov, Newey, and Singh (2018) for $v(u)=u^{2}/2$ to any twice differentiable convex function $v(u).$ It has the nice feature that an explicit estimator of $\bar{v}_{ \operatorname{u} \operatorname{u} }(X)$ is not required to be places in a denominator. Instead the presence of $\bar{v}_{ \operatorname{u} \operatorname{u} }(X)$ is accounted for in the weighted second moment estimator $\hat{Q}_{\ell }.$
Example 2: When $\gamma_{0}(X)$ is a conditional quantile $v_{u}(u)$ is not differentiable so a different $\hat{Q}_{\ell}$ must be used. The $v_{ \operatorname{u} \operatorname{u} }(Y_{i}-\hat{\gamma}_{\ell}(X_{i}))$ in $\hat{Q}_{\ell}$ must be replaced with something such that $\hat{Q}_{\ell}$ estimates $Q=E[\bar{v}_{ \operatorname{u} \operatorname{u} }(X)b(X)b(X)^{\prime}].$ For quantiles $\bar{v}_{ \operatorname{u} \operatorname{u} }(X)=f(0|X)$ where $f(0|X)$ is the conditional pdf of $U=Y-\gamma_{0}(X)$ given $X$ at $U=0$. The $\hat{Q}_{\ell}$ given in Example 2 in Section 2 takes account of this in the presence of the kernel term in \[ \hat{Q}_{\ell}=\frac{1}{n-n_{\ell}}\sum_{\ell^{\prime}\neq\ell}\sum_{i\in I_{\ell^{\prime}}}\frac{1}{h}K(\frac{Y_{i}-\hat{\gamma}_{\ell,\ell^{\prime} }(X_{i})}{h})b(X_{i})b(X_{i})^{\prime}, \] where $\hat{\gamma}_{\ell,\ell^{\prime}}(X_{i})$ uses only observations not in $I_{\ell}$ or $I_{\ell}^{\prime}.$ Here $\hat{Q}_{\ell}$ estimates $Q=E[f(0|X)b(X)b(X)^{\prime}]$. We also use nested sample splitting to construct $\hat{\gamma}_{\ell,\ell^{\prime}}(X_{i})$ in order to obtain asymptotic results for $\hat{\theta}$ in Section 8 using just a mean square convergence rate for $\hat{\gamma}_{\ell\ell^{\prime}}$.
It would be interesting to use the moment functions ((ref)) to construct $\hat{\alpha}$ for first steps other than a conditional location function $\gamma_{0}(X)$ in equation ((ref)). That is beyond the scope of this paper and is reserved to future work, including identification of $\alpha_{0}$ and asymptotic theory for $\hat{\alpha}$.
The construction of estimating equations for $\alpha$ from the Gateaux derivative of orthogonal sample moments with respect to variation in the first step away from $\hat{\gamma}$ in equation ((ref)) generalizes the estimating equations of Chernozhukov, Newey, and Singh (2018) for linear functionals of a conditional expectation to any orthogonal moment function and first step $\hat{\gamma}$. These also generalize estimators for $\alpha$ for average treatment effects that were proposed and analyzed in Vermeulen and Vansteelandt (2015), Avagyan and Vansteelandt (2017), and Tan (2018).
This approach of estimating the nonparametric influence function uses its form $\phi(w,\gamma,\alpha,\theta)$ to construct an estimator of $\alpha_{0}$. Using the form of $\phi(w,\gamma,\alpha,\theta)$ seems good in high dimensional settings where $\phi(w,\gamma,\alpha,\theta)$ may be a high dimensional function. It is also possible to estimate the entire nonparametric influence function using just the first step and the identifying moments. Such estimators are available for first step series and kernel estimation. For first step series estimation an estimator of $\phi(w,\gamma,\alpha,\theta)$ can be constructed by treating the first step estimator as if it were parametric and applying a standard formula parametric two-step estimators, e.g. as in Newey (1984) and Murphy and Topel (1985). Newey (1994a, 1997) and Ackerberg, Chen, and Hahn (2012) used this approach for estimating the asymptotic variance of functions of series estimators, while we refer here to another use in constructing debiased GMM. Debiased GMM estimators can be constructed by adding the nonparametric influence function obtained by treating the first step as if it were parametric. For parametric maximum likelihood the resulting orthogonal moment functions are the basis of Neyman's (1959) C-alpha test. Wooldridge (1991) generalized such moment conditions to parametric nonlinear least squares and Lee (2005), Bera, Motes-Rohas, and Sosa-Escudero. (2010), and Chernozhukov, Hansen, and Spindler to GMM.
For first step kernel estimation one can use the numerical influence function estimator of Newey (1994b) to estimate $\phi(w,\gamma,\alpha,\theta),$ as suggested in a previous version of this paper and proven to work in a low dimensional nonparametric setting in Bravo, Escanciano, and van Keilegom (2020). The idea is to differentiate with respect to the effect of the $i^{th}$ observation on sample moments. Kernel estimators are not well suited to high dimensional settings with machine learning so we do not consider them here.
It is also possible to estimate the nonparametric influence function using a numerical derivative version of equation ((ref)). This approach has been given in Carone, Luedtke, and van der Laan (2016) and Bravo, Escanciano, and van Keilegom (2020) for construction of orthogonal moment functions. We focus here on estimating $\alpha_{0}$ where there is a known form $\phi(w,\gamma,\alpha,\theta)$ because that information is widely available and using it seems likely to be good in high dimensional settings.
The zero derivative condition in equation ((ref)) is an appealing robustness property. This condition can be interpreted as local insensitivity of the moments to the value of $\gamma$, with the moments remaining close to zero as $\gamma$ varies away from its true value. Because it is difficult to get nonparametric functions exactly right, especially in high dimensional settings, this property is an appealing one.
Such robustness considerations, well explained in Robins and Rotnitzky (2001), have motivated the development of doubly robust moment conditions. Doubly robust moment conditions have expectation zero if one first step component is incorrect. Doubly robust moment conditions allow two chances for the moment conditions to hold, an appealing robustness feature. Also, doubly robust moment conditions have simpler conditions for asymptotic normality than general debiased GMM, as discussed in Section 8.
In this Section we characterize double robustness and derive several novel classes of doubly robust moment conditions. We construct doubly robust moment functions by adding to identifying moment functions the nonparametric influence functions to obtain orthogonal moments. In this way the derivation of new doubly robust moment functions is aided by the construction of orthogonal moment functions from adding the nonparametric influence function.
Double robustness is that for all $\gamma\in\Gamma,$ $\alpha$, and $\theta$ \[ 0=\bar{\psi}(\gamma,\alpha_{0},\theta_{0})=\bar{\psi}(\gamma_{0},\alpha ,\theta), \] where $\gamma\in\Gamma$ is a set of possible first steps. The second equality already follows from Theorem 2. The first conclusion of Theorem 3 gives a local version of the first equality. If $\bar{\psi}(\gamma,\alpha_{0} ,\theta_{0})$ is affine in $\gamma$ then this local property becomes global so that double robustness holds. Clearly doubly robust moment conditions are also affine, so that $\bar{\psi}(\gamma,\alpha_{0},\theta_{0})$ being affine is a complete characterization of double robustness. The following result gives this characterization for a zero Gateaux derivative, a condition easier to check than zero Hadamard derivative in Theorem 3.
Theorem 7: If $\Gamma$ is linear then $\psi(w,\gamma,\alpha,\theta)$ is doubly robust if and only if for every $\gamma\in\Gamma$ \[ \left. \frac{\partial\bar{\psi}((1-\tau)\gamma_{0}+\tau\gamma,\alpha _{0},\theta_{0})}{\partial\tau}\right\vert _{\tau=0}=0, \] and $\bar{\psi}(\gamma,\alpha_{0},\theta_{0})$\textit{ is affine in }$\gamma$\textit{.}
This characterization can be used to construct doubly robust moment conditions from identifying moment conditions that are affine in $\gamma$. If $g(W,\gamma,\theta_{0})$ is affine in $\gamma$ and $\phi(W,\gamma,\alpha _{0},\theta_{0})$ is also affine in $\gamma$ then $\bar{\psi}(\gamma ,\alpha_{0},\theta_{0})$ will also be affine in $\gamma.$ In addition the zero Gateaux derivative condition in Theorem 6 will hold by Theorem 3, so that $\bar{\psi}(\gamma,\alpha,\theta)$ will be doubly robust. We use this construction to obtain doubly robust moment conditions for first steps that satisfy conditional moment restrictions and for a first step that is a density function.
Robins and Rotnitzky (2001) gave conditions for the existence of doubly robust moment conditions in semiparametric models. Theorem 5 is complementary to those results in giving a complete characterization of doubly robust moments when $\Gamma$ is linear.
A novel class of doubly robust moment functions are those where the first step $\gamma_{0}$ satisfies a conditional moment restriction
where $\lambda(W,\gamma)$ is a scalar functional of $\gamma$ that is affine in $\gamma$ and $X$ are regressors or instrumental variables. Suppose that $\hat{\gamma}$ is the nonparametric 2SLS estimator of Newey and Powell (1989, 2003) and Newey (1991) where $\gamma(F)=\arg\min_{\Gamma}E_{F}[\{E_{F} [\lambda(W,\gamma)|X]\}^{2}].$ It follows from Newey (1994a), Ai and Chen (2007, p. 40), and Ichimura and Newey (2017) that when equation ((ref)) is satisfied and $\phi(w,\gamma,\alpha,\theta)$ exists there is $\alpha(x,\theta)$ such that \[ \phi(w,\gamma,\alpha,\theta)=\alpha(x,\theta)\lambda(W,\gamma). \] The following result characterizes double robustness in this setting.
Theorem 8:\ $\psi(W,\gamma,\alpha,\theta)=g(W,\gamma ,\theta)+\alpha(X)\lambda(W,\gamma)$ is doubly robust over $\gamma \in\Gamma$ if and only if \[ E[g(W,\gamma,\theta_{0})]=-E[\alpha_{0}(X)\lambda(W,\gamma)]\text{, \textit{for all} }\gamma\in\Gamma. \]
When $E[g(W,\gamma,\theta_{0})]$ is affine in $\gamma$ the condition in Theorem 8 is an expected outer product representation of $E[g(W,\gamma ,\theta_{0})].$ This characterization of a doubly robust moment function with a nonparametric 2SLS first step has several interesting special cases. The form of the doubly robust moment function in Theorem 8 is similar to the efficient score in some cases in Ai and Chen (2012).
Example 5: An important example is a linear functional $\theta _{0}=E[m(W,\gamma_{0})]$ of a regression function $\gamma$, where $\lambda(W,\gamma)=Y-\gamma(X)$ for some outcome variable $Y$ and $m(w,\gamma)$ is linear in $\gamma$. Here the identifying moment function is $g(w,\gamma,\theta)=m(w,\gamma)-\theta$, which is affine in $\gamma$. Also $\lambda(w,\gamma)$ is also affine in $\gamma$, so that the conditions of Theorem 8 are satisfied. The next result follows from Theorem 8.
Corollary 9: If $m(w,\gamma)$ is linear in $\gamma $ and there is $\alpha_{0}(x)$ such that $E[\alpha _{0}(X)^{2}]<\infty$ and $E[m(W,\gamma)]=E[\alpha_{0}(X)\gamma (X)]$\textit{ for all }$E[\gamma(X)^{2}]<\infty$\textit{ then }$\psi (w,\gamma,\alpha,\theta)=m(w,\gamma)-\theta+\alpha(x)[y-\gamma(x)]$\textit{ is doubly robust.}
In Corollary 9 $\alpha_{0}(x)$ is the Riesz representer of the functional $E[m(W,\gamma)]=E[\alpha_{0}(X)\gamma(X)]$ for all $\gamma$ with $E[\gamma(X)^{2}]<\infty$, as in Proposition 4 of Newey (1994a). Many important doubly robust moment functions are special cases of Corollary 9, including average treatment effects, policy effects, and average derivatives, as discussed in Newey and Robins (2017), Chernozhukov, Newey, and Robins (2018), Hirshberg and Wager (2018), and Chernozhukov, Newey, and Singh (2018). In these papers Corollary 9 is also used to derive new doubly robust moment functions.
Example 6: An interesting generalization allowing for endogeneity has $\lambda(W,\gamma)=Y-\gamma(Z)$ where $Z$ need not be equal to $X$. Here the conditional moment restriction ((ref)) is a nonparametric instrumental variables model as in Newey and Powell (1989, 2003) and Newey (1991). Theorem 8 can be applied to derive doubly robust moment functions since $\lambda(W,\gamma)=Y-\gamma(Z)$ is affine in $\gamma.$
Corollary 10: If $m(w,\gamma)$ is linear in $\gamma$ and there is $\alpha_{0}(x)$ such that $E[\alpha_{0}(X)^{2}]<\infty$ and $E[m(W,\gamma)]=E[\alpha _{0}(X)\gamma(Z)]$\textit{ for all }$E[\gamma(Z)^{2}]<\infty$\textit{ then }$\psi(w,\gamma,\alpha,\theta)=m(w,\gamma)-\theta+\alpha(x)[y-\gamma (z)]$\textit{ is doubly robust.}
As discussed in Ichimura and Newey (2017), if there is $v(Z)$ with $E[v(Z)^{2}]<\infty$ and $E[m(W,\gamma)]=E[v(Z)\gamma(Z)]$ then existence of $\alpha_{0}(X)$ satisfying the condition of Corollary 10 requires $v(Z)=E[\alpha_{0}(X)|Z]$, which is necessary for root-n consistent estimability of $\theta_{0}$, as shown by Severini and Tripathi (2012).
Example 7: Many novel examples of doubly robust moment functions can be derived from Corollary 10, including policy effects and average derivatives. A weighted average derivative example has $m(w,\gamma)=\bar {v}(z)\partial\gamma(z)/\partial z_{1}$ for some known $\bar{v}(z).$ A doubly robust moment function is \[ \psi(w,\gamma,\alpha,\theta)=\bar{v}(z)\frac{\partial\gamma(z)}{\partial z_{1}}-\theta+\alpha(x)[y-\gamma(z)],\text{ }E[\alpha_{0}(X)|Z]=-\frac {\partial\{f_{0}(Z)\bar{v}(Z)\}/\partial z_{1}}{f_{0}(Z)}, \] where $f_{0}(z)$ is the marginal pdf of $Z$. This is a doubly robust moment function that could be used to construct a doubly robust version of the plug-in estimator of Ai and Chen (2007).
Using Theorem 7 to construct doubly robust moment functions can depend on specifying $\gamma$ to make $g(W,\gamma,\theta_{0})$ and $\phi(W,\gamma ,\alpha_{0},\theta_{0})$ affine in $\gamma.$ We illustrate with a well known example.
Example 8: Suppose that the object of interest is $\theta _{0}=E[Y^{\ast}]$ where $Y=1(D=1)Y^{\ast}$ is observed for a observed completed data indicator $D\in\{0,1\}$ and the data are missing at random with $E[Y^{\ast}|X,D=1]=E[Y^{\ast}|X]$ for observed covariates $X.$ Inverse probability weighting gives $\theta_{0}=E[DY/P_{0}(X)]=E[P_{0}(X)^{-1} E[DY|X]],$ which is nonlinear in the unknown propensity score $P_{0} (X)=\Pr(D=1|X)$. A corresponding affine in $\gamma$ identifying moment function is $g(w,\gamma,\theta)=g(w,\gamma,\theta)=\gamma(x)dy-\theta$ with true first step $\gamma_{0}(X)=P_{0}(X)^{-1}.$ This $\gamma_{0}$ satisfies the conditional moment restriction in equation ((ref)) for $\lambda(w,\gamma)=1-\gamma(x)d$ that is affine in $\gamma.$ Also, for $\alpha_{0}(X)=E[Y|X,D=1]=E[DY|X]\gamma_{0}(X)$ we have
The doubly robust moment function from Theorem 8 is then $\psi(w,\gamma ,\alpha,\theta)=\gamma(x)dy-\theta+\alpha(x)(1-\gamma(x)d),$ which is the doubly robust moment function of Robins, Rotnitzky, and Zhao (1994). This example shows how that classic doubly robust moment function is a special case of Theorem 7, with moment condition that is affine in a first step $\gamma$ for $\gamma_{0}$ equal to the inverse propensity score. The only if part of Theorem 7 states that every doubly robust moment function will have expectation that is affine in $\gamma$.
Rotnitzky, Smucler, and Robins (2019) give a general class of robust estimators that includes interesting examples not treated here.
Another novel class of doubly moment conditions are those where the first step $\gamma$ is a pdf of a function $X$ of the data observation $W.$ By Proposition 5 of Newey (1994a), the first step influence function is \[ \phi(w,\gamma,\alpha,\theta)=\alpha(x)-\int\alpha(u)\gamma(u)du, \] which is affine in $\gamma$. When the identifying moment function is affine adding this nonparametric influence function gives a doubly robust moment function.
Theorem 11:\ If there exists $\alpha_{0}(x)$ with $\int\alpha_{0}(u)^{2}du$ $<\infty$ and $E[g(W,\gamma,\theta _{0})]=\int\alpha_{0}(u)[\gamma(u)-\gamma_{0}(u)]du$ for all $\gamma$ with $\int\gamma(u)^{2}du<\infty$ then $\psi(W,\gamma,\alpha ,\theta)=g(W,\gamma,\theta)+\alpha(X)-\int\alpha(u)\gamma(u)du$\textit{ is doubly robust.}
Here $\alpha_{0}(x)$ is the Riesz representer of Proposition 5 of Newey (1994a) for the Lebesgue inner product
Example 9: An example is the density weighted average derivative of Powell, Stock, and Stoker (1989), where $g(w,\gamma,\theta)=-2y\cdot \partial\gamma(x)/\partial x-\theta$ and $\alpha_{0}(x)=\partial \{E[Y|X=x]\gamma_{0}(x)\}/\partial x.$ Because $g(w,\gamma,\theta)$ is affine in $\gamma$ Theorem 11 implies \[ \psi(W,\gamma,\alpha,\theta)=-2Y\frac{\partial\gamma(X)}{\partial x} -\theta+\alpha(X)-\int\alpha(u)\gamma(u)du, \] is doubly robust. Double robustness of this moment function seems to be a novel result.
Doubly robust moment conditions can be used to identify parameters of interest.
Theorem 12: If Assumption 1 is satisfied, $\alpha_{0} $ is identified, and for some $\bar{\gamma}$ the equation $E[\psi(W,\bar{\gamma},\alpha_{0},\theta)]=0$ has a unique solution at $\theta=$ $\theta_{0}$\textit{ then }$\theta_{0}$\textit{ is identified as that solution.}
Example 10: Applying this result to the nonparametric instrumental variables setting of Assumption 6 leads to identification of functionals of $\gamma_{0}$ without requiring that $\gamma_{0}$ be identified. Focusing on Example 7, note that $\alpha_{0}(X)$ is identified as a solution to $-f_{0}(Z)^{-1}\partial\{f_{0}(Z)\bar{v}(Z)\}/\partial z_{1}=E[\alpha _{0}(X)|Z].$ Setting $\bar{\gamma}=0$ in Theorem 7 then identifies $\theta _{0}=E[\alpha_{0}(X)Y]$, extending Santos (2011) and Severini and Tripathi (2006, 2012), to the weighted average derivative.
Partial robustness refers to identifying moments where $E[g(W,\theta_{0} ,\bar{\gamma})]=0$ for some $\bar{\gamma}\neq\gamma_{0}$. This is a weaker property for identifying moment function than double robustness for the associated orthogonal moment function. Also, partial robustness for identifying moments that are affine in $\gamma$ with $E[\phi(W,\gamma ,\alpha_{0},\theta_{0})]$ affine in $\gamma$ can be characterized by the nonparametric influence function, since double robustness implies \[ E[g(W,\theta_{0},\gamma)]=-E[\phi(W,\gamma,\alpha_{0},\theta_{0})]. \] We give two examples of partial robustness results that follow from double robustness.
Example 5: For a linear functional $\theta_{0}=E[m(W,\gamma_{0})]$ of a regression function $\gamma_{0}(X)=E[Y|X]$, let $b(X)$ be a $p\times1$ vector of functions of $X$ and $\bar{\gamma}(X)=$ $b(X)^{\prime}\delta,$ $\delta=E[b(X)b(X)^{\prime}])^{-1}E[b(X)Y]$, be the best linear predictor of $\gamma_{0}(X)$ by $b(X)$.
Theorem 13: If $E[b(X)b(X)^{\prime}]$ is nonsingular and $\alpha_{0}(X)=\rho_{0}^{\prime}b(X)$ for some $\rho_{0} $ then $\theta_{0}=E[m(W,\bar{\gamma})]$.
This result generalizes Stoker's (1986) result that linear regression coefficients equal average derivatives when the regressors are multivariate Gaussian to any linear functional $m(w,\gamma)$ and nonlinear $b(X)$.
Example 7: Consider the average derivative $\theta_{0}=E[\partial \gamma_{0}(Z)/\partial z_{1}]$ where $g(w,\gamma,\theta)=\partial \gamma(z)/\partial z_{1}-\theta.$ Let $\delta=(E[a(X)b(Z)^{\prime} ])^{-1}E[a(X)Y]$ be the limit of the linear instrumental variables estimator with right hand side variables $b(Z)$ and the same number of instruments $a(X)$, and $\bar{\gamma}(Z)=b(Z)^{\prime}\delta$ the linear instrumental variables estimand.
Theorem 14: If $-\partial\ln f_{0}(Z)/\partial z_{r}=c^{\prime}b(Z)$ for a constant vector $c$, $E[b(Z)b(Z)^{\prime}]$ is nonsingular, and $E[a(X)|Z]=\Pi b(Z)$ for a square nonsingular $\Pi$\textit{ then }$\theta _{0}=E[\partial\bar{\gamma}(Z)/\partial z_{1}].$
This is a generalization to nonparametric instrumental variables of Stoker's (1986) result.
In this Section we give simple and general asymptotic theory for debiased GMM. The results differ from Chernozhukov et al. (2018) in the use of Theorem 2, a different remainder decomposition that leads to simpler conditions, and incorporation of double robustness in the general conditions. We begin with conditions for the key property
Assumption 1: $E[\left\Vert \psi(W_{i},\theta_{0},\gamma_{0} ,\alpha_{0})\right\Vert ^{2}]<\infty$ and
These are mild mean square consistency conditions for $\hat{\gamma}_{\ell}$ and $(\hat{\alpha}_{\ell},\tilde{\theta}_{\ell})\,$separately. They differ from Chernozhukov et al. (2018) in being separate conditions for $\hat{\gamma }_{\ell}$ and $(\hat{\alpha}_{\ell},\tilde{\theta}_{\ell})$ and for $g(w,\gamma,\theta)$ and $\phi(w,\gamma,\alpha,\theta).$ Let \[ \hat{\Delta}_{\ell}(w)=\phi(w,\hat{\gamma}_{\ell},\hat{\alpha}_{\ell} ,\tilde{\theta}_{\ell})-\phi(w,\gamma_{0},\hat{\alpha}_{\ell},\tilde{\theta }_{\ell})-\phi(w,\hat{\gamma}_{\ell},\alpha_{0},\theta_{0})+\phi(w,\gamma _{0},\alpha_{0},\theta_{0}) \]
Assumption 2: For each $\ell=1,...,L$, either i) \[ \sqrt{n}\int\hat{\Delta}_{\ell}(w)F_{0}(dw)\overset{p}{\longrightarrow }0,\text{ }\int\left\Vert \hat{\Delta}_{\ell}(w)\right\Vert ^{2} F_{0}(dw)\overset{p}{\longrightarrow}0, \] or ii) $\sum_{i\in I_{\ell}}\left\Vert \hat{\Delta}_{\ell} (W_{i})\right\Vert /\sqrt{n}\overset{p}{\longrightarrow}0,$ or iii) $\sum_{i\in I_{\ell}}\hat{\Delta}_{\ell}(W_{i})/\sqrt{n} \overset{p}{\longrightarrow}0.$
This condition imposes a rate condition on the interaction remainder $\hat{\Delta}_{\ell}(w)$, that its average must go to zero faster than $1/\sqrt{n}.$ It differs from Chernozhukov et al. (2018) in applying only to the nonparametric influence function and allowing for the sample average rate condition in iii), which is helpful for obtaining weak regularity conditions of Newey and Robins (2017).
Assumption 3: For each $\ell=1,...,L$, i) $\int \phi(w,\gamma_{0},\hat{\alpha}_{\ell},\tilde{\theta}_{\ell})F_{0}(dw)=0;$ and either ii) $\bar{\psi}(\gamma,\alpha_{0},\theta_{0})$ is affine in $\gamma;$ or iii) $\left\Vert \hat{\gamma}_{\ell} -\gamma_{0}\right\Vert =o_{p}(n^{-1/4})$ \textit{and }$\left\Vert \bar{\psi }(\gamma,\alpha_{0},\theta_{0})\right\Vert \leq C\left\Vert \gamma-\gamma _{0}\right\Vert ^{2}$\textit{ for all }$\gamma$ \textit{with }$\left\Vert \gamma-\gamma_{0}\right\Vert $\textit{ small enough; or} iv)$\sqrt{n}\bar {\psi}(\hat{\gamma}_{\ell},\alpha_{0},\theta_{0})\overset{p}{\longrightarrow }0.$
Assumption 3 incorporates Theorem 2 in i) and doubly robust moment functions through ii), in which case Assumption 3 imposes no conditions additional to Assumptions 1 and 2. Conditions iii) and iv) are alternative small bias conditions that are only required to hold for $\hat{\gamma}_{\ell},$ and not for $\hat{\alpha}_{\ell}.$ Condition iii) requires a faster than $n^{-1/4}$ rate for $\hat{\gamma}$ as is familiar from the semiparametric estimation literature. In many cases iii) will be satisfied for a mean square norm $\left\Vert \cdot\right\Vert $ so that Assumptions 1-3 will only require mean-square convergence rates, as is important in many machine learning contexts where only mean square rates are available.
Lemma 15: If Assumptions 1-3 are satisfied then equation ((ref)) is satisfied.
This key asymptotic result differs from previous results of Andrews (1994), Newey (1994a), Newey and McFadden (1994), Pakes and Olley (1995), Chen, Linton, and van Keilegom (2003), Ichimura and Lee (2010), Escanciano et al. (2016), and others in requiring no Donsker conditions. This feature is made possible by the use of cross-fitting in the moment conditions. It is important to not impose Donsker conditions for machine learning first steps which generally do not, or are not known to, satisfy Donsker conditions, as previously discussed in Chernozhukov et al. (2018).
This result improves upon Chernozhukov et al. (2018) in allowing $\hat{\alpha }_{\ell}$ to converge slower than $n^{-1/4}$ in general, in Assumption 1 applying separately to $\hat{\gamma}_{\ell}$ and $\hat{\alpha}_{\ell}$, and having weaker conditions for terms that involve both $\hat{\gamma}_{\ell}$ and $\hat{\alpha}_{\ell}$ in Assumption 2. These improvements result from Theorem 2 and the structure of orthogonal moments as the sum of identifying moment functions and the nonparametric influence function.
With additional conditions we obtain consistency of the estimator $\hat{\Psi}$ of the variance of the orthogonal moment functions given in Section 2. Let $\Psi:=E[\psi(W,\gamma_{0},\alpha_{0},\theta_{0})\psi(W,\gamma_{0},\alpha _{0},\theta_{0})^{\prime}]$.
Lemma 16: If Assumption 1 is satisfied and $\int\left\Vert g(w,\hat{\gamma},\tilde{\theta}_{\ell})-g(w,\hat{\gamma},\theta_{0} )\right\Vert ^{2}F_{0}(dw)\overset{p}{\longrightarrow}0$ and $\int\left\Vert \hat{\Delta}_{\ell}(w)\right\Vert ^{2}F_{0} (dw)\overset{p}{\longrightarrow}0$ for each $(\ell=1,...,L),$ then $\hat{\Psi}\overset{p}{\longrightarrow}\Psi.$
It is also important to have conditions for convergence of the Jacobian of the identifying sample moments $\partial\hat{g}(\bar{\theta})/\partial \theta\overset{p}{\longrightarrow}G=E[\partial g(W,\gamma_{0},\theta _{0})/\partial\theta]$ for any $\bar{\theta}\overset{p}{\longrightarrow} \theta_{0}$. To that end we impose the following condition:
Assumption 4: $G$\ exists and there is a neighborhood $\mathcal{N}$ of $\theta_{0}$ and $\left\Vert \cdot\right\Vert $ such that i) for each $\ell,$ $\left\Vert \hat{\gamma}_{\ell}-\gamma_{0}\right\Vert \overset{p}{\longrightarrow}0;$ ii) for all $\left\Vert \gamma-\gamma_{0}\right\Vert $ small enough $g(W,\gamma ,\theta)$ is differentiable in $\theta$ \textit{on }$\mathcal{N} $\textit{ with probability approaching }$1$\textit{ and there is } $C>0$\textit{ and }$d(W,\gamma)$\textit{ such that for }$\theta\in\mathcal{N} $\textit{ and }$\left\Vert \gamma-\gamma_{0}\right\Vert $\textit{ small enough } \[ \left\Vert \frac{\partial g(W,\gamma,\theta)}{\partial\theta}-\frac{\partial g(W,\gamma,\theta_{0})}{\partial\theta}\right\Vert \leq d(W,\gamma)\left\Vert \theta-\theta_{0}\right\Vert ^{1/C};\text{ }E[d(W,\gamma)]<C. \] \textit{iii) For each }$\ell=1,...,L,$ $j,$ and $k$, $\int\left\vert \partial g_{j}(w,\hat{\gamma}_{\ell},\theta_{0})/\partial\theta_{k}-\partial g_{j}(w,\gamma_{0},\theta_{0})/\partial\theta_{k}\right\vert F_{0} (dw)\overset{p}{\longrightarrow}0.$
Lemma 17: If Assumption 4 is satisfied and $\bar{\theta }\overset{p}{\longrightarrow}\theta_{0}$ then $\partial\hat{g} (\bar{\theta})/\partial\theta\overset{p}{\longrightarrow}G.$
With these results in place the asymptotic normality of semiparametric GMM follows in a standard way.
Theorem 18: If Assumptions 1-4 are satisfied, $\hat{\theta }\overset{p}{\longrightarrow}\theta_{0},$ $\hat{\Upsilon }\overset{p}{\longrightarrow}\Upsilon$, and $G^{\prime}\Upsilon G$ is nonsingular, then \[ \sqrt{n}(\hat{\theta}-\theta_{0})\overset{d}{\longrightarrow}N(0,V),\text{ }V=(G^{\prime}\Upsilon G)^{-1}G^{\prime}\Upsilon\Psi \Upsilon G(G^{\prime }\Upsilon G)^{-1}. \] If also the conditions of Lemma 16 are satisfied then $\hat{V} =(\hat{G}^{\prime}\hat{\Upsilon}\hat{G})^{-1}\hat{G}^{\prime}\hat{\Upsilon }\hat{\Psi}\hat{\Upsilon}\hat{G}(\hat{G}^{\prime}\hat{\Upsilon}\hat{G} )^{-1}\overset{p}{\longrightarrow}V.$
This result and the Lemmas 15-17 are both general and simple. They are general in applying to any first step estimators $\hat{\gamma}_{\ell}$ and $\hat{\alpha}_{\ell}.$ They are simple in requiring only a few mean-square convergence conditions for $\hat{\alpha}$ and $\hat{\gamma}$ when $\left\Vert \cdot\right\Vert $ denotes the mean square norm. This generality and simplicity results from the use of orthogonal moment functions and cross-fitting. The orthogonality of the moment functions leads to Assumption 3, as shown by Theorems 2 and 3. The cross-fitting and a specific remainder decomposition used in the proof of Lemma 15 motivate Assumptions 1 and 2, with separate treatment of the identifying moment functions and the nonparametric influence function.
Functionals of a first step satisfying a conditional moment restriction as in Section 7.2 are of wide interest, including for Example 2. For an identifying moment function $m(w,\gamma)-\theta$ and nonparametric influence function $\alpha(x,\theta)\lambda(W,\gamma)$ debiased GMM $\hat{\theta}$ and $\hat{V}$ are
Let $\bar{\lambda}(X,\gamma)=E[\lambda(W,\gamma)|X]$, $\bar{\psi}(\gamma )=\int[m(w,\gamma)+\alpha_{0}(x)\bar{\lambda}(x,\gamma)]F_{0}(dw)-$ $\theta_{0}$, $V=Var(m(W,\gamma_{0})+\alpha_{0}(X)\lambda(W,\gamma_{0}))$, and $\left\Vert a\right\Vert =\sqrt{\int a(w)^{2}F_{0}^{{}}(w)}$ denote the mean square norm.
Theorem 19 If i) $E[\lambda(W,\gamma_{0})|X]=0$; ii) $\alpha_{0}(X)$ and $E[\lambda(W,\gamma_{0})^{2}|X]$ are bounded and $E[m(W,\gamma_{0})^{2}]<\infty$; for $(\ell =1,...,L),$\textit{ iii) }$\int[m(w,\hat{\gamma})-m(w,\gamma_{0})]^{2} F_{0}(dw)\overset{p}{\longrightarrow}0$\textit{, }$\int[\lambda(w,\hat{\gamma }_{\ell})-\lambda(w,\gamma_{0})]^{2}F_{0}(dw)\overset{p}{\longrightarrow} 0,$\textit$\left\Vert \hat{\alpha}_{\ell}-\alpha_{0}\right\Vert \overset{p}{\longrightarrow}0;$\textit{ iv) }$\sqrt{n}\bar{\psi}(\hat{\gamma }_{\ell})\overset{p}{\longrightarrow}0$; \textit{v) Either a) }$\int [\hat{\alpha}_{\ell}(x)-\alpha_{0}(x)]^{2}[\lambda(w,\hat{\gamma}_{\ell })-\lambda(w,\gamma_{0})]^{2}\overset{p}{\longrightarrow}0$\textit{ and }$\sqrt{n}\left\Vert \hat{\alpha}_{\ell}-\alpha_{0}\right\Vert \left\Vert \bar{\lambda}(\hat{\gamma}_{\ell})-\bar{\lambda}(\gamma_{0})\right\Vert \overset{p}{\longrightarrow}0,$ \textit{or b)} $\sqrt{n}\left\Vert \hat {\alpha}_{\ell}-\alpha_{0}\right\Vert \left\Vert \lambda(\hat{\gamma}_{\ell })-\lambda(\gamma_{0})\right\Vert \overset{p}{\longrightarrow}0$ \textit{and }$\hat{\alpha}_{\ell}(x)$\textit{ in }$\hat{\psi}_{i\ell}$\textit{ is replaced by }$\bar{\alpha}_{\ell}(x)=\hat{\alpha}_{\ell}(x)1(\left\vert \hat{\alpha }_{\ell}(x)\right\vert \leq M)+sgn(\hat{\alpha}_{\ell}(x))M1(\left\vert \hat{\alpha}_{\ell}(x)\right\vert >M)$\textit{ and} $M\left\Vert \lambda (\hat{\gamma}_{\ell})-\lambda(\gamma_{0})\right\Vert \overset{p}{\longrightarrow}0,$ \textit{then} \[ \sqrt{n}(\hat{\theta}-\theta_{0})\overset{d}{\longrightarrow}N(0,V)\text{, }\hat{V}\overset{p}{\longrightarrow}V. \]
For functionals of a nonparametric 2SLS estimator this result is about debiased GMM rather than the plug-in series estimator without cross-fitting considered in Ai and Chen (2007). Theorem 19 is more general than Theorem 4.1 of Ai and Chen (2007) in applying to any first step estimator rather than just a series estimator. Theorem 19 is simpler in only requiring mean square convergence rates rather than the many Assumptions 3.1-3.8 and 4.1-4.6 of Ai and Chen (2007). The estimator $\hat{\theta}$ and Theorem 19 is more complicated in involving construction of and properties for $\hat{\alpha }_{\ell},$ but some such $\hat{\alpha}_{\ell}$ is needed in any case for $\hat{V}$. This comparison would also apply to a host of plug-in estimators and root-n consistency and asymptotic normality results in the literature, including early contributions by Powell, Stock, and Stoker (1989) for kernel density estimators and Newey (1994a) for series estimators of least squares projections.
We will specify conditions that allow us to apply Theorem 19 to this problem. The next condition will be sufficient for condition iv) of Theorem 19 with $\alpha_{0}(x)$ for $\lambda(w,\gamma)=v_{u}(Y-\gamma).$
Assumption 5: i) There exists bounded $\bar{\alpha} (x)$ such that $E[m(W,\gamma)]=E[\bar{\alpha}(X)\gamma(X)]$ for all $\gamma(X)$ with $E[\gamma(X)^{2}]<\infty;$ ii) $U=Y-\gamma_{0}(X)$\textit{ is continuously distributed and there is } $C>0$\textit{ such that the conditional density }$f(u|X)$\textit{ of conditional on }$X$\textit{ satisfies }$C^{-1}\leq f(0|X)\leq C$\textit{ and is twice continuously differentiable in }$u$\textit{ with probability one with }$\left\vert \partial^{j}f(u|X)/\partial u^{j}\right\vert \leq C,$ $(j=1,2).$
This condition specifies that $E[m(W,\gamma)]$ is a mean square continuous functional of $\gamma$ with Riesz representer $\bar{\alpha}(X)$ and imposes some restrictions on the conditional pdf of $U$ given $X$.
Lemma 20: If Assumption 5 is satisfied then $\left\vert \bar{\psi}(\gamma)\right\vert \leq C\left\Vert \gamma-\gamma_{0}\right\Vert ^{2}$ for $\left\Vert \gamma-\gamma_{0}\right\Vert ^{2}=\int [\gamma(x)-\gamma_{0}(x)]^{2}F_{0}(dx)$ and \[ \alpha_{0}(X)=f(0|X)^{-1}\bar{\alpha}(X),\text{ }\bar{\psi}(\gamma )=E[m(W,\gamma)+\alpha_{0}(X)v_{u}(Y-\gamma(X))]-\theta_{0}. \]
Here we see that $\alpha_{0}(X)=f(0|X)^{-1}\bar{\alpha}(X)$ is the ratio of the Riesz representer $\bar{\alpha}(X)$ to the conditional pdf $f(0|X)$. This formula for $\alpha_{0}(X)$ differs from that of Section 7 for functionals of conditional means where $\alpha_{0}(X)$ is the Riesz representer of the linear functional.
The $\hat{\alpha}_{\ell}(X)$ given in Section 2 will estimate $\alpha_{0}(X)$ because weighting by $f(0|X)$ is incorporated in the kernel weighting included in $\hat{Q}_{\ell}.$ This weighting allows us to avoid inverting an estimator of $f(0|X).$ We obtain a mean square convergence rate for this $\hat{\alpha }_{\ell}(x)$ by extending the results of Chernozkukov, Newey, and Singh (2018) to allow kernel weighting in $\hat{Q}_{\ell}$. Because this paper is focused on the properties of $\hat{\theta}$ we reserve the full conditions to Appendix B, only stating here the conditions required of the kernel $K(u),$ the bandwidth $h,$ and the regularization factor $r_{\lambda}$ in the Lasso minimum distance estimator in equation ((ref)).
Assumption 6: i) $K(u)$ is a symmetric bounded kernel of order $\kappa$ with bounded support; ii) $h\sqrt{n}\longrightarrow \infty;$ iii) for each $\ell,\ell^{\prime},$ $\left\Vert \hat{\gamma}_{\ell,\ell^{\prime}}-\gamma_{0}\right\Vert =O_{p}(n^{-d_{\gamma} })$; iv) $\sqrt{\ln(p)/(hn)}+h^{2}+n_{{}}^{-d_{\gamma}}=o(r_{\lambda });$\textit{ v) }$r_{\lambda}\longrightarrow0.$
Lemma 21: If Assumptions 6 and B1 are satisfied then $\left\Vert \hat{\alpha}_{\ell}-\alpha_{0}\right\Vert =O_{p}(\sqrt {r_{\lambda}})$. If Assumption B2 is also satisfied then for the sparse approximation rate $\xi\geq1/2$ from Assumption B2 we have $\left\Vert \hat{\alpha}_{\ell}-\alpha_{0}\right\Vert =O_{p}(r_{\lambda }^{2\xi/(1+2\xi)})$.
The following result gives conditions for asymptotic inference for the estimator of a linear functional of a regression quantile estimator.
Theorem 22: If i) Assumptions 5 and 6 are satisfied; ii) $E[m(W,\gamma_{0})^{2}]<\infty$ and $\int[m(w,\hat{\gamma })-m(w,\gamma_{0})]^{2}F_{0}(dw)\overset{p}{\longrightarrow}0$; iii) $\left\Vert \hat{\gamma}_{\ell}-\gamma_{0}\right\Vert =O_{p}(n^{-d_{\gamma} })$ for $1/4<d_{\gamma}<1/2;$ either iv) Assumption B1 is satisfied and $\sqrt{n}\sqrt{r_{\lambda}}n^{-d_{\gamma}}\longrightarrow 0$\textit{ or v) Assumptions B1 and B2 are satisfied and }$\sqrt{n}r_{\lambda }^{2\xi/(1+2\xi)}n^{-d_{\gamma}}\longrightarrow0$\textit{ then} \[ \sqrt{n}(\hat{\theta}-\theta_{0})\overset{d}{\longrightarrow}N(0,V)\text{, }\hat{V}\overset{p}{\longrightarrow}V. \]
This result depends on the conditional quantile estimator converging at a mean square rate that is faster than $n^{-1/4}.$ Such a rate for an $L_{1}$ regularized conditional quantile estimator is derived by Belloni and Chernozhukov (2011).
An result that is important for the properties of $\hat{\theta}$ for dynamic discrete choice and more generally for economic structural model is a convergence rate for the estimator $\hat{\gamma}_{2}(x)$ of the value function term in the choice probability. We continue to let $\left\Vert a\right\Vert =\sqrt{\int a(w)^{\prime}a(w)F_{0}(dw)}$ denote the mean square norm. We continue to maintain independence of observations across $i$ but allow arbitrary dependence across $t$.
Assumption 7: i) There is $\varepsilon>0$ such that $\gamma_{10}(X)\in\lbrack\varepsilon,1-\varepsilon]$, for all $\ell,\ell^{\prime},$ $\hat{\gamma}_{1\ell\ell^{\prime}}(X_{t})\in \lbrack\varepsilon,1-\varepsilon]$, and $H(p)$ is twice continuously differentiable on $[\varepsilon,1-\varepsilon];$\textit{ ii) For all }$\ell,\ell^{\prime},$ $\left\Vert \hat{\gamma}_{1\ell\ell^{\prime} }-\gamma_{0}\right\Vert =O_{p}(n^{-d_{1}})$, $0<d_{1}<1/2;$ \textit{iii) Assumptions B1 and B2 are satisfied with }$\alpha_{0}(x)=\gamma_{20}(x)$ \textit{and sparse approximation rate }$\xi_{1}>1/2;$\textit{ iv) }$n^{-d_{1} }=o(r_{1})$\textit{ and }$r_{1}=O(n^{-d_{1}}\ln(n));$ v) $\gamma_{20} (X)=\sum_{j=1}^{\infty}\beta_{j0}b_{j}(X)$ \textit{with} $\sum_{j>p}\left\vert \beta_{j0}\right\vert =O_{p}(n^{-d_{1}(2\xi_{1}-1)/(2\xi_{1}+1)}\ln(n)).$
In practice condition i) requires fixed trimming where $\hat{\gamma} _{1\ell\ell^{\prime}}(X_{t})$ is censored below by $\varepsilon$ and above by $1-\varepsilon$, with $\varepsilon$ being known. Here and in the Theorem 24 below we impose tighter restrictions on the penalty sizes $r_{1}$, $r_{2}$, and $r_{3}$ than needed in order to allow smaller sparse approximation rates, e.g. $\xi_{1}$ in Assumption 7.
Lemma 23: If Assumption 7 is satisfied then for $\gamma _{20}(X_{t})=E[H(\gamma_{10}(X_{t+1}))|X_{t}]$ and $\ell$
We expect this result to be useful more generally for dynamic structural models to provide a machine learner of expected value differences.
Theorem 24: If i) Assumption 7 is satisfied, ii) $\Lambda(a)>0$ for all $a\in\Re,$ $\ln\Lambda_{a} (a)$ is concave, $\Lambda(a)$ is twice differentiable with uniformly bounded derivatives, $D(x)$ \textit{is bounded,} $E[D(X)D(X)^{\prime}]$ \textit{is nonsingular; iii) Assumptions B1 and B2 are satisfied for }$\alpha_{0}(x)$\textit{ equal to each element of } $E[D(X_{t})\pi(a_{0}(X_{t}))\Lambda_{a}(a(X_{t})Y_{2t}/\Lambda(a(X_{t} ))|X_{t+1}=x]$ with sparse approximation rate $\xi_{2}$ and for $E[Y_{1t} |X_{t+1}=x]$\textit{ with sparse approximation rate }$\xi_{3};$\textit{ and iv) }$d_{1}>1/4;$ v)\textit$1+[(2\xi_{1}-1)/(2\xi_{1}+1)]2\xi_{2}/[2\xi _{2}+1]>1/2d_{1},$ $n^{-d_{1}(2\xi_{1}-1)/(2\xi_{1}+1)}=o(r_{2}),$\textit{ and }$r_{2}=O(n^{-d_{1}(2\xi_{1}-1)/(2\xi_{1}+1)}\ln(n));$ \textit{vi) }$\xi _{3}/[2\xi_{3}+1]+d_{1}>1/2$, $\sqrt{\ln(p)/n}=o(r_{3}),$ \textit{and } $r_{3}=O(\sqrt{\ln(p)/n}\ln(n))$; \textit{vii) }$(4\xi_{1}-1)/(2\xi _{1}+1)>1/2d_{1};$\textit \textit{then for} $V=G^{-1}E[\psi_{0}(W)\psi _{0}(W)^{\prime}]G^{-1}$ \[ \sqrt{n}(\hat{\theta}-\theta_{0})\overset{d}{\longrightarrow}N(0,V),\text{ }\hat{V}\overset{p}{\longrightarrow}V. \]