Learning Causal Relationships from Conditional Moment Restrictions by Importance Weighting

1 INTRODUCTION . Consider learning the causal relationship between airline ticket prices and demand . As one might expect , prices and demand rise and fall through the seasons , being affected by other events like vacation periods , which are called confounders and may or may not be observable . Due to confounders , naively inferring from this pattern that higher ( lower ) prices increase ( decrease ) demand would be incorrect , and potentially detrimental . Thus , controlling for confounding effects is essential . This issue frequently arises in practice , especially when learning causal ( structural ) relationships is essential to answer counterfactual questions regarding policy intervention and outcome . One approach to deal with confounding effects ( like in the airline example above ) is the instrumental variable ( IV ) approach ( Wooldridge , 2002 ; Greene , 2003 ) . In the IV approach , the conditional moment restriction is defined as such that the causal model satisfies the restriction of zero expected value given IVs , thus conditioning out the confounding effect . The simplest representation of this idea is the two-step least squares ( 2SLS ) for linear models ( Wooldridge , 2002 ; Greene , 2003 ) . However , given the complex nature of the causal effect and confounding effect ( and their relation ) , assuming a linear relation can be too strong . Thus , in this paper , we focus on nonparametric IV ( NPIV ) regressions , allowing for much more flexible estimation ( Newey & Powell , 2003 ) . NPIV can be viewed as an instance of a more general framework of causal inference under conditional moment restrictions . In this light , the machinery for inference under conditional moment restrictions also applies to NPIV , as well as its shortcomings . One major issue with using the conditional moment restrictions for causal inference is that one must approximate the conditional expectation , which is often difficult to do ( see , e.g. , Newey ( 1993 ) ; Donald et al . ( 2003 ) for parametric and Newey & Powell ( 2003 ) ; Ai & Chen ( 2003 ) for nonparametric IVs using sieves ) . For instance , Lewbel ( 2007 ) and Otsu ( 2007 ) estimate the conditional expectation by local kernel density estimation . However , local kernel density estimation suffers under high dimensionality . For this problem , recent methods suggest the use of machine learning methods , such as neural networks ( Hartford et al. , 2017 ) . In this paper , we propose transforming conditional moment restrictions into unconditional moment restrictions by importance weighting using the conditional density ratio , which is defined as the ratio of the conditional probability density , conditioned on the IVs , to the unconditional probability density . We show that the unconditional expectation of a random variable weighted by the conditional density ratio is equal to the conditional expectation . Further , we show that it is possible to estimate the conditional density ratio with the least-squares method with a neural network . Once the conditional density ratio is estimated , the usual method of moments , such as GMM , can be used straightforwardly . The contribution of this paper is as follows : ( i ) we propose a novel approach to convert conditional moment restrictions to unconditional moment restrictions by importance weighting ; ( ii ) using our proposed transformation , we develop methods for NPIV ; ( iii ) we analyze the estimation error 2 SETUP AND NOTATION . Among various problems of learning causal relationships from conditional moment restrictions , we focus on the NPIV regression for ease of discussion . Note that our proposed method can be applied to more general settings , similar to Ai & Chen ( 2003 ) . Suppose that the observations { ( Yi , Xi , Zi ) } ni=1 are i.i.d. , where Yi ∈ Y ⊆ R is an observable scalar random variable , Xi ∈ X ⊆ RdX is a dX dimensional explanatory variable , Zi ∈ Z ⊆ RdZ is a dZ dimensional random variable , called an IV , and X and Z are compact with nonempty interior . We also assume that the probability densities of ( Yi , Xi , Zi ) , ( Yi , Xi ) , Zi exist and denote them by p ( y , x , z ) , p ( y , x ) , and p ( z ) , respectively . Let us define the causal relationships between Yi and Xi as Yi = f ∗ ( Xi ) + εi , where f∗ : X → Y is a structural function , εi is the sub-Gaussian error term with mean zero . To learn f∗ , suppose the IV Zi satisfies the following conditional moment restrictions : E [ εi|Zi ] = 0 ∀i ∈ { 1 , 2 , . . . , n } . ( 1 ) Then , we also assume that under the conditional moment restriction , we can uniquely identify f∗ . Our goal is to learn f∗ from the conditional moment restrictions in ( 1 ) . If Zi = Xi , this problem boils down to the estimation of the conditional expectation ( regression function ) E [ Yi|Xi ] . However , when E [ εi|Xi ] 6= 0 , E [ Yi|Xi ] is not equivalent to f∗ ( Xi ) ; that is , typical regression analysis , such as least squares , may not return the correct estimate of f∗ ( Xi ) . 3 PRELIMINARIES AND LITERATURE REVIEW . In this section , we briefly review causal inference under moment restrictions . 3.1 IV METHOD FOR LINEAR STRUCTURAL FUNCTIONS . One of the basic cases of using the IV is when f∗ is a linear model X > i θ ∗ with dX dimensional vector θ∗ and the error term εi is correlated with the explanatory variable Xi . In this case , the parameter θ∗ of the linear model can be estimated if there are IVs of dimension dX or more that satisfy the unconditional moment restrictions , E [ Ziεi ] = 0dZ , where 0d is a d dimensional zero vector . In the just-identified case ( dX = dZ ) , we can estimate θ∗ as θ̂IV = ( 1 n ∑n i=1 ZiX > i ) −1 1 n ∑n i=1 ZiYi . In the over-identified case ( dX ≤ dZ ) , we can estimate it by the 2SLS . In the 2SLS , we first regress Xi by Zi ; then using a dX dimensional function ĝ ( Zi ) ob- tained from the first stage regression , we estimate θ∗ as θ̂2SLS = ( 1 n ∑n i=1 X̂iX̂ > i ) −1 1 n ∑n i=1 X̂iYi , where X̂d , i = Z > i ( 1 n ∑n j=1 ZjZ > j ) −1 1 n ∑n j=1 ZjXd , j and X̂i = ( X̂1 , i , . . . , X̂dX , i ) > . More generally , we can formulate the estimation of the linear structural function by unconditional moment restrictions defined using the IV ; that is , E [ m ( θ∗ ; Yi , Xi , Zi ) ] = 0dm , where θ∗ ∈ Θ is a parameter representing the causal effect , Θ is the parameter space , and m : Θ×R×X×Z → Rdm is a dm dimensional moment function . Then , the GMM estimator is defined as θ̂GMM = arg minθ∈Θ ( 1 n ∑n i=1m ( θ ; Yi , Xi , Zi ) ) > Wdm ( 1 n ∑n i=1m ( θ ; Yi , Xi , Zi ) ) , where Wdm is a dm × dm weight matrix . If Wdm is chosen as W ∗dm ∝ E [ m ( θ ∗ ; Yi , Xi , Zi ) > m ( θ∗ ; Yi , Xi , Zi ) ] , the estimator θ̂GMM is efficient under the posited moment conditions . Note that the GMM includes the 2SLS as a special case where εi has the same variance among i ∈ { 1 , . . . , n } and the zero covariance . Three methods have been proposed to obtain W ∗dm ; two-step GMM , iterative GMM ( Hayashi , 2000 ) , and continuous updating GMM ( CU estimator ; CUE , Hansen et al. , 1996 ) . These estimators have the same asymptotic distribution , but different non-asymptotic properties . In particular , CUE is known as a special case of the generalized empirical likelihood ( GEL ) estimator ( Owen , 1988 ; Smith , 1997 ) , which plays an important role in estimation with many moment restrictions ( Newey & Smith , 2004 ) . 3.2 NPIV AND LEARNING FROM CONDITIONAL MOMENT RESTRICTIONS . In linear structural functions , zero covariance between the IV and error term suffices for identification . However , in nonparametric structural functions , we require a stronger restriction : the error term has conditional expectation zero given the IVs . Then , we can characterize the solution of NPIV as that of an integral equation K ( z ) = E [ Yi|Zi = z ] = E [ f∗ ( Xi ) |z ] = ∫ f∗ ( x ) dF ( x|z ) with F denoting the conditional c.d.f of x given z . The identification also results in the uniqueness of the solution . Several estimators have been proposed . Newey & Powell ( 2003 ) proposes a nonparametric analogue of the 2SLS for linear structural functions . They first define the linear-in-parameter model as a model to approximate the nonparametric structural function f∗ , where they use a linear approximation with basis expansion , such as sieve ( series ) regression . Let us denote the approximation as f∗ ( Xi ) ≈ ψ ( Xi ) > θ∗ , where ψ : X → Rdψ , dψ ≤ n is a vector-valued function consisting of outputs of basis functions . Then , they conduct the 2SLS as follows : ( i ) they define a linear-in-parameter model as an approximation to the nonparametric model of Zi , and estimate E [ ψ ( Xi ) |Zi ] using that model ; ( ii ) they regress Yi by E [ ψ ( Xi ) |Zi ] to estimate θ∗ . In contrast , Ai & Chen ( 2003 ) proposes a nonparametric analogue of the GMM . Ai & Chen ( 2003 ) also approximate the nonparametric structural function f∗ by a linear-in-parameter model . However , unlike Newey & Powell ( 2003 ) , Ai & Chen ( 2003 ) estimates E [ ( Yi − f ( Xi ) ) |Zi ] , instead of estimating E [ ψ ( Xi ) |Zi ] , by another linear-in-parameter model . Using the estimator of E [ ( Yi−f ( Xi ) ) |Zi ] , Ai & Chen ( 2003 ) transforms conditional moment restrictions into unconditional ones and apply the GMM to estimate f∗ . In addition to these two typical methods , Darolles et al . ( 2011 ) , Carrasco et al . ( 2007 ) propose their own methods for NPIV . This is how the NPIV problem is typically cast into the framework of conditional moment restrictions , where a parameter representing the causal relationship satisfies E [ m ( θ∗ ; Yi , Xi , Zi ) |Zi ] = 0dm . As with NPIV , we often define the moment function as a function of the nonparametric function f instead of the parameter θ . We can estimate f∗ defined under conditional moment restrictions using variants of GMM or empirical likelihood , e.g. , Ai & Chen ( 2003 ) ; Domínguez & Lobato ( 2004 ) ; Otsu ( 2007 ) ; Lewbel ( 2007 ) , by transforming the conditional moment restrictions to unconditional ones .

Summary: The present paper proposes an importance weighting approach to address the issue of regression under conditional moment restrictions in the context of non-parametric instrumental variables. The authors provide error bounds on the learned structural function. They show that their approach has a convergence rate of $\mathcal{O}(1/\sqrt{n})$. The main contributions of this work are presented in Sections 4 and 5, where they first introduce their method based on re-weighting by an estimated conditional density ratio function and then provide an estimation error analysis. The theoretical results are complemented with two synthetic experiments which show that the proposed method can compete with and in certain cases improves upon state of the art.