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 paper presents a method for estimating causal effects (or more generally non/parametric functions) under conditional moment restrictions, focusing in this case on nonparametric IVs. The main idea is casting these conditional restrictions to an unconditional version through importance resampling using a conditional density estimator (e.g. based on a least-squares method with a NN). The paper also provides a theoretical analysis of the estimation error, providing am error bound (Thm1). The method is evaluated based on three econometrics datasets from literature, showing a smaller MSE than the compared methods.