Econometric analysis of potential outcomes time series: instruments, shocks, linearity and the causal response function
aa r X i v : . [ ec on . E M ] F e b Econometric analysis of potential outcomes time series:instruments, shocks, linearity and the causal response function ∗ Ashesh Rambachan
Department of Economics,Harvard University [email protected]
Neil Shephard
Department of Economics andDepartment of Statistics,Harvard University [email protected]
February 27, 2020
Abstract
Bojinov and Shephard (2019) defined potential outcome time series to nonparametrically mea-sure dynamic causal effects in time series experiments. Four innovations are developed in thispaper: “instrumental paths”, treatments which are “shocks”, “linear potential outcomes” and the“causal response function.” Potential outcome time series are then used to provide a nonparametriccausal interpretation of impulse response functions, generalized impulse response functions, localprojections and LP-IV.
Keywords:
Dynamic causality, instrumental variables, linearity, potential outcomes, time series,shocks. ∗ This is a revised version of “A nonparametric dynamic causal model for macroeconometrics.” We thank Iavor Bojinov,Gary Chamberlain, Fabrizia Mealli, James M. Robins and James H. Stock for conversations that have developed ourthinking on causality. We are also grateful to Isaiah Andrews, John Campbell, Peng Ding, Avi Feller, Ron Gallant,Peter R. Hansen, Kosuke Imai, Guido Kuersteiner, Daniel Lewis, Luke Miratrix, Sendhil Mullainathan, Ulrich Muller,Susan A. Murphy, Andrew Patton, Natesh Pillai, Mikkel Plagborg-Moller, Julia Shephard and Christopher Sims forvaluable feedback on an earlier draft. Participants at the Harvard Econometrics Workshop, Workshop on Score-drivenTime-series Models at the University of Cambridge, NBER-NSF Time Series Conference 2019, the Conference in Honorof George Tauchen at Duke University, the Workshop on Causal Inference with Interactions at UCL and the (EC) Introduction
Bojinov and Shephard (2019) developed potential outcome time series to nonparametrically measuredynamic causal effects from time series randomized experiments conducted in financial markets. How-ever, most time series data used in economics is observational. In this paper we develop the toolsneeded to use the potential outcome time series framework on observational data, yielding an ob-servational, nonparametric framework for measuring dynamic causal effects. It provides a flexiblefoundation upon which to build new methods and interpret existing methods for causal inference oneconomic time series.Our analysis is based on four new ideas beyond Bojinov and Shephard (2019). The first threeare special cases of the potential outcome time series: “instrumented potential outcome time series”,treatments which are “shocks” and “linear potential outcomes”. The fourth innovation is the “causalresponse function,” which is a new dynamic causal estimand. To illustrate the power of these fourideas, we provide a nonparametric causal interpretation to four tools commonly used in the time seriesliterature: impulse response functions, generalized impulse response functions, local projections andlocal projections with an instrumental variable (LP-IV). Our results show that a tightly parameterizedmodel such as the structural moving average is not needed to provide a causal interpretation to theseobjects.Of course, there is a storied history of economists trying to learn dynamic causal effects from timeseries data. Modern reviews include Ramey (2016) and Stock and Watson (2018). As this vast bodyof research emphasizes, conceptualizing and estimating dynamic causal effects is quite challenging.Dynamic feedback between treatments and observed outcomes makes it difficult to disentangle causesfrom effects. Additionally, in many important applications, only several hundred time series observa-tions are available. Given these challenges, much of the literature on dynamic causal effects in timeseries relies on parameterized linear models. Canonical examples are structural vector autoregressions(Sims, 1980) , local projections (Jord´a, 2005) and LP-IV (Jord´a et al., 2015; Stock and Watson, 2018).However, there are exceptions such as Priestley (1988), Engle et al. (1990) and Gallant et al. (1993).Influentially, Koop et al. (1996) defined a “generalized impulse response function” for non-linear, non-causal models.While tractable, the heavy emphasis on linear models has drawbacks. The role of particular setsof assumptions are often unclear in existing approaches. For example, it is common in economics to Structural vector autoregressions are typically motivated as a linear approximation to an equilibrium arising from anunderlying dynamic stochastic general equilibrium model such as Christiano et al. (1999, 2005), Smets and Wouters(2003, 2007). T , large- N panels. The groundbreaking panel work of Robins (1986) led to an enormous lit-erature on dynamic causal effects (Murphy et al., 2001; Murphy, 2003; Abbring and Heckman, 2007;Heckman and Navarro, 2007; Lechner, 2011; Heckman et al., 2016; Boruvka et al., 2018; Blackwell and Glynn,2018; Hernan and Robins, 2019). However, the four new ideas in this paper are not the focus of thosepapers.Inference on dynamic causal effects is one of the great themes of the broader time series literature.Researchers quantify causality in time series in a variety of ways such as using “Granger causality”(Wiener, 1956; Granger, 1969), highly structured models such as DSGE models (Herbst and Schorfheide,2015), behavioral game theory (Toulis and Parkes, 2016), state space modelling (Harvey and Durbin,1986; Harvey, 1996; Bondersen et al., 2015), Bayesian structural models (Brodersen et al., 2015) aswell as intervention analysis (Box and Tiao, 1975) and regression discountinuity (Kuersteiner et al.,2018). The potential outcome time series is distinct from each of those approaches.The closest work to the potential outcome time series framework is Angrist and Kuersteiner (2011)and Angrist et al. (2018), which also studies time series using potential outcomes (see also White and Lu(2010) and Lu et al. (2017)). That work is importantly different from Bojinov and Shephard (2019),as it avoids discussion of treatment paths, defining potential outcomes as a function of a single priortreatment — this difference will be detailed in Section 2. More importantly, Angrist and Kuersteiner(2011) and Angrist et al. (2018) do not discuss the main contribution of this paper which are thespecial cases of instruments, shocks and linear potential outcomes and the establishment of the causalresponse function. Also related to the framework is Robins et al. (1999) who used potential out-3ome paths for binary time series and Bondersen et al. (2015) who used them for state space models.Recently, Bojinov and Shephard (2019) and Blackwell and Glynn (2018) used them in more generalsettings. Overview of the paper:
Section 2 recalls the definition of a potential outcome time series. Thentwo examples of this setup are given, before developing the three important special cases that dealwith instrumental variables, shocks and linear potential outcomes. Section 3 defines causal effects,introducing a weighted causal effect and a causal response function. We provide definitions that allowus to link them with the economics literature and are more general than those in Bojinov and Shephard(2019). We show that the causal response function is closely related to the impulse response function.We also analyze the properties of these causal estimands under the assumptions of linear potentialoutcomes and shocked treatments. Section 3 finishes with a nonparametric causal interpretation ofthe local projection and its instrumental variables version. Section 4 concludes the paper. Longerproofs and a series of additional results are collected in our Web Appendix.
Notation:
The mathematics of this paper is written using standard path notation: for a time series { A t : t = 1 , , . . . T } , let A t := ( A , . . . , A t ). Here := denotes a definition of the left hand side of theequation. Further, A ⊥⊥ B generically means that the random variable A is stochastically independentof B . A B denotes A and B not being independent, while A L = B means A and B have the samelaw or distribution. For a matrix A , A ⊺ is the transpose of A . We recall the definition of the potential outcome time series developed by Bojinov and Shephard(2019) in the context of time series experiments seen in financial economics. There is nothing novel inthis first subsection.There is a single unit that is observed over t = 1 , . . . , T periods. At each time period, the unitreceives a new K -dimensional treatment W t and we observe a scalar outcome Y t . The potential outcometime series links treatments and outcomes using four foundation stones: (i) the definition of treatmentand potential outcome paths, (ii) an assumption of non-anticipating outcomes, (iii) an assumptionthat generates outcomes by linking potential outcomes to treatments and (iv) an assumption of non-anticipating treatments. 4 potential outcome describes what would be observed at time t for a particular path of treatments.Its formal definition is given below. Definition 1. A treatment path W T is a stochastic process where each random variable W t hascompact support W ⊂ R K . The potential outcome path is, for any deterministic w T ∈ W T , thestochastic process Y T ( w T ) := ( Y ( w T ) , Y ( w T ) , ..., Y T ( w T )) ⊺ , where the time- t potential outcome Y t ( w T ) : W T → R . In the definition above, the potential outcomes can depend on future treatments. Now, we employour second foundation stone: restricting the potential outcomes to only depend on past and currenttreatments.
Assumption 1 (Non-anticipating potential outcomes) . For each t = 1 , . . . , T , Y t ( w t , w t +1: T ) = Y t ( w t , w ′ t +1: T ) almost surely, for all deterministic w T ∈ W T , w ′ t +1: T ∈ W T − t . Assumption 1 is the time series analogue of SUTVA (Cox, 1958; Rubin, 1980). For convenience, wewill drop references to future treatments and write the time- t potential outcome random variable Y t ( w t ) : W t → R , while the stochastic process version is written as Y T ( w T ) = ( Y ( w ) , Y ( w ) , ..., Y T ( w T )) ⊺ . We link the potential outcomes and treatments to deliver outcomes through our third stone: Assumption 2 (Outcomes) . The time- t outcome is the random variable Y t := Y t ( W t ) , while theoutcome stochastic process is Y T := ( Y ( W ) , Y ( W ) , ..., Y T ( W T )) ⊺ . Let F t stand for the natural filtration generated by the observed stochastic process { Y t , W t } .The final foundation stone is that W T is non-anticipating: the assignment of the treatmentdepends only on past outcomes and past treatments. This is a probabilistic assumption involving the Angrist and Kuersteiner (2011); Angrist et al. (2018) allow treatments to stochastically depend on past outcomes andtreatments but define their potential outcomes as { Y t,p ( w ) , w ∈ W} , for each lag p ≥
0. This latter step limits thedependence of the potential outcomes on the full treatment path, e.g. for p = 1, { Y t, ( w ) , w ∈ W} only depends onthe treatment assigned at period t − t . In principle, the 1-step aheadcausal effect of the treatment on the outcome may differ depending on what treatments are assigned at period t but thisnotation rules this out. As we next discuss in detail in Section 3, introducing explicit dependence on the full treatmentpath leads to a rich set of interesting causal estimands. { W t , Y t : T ( W t − , w t : T ) }|F t − . The associated (conditional) probability triple of this jointconditional distribution is written as (Ω , G , Pr), hiding the implicit dependence on w t : T and F t − . Assumption 3 (Non-anticipating treatment paths) . For each t = 1 , . . . , T {{ Y t : T ( W t − , w t : T ) , w t : T ∈ W T − t +1 } ⊥⊥ W t } | F t − . Assumption 3 is the time-series analogue of unconfoundedness. It says that the future potentialoutcomes { Y t : T ( W t − , w t : T ) , w t : T ∈ W T − t +1 } do not Granger-cause the current treatment W t (Sims,1972; Chamberlain, 1982; Engle et al., 1983; Kuersteiner, 2010; Lechner, 2011; Hendry, 2017).With our four foundation stones in place we can now define a potential outcome time series . Definition 2 (Potential outcome time series) . A stochastic process of potential outcomes and treat-ments { Y t , W t } that satisfies Assumptions 1, 2 and 3 is a potential outcome time series . We now illustrate the potential outcome time series through two examples.
Example 1 (Autoregression) . Consider a bivariate time series, where for all w t ∈ W t , Y t ( w t ) W t = µ + φY t − ( w t − ) + β w t γ + θW t − + δY t − ( W t − ) + ǫ t η t , ǫ t η t iid ∼ N , σ ǫ ρσ ǫ σ η ρσ ǫ σ η σ η . (1) The resulting { Y t , W t } is a Gaussian process. However, in general this system is not a potential out-come time series, as ǫ t and η t are contemporaneously correlated which disallows the use of Assumption3. If ρ = 0 then this is a potential outcome time series. More generally, if we replace the assumptionabout the joint law of ǫ t , η t in (1) entirely with the assumption { ǫ t ⊥⊥ W t }|F t − , then this is a potentialoutcome time series. Example 2 (Expectations of future treatments and non-anticipation) . In economics, consumers andfirms are often modelled as forward-looking, with the distribution of futures outcomes influencing to-day’s treatment choice. A simple version of this (e.g. in the tradition of Muth (1961), Lucas (1972), In panel data settings, Robins (1994), Robins et al. (1999) and Abbring and van den Berg (2003) use this type of“selection on observables” assumption for the treatment paths W T . When T = 2 this assumption is equivalent to the“latent sequential ignorability” assumption of Ricciardi et al. (2020). More broadly, Frangakis and Rubin (1999) callthis type of assumption “latent ignorable”. argent (1981)) is: W t = arg max w t (cid:18) max w t +1: T E [ U ∗ ( Y t : T ( W t − , w t : T ) , w t : T ) | F t − ] (cid:19) , (2) where U ∗ is a utility function of future outcomes and treatments. This decision rule delivers W t andthus Y t ( W t ) . This is a potential outcome time series. We now focus on three, new special cases of the nonparametric potential outcome time series whichallow the formal definitions of an instrumental path, a linear potential outcome and a nonparametricshock in causal time series models. These three cases were not in Bojinov and Shephard (2019).The first special case of the potential outcome time series connects this framework to the literatureon instrumental variables (Angrist et al. (1996); Angrist and Krueger (2001)).
Definition 3 (Instrumented potential outcome time series) . Partition the treatment path V t = ( W ′ t , Z ′ t ) ′ ,where W t ∈ W W and Z t ∈ W Z . Assume { Y t , V t } is a potential outcome time series and additionallythat:1. Exclusion condition: Y t ( w , z , ..., w t , z t ) = Y t ( w , z ′ , ..., w t , z ′ t ) for all w t ∈ W tW , z t , z ′ t ∈ W tZ .2. Relevance condition: Z t W t | F t − . Then { Y t , V t } is an instrumented potential outcome time series , where Z t is labelled an instrument path . The lack of dependence of the potential outcomes on the instrument means it is convenient to refer toit as Y t ( w t ) : W tW → R , while Y T ( w T ) = ( Y ( w ) , Y ( w ) , ..., Y T ( w T )) ⊺ . Example 1 (continuing from p. 6) . In economics, it is often difficult to measure accurately the treat-ment W t , so instead, researchers use an estimator, ˆ W t , of the treatment. We take the instrument Z t = ˆ W t , following the statistical measurement error tradition of Durbin (1954), which is used inthe context of dynamic linear causal models by Jord´a et al. (2015). An empirical example of thisis Stock and Watson (2018) where W t is a monetary policy movement and ˆ W t is an estimator of The non-anticipation assumptions are similarly plausible if a different model for expectations is used. “Natural ex-pectations” as in Fuster et al. (2010) or “diagnostic expectations” as in Bordalo et al. (2018) both only allow currentdecisions to depend on (possibly biased) beliefs about future outcomes, not the exact realizations along alternative paths Y t ( w t ). t constructed from high-frequency movements in the rates on federal funds contracts around policyannouncements. A simple time series example of this extends Example 1 with ˆ W t = α + α W t + ζ t , where ǫ t η t ζ t iid ∼ N , σ ǫ σ η
00 0 σ ζ . Hence the estimated treatment is biased but not independent of the treatment. As ˆ W t does not movearound the potential outcomes, this system is an instrumented potential outcome time series. Remark 2.1.
The non-anticipation of treatments means that instrumented potential outcome timeseries has { Z t − p ⊥⊥ Y t ( W t − p − , w t − p : t ) } | F t − p − for all w t − p : t ∈ W p +1 W . Our second special case of the potential outcome time series bridges this framework to the literatureon linear dynamic causal models (e.g. the survey of Ramey (2016)).
Definition 4 (Linear potential outcome time series) . Assume a potential outcome time series. If, forevery w t ∈ W t , Y t ( w t ) = U t + t − X s =0 β t,s w t − s , almost surely , where β t,s are non-stochastic, then { Y t , W t } is called a linear potential outcome time series . If β t,s = β s for every t , then the linear potential outcome time series is time-invariant . Here, { U t } is an arbitrary stochastic process whose only constraint is that it does not vary with w t and ( U t ⊥⊥ W t ) |F t − . For example, { U t } may be an ARCH process, which is non-linear, or a randomwalk, which is non-stationary.Our last special case bridges the potential outcome time series framework to the literature onshocks in economics (e.g. the surveys of Ramey (2016) and Stock and Watson (2018)). Definition 5 (Shocked potential outcome) . For a potential outcome time series, if, E [ W t | F t − ] = 0 , then W t is called a shock and we label { Y t , W t } a shocked potential outcome time series . Moreover, constructed measures of changes in government spending and tax policy have also recently been used asinstruments to study the effects of fiscal policy on macroeconomic outcomes using time series data (Ramey and Zubairy,2018; Fieldhouse et al., 2018; Mertens and Montiel Olea, 2018).
Example 1 (continuing from p. 6) . If Y t ( w t ) = µ + φY t − ( w t − ) + β w t + ǫ t , where E ( W t |F t − ) = 0 ,and { ǫ t ⊥⊥ W t }|F t − , then the system is a shocked potential outcome time series. The class of shocked potential outcome time series provides the formal definition of a sequence ofnonparametric shocks within a causal framework. To our knowledge, this formalization of a causalshock is novel. Shocks are often described heuristically or precisely with respect to a model such asa structural moving average. For example, Stock and Watson (2018) describe macroeconomic shocksas “unanticipated structural disturbances” that produce “unexpected changes” in the macroeconomicoutcomes of interest. Ramey (2016) also describes shocks as: (1) “exogenous with respect to the othercurrent and lagged endogenous variables,” (2) “uncorrelated with other exogenous shocks” and (3)“either unanticipated movements in exogenous variables or news about future movements in exogenousvariables.”Shocks are central to modern macroeconomics and financial economics. Leading empirical exam-ples of macroeconomic shocks include “oil price shocks,” (Hamilton, 2003, 2013) and sudden changesin national defense spending (Ramey, 2011; Barro and Redlick, 2011; Ramey and Zubairy, 2018). Ex-amples of shocks in financial economics include “earnings surprises” (Kothari, 2001; Kothari et al.,2006; Patton and Verardo, 2012) and “news impact” (Engle and Ng, 1993; Anatolyev and Petukhov,2016). L projections of potential outcomes In economics, it is common to use best linear approximations or representations of potentially non-linear systems or expectations (Rudd, 2000; Plagborg-Møller and Wolf, 2019). That tradition gener-ates two superpopulation L projections of potential outcomes on lagged treatments. Definition 6.
Suppose { Y t , W t } is a shocked potential outcome time series where K = 1 , E ( Y t ) < ∞ , < E ( W t − p ) < ∞ , and p = 0 , , , ... . Define the time- t projection β Lt,p := arg min β h min α E ( Y t − α − βW t − p ) i , nd the “universal” β Up := arg min β h min α S p ( α, β ) i where S p ( α, β ) := lim T →∞ E T − p T X t = p +1 ( Y t − α − βW t − p ) . Also define the L projections of the time- t potential outcomes Y Lt ( w t ) := α t + t − X s =0 β Lt,s w t − s , and Y Ut ( w t ) := α + t − X s =0 β Ls w t − s , where α t = E ( Y t ) and α = lim T →∞ T P Tt =1 E ( Y t ) . Then β Lt,p = E ( Y t W t − p ) E ( W t − p ) , and β Up = lim T →∞ T − p P Tt = p +1 E ( Y t W t − p ) lim T →∞ T − p P Tt = p +1 E ( W t − p ) , since the martingale difference treatments implies E ( W t − p ) = 0. The two terms are related to oneanother through β Up = lim T →∞ T − p P Tt = p +1 β Lt,p E ( W t − p ) lim T →∞ T − p P Tt = p +1 E ( W t − p ) . (3)Hence β Up is a weighted average of { β Lt,p } , where the weights are the time-varying variance of thetreatments. If E ( W t ) is time-invariant, then the simplification β Up = lim T →∞ T − p P Tt = p +1 β Lt,p holds.These quantities are important in modern dynamic econometrics. In Section 3.5, we will show that β Up is the implicit estimand for the “Local Projection” estimator of the lag- p dynamic causal effect fora shocked potential outcome time series.To link { β Lt,p } and β Up directly to definitional terms { β t,p } under the linear potential outcomes(Definition 4), we combine the shocked assumption with linearity. Theorem 2.1. If { Y t , W t } is a shocked, linear potential outcome time series where K = 1 , E ( Y t ) < ∞ , < E ( W t − p ) < ∞ , and p = 0 , , , ... , then β Lt,p = β t,p , and β Up = β U ∗ p , where β U ∗ p := lim T →∞ T − p P Tt = p +1 β t,p E ( W t − p ) lim T →∞ T − p P Tt = p +1 E ( W t − p ) . (4) Proof.
Given in the Appendix A. 10
Dynamic causal effects p causal effects Dynamic causal effects are comparisons of potential outcomes at a particular point in time alongdifferent treatment paths. In particular, for a potential outcome time series, the time- t causal effecton Y t of treatment path w t , compared to counterfactual path w ′ t , is Y t ( w t ) − Y t ( w ′ t ) . The time- t , lag- p causal effect measures how the outcome at time t changes if the treatment at time t − p changes, where p ≥
0, fixing the treatment path up to time t − p − W t − p − . Definition 7 (Lag- p causal effect) . For a potential outcome time series and scalars w, w ′ , then τ t,p ( w, w ′ ) := Y t ( W t − p − , w, w t − p +1: t ) − Y t ( W t − p − , w ′ , w ′ t − p +1: t ) , is a lag- p , time- t causal effect for all w t − p +1: t , w ′ t − p +1: t ∈ W p . Bojinov and Shephard (2019) introduced and studied the case where the treatment and counterfac-tual at time t − p varies but w ′ t − p +1: t = w t − p +1: t . The more general time- t , lag- p τ t,p ( w, w ′ ) is newand our focus. This generalization is essential to link existing model-based dynamic causal methodsdeveloped in economics to the potential outcome framework. Its development below is the fourth maincontribution of this paper.We can similarly define the projection versions of the lag- p , time- t causal effect as τ Lt,p ( w, w ′ ) := Y Lt ( W t − p − , w, w t − p +1: t ) − Y Lt ( W t − p − , w ′ , w ′ t − p +1: t ) and τ Ut,p ( w, w ′ ) := Y Ut ( W t − p − , w, w t − p +1: t ) − Y Ut ( W t − p − , w ′ , w ′ t − p +1: t ). Example 3.
Assume a linear potential outcome time series, then τ It,p ( w, w ′ ) = β t,p ( w − w ′ ) , and τ t,p ( w, w ′ ) = β t,p ( w − w ′ ) + p − X s =0 β t,s ( w t − s − w ′ t − s ) . For a shocked potential outcome time series: τ Lt,p ( w, w ′ ) = β Lt,p ( w − w ′ ) + P p − s =0 β Lt,s ( w t − s − w ′ t − s ) , and τ Ut,p ( w, w ′ ) = β Up ( w − w ′ ) + P p − s =0 β Us ( w t − s − w ′ t − s ) are, respectively, the time- t and universal L projections of the lag- p , time- t causal effect. Under a shocked, linear potential outcomes, notice that Our approach follows the finite sample tradition that manipulates causal effects without reference to superpop-ulations (Imbens and Rubin, 2015). It contrasts with the superpopulation approach used by Robins (1986),Angrist and Kuersteiner (2011) and Boruvka et al. (2018) in the context of panel data and Angrist et al. (2018) fortime series. Lt,p ( w, w ′ ) = τ t,p ( w, w ′ ) = τ Ut,p ( w, w ′ ) and that Y Lt ( w t ) , Y Ut ( w t ) and Y t ( w t ) all differ, recalling thedefinitions of Y Lt ( w t ) , Y Ut ( w t ) from Definition 6 (e.g. α , α t and U t all differ). We now introduce causal estimands built from the lag- p , time- t causal effect τ t,p ( w, w ′ ).Many possible w t − p : t and w ′ t − p : t are consistent with passing through w t − p = w and w ′ t − p = w ′ . Eachpossible path leads to a valid lag- p , time- t causal effect. We weight these different paths, selecting aweight function which will eventually lead to existing model-based dynamic causal methods developedin economics. The weights we choose will be generated by using distributions of W t − p : t , W ′ t − p : t givenpast data. Definition 8.
Let Y t := Y t ( W t ) , Y ′ t := Y t ( W t − p − , W ′ t − p : t ) . Then, the weighted causal effect is τ ∗ t,p ( w, w ′ ) := E (cid:2)(cid:0) Y t − Y ′ t (cid:1) | F t − p − , W t − p = w, W ′ t − p = w ′ , (cid:8) Y t − p : t ( W t − p − , w t − p : t ) , w t − p : t ∈ W p +1 (cid:9)(cid:3) , (5) if it exists, where the expectation is generated by { W t − p : t , W ′ t − p : t }| F t − p − , { Y t − p : t ( W t − p − , w t − p : t ) , w t − p : t ∈W p +1 } . The causal response function is, if it exists,
CRF t,p ( w, w ′ ) := E (cid:2)(cid:0) Y t − Y ′ t (cid:1) | W t − p = w, W ′ t − p = w ′ , F t − p − (cid:3) , (6) where the expectation is generated by { Y t , W t − p : t , Y ′ t , W ′ t − p : t }|F t − p − . Temporally averaging these causal effects produces the estimands:¯ τ ∗ p ( w, w ′ ) = 1 T − p T X t = p +1 τ ∗ t,p ( w, w ′ ) , CRF p ( w, w ′ ) = 1 T − p T X t = p +1 CRF t,p ( w, w ′ ) , (7)which we label the lag- p average weighted causal effect and the lag- p average causal responsefunction , respectively.The lag- p average weighted causal effect ¯ τ ∗ p ( w, w ′ ) is a finite sample dynamic causal estimand,invoking no stochastic model for the potential outcomes. Intuitively, it describes the observed, his-torical causal effects. CRF p ( w, w ′ ) is a superpopulation quantity. The difference between superpop-ulation and finite sample causal estimands is subtle and increasingly emphasized in microeconomics(Aronow and Samii, 2016; Abadie et al., 2020). Here we introduce this distinction into time series.12e now make an additional assumption about the selected weights that places restrictions on therelationship between the counterfactual and treatment paths, enabling us to simplify the expressionsfor the weighted causal effect and the causal response function. Assumption 4.
For a potential outcome time series assume that:1. { Y ′ t , W ′ t − p : t }|F t − p − L = { Y t , W t − p : t }|F t − p − , where Y t := Y t ( W t ) , Y ′ t := Y t ( W t − p − , W ′ t − p : t ) , { Y t , W t − p : t } ⊥⊥ W ′ t − p |F t − p − , and { Y ′ t , W ′ t − p : t } ⊥⊥ W t − p |F t − p − . Assumption 4.2 means that the treatment path and outcome is independent from the t − p coun-terfactual, given the past. Lemma 3.1.
For a potential outcome time series, if Assumption 4 holds and the expectations exist,then τ ∗ t,p ( w, w ′ ) = E [ Y t | F t − p − , W t − p = w, { Y t − p : t ( W t − p − , w t − p : t ) , w t − p : t ∈ W p +1 } ] − E [ Y t | F t − p − , W t − p = w ′ , { Y t − p : t ( W t − p − , w t − p : t ) , w t − p : t ∈ W p +1 } ] , where the expectations are from W t − p : t |F t − p − , { Y t − p : t ( W t − p − , w t − p : t ) , w t − p : t ∈ W p +1 } . Likewise,
CRF t,p ( w, w ′ ) = E [ Y t | F t − p − , W t − p = w ] − E [ Y t | F t − p − , W t − p = w ′ ] , where the expectations are from the law of { Y t ( W t ) , W t − p }|F t − p − . Proof.
Given in the Appendix A.Lemma 3.1 shows that under Assumption 4 the
CRF t,p ( w, w ′ ) is the same as the “generalizedimpulse response function” of Koop et al. (1996) when w ′ = 0, but those authors have no broaddiscussion of causality. The CRF t,p ( w, w ′ ) is also similar in spirit to the “average policy effect” inAngrist et al. (2018) where w, w ′ are discrete. However, the “average policy effect” is not explicitlydefined in terms of treatment paths.A simple T / -consistent and asymptotically Gaussian kernel estimator of the finite sample, averageweighted causal effect ¯ τ ∗ p ( w, w ′ ) is developed in Appendix B for continuous w, w ′ . This means that theaverage dynamic causal effects can be nonparametrically identified solely from assuming a potentialoutcome time series. No further assumptions on the potential outcomes, such as stationarity, linearityor shocks, are needed. Those auxiliary assumptions on the potential outcomes may improve the13fficiency of estimation, but they are not fundamental to causal identification of the average weightedcausal effect ¯ τ ∗ p ( w, w ′ ). This is a conceptually important point. We now link the CRF to the impulse response function (IRF), which was introduced by Sims (1980)for vector autoregressions (Ramey, 2016; Stock and Watson, 2016; Kilian and Lutkepohl, 2017). Wefirst give the IRF definition.
Definition 9 (Impulse response function) . Assume { Y t , W t } is strictly stationary and IRF p ( w, w ′ ) := E [ Y t | W t − p = w ] − E [ Y t | W t − p = w ′ ] , exists, where here E [ · ] is calculated from the joint law of Y t , W t − p .Then, IRF p ( w, w ′ ) is an impulse response function (IRF). The IRF is commonly viewed as tracing out the dynamic causal effect of the treatment on the outcome.However, the IRF does not have causal meaning without additional assumptions as it is just thedifference of two conditional expectations. In contrast, a causal effect measures what would happenif W t − p is moved from w to w ′ . This is well known, as IRFs are typically used in the context ofparametrized causal models such as the structural vector moving average.With that said, Theorem 3.1 gives the IRF a nonparametric causal meaning by linking it to theCRF. Theorem 3.1.
Assume { Y t , W t } is a stationary potential outcome time series and that Assumption4 holds. Then, if the expectations exist, E [ CRF t,p ( w, w ′ )] = IRF p ( w, w ′ ) , where the expectation is generated by the stationary distribution of treatments and outcomes.Proof. If the expectations exist, then E [ CRF t,p ( w, w ′ )] = E [ Y t ( W t ) | W t − p = w ] − E [ Y t ( W t ) | W t − p = w ′ ] , and the RHS is the IRF.Here, F t − p − is averaged out by stationarity, implying the causal measure holds universally. Hence,if we add stationarity to the potential outcome time series assumption, we can nonparametricallyestimate the impulse response function by the difference of a kernel regression of Y t on W t − p (Robinson,1983; Fan and Yao, 2006) evaluated at w and w ′ , respectively, converging at, again, T / , the standardnonparametric rate. However, this rate is not an improvement over what could be obtained for theaverage weighted causal effect ¯ τ ∗ p ( w, w ′ ) without stationarity.14 .4 Example: linear potential outcomes and shocked treatments Here we detail the properties of the weighted causal effect and the causal response function underspecial features such as a linear potential outcomes and shocked treatments. These two assumptionsare crucial, as most empirical dynamic causal work in economics is carried out using linear modelsunder the assumption that treatments are shocks. It is this restriction that will eventually allow aparametric rate of convergence.
Example 3 (continuing from p. 11) . Under the linear potential outcomes, then the weighted causaleffect becomes τ ∗ t,p ( w, w ′ ) = β t,p ( w − w ′ ) + p − X s =0 β t,s { µ t − s | t − p − ( w ) − µ t − s | t − p − ( w ′ ) } , where µ t − s | t − p − ( w ) = E (cid:2) W t − s |F t − p − , W t − p = w, { Y t − p : t ( W t − p − , w t − p : t ) , w t − p : t ∈ W p +1 } (cid:3) and thecausal response function becomes CRF t,p ( w, w ′ ) = β t,p ( w − w ′ ) + p − X s =0 β t,s { E [ W t − s |F t − p − , W t − p = w ] − E [ W t − s |F t − p − , W t − p = w ′ ] } , assuming all the relevant moments exist. Likewise for a linear, shocked potential outcome time series τ ∗ t,p ( w, w ′ ) = CRF t,p ( w, w ′ ) = β t,p ( w − w ′ ) = τ It,p ( w, w ′ ) . Under time-invariant, linear, stationary potential outcome time series
IRF p ( w, w ′ ) = β p ( w − w ′ ) + P p − s =0 β s { E [ W t − s | W t − p = w ] − E [ W t − s | W t − p = w ′ ] } . For a time-invariant, linear, stationary, shockedpotential outcome time series, then
IRF p ( w, w ′ ) = τ ∗ t,p ( w, w ′ ) = CRF t,p ( w, w ′ ) = β p ( w − w ′ ) . This example shows that if treatments are shocks and potential outcomes are linear, then
CRF p ( w, w ′ ) = ¯ τ ∗ p ( w, w ′ ) = ( w − w ′ ) 1 T − p T X t = p +1 β t,p . Thus estimating
CRF p ( w, w ′ ) or ¯ τ ∗ p ( w, w ′ ) will be estimating the temporal average of β t,p . The timeseries properties of the outcomes (which includes { U t } ) do not drive this result, it is the properties ofthe treatments and the linear potential outcomes which determine it.15 .5 Local projection estimator of causal estimands Here we use the shocked potential outcome time series to provide a formal, causal interpretationto the “local projections” estimator, which is commonly used in economics. This estimator directlyregresses the observed outcome on the observed treatment at a variety of lags, interpreting the coef-ficients on the lagged treatments as estimates of dynamic causal effects (Jord´a, 2005; Ramey, 2016;Stock and Watson, 2018). Theorem 3.2 (Local projection) . Assume { Y t , W t } is a shocked potential outcome time series where K = 1 , E ( Y t ) < ∞ , < E ( W t − p ) < ∞ , p = 0 , , , ... . Construct β Lt,p = E ( Y t W t − p ) / E ( W t − p ) and themean-zero error U Lt := Y t W t − p − β Lt,p W t − p . Assume that { U Lt } , { W t − p } are ergodic processes and β Up (Definition 6) exists. If T → ∞ , then ˆ β OLSp = P Tt = p +1 Y t W t − p P Tt = p +1 W t − p p −→ β Up . If T − / P Tt = p +1 U Lt = O p (1) , T − P Tt = p +1 W t − p p −→ σ W > , then ˆ β OLSp is T / -consistent for β Up .Proof. The probability limit is by construction. The convergence rate is a standard calculation.By construction, ˆ β OLSp estimates, at the parametric rate, the universal β Up from Definition 6. However, β Up only has indirect causal meaning, through the definition τ Ut,p ( w, w ′ ) in Example 3. If we furtherassume a linear potential outcome time series, then this has a direct causal meaning. Corollary 3.1.
Maintain the same conditions as Theorem 3.2 and strengthen { Y t , W t } to a shocked,linear potential outcome time series. If T → ∞ , then ˆ β OLSp p −→ β U ∗ p = lim T →∞ T − p P Tt = p +1 CRF t,p (1 , E ( W t − p ) lim T →∞ T − p P Tt = p +1 E ( W t − p ) . If T − / P Tt = p +1 U Lt = O p (1) , T − P Tt = p +1 W t − p p −→ σ W > , then ˆ β OLSp is T / -consistent for β U ∗ p .Proof. The strenghtening to linearity implies β Lt,p = β t,p = CRF t,p (1 ,
0) and β Up = β U ∗ p , so resultfollows from Theorem 3.2.Under a shocked, linear potential outcome time series β U ∗ p is the temporal weighted average of This is related to, but different from, the literature on direct forecasting, which forecasts Y t by regressing on Y t − p ratherthan iterating one step ahead forecasts p times (Cox, 1961; Marcellino et al., 2006). RF t,p (1 , E ( W t − p ). It is lim T →∞ CRF p (1 ,
0) if E ( W t − p ) is time-invariant. If β t,p = β p ,then ˆ β OLSp p −→ β p irrespective of the variation of E ( W t − p ). A major concern is that precisely measuring the treatment may be very difficult (Jord´a et al., 2015;Stock and Watson, 2018; Plagborg-Møller and Wolf, 2018). Here, we use the instrumented potentialoutcome time series to provide a causal interpretation of LP-IV.
Theorem 3.3 (LP-IV) . Suppose { Y t , V t } is a shocked, linear, instrumented potential outcome timesseries, where for each t = 1 , , ..., T , that V t = ( W t , ˆ W t ) , E ( Y t ) < ∞ , < E ( W t ) < ∞ , < E ( ˆ W t ) < ∞ . For each t = 1 , , ...T, and p = 0 , , ..., t − construct β Lt,p = E ( Y t W t − p ) / E ( W t − p ) , η Lt := ( Y t − β Lt,p W t − p ) ˆ W t − p and ζ Lt := β Lt,p { W t − p ˆ W t − p − E ( W t − p ˆ W t − p ) } and assume that β γp := lim T →∞ T − p P Tt = p +1 β Lt,p E ( W t − p ˆ W t − p ) exists. If { η Lt } and { ζ Lt } are ergodic and β γ = 0 , then ˆ β IVp = P Tt = p +1 Y t ˆ W t − p P Tt = p +1 Y t − p ˆ W t − p p −→ β IVp := lim T →∞ T − p P Tt = p +1 CRF t,p (1 , E ( W t − p ˆ W t − p ) lim T →∞ T − p P Tt = p +1 CRF t, (1 , E ( W t − p ˆ W t − p ) . If, additionally, T − / P Tt = p +1 η Lt = O p (1) , T − P Tt = p +1 Y t − p ˆ W t − p p −→ β γ , then ˆ β IVp is T / -consistentfor β IVp .Proof.
Given in the Appendix A.Under a shocked, linear, instrumented potential outcome time series β IVp is the ratio of theweighted-average of the
CRF t,p (1 , E ( W t − p ˆ W t − p ) to the weightedaverage of CRF t, (1 , E ( W t − p ˆ W t − p ) is time-invariant, then β IVp = lim T →∞ CRF p (1 , /lim T →∞ CRF (1 , . In the LP-IV literature it is conventional to take β t, = 1 (e.g. Stock and Watson(2018)), which would mean that β IVp = lim T →∞ CRF p (1 , β IVp . A sufficientcondition to rule out such behavior is E ( W t − p ˆ W t − p ) ≥ t , which is a signrestriction and is similar in spirit to the “monotonicity” assumption found in the LATE literature oncross-sectional instrumental variables (Imbens and Angrist, 1994; Angrist et al., 1996). Whether sucha restriction is reasonable will depend on the empirical application.17emark 2.1 says that the instrumented potential outcome times series implies ˆ W t − p is uncorrelatedfrom the counterfactual. This lack of correlation is needed for the LP-IV to be causal. Otherwise,ˆ β IVp p −→ β γp + β ′ p β γ , where T − p P Tt = p +1 E ( U t ˆ W t − p ) → β ′ p . Remark 3.1 (Lead-lag exogeneity) . The need for the condition that
Cov ( ˆ W t − p , Y t ( W t − p − , w t − p : t )) =0 is seemingly missing from the LP-IV literature. Instead the existing literature typically uses a “lead-lag exogeneity” assumption that Cov ( W t , ˆ W s ) = 0 for all t = s . Unfortunately, lead-lag exogeneityplus the assumption that { Y t , W t } is a shocked potential outcome time series does not imply that Cov ( ˆ W t − p , Y t ( W t − p − , w t − p : t )) = 0 . This assumption is implied by lead-lag exogeneity assumptionin the tightly parameterized setting studied by existing literature on LP-IV (i.e., outcomes that aregenerated by a structural moving average in the treatment, where the treatments are white noise). Ouranalysis shows that lead-lag exogeneity is not sufficient in more general causal models. In this paper, we adapted the nonparametric potential outcomes time series framework for experimentsto formalize dynamic causal effects in observational time series data. We did so by introducing threecrucial special cases of the potential outcome time series: instruments, shocks and linearity. Further,we deepened our understanding of dynamic causal effects by developing a fourth idea: the finite sampleweighted causal effect and its superpopulation analogue, the causal response function.These four ideas give nonparametric causal meaning to the impulse response function, which is amajor device for economists to measure dynamic causal effects. Further, we used this framework toprovide a causal interpretation to the implicit estimand of the local projections estimator. Finally,we made two important contributions to literature on LP-IV. We showed that the LP-IV estimatoridentifies a weighted average of dynamic causal effects, where the weights depend on the possiblytime-varying relationship between the instrument and the treatment. We also showed that typicalassumptions (i.e. lead-lag exogeneity) are not sufficient to identify a causally interpretable estimandbecause it does not enforce that the instrument is independent of the counterfactual given the past.18 eferences
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Biometrics 72 , 1055–1065.24 conometric analysis of potential outcomes time series:instruments, shocks, linearity and the causal response function
Online Appendix
Ashesh Rambachan Neil Shephard
A Appendix: a collection of proofs
Proof of Theorem 2.1.
Under a linear potential outcome time series Y t ( w t ) = U t + t − X s =0 β t,s w t − s , so if { W t } is a MD sequence, then E ( Y t W t − p ) = E ( U t W t − p ) E ( Y t ) = E ( U t ) = α t . By non-antipicapting treatments of the potential outcome time series, E ( U t W t − p ) = 0 so long as themoment exists. This delivers the required result using conventional arguments. Proof of Lemma 3.1.
As the moments exist, so
CRF t,p ( w, w ′ ) simplifies to E [ { Y t ( W t − p − , w, W t − p +1: t ) | ( W t − p = w, W ′ t − p = w ′ , F t − p − )] − E [ { Y t ( W t − p − , w ′ , W ′ t − p +1: t ) } | ( W t − p = w, W ′ t − p = w ′ , F t − p − )] . Due to property 2 of the causal predictive weight, E [ Y t ( W t ) | F t − p − , W t − p = w, W ′ t − p = w ′ ] = E [ Y t ( W t ) | F t − p − , W t − p = w ] . Due to property 1 of the causal predictive weights, E [ Y t ( W t − p − , w ′ , W ′ t − p +1: t ) | F t − p − , W ′ t − p = w ′ ] = E [ Y t ( W t ) | F t − p − , W t − p = w ′ ] . The corresponding results for the weighted causal effect follow using the same logical arguments.25 roof of Theorem 3.3.
Define ǫ Lt,p := Y t − β Lt,p W t − p , then by the shock and instrument property ofthe time series, E ( ǫ Lt,p ˆ W t − p ) = 0 . So construct the zero mean time series η Lt := ǫ Lt,p ˆ W t − p and ζ Lt := β Lt,p { W t − p ˆ W t − p − E ( W t − p ˆ W t − p ) } . Then1 T − p T X t = p +1 Y t ˆ W t − p = 1 T − p T X t = p +1 β Lt,p W t − p ˆ W t − p + 1 T − p T X t = p +1 η Lt . If { η Lt } is ergodic, the latter sum disappears, while if { ζ Lt } is ergodic then the former term converges tothe limit of the expectations as expected. Shocks plus linearity implies β Lt,p = β t,p = CRF t,p (1 , Appendix: estimation of ¯ τ ∗ p ( w, w ′ ) B.1 Conditioning on the potential outcomes
Throughout this Section F T,t denotes the triangular filtration (pg. 53 of Hall and Heyde (1980))generated by { W t , Y t , { Y t +1: T ( W t , w t +1: T ) , w t +1: T ∈ W T − t }} . Recall ¯ τ ∗ p ( w, w ′ ) = 1 T − p T X t = p +1 τ ∗ t,p ( w, w ′ )where τ ∗ t,p ( w, w ′ ) = E [ Y t | F T,t − p − , W t − p = w ] − E [ Y t | F T,t − p − , W t − p = w ′ ] . The expectations are over the treatment path, holding fixed the potential outcomes. Fixing thepotential outcomes follows the microeconometrics tradition discussed by Imbens and Rubin (2015),Abadie et al. (2017, 2020) and traces back to Fisher (1925, 1935) and Cox (1958). Bojinov and Shephard(2019) first introduced this type of approach into time series experiments.Our task is to estimate τ ∗ t,p ( w, w ′ ) and ¯ τ ∗ p ( w, w ′ ). B.2 When W is discrete B.2.1 Estimator
We start by assuming that W is discrete and that the treatment is probabilistic . Assumption 5 (Probabilistic treatment) . For all t ≥ , F T,t − and w ∈ W , p t ( w ) := Pr( W t = w | F T,t − ) > . Assumption 5 is the analogue of the “overlap” assumption made in cross-sectional settings. Through-out we will regard p t ( w ) as known, which will be true in experimental settings and unlikely in obser-vational ones where p t ( w ) would need to be estimated.27efine a time series version of the classic Horvitz and Thompson (1952) style estimatorˆ¯ τ ∗ p ( w, w ′ ) := 1 T − p T X t = p +1 ˆ τ ∗ t,p ( w, w ′ ) , ˆ τ ∗ t,p ( w, w ′ ) := Y t (cid:26) ( W t − p = w ) − ( W t − p = w ′ ) (cid:27) p t − p ( W t − p ) . (8)This estimator appears in Angrist et al. (2018), but for a superpopulation estimand. Bojinov and Shephard(2019) also use a Horvitz and Thompson (1952) style estimator, but differently setup and for a differentfinite sample estimand. The results which follow are roughly inline with those in Bojinov and Shephard(2019), although the details differ. No new ideas are needed to generate the results. B.2.2 Properties of ˆ τ ∗ t,p ( w, w ′ ) and ˆ¯ τ ∗ p ( w, w ′ )The following theorem shows that ˆ τ ∗ t,p ( w, w ′ ) − τ ∗ t,p ( w, w ′ ) has martingale difference errors and henceˆ¯ τ ∗ p ( w, w ′ ) is unbiased, conditional on the potential outcomes. Theorem B.1 (Properties of ˆ τ ∗ t,p ( w, w ′ )) . Assume a potential outcome time series and Assumption5. Let u t − p ( w, w ′ ) := ˆ τ ∗ t,p ( w, w ′ ) − τ ∗ t,p ( w, w ′ ) . Then, over the non-anticipating treatment path, E [ u t − p ( w, w ′ ) | F T,t − p − ] = 0 , and E [ˆ¯ τ ∗ p ( w, w ′ )] = ˆ¯ τ ∗ p ( w, w ′ ) . (9) Further η t − p ( w, w ′ ) := V ar [ u t − p ( w, w ′ ) |F T,t − p − ] , is E (cid:18) Y t ( W t − p − , w, W t − p +1: t ) p t − p ( w ) | F T,t − p − , W t − p = w (cid:19) (10)+ E (cid:18) Y t ( W t − p − , w ′ , W t − p +1: t ) p t − p ( w ′ ) | F T,t − p − , W t − p = w ′ (cid:19) − τ ∗ t,p . (11) Proof.
We produce equation (9) by noting that E (cid:18) Y t ( W t − p − , w, W t − p +1: t )1( W t − p = w ) p t − p ( w ) |F T,t − p − (cid:19) = E { Y t ( W t − p − , w, W t − p +1: t ) |F T,t − p − , W t − p = w } = E { Y t |F T,t − p − , W t − p = w } The form of η t − p ( w, w ′ ) is expected from the cross-sectional literature, and can be derived using thevariance of a Bernoulli trial. 28hus, over the treatment path, conditioning on the entire path of all potential outcomes,( T − p ) V ar (ˆ¯ τ ∗ p ( w, w ′ ) − ¯ τ ∗ p ( w, w ′ ) |{ Y T ( w T ) , w T ∈ W T } ) = ¯ η T ( w, w ′ ) (12)where ¯ η T ( w, w ′ ) = 1 T − p T X t = p +1 E ( η t − p ( w, w ′ ) |{ Y T ( w T ) , w T ∈ W T } ) . So long as the conditional mean of η t − p ( w, w ′ ) is bounded, then this the conditional variance ofˆ¯ τ ∗ p ( w, w ′ ) will contract with T .The following Theorem, which just applies a triangular martingale difference central limit theorem,extends these results to where T → ∞ . It shows that ˆ¯ τ ∗ t,p ( w, w ′ ) is consistent for ¯ τ ∗ p ( w, w ′ ) and theestimator’s error is asymptotically normal under weak conditions. Theorem B.2.
Under the conditions of Theorem B.1, additionally assume that lim T →∞ ¯ η T ( w, w ′ ) < ∞ . Then ˆ¯ τ ∗ p ( w, w ′ ) − ¯ τ ∗ p ( w, w ′ ) p → as T → ∞ . Finally, if T − p P Tt = p +1 η t − p ( w, w ′ ) p → η ( w, w ′ ) > , then, over the non-anticipating treatment path, as T → ∞ , √ T { ˆ¯ τ ∗ p ( w, w ′ ) − ¯ τ ∗ p ( w, w ′ ) } η ( w, w ′ ) d → N (0 , . (13) Proof.
The first result follows from (12) as ¯ η T ( w, w ′ ) is bounded. The second follows from a martingalearray CLT of Theorem 3.2 in Hall and Heyde (1980) as the potential outcomes are bounded whichmeans the Lindeberg condition hold.Again, the only source of randomness here is the path of the treatments. B.3 When W is continuous B.3.1 Estimator
There is a modest literature on the nonparametric estimation of causal effects when treatments arecontinuous in cross-sectional and panel settings. For example, Hirano and Imbens (2004) study con-tinuous treatments using “generalized propensity scores.” Marginal structural models of Robins et al.(2000) provide parametric and series based nonparametric strategies to deal with continuous treat-ments. Cattaneo (2010) provides an extensive discussion of the multivalued case and the relatedliterature. Yang et al. (2016) is a recent paper on this topic.29rite F t ( w ) := Pr( W t ≤ w |F T,t − ) , and f t ( w ) := ∂F t ( w ) /∂w. Then, using a bandwidth h >
0, define the time- t kernel regression estimatorˆ τ ∗ t,p ( w, w ′ ) := ˆ g t,p ( w ) − ˆ g t,p ( w ′ ) , and ˆ g t,p ( w ) := Y t k h ( W t − p − w ) f t − p ( W t − p ) , (14)where k h ( u ) = h − k ( u/h ) is a kernel weight function, and the estimand is τ ∗ t,p ( w, w ′ ) = g t,p ( w ) − g t,p ( w ′ ) , where g t,p ( W t − p ) := E ( Y t |F T,t − p − , W t − p ) . In a moment we will use the definitions g [2] t,p ( W t − p ) = E ( Y t |F T,t − p − , W t − p ) , κ j = R u j k ( u ) du , b = R k ( u ) du and k ∗ ( x ) = R k ( u ) k ( x + u ) du .Theorem B.3 quantifies the variance and bias terms of the time-t kernel regression estimator,holding the potential outcomes as fixed. The derivation of the result is entirely conventional from thekernel literature. Theorem B.3.
Assume h > and f t − p ( w ) > for all w . Define µ t,p ( w ) := E (cid:20) ˆ g t,p ( w ) |F T,t − p − (cid:21) , σ t,p ( w ) := h × V ar (cid:20) ˆ g t,p ( w ) |F T,t − p − (cid:21) ,c t,p ( w, w ′ ) := h × Cov (ˆ g t,p ( w ) , ˆ g t,p ( w ′ ) |F T,t − p − ) , where the expectations are over the treatment process W t − p : t |F T,t − p − , holding the potential outcomesfixed. If u t,p ( w ) := ˆ g t,p ( w ) − µ t,p ( w ) then E ( u t,p ( w ) |F T,t − p − ) = 0 , V ar ( u t,p ( w ) |F T,t − p − ) = h − σ t,p ( w ) Cov ( u t,p ( w ) , u t,p ( w ′ ) |F T,t − p − ) = h − c t,p ( w, w ′ ) . Further, if g t,p ( w ) is twice continuously differentiable in w , κ = 1 , κ = 0 and h ↓ , then µ t,p ( w ) ≃ g t,p ( w ) + 0 . h g ′′ t,p ( w ) κ , σ t,p ( w ) ≃ g [2] t,p ( w ) f t − p ( w ) b,c t,p ( w, w ) ≃ (cid:18) g [2] t,p ( w ) f t − p ( w ) + g [2] t,p ( w ′ ) f t − p ( w ′ ) (cid:19) k ∗ (( w − w ′ ) /h ) . inally, if as | x | → ∞ , k ∗ ( x ) = o (1) , then c t,p ( w, w ) = o (1) , if w = w ′ .Proof. All but the last 3 results are by definition. Now h t,p ( w ) := E (cid:20) Y t k h ( W t − p − w ) f t − p ( W t − p ) |F T,t − p − (cid:21) = h − Z g t,p ( x ) k (( x − w ) /h ) dx. Transforming to u = ( x − w ) /h , so x = w + hu , we have h t,p ( w ) = Z g t,p ( w + hu ) k ( u ) du ≃ g t,p ( w ) + 0 . h g ′′ t,p ( w ) κ , as κ = 1 and κ = 0. Likewise E (cid:20) Y t k h ( W t − p − w ) f t − p ( W t − p ) |F T,t − p − (cid:21) = h − Z g [2] t,p ( x ) f t − p ( x ) k (( x − w ) /h ) dx = h − Z g [2] t,p ( w + hu ) f t − p ( w + hu ) k ( u ) du ≃ h − g [2] t,p ( w ) f t − p ( w ) b, while E (cid:20) Y t k h ( W t − p − w ) f t − p ( W t − p ) k h ( W t − p − w ′ ) f t − p ( W t − p ) |F T,t − p − (cid:21) = h − Z g [2] t,p ( x ) f t − p ( x ) k (( x − w ) /h ) k (( x − w ′ ) /h ) dx = h − Z g [2] t,p ( x ) f t − p ( x ) k (( x − w ) /h ) k (( x − w ′ ) /h ) dx + h − Z g [2] t,p ( x ) f t − p ( x ) k (( x − w ) /h ) k (( x − w ′ ) /h ) dx = h − Z g [2] t,p ( w + hu ) f t − p ( w + hu ) k ( u ) k ( u + ( w − w ′ ) /h ) du + h − Z g [2] t,p ( w ′ + hu ) f t − p ( w ′ + hu ) k ( u + ( w − w ′ ) /h ) k ( u ) du ≃ h − (cid:18) g [2] t − p ( w ) f t − p ( w ) + g [2] t,p ( w ′ ) f t − p ( w ′ ) (cid:19) Z k ( u ) k ( u + ( w − w ′ ) /h ) du = h − (cid:18) g [2] t,p ( w ) f t − p ( w ) + g [2] t,p ( w ′ ) f t − p ( w ′ ) (cid:19) k ∗ (( w − w ′ ) /h ) . Then the result follows by the assumed property of k ∗ .For each T fix the bandwidth as h T . For each T , the estimation error { u t,p ( w ) } is a martingaledifference sequence but centered at µ t,p ( w ) not g t ( w ). Now assume that T − p P Tt = p +1 σ t,p ( w ) p −→ σ p ( w ),then for h T > p h ( T − p ) { ˆ¯ τ ∗ p ( w ) − ˆ¯ τ ∗ p ( w ′ ) } − { ¯ µ p ( w ) − ¯ µ p ( w ′ ) } q σ p ( w ) + σ p ( w ′ ) d −→ N (0 , , µ p ( w ) = T − p P Tt = p +1 µ t,p ( w ). Of course¯ µ p ( w ) − ¯ µ p ( w ′ ) ≃ { ¯ g p ( w ) − ¯ g p ( w ′ ) } + 0 . h κ { ¯ g ′′ p ( w ) − ¯ g ′′ p ( w ′ ) } , where ¯ g ′′ p ( w ) = T − p P Tt = p +1 g ′′ t,p ( w ). Notice that the bias involves the difference of two second deriva-tives of ¯ g p ( w ) evaluated at w and w ′ . Remark B.1.
The corresponding results when the regression kernels for ˆ¯ g p ( w ) and ˆ¯ g p ( w ′ ) use differentbandwidth, h w and h w ′ , is straightforward to write out. However, in practice this has the disadvantagethat the bias term becomes . κ { h w ¯ g ′′ p ( w ) − h w ′ ¯ g ′′ p ( w ′ ) } , which shows no sign of cancelling. Further, if we aggregate period of period mean square error, then1 T − p T X t = p +1 E (cid:20)(cid:18) { ˆ τ ∗ t,p ( w ) − ˆ τ ∗ t,p ( w ′ ) } − { g t,p ( w ) − g t,p ( w ′ ) } (cid:19) |F T,t − p − (cid:21) ≃ h ( T − p ) { σ p ( w ) + σ p ( w ′ ) } + 0 . h κ T − p T X t = p +1 (cid:18) g ′′ t,p ( w ) − g ′′ t,p ( w ′ ) (cid:19) , which is minimized by selecting h ∝ ( T − p ) − / so the mean square error declines at the usualnonparametric rate T − / , which does not vary with p . None of these results are surprising from thevast nonparametric literature. Remark B.2.
At a fundamental level it would be convenient to be able to estimate individual finitesample terms like Y t ( W t − p − , w ) − Y t ( W t − p − , w ′ ) , where w, w ′ ∈ W p +1 , or their temporal average.Can these terms be nonparametrically identified just using the structure of the potential outcome timeseries, conditioning on all of the potential outcomes? We sketch out below that the answer to this isyes, but that the result is of little immediate practice use due to the slow rate of convergence. Writethe intermediate estimand as g t,p ( w ) := Y t ( W t − p − , w ) , where w ∈ W p +1 , while write g ′′ it,p ( w ) := ∂ g t,p /∂w i , k h,r ( u ) := h − r k ( u ) ...k ( u r ) , and F t − p : t ( w ) := Pr( W t − p : t ≤ w |F T,t − p − ) , and f t − p : t ( w ) := ∂F t − p : t ( w ) /∂w. The corresponding intermediate estimator is ˆ g t,p ( w ) := Y t f t − p : t ( W t − p : t ) k h,p +1 ( W t − p : t − w ) . The eventual goal is to use ˆ g t,p ( w ) − ˆ g t,p ( w ′ ) to estimate g t,p ( w ) − g t,p ( w ′ ) . Now E (ˆ g t,p ( w, w ′ ) |F T,t − p − ) = h − ( p +1) Z g t,p ( x ) k h,r (( x − w ) /h ) dx ...dx p +1 Z g t,p ( w + hu ) k ( u ) ...k ( u p +1 ) du ...du p +1 = g t,p ( w ) + h ( p + 1)0 . κ p + 1 p +1 X i =1 g ′′ it,p ( w ) , while E (ˆ g t,p ( w, w ′ ) |F T,t − p − ) = h − p +1) Z g t,p ( x ) f t − p : t ( x ) K (( x − w ) /h ) dx ...dx p +1 ≃ h − ( p +1) g t,p ( w ) f t − p : t ( w ) . As before the covariance between ˆ g t,p ( w ) and ˆ g t,p ( w ′ ) is comparatively unimportant. Hence, averagingover T data points, in terms of mean square the best bandwidth choice would be h ∝ T − / ( p +5) so themean square error declines at the usual multivariate rate of T ( p +4) / ( p +5) . Hence g t,p ( w ) − g t,p ( w ′ ) isnonparametrically identified, but at its core it is a very nasty result empirically. As the length of thelags increases the rate of convergence slows.isnonparametrically identified, but at its core it is a very nasty result empirically. As the length of thelags increases the rate of convergence slows.