Supplementary Appendix for "Inference on Treatment Effects After Selection Amongst High-Dimensional Controls"

Intuition for the Importance of Double Selection

To build intuition, we discuss the case where there is only one control; that is, $p=1$. This scenario provides the simplest possible setting where variable selection might be interesting. In this case, Lasso-type methods act like conservative $t$-tests which allows the properties of selection methods to be explained easily.

For simplicity, all errors and controls are taken as normal,

The standard post-single-selection method for inference proceeds by applying model selection methods – ranging from standard $t$-tests to Lasso-type selectors – to the first equation only, followed by applying OLS to the selected model. In the model selection stage, standard selection methods would necessarily omit $x_{i}$ wp $\to 1$ if

where $\sigma^{*2}_{n}=\sigma^{2}_{\zeta}(\sigma^{2}_{d})^{-1}$. The variance $\sigma^{*2}_{n}$ is smaller than the variance $\sigma^{2}_{n}$ from estimation with $x_{i}$ included if $\beta_{m}\neq 0$. The potential reduction in variance is often used as a “motivation” for the standard selection procedure. The estimator is super-efficient, achieving a variance smaller than the semi-parametric efficiency bound under homoscedasticity. That is, the estimator is “too good.”

That is, the standard post-selection estimator is not asymptotically normal and even fails to be uniformly consistent at the rate of $\sqrt{n}$. This poor behavior occurs because the omitted variable bias created by dropping $x_{i}$ may be large even when the magnitude of the regression coefficient, $|\beta_{m}|$, in the confounding equation (1.2) is small but is not exactly zero. To see this, note

The term $i^{*}$ has standard behavior; namely $i^{*}\rightsquigarrow N(0,1)$. The term $ii$ generates the omitted variable bias, and it may be arbitrarily large, since wp $\to 1$,

In contrast to the standard approach, our post-double-selection method for inference proceeds by applying model selection methods, such as standard $t$-tests or Lasso-type selectors, to both equations and taking the selected controls as the union of controls selected from each equation. This selection is than followed by applying OLS to the selected controls. Thus, our approach drops $x_{i}$ only if the omitted variable bias term $ii$ is small. To see this, note that the double-selection-methods include $x_{i}$ wp $\to 1$ if its coefficient in either (1.1) or (1.2) is not very small. Mathematically, $x_{i}$ is included if

Given the discussion in the preceding paragraph, it is immediate that the post-double selection estimator satisfies

i.e. coefficients in front of $x_{i}$ in both equations are small. In this case,

Once again, the term $ii$ is due to omitted variable bias, and it obeys wp $\to 1$ under (1.13)

since $(\sigma_{v}/\sigma_{d})^{2}=1-\rho^{2}$. Moreover, we can show $i^{*}-i=o_{P}(1)$ under such sequences, so the first order asymptotics of $\check{\alpha}$ is the same whether $x_{i}$ is included or excluded.

Extensions: Other Problems and Heterogeneous Treatment Effects

In order to discuss extensions in a very simple manner, we assume i.i.d sampling as well as assume away approximation errors, namely $g(z_{i})=x_{i}^{\prime}\beta_{g0}$ and $m(z_{i})=x^{\prime}_{i}\beta_{m0}$, where parameters $\beta_{g0}$ and $\beta_{m0}$ are high-dimensional and that $x_{i}=P(z_{i})$ as before. In this paper we considered a moment condition:

where $\varphi(u)=u$ and $v_{i}$ are measurable functions of $z_{i}$, and the target parameter is $\alpha_{0}$. We selected the instrument $v_{i}$ such that the equation is first-order insensitive to the parameter $\beta$ at $\beta=\beta_{g0}$:

Note that $\varphi(u)=u$ and $v_{i}=d_{i}-m(z_{i})$ implement this condition. If (2.15) holds, the estimator of $\alpha_{0}$ gets “immunized” against nonregular estimation of $\beta_{0}$, for example, via a post-selection procedure or other regularized estimators. Such immunization ideas are in fact behind the classical Frisch-Waugh-Robinson partialling out technique in the linear setting and the ?’s $C(\alpha)$ test in the nonlinear setting. Our contribution here is to recognize the importance of this immunization in the context of post-selection inference, To the best of our knowledge, all prior theoretical and empirical work uses the standard post-selection approach based on the outcome equation alone, which is highly non-robust way of conducting inference, as shown in extensive monte-carlo, in Section 2.4, and in a sequence of fundamental critiques by Leeb and Potscher, see ? . to develop a robust post-selection approach to inference on the target parameter, and characterize the uniformity regions of this procedure. Our approach uses modern selection methods to estimate $g$ and the function $m$ defining the intstrument $v$. In an ongoing work, we explore other regularization methods, such as the ridge method or combination of ridge method with Lasso methods, and characterize uniformity regions of the resulting procedures. Also, generalizations to nonlinear models, where $\varphi$ is non-linear and can correspond to a likelihood score or quantile check function are given in ? and ?; in these generalizations achieving (2.15) is also critical.

Within the context of this paper, a potentially important extension is to consider a general treatment effect model, where $d_{i}$ is interacted with transformations of $z_{i}$. As long as the interest lies in a particular regression coefficient, the current framework covers this implicitly since $x_{i}$ could contain interactions of $d_{i}$ with transformations of controls $z_{i}$. In the case a fixed number of such regression coefficients is of interest, we can estimate each of the coefficients by re-labeling the corresponding regressor as $d_{i}$ and other regressors as $x_{i}$ and then applying our procedure the fixed number of times. Such component-wise procedure is valid as long as our regularity conditions hold for each of the resulting regression models in this manner.

A related research direction being pursued is the study of estimation of average treatment effects when treatment effects are fully heterogeneous. When the treatment variable $d_{i}\in\{0,1\}$ is binary (or discrete more generally) our approach is readily amenable to this problem. In this case the parameter of interest is the average treatment effect

where $\varphi(\alpha,y,d,g_{0},g_{1},m)=\alpha-\frac{d(y-g_{1})}{m}+\frac{(1-d)(y-g_{0})}{1-m}-(g_{1}-g_{0}).$ It is straightforward to check that for each $j\in\{1,2,3\}$:

where the latter is the semiparametric efficiency bound of ?. Formal result is proven and stated below, and other formal results along these lines are given in an ongoing work that studies this as well as other types of effects.

2. Theoretical Results on ATE with Heterogeneity

The confounding factors $z_{i}$ affect the policy variable via the propensity score $m(z_{i})$ and the outcome variable via the function $g(d_{i},z_{i})$. Both of these functions are unknown and potentially complicated. As in the main text, we use linear combinations of control terms $x_{i}=P(z_{i})$ to approximate $g(z_{i})$ and $m(z_{i})$, writing (2.19) and (2.20) as

where $r_{gi}$ and $r_{mi}$ are the approximation errors, and

where $x_{i}^{\prime}\beta_{g0,1}$, $x_{i}^{\prime}\beta_{g0,0}$, and $x_{i}^{\prime}\beta_{m0}$ are approximations to $g(1,z_{i})$, $g(0,z_{i})$, and $m(z_{i})$, and $\Lambda(u)=u$ for the case of linear link and $\Lambda(u)=e^{u}/(1+e^{u})$ for the case of the logistic link. In order to allow for a flexible specification and incorporation of pertinent confounding factors, the vector of controls, $x_{i}=P(z_{i})$, we can have a dimension $p=p_{n}$ which can be large relative to the sample size.

The efficient moment condition, derived by ?, for parameter $\alpha_{0}$ is as follows:

The post-double-selection estimator $\check{\alpha}$ that solves

where $\widehat{g}(d_{i},z_{i})$ and $\widehat{m}(z_{i})$ are post-Lasso estimators of functions $g$ and $m$ based upon equations (2.21)-(2.22). In case of the logistic link $\Lambda$, Lasso for logistic regression is as defined in ? and ?, and the associated post-Lasso estimators are as those defined in ?.

These conditions are simple high-level conditions, which encode both the approximate sparsity of the models as well as impose some reasonable behavior on the post-selection estimators of $m$ and $g$ (or other sparse estimators). These conditions are implied by other more primitive conditions in the literature. Sufficient conditions for the equivalence between population and empirical sparse eigenvalues are given in ? and ?. The boundedness conditions are made to simplify arguments, and they could be dealt away with more complicated proofs, under more stringent side conditions.

The next target parameter is the average treatment effect on the treated:

The efficient moment condition, derived by ?, for parameter $\gamma_{0}$ is as follows:

In this case the post-double-selection estimator $\check{\gamma}$ that solves

3. Proof of Theorems 1 and 2.

The two results have identical structure and have nearly the same proof, and so we present the proof of the Proof of Theorems 1 only.

Step 1. In this step we establish claim (1).

(a) We begin with a preliminary observation. Define, for $t=(t_{1},t_{2},t_{3})$,

where $L$ depends only on $c^{\prime}$ and $C$, $|k|=\sum_{j=1}^{3}k_{j},$ and

We observe that with probability no less than $1-\Delta_{n}$,

for $\beta=\widehat{\beta}_{g}$, with evaluation after computing the norms, and noting that for any $\beta$

under condition (iii). Furthermore, for $n\geqslant n_{0}=\min\{j:\delta_{j}\leqslant 1/2\}$:

for $\beta=\widehat{\beta}_{m0}$, with evaluation after computing the norms.

Hence with probability at least $1-\Delta_{n}$,

with $h$ evaluated at $h=\widehat{h}$. By Liapunov central limit theorem,

(d) Note that for $\Delta_{i}=h(z_{i})-h_{0}(z_{i})$,

(with $h$ evaluated at $h=\widehat{h}$). By the law of iterated expectations and because

Moreover, uniformly for any $h\in\mathcal{H}_{n}$ we have that

Since $\widehat{h}\in\mathcal{H}_{n}$ with probability $1-\Delta_{n}$, we have that once $n\geqslant n_{0}$,

The class of functions $\mathcal{G}_{d}$ for $d\in\{0,1\}$ is a union of at most $\binom{p}{Cs}$ VC-subgraph classes of functions with VC indices bounded by $C^{\prime}s$. The class of functions $\mathcal{M}$ is a union of at most $\binom{p}{Cs}$ VC-subgraph classes of functions with VC indices bounded by $C^{\prime}s$ (monotone transformation $\Lambda$ preserve the VC-subgraph property). These classes are uniformly bounded and their entropies therefore satisfy

Finally, the class $\mathcal{F}_{n}=\{f_{h}-f_{h_{0}}:h\in\mathcal{H}_{n}\}$ is a Lipschitz transform of $\mathcal{H}_{n}$ with bounded Lipschitz coefficients and with a constant envelope. Therefore, we have that

We shall invoke the following lemma derived in ?.

Let $\mathcal{F}$ be a measurable function class on a sample space. Let $F=\sup_{f\in\mathcal{F}}|f|$, and suppose that there exist some constants $\omega_{n}>3$ and $\upsilon>1$, such hat

Then for every $\delta\in(0,1/6)$ we have

with probability at least $1-\delta$ for some constant that $C_{\upsilon}$.

Then by Lemma 1 together and some simple calculations, we have that

Step 3. Claim (3) is immediate from claims (2) and (3) by the way of contradiction. ∎

Deferred Proofs: Proof of Lemma 1

We establish the result for Lasso (the proof for other feasible Lasso estimators is similar).

By Condition SE, with probability $1-o(1)$ for $n$ sufficiently large we have $\kappa_{\bar{c}}>\kappa^{\prime}/2\|\widehat{\Psi}\|_{\infty}$ so that with the same probability

Moreover, by condition RF we have with probability $1-o(1)$ that

Finally, since $\lambda\gtrsim\sqrt{n\log(p\vee n)}$ we have

since $c_{s}\lesssim_{P}\sqrt{s/n}$ by condition ASM and Chebyshev inequality.

To show the second statement in (i), note that

where $\widehat{\beta}$ is the Lasso estimator. Again by Lemma 7 in ? we have that the assumptions of Lemma 6 in ? hold with probability $1-o(1)$. Using Condition SE to bound $\kappa_{\bar{c}}$ from below and Condition RF to bound $\|\widehat{\Psi}\|_{\infty}$ from above with probability $1-o(1)$ as before, and $\lambda\lesssim\sigma\sqrt{n\log(p\vee n)}$, it follows from Lemma 6 in ? that with probability $1-o(1)$ that

The results regarding Post-Lasso in (ii) follow similarly by invoking Lemma 8 in ?.

Split-Sample Estimation and Inference

In this section we discuss a variant of the double selection estimator based on sample splitting. The motivation for the split-sample estimator is that its use allows us to relax the requirement $s^{2}\log^{2}(p\vee n)=o(n)$ that is assumed in the full-sample counterpart to the milder condition

For each subsample $k=a,b$, the model $\widehat{I}^{k}$ is selected based on the subsample $k$ independently from the subsample $k^{c}$. In what follows the model $\widehat{I}^{k}$ is used to fit the subsample $k^{c}$. A constructive way to obtain $\widehat{I}^{a}$ and $\widehat{I}^{b}$ is to apply the double selection method for each subsample to select the sets of controls $\widehat{I}^{a}:=\widehat{I}_{1}^{a}\cup\widehat{I}_{2}^{a}\cup\widehat{I}_{3}^{a}$ and $\widehat{I}^{b}:=\widehat{I}_{2}^{b}\cup\widehat{I}_{2}^{b}\cup\widehat{I}_{3}^{b}$.

Then we form estimates in the two subsamples

For an index $i$ in the subsample $k$, we define the residuals

Finally, we combine the estimates into the split-sample estimator based on $\widehat{I}^{a}$ and $\widehat{I}^{b}$ is defined as

where $\Upsilon^{k}=D^{k}{}^{\prime}\mathcal{M}_{\widehat{I}^{k^{c}}}D^{k}/n_{k}$.

We state below sufficient conditions for the analysis of the split-sample method.

The Conditions ASTESS(i)-(iii) agree with the corresponding conditions in ASTE. The remaining conditions ASTESS(iv)-(v) are implied by Condition ASTE. We note that Condition ASTESS(vi) is needed only for obtaining consistent estimates of the asymptotic variance. Such conditions are mild since they do not require uniform estimation of the functions $g$ and $m$.

The next result establishes that the split-sample estimator $\check{\alpha}_{ab}$ has similar large sample properties to the full-sample double-selection estimator under weaker growth condition.

Step 0.(Combining) In this step we combine both subsample estimators. Letting $\Upsilon^{k}=D^{k}{}^{\prime}\mathcal{M}_{\widehat{I}^{k^{c}}}D^{k}/n_{k}$, for $k=a,b$, so that we have

which follows similarly to the proofs given in Step 5.

where $z_{i,n}=\sigma_{n}^{-1}v_{i}\zeta_{i}/\sqrt{n}$ are i.n.i.d. with mean zero. We have that for some small enough $\delta>0$

This condition verifies the Lyapunov condition and thus implies that $Z_{n}\to_{d}N(0,1)$.

Step 1.(Main) For the subsample $k=a,b$ write $\check{\alpha}_{k}=\left[D^{k}{}^{\prime}\mathcal{M}_{\widehat{I}^{k^{c}}}D^{k}/n_{k}\right]^{-1}[D^{k}{}^{\prime}\mathcal{M}_{\widehat{I}^{k^{c}}}Y^{k}/n_{k}]$ so that

First, note that by Condition ASTESS we have

Second, by the split sample construction, we have that $\widehat{I}^{k^{c}}$ is independent from $\zeta^{k}$, and by assumption of the model $m^{k}$ is also independent of $\zeta^{k}$. Thus by Chebyshev inequality

where the last relation follows by ASTESS.

Third, using similar independence arguments, by Chebyshev and Condition ASTESS, conclude

Fourth, using that $\widehat{s}^{k^{c}}\lesssim_{P}s$ by ASTESS so that $\phi^{-1}_{\min}(\widehat{s}^{k^{c}})\lesssim_{P}1$ by condition SE, we have that

Step 3.(Behavior of $ii_{k}$.) Since $ii_{k}=(m^{k}+V^{k})^{\prime}\mathcal{M}_{\widehat{I}^{k^{c}}}(m^{k}+V^{k})/n_{k}$, decompose

Then $|ii_{k,a}|=o_{P}(1)$ by Condition ASTESS, $|ii_{k,b}|=o_{P}(1)$ by reasoning similar to deriving the bound for $|i_{k,b}|$, and $|ii_{k,c}|=o_{P}(1)$ by reasoning similar to deriving the bound for $|i_{k,d}|$.

By condition ASTESS $\|\mathcal{M}_{\widehat{I}^{k^{c}}}g^{k}\|=o_{P}(n^{1/4})$ and by condition SM(ii) we have $\|\mathcal{P}_{\widehat{I}^{k^{c}}}D^{k}/\sqrt{n_{k}}\|\leqslant\|D^{k}/\sqrt{n_{k}}\|\lesssim_{P}1$, and by Step 1 we have $|\check{\alpha}_{k}-\alpha_{0}|\lesssim_{P}n^{-1/2}$. Moreover,

We have $\sqrt{\phi_{\max,k}(\widehat{s}^{k^{c}})}/\phi_{\min,k}(\widehat{s}^{k^{c}})\lesssim_{P}1$ by condition SE, and $\|X^{k}[I^{k^{c}}]^{\prime}\zeta^{k}/\sqrt{n_{k}}\|\lesssim_{P}\sqrt{\widehat{s}^{k^{c}}}$ by condition SM(ii), the independence between the selected components $\widehat{I}^{k^{c}}$ and $\zeta^{k}$ since they are based on different subsamples, and applying Chebyshev inequality.

Similarly, we have $\|m^{k}-X^{k}\widehat{\beta}_{k}\|/\sqrt{n^{k}}\lesssim_{P}o(n^{-1/4})+\sqrt{\widehat{s}^{k^{c}}/n_{k}}$.

Step 5.(Variance Estimation.) Since $\widehat{s}^{k}\lesssim_{P}s=o(n)$, $(n_{k}-\widehat{s}^{k}-1)/n_{k}=o_{P}(1)$, so we can use $n$ as the denominator. Recall the definitions $\widehat{\zeta}_{i}^{o}=y_{i}-d_{i}\check{\alpha}_{k}-x_{i}^{\prime}\check{\beta}_{k}$, $\widehat{v}_{i}=d_{i}-x_{i}^{\prime}\widehat{\beta}_{k}$ and $\widehat{\zeta}_{i}=\widehat{\zeta}_{i}^{o}1\{|\widehat{\zeta}_{i}^{o}|\vee|\widehat{v}_{i}|\leqslant H_{k}\}$ if $i$ belongs to subsample $k$ where $H_{k}=C\sqrt{n/[(\widehat{s}^{k^{c}}\vee n^{1/2})\log n]}$. For notational convenience let $A_{i}=\{|\widehat{\zeta}_{i}^{o}|\vee|\widehat{v}_{i}|\leqslant H_{k}\}$. Since $q>4$, $\widehat{s}^{k^{c}}\lesssim_{P}s$, and $n^{2/q}s\log(n\vee p)=o(n)$, we have $n^{1/q}=o_{P}(H_{k})$. Hence consider

Finally, since $\max_{i\leqslant n}\|1\{A_{i}\}(\widehat{v}_{i},\widehat{\zeta}_{i},\zeta_{i},v_{i})^{\prime}\|_{\infty}^{2}\lesssim_{P}(H_{k}^{2}\vee n^{2/q})\lesssim_{P}H_{k}^{2}$, we have

Step 6.(Controlling large terms) By definition of the event $A_{i}$ we have

Additional Simulation Results

In this section, we present additional simulation results. All of the simulation results are based on the structural model

where $p=\dim(x_{i})=200$, the covariates $x\sim N(0,\Sigma)$ with $\Sigma_{kj}=(0.5)^{|j-k|}$, $\alpha_{0}=.5$, and the sample size $n$ is set to $100$. In each design, we generate

with E[$\zeta_{i}v_{i}]=0$. Inference results for all designs are based on conventional t-tests with standard errors calculated using the heteroscedasticity consistent jackknife variance estimator discussed in ?. We set $\lambda$ according to the algorithm outlined in Appendix A with $1-\gamma=.95$. We draw new $x$’s, $\zeta$’s and $v$’s at every replication and draw new $\beta_{0}$’s and $\beta_{1}$’s at every replication in the random coefficient designs.

In the first thirteen designs, $\beta_{1}=\beta_{0}$. We set the constants $c_{y}$ and $c_{d}$ to generate desired population values for the reduced form $R^{2}$’s, i.e. the $R^{2}$’s for equations (5.41) and (5.42). Let $R^{2}_{y}$ be the desired $R^{2}$ for the regression of $y$ on $x$ and $R^{2}_{d}$ be the desired $R^{2}$ from the regression of $d$ on $x$. For each equation, we choose $c_{y}$ and $c_{d}$ to generate $R^{2}=0,.2,.4,.6,$ and $.8$. In the heteroscedastic and binary designs discussed below, we choose $c_{y}$ and $c_{d}$ based on $R^{2}$ as if (5.41) held with $d_{i}=d_{i}^{*}$ and $v_{i}$ and $\zeta_{i}$ were homoscedastic with variance equal to the average variance and label the results by $R^{2}$ as in the other cases. In the homoscedastic cases, we set $\sigma_{y}=\sigma_{d}=1$; and in the heteroscedastic cases, the average of $\sigma_{d}(x_{i})$ and the average of $\sigma_{y}(d_{i},x_{i})$ are both one. We set

Design 1. $d_{i}=d_{i}^{*}$, $\beta_{0}=(1,1/2,1/3,1/4,1/5,0,0,0,0,0,1,1/2,1/3,1/4,1/5,0,...,0)^{\prime}$, $\sigma_{y}=\sigma_{d}=1$.

Design 2. $d_{i}=d_{i}^{*}$, $\beta_{0}=(1,1/4,1/9,1/16,1/25,0,0,0,0,0,1,1/4,1/9,1/16,1/25,0,...,0)^{\prime}$, $\sigma_{y}=\sigma_{d}=1$.

Design 22. $d_{i}=d_{i}^{*}$, $\beta_{0,j}=(1/j)^{2}$, $\sigma_{y}=\sigma_{d}=1$.

Design 3. $d_{i}=d_{i}^{*}$, $\beta_{0}=(1,1/2,1/3,1/4,1/5,0,0,0,0,0,1,1/2,1/3,1/4,1/5,0,...,0)^{\prime}$, $\sigma_{d}=\sqrt{\frac{(1+x_{i}^{\prime}\beta_{0})^{2}}{\frac{1}{n}\sum_{i=1}^{n}(1+x_{i}^{\prime}\beta_{0})^{2}}}$, $\sigma_{y}=\sqrt{\frac{(1+\alpha_{0}d_{i}+x_{i}^{\prime}\beta_{0})^{2}}{\frac{1}{n}\sum_{i=1}^{n}(1+\alpha_{0}d_{i}+x_{i}^{\prime}\beta_{0})^{2}}}$.

Design 4. $d_{i}=d_{i}^{*}$, $\beta_{0}=(1,1/4,1/9,1/16,1/25,0,0,0,0,0,1,1/4,1/9,1/16,1/25,0,...,0)^{\prime}$, $\sigma_{d}=\sqrt{\frac{(1+x_{i}^{\prime}\beta_{0})^{2}}{\frac{1}{n}\sum_{i=1}^{n}(1+x_{i}^{\prime}\beta_{0})^{2}}}$, $\sigma_{y}=\sqrt{\frac{(1+\alpha_{0}d_{i}+x_{i}^{\prime}\beta_{0})^{2}}{\frac{1}{n}\sum_{i=1}^{n}(1+\alpha_{0}d_{i}+x_{i}^{\prime}\beta_{0})^{2}}}$.

Design 44. $d_{i}=d_{i}^{*}$, $\beta_{0,j}=(1/j)^{2}$, $\sigma_{d}=\sqrt{\frac{(1+x_{i}^{\prime}\beta_{0})^{2}}{\frac{1}{n}\sum_{i=1}^{n}(1+x_{i}^{\prime}\beta_{0})^{2}}}$, $\sigma_{y}=\sqrt{\frac{(1+\alpha_{0}d_{i}+x_{i}^{\prime}\beta_{0})^{2}}{\frac{1}{n}\sum_{i=1}^{n}(1+\alpha_{0}d_{i}+x_{i}^{\prime}\beta_{0})^{2}}}$.

Design 5. $d_{i}=\mathbf{1}\{d_{i}^{*}>0\}$, $\beta_{0}=(1,1/2,1/3,1/4,1/5,0,0,0,0,0,1,1/2,1/3,1/4,1/5,0,...,0)^{\prime}$, $\sigma_{y}=\sigma_{d}=1$.

Design 6. $d_{i}=d_{i}^{*}$, $\beta_{0,j}\sim N(0,1)$, $\sigma_{y}=\sigma_{d}=1$.

Design 7. $d_{i}=d_{i}^{*}$, $\widetilde{\beta}_{0}=(1,1/2,1/3,1/4,1/5,0,0,0,0,0,1,1/2,1/3,1/4,1/5,0,...,0)^{\prime}$, $\beta_{0,j}\sim N(0,\widetilde{\beta}_{0,j}^{2})$, $\sigma_{y}=\sigma_{d}=1$.

Design 72. $d_{i}=d_{i}^{*}$, $\widetilde{\beta}_{0}=(1,1/4,1/9,1/16,1/25,0,0,0,0,0,1,1/4,1/9,1/16,1/25,0,...,0)^{\prime}$, $\beta_{0,j}\sim N(0,\widetilde{\beta}_{0,j}^{2})$, $\sigma_{y}=\sigma_{d}=1$.

Design 722. $d_{i}=d_{i}^{*}$, $\widetilde{\beta}_{0,j}=(1/j)^{2}$, $\beta_{0,j}\sim N(0,\widetilde{\beta}_{0,j}^{2})$, $\sigma_{y}=\sigma_{d}=1$.

Design 8. $d_{i}=d_{i}^{*}$, $\widetilde{\beta}_{0,j}=u_{j}z_{1,j}+(1-u_{j})z_{2,j}$, $u_{j}\sim\textnormal{Bernoulli}(.05)$, $z_{1,j}\sim N(0,25)$, $z_{2,j}\sim N(0,.0025)$, $\sigma_{y}=\sigma_{d}=1$

Design 1001. $d_{i}=d_{i}^{*}$, $\beta_{0,j}=\mathbf{1}\{j\in\{2,4,6,...,38,40\}\}$, $\sigma_{y}=\sigma_{d}=1$.

In the last thirteen designs, we set the constants $c_{y}$ and $c_{d}$ according to

for $R^{2}_{d}=0,.2,.4,.6,$ and $.8$ and $R^{2}_{y}=0,.2,.4,.6,$ and $.8$.

Design 1a. $d_{i}=d_{i}^{*}$, $\beta_{0}=(1,1/2,1/3,1/4,1/5,0,0,0,0,0,1,1/2,1/3,1/4,1/5,0,...,0)^{\prime}$, $\beta_{1}=(1,1/2,1/3,1/4,1/5,1/6,1/7,1/8,1/9,1/10,0,...,0)^{\prime}$, $\sigma_{y}=\sigma_{d}=1$.

Design 2a. $d_{i}=d_{i}^{*}$, $\beta_{0}=(1,1/4,1/9,1/16,1/25,0,0,0,0,0,1,1/4,1/9,1/16,1/25,0,...,0)^{\prime}$, $\beta_{1}=(1,1/4,1/9,1/16,1/25,1/36,1/49,1/64,1/81,1/100,0,...,0)^{\prime}$, $\sigma_{y}=\sigma_{d}=1$.

Design 22a. $d_{i}=d_{i}^{*}$, $\beta_{0,j}=(1/j)^{2}$, $\beta_{1,j}=(1/j)^{2}$, $\sigma_{y}=\sigma_{d}=1$.

Design 3a. $d_{i}=d_{i}^{*}$, $\beta_{0}=(1,1/2,1/3,1/4,1/5,0,0,0,0,0,1,1/2,1/3,1/4,1/5,0,...,0)^{\prime}$, $\beta_{1}=(1,1/2,1/3,1/4,1/5,1/6,1/7,1/8,1/9,1/10,0,...,0)^{\prime}$, $\sigma_{d}=\sqrt{\frac{(1+x_{i}^{\prime}\beta_{1})^{2}}{\frac{1}{n}\sum_{i=1}^{n}(1+x_{i}^{\prime}\beta_{1})^{2}}}$, $\sigma_{y}=\sqrt{\frac{(1+\alpha_{0}d_{i}+x_{i}^{\prime}\beta_{0})^{2}}{\frac{1}{n}\sum_{i=1}^{n}(1+\alpha_{0}d_{i}+x_{i}^{\prime}\beta_{0})^{2}}}$.

Design 4a. $d_{i}=d_{i}^{*}$, $\beta_{0}=(1,1/4,1/9,1/16,1/25,0,0,0,0,0,1,1/4,1/9,1/16,1/25,0,...,0)^{\prime}$, $\beta_{1}=(1,1/4,1/9,1/16,1/25,1/36,1/49,1/64,1/81,1/100,0,...,0)^{\prime}$, $\sigma_{d}=\sqrt{\frac{(1+x_{i}^{\prime}\beta_{1})^{2}}{\frac{1}{n}\sum_{i=1}^{n}(1+x_{i}^{\prime}\beta_{1})^{2}}}$, $\sigma_{y}=\sqrt{\frac{(1+\alpha_{0}d_{i}+x_{i}^{\prime}\beta_{0})^{2}}{\frac{1}{n}\sum_{i=1}^{n}(1+\alpha_{0}d_{i}+x_{i}^{\prime}\beta_{0})^{2}}}$.

Design 44a. $d_{i}=d_{i}^{*}$, $\beta_{0,j}=(1/j)^{2}$, $\beta_{1,j}=(1/j)^{2}$, $\sigma_{d}=\sqrt{\frac{(1+x_{i}^{\prime}\beta_{1})^{2}}{\frac{1}{n}\sum_{i=1}^{n}(1+x_{i}^{\prime}\beta_{1})^{2}}}$, $\sigma_{y}=\sqrt{\frac{(1+\alpha_{0}d_{i}+x_{i}^{\prime}\beta_{0})^{2}}{\frac{1}{n}\sum_{i=1}^{n}(1+\alpha_{0}d_{i}+x_{i}^{\prime}\beta_{0})^{2}}}$.

Design 5a. $d_{i}=\mathbf{1}\{d_{i}^{*}>0\}$, $\beta_{0}=(1,1/2,1/3,1/4,1/5,0,0,0,0,0,1,1/2,1/3,1/4,1/5,0,...,0)^{\prime}$, $\beta_{1}=(1,1/2,1/3,1/4,1/5,1/6,1/7,1/8,1/9,1/10,0,...,0)^{\prime}$, $\sigma_{y}=\sigma_{d}=1$.

Design 6a. $d_{i}=d_{i}^{*}$, $\beta_{0,j}\sim N(0,1)$, $\beta_{1,j}\sim N(0,1)$, E$[\beta_{0,j}\beta_{1,j}]=.8$, $\sigma_{y}=\sigma_{d}=1$.

Design 7a. $d_{i}=d_{i}^{*}$, $\widetilde{\beta}_{0}=(1,1/2,1/3,1/4,1/5,0,0,0,0,0,1,1/2,1/3,1/4,1/5,0,...,0)^{\prime}$, $\widetilde{\beta}_{1}=(1,1/2,1/3,1/4,1/5,1/6,1/7,1/8,1/9,1/10,0,...,0)^{\prime}$, $\beta_{0,j}=\widetilde{\beta}_{0,j}z_{0,j}$, $\beta_{1,j}=\widetilde{\beta}_{1,j}z_{1,j}$, $z_{0,j}\sim N(0,1)$, $z_{1,j}\sim N(0,1)$, E$[z_{0,j}z_{1,j}]=.8$, $\sigma_{y}=\sigma_{d}=1$.

Design 72a. $d_{i}=d_{i}^{*}$, $\widetilde{\beta}_{0}=(1,1/4,1/9,1/16,1/25,0,0,0,0,0,1,1/4,1/9,1/16,1/25,0,...,0)^{\prime}$, $\widetilde{\beta}_{1}=(1,1/4,1/9,1/16,1/25,1/36,1/49,1/64,1/81,1/100,0,...,0)^{\prime}$, $\beta_{0,j}=\widetilde{\beta}_{0,j}z_{0,j}$, $\beta_{1,j}=\widetilde{\beta}_{1,j}z_{1,j}$, $z_{0,j}\sim N(0,1)$, $z_{1,j}\sim N(0,1)$, E$[z_{0,j}z_{1,j}]=.8$, $\sigma_{y}=\sigma_{d}=1$.

Design 722a. $d_{i}=d_{i}^{*}$, $\widetilde{\beta}_{0,j}=(1/j)^{2}$, $\widetilde{\beta}_{1,j}=(1/j)^{2}$, $\beta_{0,j}=\widetilde{\beta}_{0,j}z_{0,j}$, $\beta_{1,j}=\widetilde{\beta}_{1,j}z_{1,j}$, $z_{0,j}\sim N(0,1)$, $z_{1,j}\sim N(0,1)$, E$[z_{0,j}z_{1,j}]=.8$, $\sigma_{y}=\sigma_{d}=1$.

Design 8a. $d_{i}=d_{i}^{*}$, $\widetilde{\beta}_{0,j}=5u_{j}z_{11,j}+.05(1-u_{j})z_{12,j}$, $\widetilde{\beta}_{1,j}=5u_{j}z_{21,j}+.05(1-u_{j})z_{22,j}$, $u_{j}\sim\textnormal{Bernoulli}(.05)$, $z_{11,j}\sim N(0,1)$, $z_{12,j}\sim N(0,1)$, $z_{21,j}\sim N(0,1)$, $z_{22,j}\sim N(0,1)$, $\sigma_{y}=\sigma_{d}=1$

Design 1001a. $d_{i}=d_{i}^{*}$, $\beta_{0,j}=\mathbf{1}\{j\in\{2,4,6,...,38,40\}\}$, $\beta_{1,j}=\mathbf{1}\{j\in\{1,3,5,...,37,39\}\}$, $\sigma_{y}=\sigma_{d}=1$.

Results are summarized in figures and tables below. In the tables, we report results for the four estimators considered in the main text (Oracle, Double-Selection Oracle, Post-Lasso, and Double-Selection). We also report results for regular Lasso (Lasso), the union of the Double-Selection interval with the Post-Lasso interval (Double-Selection Union ADS), using the union of the set of variables selected by Double-Selection and the set of variables selected by running Lasso of $y$ on $d$ and $x$ without penalizing $d$ (Double-Selection + I3), and the split-sample procedure discussed in the text (Split-Sample). For Double-Selection Union ADS, the point estimate is taken as the midpoint of the union of the intervals.

References