3.1 What we know and what we want to know

While we are interested in the average treatment effects of always-takers and never-takers, these quantities are not directly observed and require extrapolation. In order to identify these ATEs, we need to know the following four quantities in Table 1:

Table 1. Quantities in the ATEs of always-takers and Never-takers

Under the assumptions described in the previous section, two of these four quantities are identifiable from the data. That is, the potential outcome of Y(1) is point identified for always-takers and the potential outcome of Y(0) is point identified for never-takers. We state the result in lemma 1.

Given these results, we can show that the ATEs of always-takers and never-takers are weighted averages of the MTE functions and that the weights are identifiable from the data. This result is a special case of the more general results developed in Heckman and Vytlacil (Reference Heckman and Vytlacil2005) which asserted but did not formally demonstrate the following theorem.

Theorem 1: The ATE of always-takers is:

$${\mathbb E}[ Y( 1) - Y( 0) \vert D( 1) = D( 0) = 1] = \displaystyle{\int}_{0}^{1} {\mathbb E}[ Y( 1) - Y( 0) \vert U = u] \displaystyle{\mathbb{1}\{ u \in [ 0,\; p( 0) ] \} \over p( 0) } \, {\rm d}u.$$

The ATE of never-takers is:

$${\mathbb E}[ Y( 1) - Y( 0) \vert D( 1) = D( 0) = 0] = \displaystyle{\int_{0}^{1}} {\mathbb E}[ Y( 1) - Y( 0) \vert U = u] \displaystyle{\mathbb{1}\{ u \in [ p( 1) ,\; 1] \} \over 1 - p( 1) } \, {\rm d}u.$$

Proof. We first prove the case for always-takers:

$$\eqalign{&{\mathbb E}[ Y( 1) - Y( 0) \vert D( 1) = D( 0) = 1] \cr &\quad = {\mathbb E} [ Y( 1) - Y( 0) \vert D( 0) = 1] \cr &\quad = {\mathbb E} [ Y( 1) - Y( 0) \vert U \leq p( 0) ] \cr &\quad = {\int}_{0}^{\,p( 0) } {\mathbb E} [ Y( 1) - Y( 0) \vert U = u] {1\over p( 0) } \, {\rm d}u \cr &\quad = { \int}_{0}^{\,1} {\mathbb E} [ Y( 1) - Y( 0) \vert U = u] {\mathbb{1}\{ u \in [ 0,\; p( 0) ] \} \over p( 0) } \, {\rm d}u \cr &\quad = { \int}_{0}^{\,1}m_{1}( u) {\mathbb{1}\{ u \in [ 0,\; p( 0) ] \} \over p( 0) } \, {\rm d}u + {\int}_{0}^{\,1}m_{0}( u) {- \mathbb{1}\{ u \in [ 0,\; p( 0) ] \} \over p( 0) } \, {\rm d}u}$$

where the first equality uses the IV monotonicity assumption; the second equality uses the selection equation; the third equality uses the fact that U|U ≤ p(0) ~ unif[0,  p(0)]. The case for never-takers can be proved analogously.  □

Corollary 1: From the derivation in theorem 1, for always-takers, we have:

$${\mathbb E}[ Y( 0) \vert D( 1) = D( 0) = 1] = \displaystyle{ \int}_{0}^{\,1} {\mathbb E}[ Y( 0) \vert U = u] \displaystyle{\mathbb{1}\{ u \in [ 0,\; p( 0) ] \} \over p( 0) } \, {\rm d}u.$$

Similarly, for never-takers, we have:

$${\mathbb E}[ Y( 1) \vert D( 1) = D( 0) = 0] = \displaystyle{\int}_{0}^{\,1} {\mathbb E}[ Y( 1) \vert U = u] \displaystyle{\mathbb{1}\{ u \in [ p( 1) ,\; 1] \} \over 1 - p( 1) } \, {\rm d}u.$$

The following two sections develop strategies for extrapolation of ATEs of always-takers and never-takers. The first, is a point identification result, but requires strong behavioral assumptions. The second is a partial identification result, which allows for more flexibility in modeling strategies. Our empirical analysis demonstrates each method.

3.2 Point identification of ATEs of always-takers and never-takers

Under strong assumptions, the MTE framework developed by Brinch et al. (Reference Brinch, Mogstad and Wiswall2017) can guarantee point identification of ATEs of always-takers and never-takers. Recall Remark 4, the task of point identification requires calculating two marginal treatment response pairs, i.e., ${\mathbb E}[ Y( 1) \vert U = u]$ and ${\mathbb E}[ Y( 0) \vert U = u]$. Further assumptions are unnecessary if P(Z) has continuous support from zero to one (Heckman and Vytlacil, Reference Heckman and Vytlacil1999, Reference Heckman and Vytlacil2007). However, in practice, instruments are often discrete and many are binary, violating this condition. In such conditions, Brinch et al. (Reference Brinch, Mogstad and Wiswall2017) invoke parametric assumption on the MTE and MTR functions, showing that at most N parameters can when p(Z) takes N different values. An implication of the identification result is that a linear MTE model can be identified with a single binary instrument. We formally state the identification result based on invoking parametric assumption of MTE and MTR functions in proposition 1.

Parametric assumptions on the MTE functions can be informed by prior research on political behavior and the institutional context. Consider the empirical study of the persuasion effect of West German media on popular support of communism among East Germans. We might have a theory which indicates behavior is well approximated by a bivariate normal distribution, which, in turn, would imply a linear MTE function. We might further assume that the MTE function is downward sloping if we expect that people with a stronger effect are more likely to consume foreign media. This sort of “selection on the gains” would only be plausible if people can anticipate how media consumption affects themselves. This self-selection behavior has been found in the literature of political economics of media (Gentzkow, Reference Gentzkow2007; Martin and Yurukoglu, Reference Martin and Yurukoglu2017).

The primary limitation of this approach is the need for such strong parametric assumptions. Namely, while proposition 1 can allow treatment effects to be heterogeneous, they must be linear with respect to any unobservables. This strong assumption is particularly unfortunate in this context as the goal of the MTE literature is to account for heterogeneity on the treatment effect with respect to unobservables across individuals.

3.3 Partial identification of ATEs of always-takers and never-takers

In this subsection, we provide partial identification results based on Mogstad et al. (Reference Mogstad, Santos and Torgovitsky2018). This approach formulates the extrapolation problem as a linear optimization problem. We then briefly compare the performance of the linear programing approach with existing partial identification strategies. Our simulations find that the linear programing approach performs better than the competing partial identification approaches.

Corollary 1 shows that the ATEs for always-takers and never-takers are weighted averages of MTR pairs.Footnote ⁷ However, we generally do not know the functional form of these MTR pairs. To bound the possible parameter space of MTR pairs, we draw on information regarding a subset of weighted averages of MTR pairs that are identifiable from the data. As shown in Heckman and Vytlacil (Reference Heckman and Vytlacil2005), many identifiable estimands, for example, OLS estimand and LATE estimand, are themselves weighted average of MTR pairs, where the weights are identifiable. Hence, the parameter space of the MTR pairs are constrained by the known values of the OLS and LATE estimands (Mogstad et al., Reference Mogstad, Santos and Torgovitsky2018; Mogstad and Torgovitsky, Reference Mogstad and Torgovitsky2018). Given these constraints, the assumption that latent utility is continuous, exogeneity of the instrument and that selection into treatment is described by Equation (4), only a subset of values of our target parameters are consistent with the limited MTR pairs’ parameter space, that is, we can partially identify the target parameter.

It turns out that this intuition applies to any target parameter that is itself a weighted average of MTR pairs. In general, we call estimands that consist of these weighted averages IV-like estimands. Any identifiable IV-like estimands can provide information about the possible parameter space of MTR pairs. The formal definition of IV-like estimands in Mogstad et al. (Reference Mogstad, Santos and Torgovitsky2018) follows.

In addition to the IV-like estimands presented in Mogstad et al. (Reference Mogstad, Santos and Torgovitsky2018), the binary instrument and binary treatment case produces cross moments between Y and 1{D = d,  Z = z} (i.e., ${\mathbb E}[ 1\{ D = d,\; \, Z = z\} Y]$, with d,  z ∈ {0,  1}) are IV-like estimands where the weights are: $s( d,\; \, z) = {\mathbb P}[ Z = z]$ and s(1 − d,  z) = 0.Footnote ⁸ As an illustration, we plot weights using the data from Kern and Hainmueller (Reference Kern and Hainmueller2009) in Figure 1, where the x-axis displays the latent utility, and the y-axis indicates the amount of weight allocated to each cross-moment.

Figure 1. Weights for Cross Moments. (a) Weights for D = 0 and (b) Weights for D = 1. Note: Sample size =3023. This figure presents weights associated with the LATE and cross moments in Kern and Hainmueller (Reference Kern and Hainmueller2009). The horizontal axis is the latent heterogeneity U in the selection equation. The vertical axis is the weights of the IV estimands in regions where they are nonzero. Figure 1(a) presents the weights for ${\mathbb E}[ Y( 0) \vert U = u]$. Figure 1(b) presents the weights for ${\mathbb E}[ Y( 1) \vert U = u]$.

In addition to using IV-like estimands to restrict the behavior of MTR pairs, researchers may additionally incorporate substantive assumptions by adding parametric or shape restrictions. Let S denote the set of IV-like specifications chosen by the researcher, and define a linear map, $\Gamma _{s}( m) \, \colon \, M \rightarrow {\mathbb R}$ for any IV-like specification s ∈ S as: $\Gamma _{s}( m) = {\mathbb E}[ {\int} _{0}^{1} m_{0}( u,\; \, X) \omega _{0s}( u,\; \, Z) \, {\rm d}u] + {\mathbb E}[ {\int} _{0}^{1} m_{1}( u,\; \, X) \omega _{1s}( u,\; \, Z) \, {\rm d}u] $. Finally, recall that the IV-like estimand has the form $\beta _{s} \equiv {\mathbb E}[ s( D,\; \, Z) Y]$. Our constrains require MTR pairs to satisfy Γ_s(m) = β _s for every s ∈ S, given the substantive assumptions.

Given these constraints, our goal is to characterize bounds on the values of our target parameter, ATEs of always-takers and never-takers, that could have been generated by MTR functions consistent with observed IV-like estimands. Note that we can define a similar linear map for the target parameter β* as $\Gamma ^{\ast }\, \colon \, M \rightarrow {\mathbb R}$, with: $\Gamma ^{\ast }( m) \equiv {\mathbb E}[ {\int} _{0}^{1} m_{0}( u) \omega _{0}^{\ast }( u,\; \, Z) \, {\rm d}\mu ^{\ast }( u) ] + {\mathbb E}[ {\int} _{0}^{1} m_{1}( u) \omega _{1}^{\ast }( u,\; \, Z) \, {\rm d}\mu ^{\ast }( u) ] $. We also note that by corollary 1, the weights in Γ*(m) for always-takers and never-takers are identifiable.

Therefore, it follows that if (Y,  D,  Z) is generated according to Equations (1), (2), assumptions 3–5, the our target parameters must belong to the identified set: $B^{\ast }_{s} \equiv \{ b \in {\mathbb R}\, \colon \, b = \Gamma ^{\ast }( m) \;\rm { for\, some } \;{\it m} \in {\it M}_{s} \}$.

To repeat the intuition mentioned at the beginning at this subsection: B*_s is the set of values for the target parameter that could have been generated by MTR pairs that satisfy the research's assumptions and are also consistent with the IV-like estimands. The next proposition formally states the identification result.

Proposition 2: Suppose that M is convex. Then either M _S is empty, hence the bounds for ATEs of always-takers and never-takers are empty. Or, the bounds for the ATEs of always-takers and never-takers can be constructed by solving following two optimization problems:

$$\underline{\beta}^{\ast } \equiv \inf_{m \in M} \Gamma^{\ast }( m)$$

subject to Γ_s(m) = β _s for all s ∈ S, and:

$$\bar{\beta}^{\ast } \equiv \sup_{m \in M} \Gamma^{\ast }( m)$$

subject to Γ_s(m) = β _s for all s ∈ S. And $\beta ^{\ast } \in [ \underline {\beta }^{\ast },\; \, \bar {\beta }^{\ast }]$.

In Appendix B, we provide alternative partial identification strategies for extrapolating LATE to ATEs of always-takers and never-takers. The competing partial identification approaches extrapolate by imposing assumptions about the direction and extent of causal effects, namely, responses have bounded support (Manski, Reference Manski1990), are monotonic (Manski, Reference Manski1997), or are ‘smooth’ (Kim et al., Reference Kim, Kwon, Kwon and Lee2018). We provide bounds of our target parameters based on those different substantive assumptions and prove their sharpness. We find that we can compute the bounds of target parameters explicitly with each of these competing partial identification strategies. Thus, we explicitly evaluate the trade-off between assumptions and identification power for those alternative strategies.

In Appendix D, we also conduct a simulation study to demonstrate different identification strategies presented in this section and Appendix B. We first demonstrate the point identification approach based on imposing parametric assumption on MTR pairs. Second, we highlight potential misspecification bias from imposing the wrong parametric assumptions on MTR pairs. Overall, our simulation study demonstrates the superior performance of linear programing approach over other competing partial identification approaches.

4.1 Empirical setting

Prior to the fall of the Berlin wall, many residents of East Germany had some access to media from the West. Opponents of Soviet rule in Europe organized propaganda, concerts, and media campaigns oriented toward German Democratic Republic (GDR) citizens. Kern and Hainmueller (Reference Kern and Hainmueller2009) addresses the question of how Western media shaped people's views of communism and contributed to the democratization of East Germany. To probe the effects of these persuasion campaigns, Kern and Hainmueller (Reference Kern and Hainmueller2009) examines surveys of support for the East German regime in the year preceding reunification. This survey solicited individual viewership of West German TV, political attitudes towards the East German political regime, and residence information between November 1988 and February 1989. In their analysis, they coded a binary variable D equal to 1 for respondents who had watched West German TV, 0 otherwise. In terms of dependent variables, they focus on the following three questions:

To what extend do you agree with the following statements probing support for the East German communist regime:

• • “I am convinced of the Leninist/Marxist worldview.”

• • “I feel closely attached to East Germany.”

• • “In East Germany, political power is exercised in ways consistent with my views.”

Each respondent was offered a choice of one of four responses, fully disagree, largely disagree, largely agree, and fully agree. East German respondents generally voiced support for each of these three statements, but between 30 percent of respondents voiced at least disagreement with at least one of these questions.

For each of these questions, Kern and Hainmueller (Reference Kern and Hainmueller2009) seek to identify the causal effect of watching West German television. A key threat to the identification of the causal effect is the endogeneity of the viewership. If, for instance, respondents with low support for Communist regime are more likely to watch West German TV, they would find a spurious association. Addressing this self-selection is key to obtaining credible inference.

To address the endogeneity problem (Kern and Hainmueller, Reference Kern and Hainmueller2009) uses an instrumental variable approach. Specifically, whether or not respondents live in the Dresden district. Dresden district was the most eastern district of the GDR, bordering Poland and Czechoslovakia. They code the binary instrument Z as 0 for respondents living in the Dresden district and 1 otherwise.

There are three main justifications for the use of this instrument. The first pertains to the assumption of exogeneity. Residents of Dresden had difficult receiving TV or radio signals from West Germany for topological reasons.Footnote ⁹ Spatial sorting was limited by restrictions on movement as well as the ill-functioning labor market in East Germany. The second and third pertain to the exclusion restriction. There are no significant differences between Dresden districts and other regions on observable characteristics and there is no significant difference in political attitudes before West German television became available between Dresden districts and other districts. Finally, we offer further evidence of the validity of the identification assumptions using a methodology developed in Mourifié and Wan (Reference Mourifié and Wan2017). This test builds on the insight that the IV assumptions imply a set of moment inequalities. While not a direct test of the validity of our extrapolation, our approach depends on the validity of the LATE assumptions. The results presented in Appendix G fail to reject the IV validity assumptions.

4.2 What we know from data

The LATE of exposure to Western media on the three measures of support for the Communist regime are presented in Table F.1 in Appendix F. Overall, the point estimates are similar to the original results with small discrepancies due to different samples. The results show that the there is a positive effect of West German TV on East Germans’ support of communism among compliers. For the remainder of the analysis, we focus on support for Leninist/Marxist ideology.

Besides LATE, there are other four types of estimands that are identifiable from the data.Footnote ¹⁰ First, under the monotonicity assumption, we can identify the proportion of compliers, always-takers, and never-takers. These proportions are the weights that we will use when we conduct the linear programing procedure. Second, we can also estimate the unconditional propensity scores by their sample analogs. Third, as shown in lemma 1, the empirical analogs of the expectation of Y(1) for always-takers and the expectation of Y(0) for never-takers are available from the data. Finally, as shown in Abadie (Reference Abadie2002), the expectation of potential outcomes for compliers are identifiable. These identifiable quantities are listed in Table 2.

Table 2. Identifiable Quantities in Kern and Hainmueller (Reference Kern and Hainmueller2009)

Note: Sample size =3023. This table presents other identifiable quantities in Kern and Hainmueller (Reference Kern and Hainmueller2009), namely, proportion of different compliance types, propensity scores, expectation of potential outcomes among compliers, and identifiable parts of target parameters. ${\mathbb P}[ NT]$, ${\mathbb P}[ AT]$, and ${\mathbb P}[ C]$ refers to the proportion of never-takers, always-takers, and compliers, respectively.

There are two interesting findings from Table 2. First, only $1.7\%$ of East Germans would not have watched Western media even if they had improved access to television signals. Second, the never-takers’ support rate for communism is high, $75\%$ support communism. By comparison, only $47.3\%$ of always-takers support communism.

4.3 Extrapolation 1: point estimates by linearizing MTR pairs

Recall proposition 1, if we impose linearity assumption on the MTR pairs, then, the ATEs of always-takers and never-takers are point identifiable. Under the linearity assumptions, it can be shown that for ${\mathbb E}_{L}[ Y \vert P( Z) = p,\; \, D = j]$, where j ∈ {0,  1}, we have:

$$\eqalign{{\mathbb E}_{L}[ Y \vert P( Z) = p,\; D = 0] & = \mu_{0} + {1\over 2} \alpha_{0} p,\; \cr {\mathbb E}_{L}[ Y \vert P( Z) = p,\; D = 1] & = \mu_{1} + {1\over 2} \alpha_{1} ( p - 1) .}$$

The quantities μ ₀, α ₀, μ ₁, and α ₁ are unknown. We can use sample analog of ${\mathbb E}[ Y \vert P( Z) = p( z) ,\; \, D = d] $, with d ∈ {0,  1}, to solve the two linear equations below to compute $\hat {\mu }_{0}$, $\hat {\alpha }_{0}$, $\hat {\mu }_{1}$, and $\hat {\alpha }_{1}$:

After solving the two equations, we have $\hat {\mu _{0}} = 0.648$, $\hat {\alpha _{0}} = 0.208$, $\hat {\mu _{1}} = 0.663$, and $\hat {\alpha _{1}} = 0.561$. Then, the approximated MTR pairs based on linearity assumptions are:

$$\left\{\matrix{ \hat{{\mathbb E}}_{L}[ Y( 0) \vert U = u] = \hat{\mu_{0}} + \hat{\alpha_{0}} u - {1\over 2} \hat{\alpha_{0}} = 0.208 \times u + 0.544 \hfill \cr \hat{{\mathbb E}}_{L}[ Y( 1) \vert U = u] = \hat{\mu_{1}} + \hat{\alpha_{1}} u - {1\over 2} \hat{\alpha_{1}} = 0.561 \times u + 0.3825. \hfill }\right.$$

Therefore, based on linearity assumptions, the extrapolated ATE of always-takers is:

$$\eqalign{\hat{{\mathbb E}}[ Y( 1) - Y( 0) \vert D( 1) = D( 0) = 1] & = \hat{{\mathbb E}}[ Y \vert D = 1,\; Z = 0] - {1\over \hat{\,p}( 0) } {\int_{0}^{\hat{\,p}( 0) }} \hat{{\mathbb E}}_{L}[ Y( 0) \vert U = u] \, {\rm d}u \cr & = -0.104.}$$

The extrapolated ATE of never-takers is:

$$\eqalign{\hat{{\mathbb E}}[ Y( 1) - Y( 0) \vert D( 1) = D( 0) = 0] & = {1\over 1 - \hat{\,p}( 1) } {\int_{\! \! \hat{\,p} \! ( 1) }^{1}} \hat{{\mathbb E}}_{L}[ Y( 1) \vert U = u] \, {\rm d}u - \hat{{\mathbb E}}[ Y \vert D = 0,\; Z = 1] \cr & = 0.189.}$$

These estimates show that the causal effect of media on attitudes depends on their propensity to consume the media. The ATE of always-takers is negative, meaning exposure to West German TV would make respondents less supportive of communism. By contrast, the ATE of never-takers is positive, reducing exposure to West German TV would make them less supportive of communism. For never-takers, the estimates support the “spiritual opium” hypothesis of Kern and Hainmueller (Reference Kern and Hainmueller2009)–perhaps watching West German TV makes their life more tolerable, increasing support for the government's preferred ideology. For always-takers, however, respondents are self-selecting the media sources according to their initial ideological positions.

The benefit of these estimates is that we can more directly speak to issues of political concern, such as the net effect of Western media exposure on support for communist ideology. We conduct a numerical experiment to compute this overall effect in the case of West German TV, taking into account the populations of compliers and the other two unobserved principal strata. In our numerical experiment, we compute the support rate of communism in two scenarios: when T = 1, i.e., when everyone consumes West German TV; when T = 0, i.e., when no one consumes West German TV. In the first scenario, we need to know following three quantities: $\hat {\mathbb {\mathbb E}}[ Y( 1) \vert D( 1) = D( 0) = 1]$, $\hat {\mathbb {\mathbb E}}[ Y( 1) \vert D( 1) > D( 0) ]$ and $\hat {\mathbb {\mathbb E}}[ Y( 1) \vert D( 1) = D( 0) = 0]$. In the second scenarios, we need to know: $\hat {\mathbb {\mathbb E}}[ Y( 0) \vert D( 1) = D( 0) = 1]$, $\hat {\mathbb {\mathbb E}}[ Y( 0) \vert D( 1) > D( 0) ]$ and $\hat {\mathbb {\mathbb E}}[ Y( 0) \vert D( 1) = D( 0) = 0]$. Each of these quantities can be found in Table 2.

The overall effect of West German TV is summarized in Table 3. These results show that exposing the entire population to West German television would, on average, produce a small increase in support for communism. The average support rate for communism when no one consumes West German TV and when everyone consumes West German TV is $64.79\%$ and $66.3\%$ respectively. However, Table 3 also shows that attitudes would be more polarized if everyone were exposed to West German TV.

Table 3. Effect of West German TV on Support for Communism

Note: Sample size = 3023. This table shows the support rate for communism under two hypothetical scenarios: either all residents in East Germany do not watch West German TV or they all do. In each cell in fourth and fifth rows, we calculate the support rate for communism across different types of residents.

4.4 Extrapolation 2: bounds by linear programming

The point estimation results require strong parametric assumptions on MTR pairs, possibly introducing severe misspecification bias. Our next step applies the linear programing approach to partially identify our target parameters, i.e., bound the ATEs of always-takers and never-takers.

The estimation of linear programing-based bounds involves inserting the sample analog into the estimating equation. The resulting maximization problem reduces to the following linear program:

(7)$$\hat{\bar{\beta }}^* = {\rm ma}{\rm x}_{\theta \in \Theta }\sum\limits_{{\rm d}\in \{ 0,1\} } {\sum\limits_{k = 0}^{K_d} {\theta _{{\rm d}k}} {\hat{\gamma }}_{s{\kern 1pt} {\rm d}k}} \;{\rm subject}\, {\rm to}\sum\limits_{{\rm d}\in \{ {\rm 0},{\rm 1}\} } {\sum\limits_{k = 0}^{K_d} {\theta _{\rm d}k{\hat{\gamma }}_{s{\kern 1pt} {\rm d}k}} } = \hat{\beta }_s{\rm for}\, {\rm all}\,{\rm s}\in {\rm S}.$$

A similar formulation is possible for the minimization problem.Footnote ¹¹

There are two ways of proceeding with the extrapolation task. First, we can treat the ATEs of always-takers and never-takers as generalized LATEs, as both are weighted average of MTEs. Second, we can extrapolate the unknown part of our target parameters: $\hat {{\mathbb E}}[ Y( 0) \vert D( 1) = \;D( 0) = 1]$ and $\hat {{\mathbb E}}[ Y( 1) \vert D( 1) = D( 0) = 0]$, then, construct the bounds from the known parts of our target parameters. We present the results using the approach treating the ATEs of always-takers and never-takers as generalized LATEs.

The extrapolation results are presented in Figure 2. The x-axis depicts the polynomial order of the basis function used in the linear programing extrapolation. The y-axis displays the estimated ATE, controlling for each respondent's mother's occupation.Footnote ¹² Each line shows the upper and lower bound for the extrapolation to each group. There two main conclusions. First, for never-takers, the bounds expand as we impose fewer restrictions on the basis functions. For always-takers, the estimated bounds are robust to difference choices of the basis functions. Second, Figure 2(b) show that the proportion of never-takers in our sample is too small to produce informative bounds. However, for always-takers, as shown in Figure 2(a), the bounds are sufficiently narrow to offer substantive insights.Footnote ¹³

Figure 2. Extrapolate ATEs of always-takers and never-takers by linear programing. (a) ATE of always-takers: Generalized LATE. (b) ATE of never-takers: Generalized LATE. Note: Sample size =3023. This figure presents the bounds on the ATEs of always-takers and never-takers when we control mother's occupation in the linear program. The constraints in the linear program are the cross-moments in proposition C.1. The x-axis displays the polynomial order of the basis functions used in the linear program. Figures 2a,b are computed from treating ATEs of always-takers and never-takers as generalized LATEs in the linear program.

The results of extrapolation from the linear program show that there is a negative effect from West German TV on always-takers’ support for communism. Note that always-takers have the highest propensity to consume the West German's media. Therefore, one interpretation is that the always-takers are acting as-if they have anticipated the negative effects and self-select to consume the West German TV that reinforces their prior opposition to communism.

Moreover, we also provide empirical results from competing partial identification strategies in Appendix J. There are two main conclusions. First, some existing partial identification strategies produce bounds for the target parameters that are too wide to be informative. Second, there is a trade-off between the strength of assumptions and the widths of bounds, that is, researchers need to invoke stronger assumptions to get narrower bounds.