20. Econometric Causality

• Statistical approach for causality: treatment effect model
• The statistical approach to causality studies effects of causes.
• Statisticians are trying to correctly describe the effect without worrying about the sources or mechanisms of those effects.
• The treatment or interventions is a black-box, whose mechanisms remain unclear.
• Econometric approach for causality: structural model
• The econometric approach to causality features economics in addition to statistics, which means that the econometric approach to causality studies the causes of effects.
• Econometricians are interested in the mechanisms behind the relationship between treatment outcomes and treatment choices, and they try to recover the causal mechanism by which outcomes are produced.
• This approach requires building explicit economic models, which helps us look into the black-box.

20.2 Structural Model

Similar to (20.1), a more general structural model, for example, could be $$Y=G(\mathbf X,\theta,U)$$ which is structural if function \(G(\cdot,\cdot,\cdot)\) is invariant to shifts in \(\mathbf X\). Then, the invariant parameters are the parameters that pin down the functional form of \(G(\cdot,\cdot,\cdot)\), and the class of manipulations is the set of all possible changes in \(\mathbf X\).

In some cases, only part of all the parameters are structural. For example, suppose in (20.1) only \(\beta_p\) is invariant to the shifts in \((\mathbf X_b,\mathbf X_p)\), while \(\beta_b\) is not. Then, the model is still structural for \(\beta_p\), but not structural for \(\beta_b\).

20.3.1 Three general steps

There are three general steps in conducting a econometric causal inference and policy analysis: 1. Define counterfactuals or parameters of interest that come out of a well-specified structure (or, call it theory). 2. Identify causal parameters, which hypothetically uses the population data to provide an algorithm or function that maps the data into the parameters of interest. Note that no estimation or sampling is involved in this step. 3. Estimation, which requires that we do inference using the actually observed finite samples.

20.3.2 Three problems of interest

1. Evaluating the impacts of implemented interventions on outcomes (could be impacts in a particular environment, e.g. consider controlling the environment variables in (20.1)).
2. (a) These could objective evaluations measured by some objective figures.
3. (b) These could also be subjective evaluations measured by some subjective reporting.
4. (c) These evaluations could be conducted ex-ante or ex-post.
5. (d) This problem only focuses on internal validity.
6. Forecasting the impacts of interventions implemented in one environment in other environments.
7. (a) This problem focuses on external validity since it take a treatment parameter or a set of parameters identified in one environment to forecast the effect in another environment.
8. (b) The external validity is also known as transportability.
9. Forecasting the impacts of interventions never historically experienced.
10. (a) This problem requires structural models with new (never previously experienced) ingredients, which is in the domain of econometric approach to causality.

See Roy model in subsection 14.3 for an example of parameters of interest in an policy evaluation problem.

20.3.3 Policy and treatment

We can better understand the relationship between policy and treatment with the following notations. - Define outcomes corresponding to state (treatment) \(s\) for an agent characterized by \(\omega\) as \(Y(s,\omega)\). - \(\omega\in\Omega\) encompasses all features of agents that affect outcome \(Y\), where \(\Omega\) is the set of all characterized agents. - \(s\in\mathcal S\), where \(\mathcal S\) is the set of all possible treatments. - For example, if \(\mathcal S=\{0,1\}\), there are only two treatments, which becomes the binary treatment case. - The functional form \(Y(\cdot,\cdot)\) may be generated by an economic theory. - \(Y(s,\omega)\) are the realized outcomes for individual \(\omega\) after state (treatment) \(s\) is chosen. - Before specifying state, \(Y(\cdot,\omega)\) is not known to agent \(\omega\) but can be estimated.

Individual objective and subjective treatment effect

Now, we can think about the individual (objective) treatment effect (treatment defined as switching from state \(s'\) to state \(s\)) for agent \(\omega\) as $$Y(s,\omega)-Y(s',\omega)\quad\text{for }s\ne s',\text{ and }s,s'\in\mathcal S$$

Suppose the agent's utility function is \(R(Y(s,\omega))\), then we can also estimate the subjective treatment effect (treatment defined as switching from state \(s'\) to state \(s\)) by $$R(Y(s,\omega))-R(Y(s',\omega))$$

Treatment assignment mechanism

We define a treatment assignment mechanism as a rule \(\tau\in\mathcal T:\Omega\to\mathcal S\), which assigns a treatment to each \(\omega\). \(\mathcal T\) is the set of all possible assignment rules.

A policy is basically a treatment assignment mechanism, which selects who gets what. More specifically, it selects individuals \(\omega\) and specifies the treatment \(s\in\mathcal S\) received by \(\omega\).

We can further complicate the treatment assignment mechanism by introducing the constraint assignment mechanism \(a\in\mathcal A:\Omega\to\mathcal B\) where \(\mathcal B\) is the set of all possible constraints faced by the agent when making a treatment decision. In words, \(a\) picks a constraint based on agent's type \(\omega\). Then, the treatment assignment mechanism becomes $$\tau\in\mathcal T:\Omega\times\mathcal A\times\mathcal B\to\mathcal S$$ which means that for a given mechanism \(a\) that picks a constraint based on agent's type, the mechanism \(\tau\) picks the agent's choice of treatment (state \(s\)). Note that the \(\mathcal B\) is redundant here since \(a\in\mathcal A\) already pins down the constraint \(b\in\mathcal B\). Inclusion of \(\mathcal B\) is just a reminder that agents do face constraints when making decision on treatment.

Here, the constraint space \(\mathcal B\) can be regarded as a set of all values of an instrument, and \(\mathcal A\) is the set of all possible assignments that assigns instrument value to individuals. So, we can analogously define full compliance as all agents are shifted by instrument (constraints), which is a one-on-one mapping between instrument values and treatment values, i.e. we have one-on-one mapping between \(\mathcal B\) and \(\mathcal S\), and between \(\mathcal A\) and \(\mathcal T\), and thus we can define \(a\in\mathcal A\) as the treatment assignment mechanism, which is true because in the case of full compliance, \(\tau\in\mathcal T\) is redundant and drops out.

There are two causal frameworks to be covered in the following two subsections: 1. Rubin-Holland Causal Model: this is a causal model based on potential outcomes but there is no choice mechanisms (structures) and no explanations of outcomes provided in this framework. 2. Autonomous Equations: this is a structural model imposing a system of equations that link both observables and unobservables.

20.5 Rubin-Holland Causal Model: the Language of Potential Outcomes

20.5.1 Recap of potential outcomes model

We have discussed potential outcome models in detail previously in section 14, 15, and 16. Here we briefly review the essence of those models. - Notations: - \(T\) is the treatment choice, whose realizations are represented by \(t\). - \(Y\) is the outcome. - Potential outcome \(Y\) of agent \(\omega\) for fixed \(T=t\) is \(Y_\omega(t)\). - The causal effect of \(t'\) versus \(t\) for \(\omega\) is \(Y_\omega(t')-Y_\omega(t)\). - The observed outcome is then $$Y=\sum_{t\in\text{supp}(T)}Y(t)\times\mathbf 1\{T=t\}$$ which is known as the switching regression in the binary treatment case. - \(X\) is baseline characteristics (background variables used as controls) whose support is denoted by \(\mathcal X\), and c.d.f. is denoted by \(F(\cdot)\). - Under Randomized Control Trial (RCT) assumption: - We assume that \(Y(t)\perp T\). - We can identify the average treatment effect (ATE) of \(t_1\) versus \(t_0\) by $$\begin{aligned}\mathbb E[Y(t_1)-Y(t_0)]&=\mathbb E[Y(t_1)]-\mathbb E[Y(t_0)]\\&=\underbrace{\mathbb E[Y(t_1)\mid T=t_1]}_{\text{Observable}}-\underbrace{\mathbb E[Y(t_0)\mid T=t_0]}_{\text{Observable}}\end{aligned}$$

• Under the matching assumption:
• We assume that \(Y(t)\perp T\mid X\).

• We can identify the average treatment effect (ATE) of \(t_1\) versus \(t_0\) by $$\begin{aligned}\mathbb E[Y(t_1)-Y(t_0)]&=\int_{x\in\mathcal X}\big(\mathbb E[Y(t_1)\mid X=x]-\mathbb E[Y(t_0)\mid X=x]\big)\,dF(x)\\&=\int_{x\in\mathcal X}\big(\mathbb E[Y(t_1)\mid T=t_1,X=x]-\mathbb E[Y(t_0)\mid T=t_0,X=x]\big)\,dF(x)\\&=\int_{x\in\mathcal X}\Big(\underbrace{\mathbb E[Y(t_1)\mid T=t_1,X=x]}_{\text{Observable}}-\underbrace{\mathbb E[Y(t_0)\mid T=t_0,X=x]}_{\text{Observable}}\Big)\,dF(x)\end{aligned}$$

• Under the mediation assumption:

• There are four random variables of interest: \(Y\), \(M\), \(T\) and \(X\).
  • \(Y\), \(T\) and \(X\) are as usual. \(T\) has a support denoted by \(\mathcal T\), and c.d.f. is denoted by \(H(\cdot)\).
  • \(M\) is the mediation variable. \(M\) has a support denoted by \(\mathcal M\), and c.d.f. is denoted by \(G(\cdot)\).
• There are two channels for \(T\) to affect \(Y\): $$T\to M\to Y$$
  • Channel 1: direct effect, i.e. \(T\) directly affects \(Y\) holding \(M\) fixed.
  • Channel 2: indirect effect, i.e. \(T\) only affects \(Y\) through affecting \(M\) and then \(M\) affecting \(Y\).
  • Two channels could coexist. Direct effect (DE) plus indirect effect (IE) is the total effect (TE).
• Sequential ignorability assumption:
  • We assume that \((Y(t',m),M(t))\perp T\mid X\), i.e. \(T\) is exogenous to both outcome variable and mediation variable conditional on \(X\).
  • We also assume that \(Y(t',m)\perp M(t)\mid(T,X)\), i.e. \(M\) is exogenous to the outcome variable conditional on \(T\) and \(X\).

• Under ignorability assumption, both the average direct and indirect effects can be identified:

  • Average Direct Effect (ADE) of \(T=t_1\) versus \(T=t_0\), for agents whose realized (only observable) treatment is \(t\), is denoted by \(\text{ADE}(t)\): $$\begin{aligned}\text{ADE}(t)&\equiv\mathbb E[Y(t_1,M(t))-Y(t_0,M(t))]\\&=\int_{\mathcal X}\big(\mathbb E[Y(t_1,M(t))\mid X=x]-\mathbb E[Y(t_0,M(t))\mid X=x]\big)\,dF(x)\\&=\int_{\mathcal X}\big(\mathbb E[Y(t_1,M(t))\mid T=t_1,X=x]-\mathbb E[Y(t_0,M(t))\mid T=t_0,X=x]\big)\,dF(x)\\&=\int_{\mathcal M}\int_{\mathcal X}\Big(\underbrace{\mathbb E[Y(t_1,M(t))\mid M=m,T=t_1,X=x]}_{\text{Observable}}-\underbrace{\mathbb E[Y(t_0,M(t))\mid M=m,T=t_0,X=x]}_{\text{Observable}}\Big)\,dF(x)\,dG(m)\end{aligned}$$
  • Average Indirect Effect (AIE) of \(T=t_1\) versus \(T=t_0\), for agents whose realized (only observable) treatment is \(t\), is denoted by \(\text{AIE}(t)\): $$\begin{aligned}\text{AIE}(t)&\equiv\mathbb E[Y(t,M(t_1))-Y(t,M(t_0))]\\&=\int_{\mathcal X}\big(\mathbb E[Y(t,M(t_1))\mid X=x]-\mathbb E[Y(t,M(t_0))\mid X=x]\big)\,dF(x)\\&=\int_{\mathcal X}\Big(\underbrace{\mathbb E[Y(t,M(t_1))\mid T=t_1,X=x]}_{\text{Observable}}-\underbrace{\mathbb E[Y(t,M(t_0))\mid T=t_0,X=x]}_{\text{Observable}}\Big)\,dF(x)\end{aligned}$$

• Under the instrumental variable assumption:

• We assume that
  • Exclusion: \(Y(t)\perp Z\mid T\);
  • IV relevance: \(Z\not\perp T\).
• We can identify the local average treatment effect (LATE) of compliers for \(z_1\) versus \(z_0\).

20.5.2 Criticisms of the language of potential outcomes - Potential outcome model is not a proper causal framework, because it does not assess any causal relationships. - For example, under the instrument variable assumption, causal relationship \(Z\to T\to Y\) is implied, but never formally articulated. - Potential outcome model does not allow for unobserved variables.

20.6 Autonomous Equations: Structural Model for Richer Causal Analysis

20.6.1 Definition of autonomous equations

Autonomous and structural are synonyms.

A causal model is defined by four components: - Random variables that are observed and/or unobserved by the analyst \(\mathcal S=\{X,V,U,Y\}\). - Error terms that are mutually independent \(\epsilon_Y,\epsilon_U,\epsilon_X,\epsilon_V\). - Structural equations that are autonomous, \(f_Y,f_U,f_X,f_V\). - By autonomy we mean deterministic functions that are invariant to changes in the arguments. - Autonomous equations are also called structural equations.

• Causal relationships that map the inputs causing each variable: $$\begin{aligned}Y&=f_Y(X,U,\epsilon_Y)&&Y\text{ observed}\\X&=f_X(V,\epsilon_X)&&X\text{ observed}\\U&=f_U(V,\epsilon_U)&&U\text{ unobserved}\\V&=f_V(\epsilon_V)&&V\text{ unobserved}\end{aligned}$$

We can also represent these functions as a Directed Acyclic Graph (DAG) (see Figure 6).

Figure 6 (DAG, paraphrased): titled "Causal Model Inside the Box." Four nodes: the two unobserved variables \(V\), \(U\) on top, the two observed variables \(X\), \(Y\) at the bottom. The directed edges are \(V\to U\), \(V\to X\), \(U\to Y\), \(X\to Y\). That is, \(V\) causes both \(U\) and \(X\); \(U\) and \(X\) jointly cause \(Y\).

If we think the functional forms \(f_Y,f_X,f_U,f_V\) are invariant to policy changes (i.e. changes in variables), then we have the desired transportability property, which means that we can estimate the functional forms with some data sets (realized policies) and then use those estimated functional forms in other data sets (counterfactual policies).

20.6.2 Children, descendants and parents

This graph has: - Children: variables directly caused by other variables: $$Ch(V)=\{U,X\}\text{ and }Ch(X)=Ch(U)=\{Y\}$$ - Descendants: $$DE(V)=\{U,X,Y\}\text{ and }D(U)=D(U)=\{Y\}$$ - Parents: variables that directly cause other variables: $$PA(Y)=\{X,U\}\text{ and }PA(X)=PA(U)=\{V\}$$

20.6.3 Local Markov condition (LMC)

20.6.4 Fixing versus conditioning

• Fixing (the do operator): model representation under fixing \(X=x\) is $$\begin{aligned}Y&=f_Y(x,U,\epsilon_Y)\\X&=x\\U&=f_U(V,\epsilon_U)\\V&=f_V(\epsilon_V)\end{aligned}$$
• Fixing is a thought experiment, a hypothetical control if \(X=x\), so we don't need to use conditional probability, and we still use the original probability.
• Fixing is a causal operator (uses unconditional distribution) but not a statistical operator.
• Fixing does not affect the distribution of its ancestors.
• Fixing has causal direction.
• Conditioning:
• Under conditioning, probability (p.d.f) is replaced by conditional probability.
• Conditioning is a statistical operator.
• Conditioning affects the distribution of all variables.
• Conditioning has no causal direction.