• Consider a situation where an action (or manipulation, treatment, or intervention) is applied to a unit.
• For example, I (=unit) take the aspirin (=action) and you (= another unit) do not take the aspirin (=another action).

• The potential outcome is the outcome that could happen when an action was applied to a unit.
• For example, the potential outcome for me when I take Aspirin, denoted by \(Y^\ast(Aspirin)\), takes a value of either headache or no headache.
• The potential outcome for me when I do not take Aspirin, denoted by \(Y^\ast(No Aspirin)\), also takes a value of either headache or no headache.
• Suppose that the true potential outcomes are like: Modified Table 1.1 of Imbens and Rubin (2015)

• The causal effect of taking the aspirin for me involves the comparison of two potential outcomes \(Y^\ast(Aspirin)\) and \(Y^\ast(No Aspirin)\).
• If \(Y^\ast(Aspirin)\) is No Headache and \(Y^\ast(NoAspirin)\) is Headache, the causal effect is labeled as the Improvement due to Aspirin.Table 1.1 of Imbens and Rubin (2015)

• The truth is either of the following scenarios but we do not know which is true:Modified Table 1.3 of Imbens and Rubin (2015)

• Suppose that one of the potential outcomes of unit \(i\) is observed when a treatment is assigned to the unit.
• For example, if \(Aspirin\) is assigned, then \(Y_i = Y_i^\ast(Aspirin)\) is observed and if \(No Aspirin\) is assigned, then \(Y_i = Y_i^\ast(No Aspirin)\) is observed.
• \(Y_i\) is called the observed outcome of unit \(i\), which differs by the assignment to unit \(i\).

• Units: \(i = 1, \cdots, N\).
• Binary treatment indicator: \(Z_i \in \{0, 1\}\).
• Potential outcomes: \(Y_i^\ast(0)\) and \(Y_i^\ast(1)\).
• Pre-treatment variables or covariates: \(W_i\).
• \(\mathbf{Y}^\ast(0)\): and \(\mathbf{Y}^\ast(1)\) be the \(N\)-component column vectors of potential outcomes with \(i\)th elements equal to \(Y_i^\ast(0)\) and \(Y_i^\ast(1)\).
• \(\mathbf{Z}\): \(N\)-component column vector with \(i\)th element equal to \(Z_i\).
• \(\mathbf{W}\): \(N \times K\) matrix of covariates with \(i\)th row equal to \(W_i\).

• The unit-level causal effects: \[ Y_i^\ast(1) - Y_i^\ast(0), i = 1, \cdots, N. \]

• The average causal effect: \[ \tau_{fs} = \frac{1}{N} \sum_{i = 1}^N [Y_i^\ast(1) - Y_i^\ast(0)]. \]

• The average causal effect conditional on the covariate:

\[ \tau_{fs}(f) = \frac{1}{N(f)} \sum_{i: W_i = f} [Y_i^\ast(1) - Y_i^\ast(0)], \] where \(N(f) \equiv \#\{i = 1, \cdots, N: W_i = f\}\).

• The average causal effect for those who were exposed to it:

\[ \tau_{fs, t} = \frac{1}{N_1} \sum_{i: Z_i = 1} [Y_i^\ast(1) - Y_i^\ast(0)], \] where \(N_1 \equiv \sum_{i = 1}^N Z_i\).

• The average causal effect for positive outcomes: \[ \tau_{fs, pos} = \frac{1}{N_{pos}} \sum_{i: Y_i^\ast(0) > 0, Y_i^\ast(1) > 0} [Y_i^\ast(1) - Y_i^\ast(0)], \] where \(N_{pos} \equiv \#\{i = 1, \cdots, N: Y_i^\ast(0) > 0, Y_i^\ast(1) > 0\}\).
  • e.g. effects on wages for employed workers.
  • The definition depends on the potential outcomes but not on observed outcomes.
• The general function of potential outcomes: \[ \tau = \tau(\mathbf{Y}^\ast(0), \mathbf{Y}^\ast(1), \mathbf{W}, \mathbf{Z}), \]
  • where \(\tau\) is a row-exchangeable function.

• The average causal effect of this population is 2.Table 1.4 of Imbens and Rubin (2015)

• The observed outcome of unit \(i\) is: \[ Y_i^{obs} = Y_i^\ast(Z_i) = \begin{cases} Y_i^\ast(0) & \text{ if } Z_i = 0\\ Y_i^\ast(1) & \text{ if } Z_i = 1. \end{cases} \]

• The missing potential outcome of unit \(i\) is: \[ Y_i^{mis} = Y_i^\ast(1 - Z_i) = \begin{cases} Y_i^\ast(1) & \text{ if } Z_i = 0\\ Y_i^\ast(0) & \text{ if } Z_i = 1. \end{cases} \]

• The average outcomes are 6 for treated and 7 for not treated.Table 1.5 of Imbens and Rubin (2015)
• The simple difference means that the treatment has the causal effect of -1, whereas the true average causal effect is 2.
• No valid conclusion can be made without knowing the treatment assignment mechanism to correct the missing data problem.

• The definition depends on the potential outcomes, but not on which outcome is actually observed.
  • For example, I take Aspirin and feel no headache can be either of Improvement due to Aspirin and Headache gone regardless of Aspirin, because the headache could have gone without taking Aspirin.
• The definition is based on the comparison of potential outcomes of the same unit.
  • For example, I take Aspirin and feel no headache and you do not take Aspirin and feel headache does not mean that Aspirin improved my headache.
• The individual causal effect is well-defined but never estimated because at most one of the potential outcomes can be realized and observed for each unit.

• The previous potential outcome model presumes the following:

• Assumption: SUTVA (Stable unit treatment value assumption):

  1. (no spillover, no interference) The potential outcomes for any unit do not vary with the treatments assigned to other units, and;
  2. (consistency, no multiple version of treatment) The observed outcome is uniquely determined by the potential outcome evaluated at the assignment to the unit.

• The assignment mechanism is a function that assigns probabilities to all \(2^N\) possible values for the \(N\)-vector of assignments \(\mathbf{Z}\), given the \(N\)-vectors of potential outcomes \(\mathbf{Y}^\ast(0)\) and \(\mathbf{Y}^\ast(1)\), and given \(N \times K\) matrix of covariates \(\mathbf{W}\).

• Definition: Assignment Mechanism:

  • Given a population of \(N\) units, the assignment mechanism is a row-exchangeable function \(\mathbb{P}[\mathbf{Z}|\mathbf{W}, \mathbf{Y}^\ast(1), \mathbf{Y}^\ast(0)]\), taking on values in \([0, 1]\), satisfying: \[ \sum_{\mathbf{Z} \in \{0, 1\}^N} \mathbb{P}[\mathbf{Z}|\mathbf{W}, \mathbf{Y}^\ast(1), \mathbf{Y}^\ast(0)] = 1, \] for all \(\mathbf{W}\), \(\mathbf{Y}^\ast(0)\), and \(\mathbf{Y}^\ast(1)\).

• In general, dependence of the assignment on the potential outcomes is not ruled out.
• The dependence of the assignment on the covariate, assignment, and the potential outcomes of other units does not contradict with SUTVA, because SUTVA is about the definition of the potential outcomes.

• Assignment mechanism is a joint probability of assignments for the entire population.
• We can derive the unit assignment probability from the assignment mechanism:
• Definition: Unit Assignment Probability: \[ p_i(\mathbf{W}, \mathbf{Y}^\ast(1), \mathbf{Y}^\ast(0)) = \sum_{\mathbf{Z}: Z_i = 1} \mathbb{P}[\mathbf{Z}|\mathbf{W}, \mathbf{Y}^\ast(1), \mathbf{Y}^\ast(0)] \]

• The average unit assignment probability for units with \(W_i = x\) is called the propensity score at \(x\).
• Definition: Finite Population Propensity Score: \[ e(x) = \frac{1}{N(x)} \sum_{i: W_i = x} p_i(\mathbf{W}, \mathbf{Y}^\ast(1), \mathbf{Y}^\ast(0)), \] where \(N(x) = \#\{i = 1, \cdots, N| W_i = x\}\).

• To classify the various types of assignment mechanisms, we present three general properties that assignment mechanisms may satisfy.

  1. Individualistic assignment.
  2. Probabilistic assignment.
  3. Unconfounded assignment.

• The first restriction limits the dependence of the treatment assignment for unit \(i\) on the outcomes and assignments for other units.
• Definition: Individualistic Assignment:
  • An assignment mechanism \(\mathbb{P}[\mathbf{Z}|\mathbf{W}, \mathbf{Y}^\ast(1), \mathbf{Y}^\ast(0)]\) is individualistic if, for some functions \(q(\cdot) \in [0, 1]\): \[ p_i(\mathbf{W}, \mathbf{Y}^\ast(1), \mathbf{Y}^\ast(0)) = q(W_i, Y_i^\ast(0), Y_i^\ast(1)) \]

• The second restriction requires that every unit has positive probability of being assigned to treatment level 0 and to treatment level 1.
• Definition: Probabilistic Assignment:
  • An assignment mechanism \(\mathbb{P}[\mathbf{Z}|\mathbf{W}, \mathbf{Y}^\ast(1), \mathbf{Y}^\ast(0)]\) is probabilistic if: \[ 0 < p_i(\mathbf{W}, \mathbf{Y}^\ast(1), \mathbf{Y}^\ast(0)) < 1, \forall \mathbf{W}, \mathbf{Y}^\ast(1), \mathbf{Y}^\ast(0), \forall i. \]

• The third condition states that the assignment does not depend on the potential outcome.
• Definition: Unconfounded Assignment:
  • An assignment mechanism \(\mathbb{P}[\mathbf{Z}|\mathbf{W}, \mathbf{Y}^\ast(1), \mathbf{Y}^\ast(0)]\) is unconfounded if: \[ \mathbb{P}[\mathbf{Z}|\mathbf{W}, \mathbf{Y}^\ast(1), \mathbf{Y}^\ast(0)] = \mathbb{P}[\mathbf{Z}|\mathbf{W}, \mathbf{Y}^\ast(1)', \mathbf{Y}^\ast(0)'] \] for all \(\mathbf{Z}, \mathbf{W}, \mathbf{Y}^\ast(1), \mathbf{Y}^\ast(0), \mathbf{Y}^\ast(1)', \mathbf{Y}^\ast(0)'\).
• If an assignment mechanism is unconfounded, we can simplify the assignment to \(\mathbb{P}(\mathbf{Z}|\mathbf{W})\).

• The assignment mechanism is written as: \[ \mathbb{P}[\mathbf{Z}|\mathbf{W}, \mathbf{Y}^\ast(1), \mathbf{Y}^\ast(0)] = \mathbb{P}(\mathbf{Z}|\mathbf{W}) = c \cdot \prod_{i = 1}^N q(W_i)^{Z_i} \cdot [1 - q(W_i)]^{1 - Z_i}, \] and the propensity score is the unit-level assignment probability: \[ e(x) = q(x). \]
• Given individualistic assignment, the combination of probabilistic and unconfounded assignment is referred to as strongly ignorable treatment assignment.

• Definition: Randomized Experiment:
  • A randomized experiment is an assignment mechanism that:
    • is probabilistic, and
    • has a known functional form that is controlled by the researcher.
• Definition: Classical Randomized Experiment:
  • A classical randomized experiment is a randomized experiment with an assignment mechanism that is:
    • individualistic, and
    • unconfounded.
• Definition: Natural Experiment:
  • A randomized experiment in which the assignment mechanism is designed by someone different from the analyst.

• A fixed number of units, say \(N_1\), is drawn at random from the population of \(N\) units to receive the active treatment, with the remaining \(N_0 = N - N_1\) assigned to the control group.
• The assignment probability: \[ \mathbb{P}[\mathbf{Z}|\mathbf{W}, \mathbf{Y}^\ast(1), \mathbf{Y}^\ast(0)] = \begin{pmatrix} N\\ N_1 \end{pmatrix}^{-1}, \] for all \(\mathbf{Z}\) such that \(\sum_{i = 1}^N Z_i = N_1\).

• First partition the population on the basis of covariate values into \(G\) strata, i.e. if the covariate space is \(\mathbb{X}\), partition \(\mathbb{X}\) into \(\mathbb{X}_1, \cdots, \mathbb{X}_G\), so that \(\bigcup_g \mathbb{X}_g = \mathbb{X}\) and \(\mathbb{X}_g \cap \mathbb{X}_{g'} = \emptyset\) if \(g \neq g'\).
• Let \(G_{ig} = 1_{W_i \in \mathbf{W}_g}\) and \(N_g\) be the number of units in stratum \(g\).
• Fix the number of treated units in each stratum as \(N_{tg}\) such that \(\sum_g N_{tg} = N_1\).
• The assignment probability is: \[ \mathbb{P}[\mathbf{Z}|\mathbf{W}, \mathbf{Y}^\ast(1), \mathbf{Y}^\ast(0)] = \prod_{g = 1}^G \begin{pmatrix} N_g\\ N_{tg} \end{pmatrix}^{-1}, \] for all \(\mathbf{Z}\) such that \(\sum_{i = 1}^N Z_i \cdot G_{ig} = N_{tg}\) for all \(g\).

• An extreme case of stratification where each stratum contains exactly one treated unit and exactly one control unit.
• There are \(G = N/2\) strata and \(N_g = 2\) and \(N_{tg} = 1\) for all \(g\).
• The assignment probability is: \[ \mathbb{P}[\mathbf{Z}|\mathbf{W}, \mathbf{Y}^\ast(1), \mathbf{Y}^\ast(0)] = \bigg(\frac{1}{2}\bigg)^{\frac{N}{2}}, \] for all \(\mathbf{Z}\) such that \(\sum_{i = 1}^N Z_i \cdot G_{ig} = 1\) for all \(g\).

• Partition the covariate space into clusters and treatments are assigned randomly to entire clusters, with all units within a cluster receiving the same level of the treatment.
• This design may be motivated by concerns that there are interactions between units.
• \(G_t\) out of \(G\) clusters are selected randomly to be assigned to the treatment group.
• Let \(\overline{W}_g = \sum_{i: G_{ig} = 1} Z_i / N_g\) be the average value of \(Z_i\).
• The assignment probability is: \[ \mathbb{P}[\mathbf{Z}|\mathbf{W}, \mathbf{Y}^\ast(1), \mathbf{Y}^\ast(0)] = \begin{pmatrix} G \\ G_t \end{pmatrix}^{-1}, \] for all \(\mathbf{Z}\) such that if \(W_{ig} = W_{i'g} = 1\) and \(\sum_g \overline{W}_g = G_t\).

• Definition: Observational Study:
  • An assignment mechanism corresponds to an observational study if the functional form of the assignment mechanism is unknown.
• Definition: Regular Assignment Mechanism:
  • An assignment mechanism is regular if:
    • the assignment mechanism is individualistic,
    • the assignment mechanism is probabilistic, and
    • the assignment mechanism is unconfounded.
• Definition: Quasi-Experiment:
  • An observational study in which some of the features of the regular assignment mechanism can be justified based on the institutional knowledge.

• Two types of uncertainty affects the inference.
• The sampling-based uncertainty refers to the randomness due to the sampling from the population.
  • The sample is finite. The population may be infinite or finite.
• The assignment-based uncertainty refers to the randomness due to the assignment over the sample.
  • Sometimes called design-based or randomization-based uncertainty.
• Make sure which of (or both of) the uncertainty matters when considering inference.

• The test score of a student when they took a tutorial.
• The grey background means the student is sampled.
• The underline means the student is assigned to the treatment group.

• Traditional approach in economics.
• Origins from Haavelmo (1943, 1944).
• All textbooks and econometric classes starting from the OLS and moving on to IV and GMM are based on the structural estimation approach.
  • Describe an economic model with shocks.
  • Derive the equilibrium condition (structural-form econometric model).
  • Solve the equilibrium condition for endogenous variables (reduced-form econometric model).
  • Use OLS to estimate parameters in the reduced-form model.
  • Use GMM to estimate parameters in the structural-form model.
  • Use IV to estimate parameters in part of the structural-form model.
• The pure form is found in Hsiao (1983)

• Anything assumed outside data distribution is a latent model.
  • The latent model in structural estimation approach is an economic model.
  • The latent model in potential outcome approach is a potential outcome model, which may or may not be derived from an economic model.
• Restrictions from latent model
  • In structural estimation approach, the economic model is used to derive restrictions.
  • In potential outcome approach, institutional knowledge is used to derive restrictions, and using economic model is avoided.
  • In potential outcome approach, an economic model just gives an interpretation of the result.

• Consider the previous table of student’s test score.
• Suppose that we are interested in the finite-sample average treatment effect.
• Model \(\mathcal{M}\): possible values of \(\mathbf{Y}^\ast(\cdot)\) and \(\mathbf{P}_Z\).
• The state of the world: a value of \(m = (\mathbf{y}^\ast(\cdot), \mathbf{p}_Z) \in \mathcal{M}\).
  • For example: \(\mathbf{y}^\ast(\cdot)\) is as in the table, \(\mathbf{p}_Z\) is a completely randomized experiment.
• This uniquely determines the value of parameter of interest \(\theta\): \[ \theta = \frac{1}{N} \sum_{i = 1}^N [y_i^\ast(1) - y_i^\ast(0)] = 174. \]

• Information \(\phi\) to the analyst: all information that the analyst has about the model.
  • Includes the basic setting of the binary potential outcome model.
  • Include the knowledge of the assignment mechanism.
  • Does not include the potential outcome.
• Structure \(s(\phi, \theta) \subset \mathcal{M}\): the value of \(m \in \mathcal{M}\) that is consistent with the information \(\phi\) and the value of \(\theta\).

• Parameter \(\theta\) and \(\tilde{\theta}\) are said to be observationally equivalent if there exist information \(\phi\) and \(s(\phi, \theta) \neq \emptyset\) and \(s(\phi, \tilde{\theta}) \neq \emptyset\).
  • If two parameters are observationally equivalent, both can be consistent with the information to the analyst.
• The parameter \(\theta\) is said to be identified if there is no other parameter \(\tilde{\theta}\) that is observationally equivalent to \(\theta\).
• The parameter space \(\Theta\) is said to be globally identified if every parameter in \(\Theta\) is identified.

• Because \(Y_i\) and the assignment mechanism \(\mathbf{p}_z\) is included in information \(\phi\), the analyst can calculate the following expected value \[ \mathbb{E}\left[\frac{1}{N_1}\sum_{i = 1}^N Z_i Y_i - \frac{1}{N_0} \sum_{i = 1}^N (1 - Z_i) Y_i\right]. \]

• Because the knowledge that the assignment mechanism is unconfounded is included in information \(\phi\), the analyst can show that the above is \[ \begin{split} &\frac{1}{N_1} \sum_{i = 1}^N \mathbb{E}[Z_i] Y_i^\ast(1) - \frac{1}{N_0} \sum_{i = 1}^N (1 - \mathbb{E}[Z_i]) Y_i^\ast(0) \\ &= \frac{1}{N_1} \sum_{i = 1}^N \frac{N_1}{N} Y_i^\ast(1) - \frac{1}{N_0} \sum_{i = 1}^N \frac{N_0}{N} Y_i^\ast(0) \\ &= \frac{1}{N} \sum_{i = 1}^N Y_i^\ast(1) - \frac{1}{N} \sum_{i = 1}^N Y_i^\ast(0) \\ &= \theta \end{split} \]

• The information uniquely determines \(\theta\): The parameter \(\theta\) is identified.

• Suppose that assignment mechanism \(\mathbf{p}_Z\) is such that assigning tutorial to three students whose individual treatment effect is the highest.
  • Units 1, 2, 7 are always treated.
  • Units 5, 9 are never treated.

• The value of \(\frac{1}{N_1} \sum_{i = 1}^N Z_i Y_i - \frac{1}{N_0} \sum_{i = 1}^N (1 - Z_i) Y_i\) is always -171.7 because treatment assignment does not change.
• Change of the model from the left to right does not change \(\phi\) including this value because it just modifies the values of potential outcome that are never observed and increases the average treatment effect by 10.

• Chapter 1-4, Guido W. Imbens and Donald B. Rubin, 2015, Causal Inference for Statistics, Social, and Biomedical Sciences, Cambridge University Press.
• Section 1-3, Athey, Susan, and Guido Imbens. 2016. “The Econometrics of Randomized Experiments.” arXiv [stat.ME]. arXiv. http://arxiv.org/abs/1607.00698.
• Abadie, Alberto, Susan Athey, Guido W. Imbens, and Jeffrey M. Wooldridge. 2020. “Sampling‐based versus Design‐based Uncertainty in Regression Analysis.” Econometrica: Journal of the Econometric Society 88 (1): 265–96.
• Haavelmo, Trygve. “The Statistical Implications of a System of Simultaneous Equations.” Econometrica 11, no. 1 (1943): 1–12.
• Haavelmo, Trygve. “The Probability Approach in Econometrics.” Econometrica 12 (1944): iii–115.
• Hsiao, Cheng. “Chapter 4 Identification.” In Handbook of Econometrics, 1:223–83. Elsevier, 1983.