Sharp null hypothesis

  • Consider the null hypothesis: \[ H_0: Y_i^\ast(0) = Y_i^\ast(1), \forall i = 1, \cdots, N. \]
  • Under this null hypothesis, we can infer all the missing potential outcomes from the observed ones.
  • A null hypothesis of this property is called the sharp null hypothesis.
  • Under a sharp null hypothesis, we can infer the exact distribution of any statistics that is a function of \(\mathbf{Y}^{obs}, \mathbf{Z}\), and \(\mathbf{W}\).

The difference in the means by treatment status

  • Consider a statistic: \[ T^{ave}(\mathbf{Z}, \mathbf{Y}^{obs}, \mathbf{W}) \equiv \overline{Y}_t^{obs} - \overline{Y}_c^{ob} = \frac{1}{N_1} \sum_{i: Z_i = 1}Y_i^{obs} - \frac{1}{N_0} \sum_{i:Z_i = 0} Y_i^{obs}. \]
  • The p-value of the observation \(\mathbf{Y}^{obs}, \mathbf{Z}^{obs}\), and \(\mathbf{W}\) (where \(\mathbf{Z}^{obs}\) is the realized treatment assignment) is: \[ p = \mathbb{P}[|T^{ave}(\mathbf{Z}, \mathbf{Y}^{obs}, \mathbf{W})| \ge |T^{ave}(\mathbf{Z}^{obs}, \mathbf{Y}^{obs}, \mathbf{W})|], \] where the probability is about \(\mathbf{Z}\).
  • It is known as the treatment assignment mechanism.

Reference

  • Chapter 5, Guido W. Imbens and Donald B. Rubin, 2015, Causal Inference for Statistics, Social, and Biomedical Sciences, Cambridge University Press.
  • Section 4.1, Athey, Susan, and Guido Imbens. 2016. “The Econometrics of Randomized Experiments.” arXiv [stat.ME]. arXiv. http://arxiv.org/abs/1607.00698.