The term multi-valued treatment effects refers to a collection of population parameters capturing the impact of a treatment variable on an outcome variable when the treatment takes multiple values. For example, in labour training programmes participants receive different hours of training or in anti-poverty programmes households receive different levels of transfers. Multi-valued treatments may be finite or infinite as well as ordinal or cardinal, and naturally extend the idea of binary treatment effects, leading to a large collection of treatment effects of interest in applications. The analysis of multi-valued treatment effects has several distinct features when compared to the analysis of binary treatment effects, including: (i) a comparison or control group is not always clearly defined, (ii) new parameters of interest arise that capture distinct phenomena such as nonlinearities or tipping points, (iii) correct statistical inference requires the joint estimation of all treatment effects (as opposed to the estimation of each treatment effect separately) in general, and (iv) efficiency gains in statistical inference may be obtained by exploiting known restrictions among the multi-valued treatment effects.
