When should one use Propensity Score Matching instead of Instrumental Variables to judge the impact? I have a dataset with a lot of covariates and we need to judge the impact of one variable on the other. Initially, I was supposed to use Propensity Score Matching, but I started wondering as to when and how IVs can also be used for the same. Is it just a theoretical difference, a matter of convenience or some other tradeoffs?
 A: The biggest difference between insturmental variable methods and methods that rely on adjusting for observed confounders (which includes propensity score methods and covariate adjustment by regression) is the assumptions that are used to identify the causal effect using them.
For instrumental variable methods, the main assumption that must be true is the exclusion restriction, which means the instrument is independent of the outcome given the treatment. The exclusion restriction can be violated when the instrument and the outcome share a common cause, the instrument and the treatment share a common cause, or there is a direct effect of the instrument on the outcome not through the treatment.
For covariate adjustment methods, the main assumption is strong ignorability (also known as conditional exchangeability or satisfaction of the backdoor criterion), which means that there exists a set of variables such that the potential outcomes are independent of the treatment conditional on the variables and those variables have been measured (i.e., are contained within the dataset). These variables form a sufficient adjustment set. This assumption can be violated when unmeasured confounding remains after including the measured variables or when one of the variables in the proposed adjustment set induces bias when conditioned upon (i.e., because it is a collider).
When the exclusion restriction is met, instrumental variable methods can be used to estimate treatment effects, and when strong ignorability is met, covariate adjustment methods can be used to estimate treatment effects. What if both assumptions are met, and either method is potentially valid?
A second consideration is the estimand of interest. The estimand the effect with respect to a target population of interest. When treatment effects differ across individuals based on variables that also cause selection into treatment, the average treatment effect will differ depending on the population under study. Covariate adjustment methods and instrumental variable methods target different estimands. Instrumental variable methods target the local average treatment effect (LATE; also known as the compiler average treatment effect), which is the effect of the treatment for those whose treatment status is affected by the instrument. Covariate adjustment methods can be used to target the average treatment effect in the whole population (ATE), the average treatment effects in the treated (ATT) or control (ATC), average treatment effects in covariate subgroups, and average treatment effects in overlap populations (ATO). These are described in Greifer and Stuart (2021).
Ideally, the estimand is decided prior to estimation based on substantive and statistical criteria and is used to guide this choice. If it is believed there is no treatment effect heterogeneity by confounders or if the estimand doesn't matter because the research question is about the effect of treatment for some population (not a specific population), then it doesn't matter which estimand is targeted.
There are several other considerations that might guide the choice between the two methods, such as the sample size; the form of the relationships between the instrument and treatment, between the covariates and treatment, and between the covariates and outcome; other more statistical assumptions like monotonicity; considerations about the desired precision; and justifiability to audiences. I highly recommend reading Matthey et al. (2020), which explicitly answers your question in more detail than this answer can provide.
