Causal inference has traditionally centered on scalar outcomes: whether a patient recovers, how much a worker earns, or how many visits a website receives. Modern studies increasingly ask causal questions about outcomes with richer form, such as clinical notes, open-ended survey responses, and images. A hospital may want to know how an AI documentation tool changes the notes physicians write, or how a nurse training program alters what patients say in survey responses. For such outcomes, the usual average treatment effect is ill-defined: one cannot meaningfully subtract one text or image from another. To this end, we propose a causal query for unstructured outcomes. The key idea is to learn what features of the outcome are most causally affected by the treatment, which we call the maximally contrasting feature (MCF). To estimate the MCF, we learn a feature-scoring function that maps each outcome to a scalar and exposes the sharpest contrast between treated and control potential outcomes. We develop identification conditions and estimation algorithms for this query, and extend it to heterogeneous effects by allowing the feature-scoring function to depend on observed covariates. We also handle settings where both the treatment and the outcome are unstructured. Empirical studies on text and images show that the algorithm recovers salient aspects of an outcome changed by a treatment.
Causal inference usually concerns a scalar treatment, yet in many problems the treatment is unstructured: a text, an image, or a sequence of clinical decisions. Consider an instructor writing a course description to attract more students: the treatment is the course description, and the outcome is enrollment. The standard target, the average treatment effect of fixing the treatment to one exact value versus another, runs into two problems. It cannot be estimated, because almost no exact description recurs across courses, leaving no comparable group from which to measure its effect; and it would be of little use even if it could, since no one wants every course to carry the same description. What the instructor actually wants to know is which features of a description raise enrollment, and which of those features can be acted on across many courses. To this end, we propose a causal query for unstructured treatments: the maximally influential feature (MIF), the feature of the treatment that most strongly influences the outcome. We formalize the MIF as a binary feature of the treatment, defined by a feature-scoring function, constrained so that both of its values stay well populated, and chosen to maximize the causal effect it induces. Turning the feature on shifts the distribution of treatments toward those that display it, turning it off shifts away, and the MIF effect contrasts the two average potential outcomes. We study identification conditions for the MIF, develop algorithms to estimate it, and make it actionable through a nudging algorithm that revises a treatment along the MIF into an outcome-improving version. We illustrate the MIF algorithm across applications in text, image, and dynamic treatment sequences.
Unstructured data, such as images and text, are increasingly used in empirical economics. Since training machine-learning models on unstructured data is costly, economists often use off-the-shelf pre-trained deep learning models developed by computer scientists to extract embeddings, which are then used as covariates in target economic analyses. Despite the popularity of this practice, its theoretical foundations remain limited. There are two main difficulties. First, the pre-trained model is usually trained on a different dataset and for a different task. Consequently, it is unclear when such a model can be used reliably for the target task. Second, the embedding function is subject to an identification problem, which makes it difficult to analyze the estimation error of the embedding function and its effect on the target task. In this paper, we provide sufficient conditions to overcome these difficulties and derive the convergence rate of machine learning models with pre-trained embeddings. We illustrate the theory through double machine learning applications for estimating parameters of interest, such as partially linear regression with unstructured controls, price elasticity in demand estimation considering the product quality measured by images and text, missing data imputation with unstructured data, and the average treatment effect with unstructured confounders.
Causal discovery from unstructured data is a challenging yet underexplored task in high-expertise domains such as healthcare, finance, and education. Existing methods typically leverage the general knowledge of large language models (LLMs) to identify causal factors from unstructured data and annotate them into structured data for causal graph construction. However, they remain limited by two key challenges (CHs): (CH1) insufficient identification of latent factors, which are implicit in the data yet essential for causal discovery, due to the lack of domain-specific knowledge; and (CH2) unreliable factor annotation, caused by the lack of domain-grounded reasoning, which propagates errors to the resulting causal graphs. To address these challenges, we introduce a novel Domain Knowledge-enhanced Causal Discovery framework (DKCD) for causal discovery from unstructured data in high-expertise domains with three interconnected components: (1) Knowledge Mining: It retrieves relevant domain knowledge based on observable factors to support subsequent causal reasoning. (2) Knowledge-guided Causal Reasoning: Reasoning with relevant knowledge, it discovers latent causal factors to address CH1 and generates key causal clues for more accurate data annotation to address CH2. (3) Causal Structure Discovery: It constructs the final causal graphs based on a more complete factor set and accurate annotations. Experiments on two domain-specific datasets show that DKCD significantly improves both causal factor identification and causal graph construction.