20.5 Causation versus Prediction

Understanding the relationship between causation and prediction is crucial in statistical modeling. Building on Kleinberg et al. (2015) and Mullainathan and Spiess (2017), consider a scenario where $Y$ is an outcome variable dependent on $X$, and we want to manipulate $X$ to maximize some payoff function $\pi(X,Y)$. Formally:

$$\pi(X,Y) = \mathbb{E}\bigl[\,U(X,Y)\bigr] \quad \text{or some other objective measure}.$$

The decision on $X$ depends on how changes in $X$ influence $\pi$. Taking a derivative:

$$\frac{d\,\pi(X,Y)}{dX} = \frac{\partial \pi}{\partial X}(Y) + \frac{\partial \pi}{\partial Y}\,\frac{\partial Y}{\partial X}.$$

We can interpret the terms:

• $\displaystyle \frac{\partial \pi}{\partial X}$: The direct dependence of the payoff on $X$, which can be predicted if we can forecast how $\pi$ changes with $X$.
• $\displaystyle \frac{\partial Y}{\partial X}$: The causal effect of $X$ on $Y$, essential for understanding how interventions on $X$ shift $Y$.
• $\displaystyle \frac{\partial \pi}{\partial Y}$: The marginal effect of $Y$ on the payoff.

Hence, Kleinberg et al. (2015) frames this distinction as one between predicting $Y$ effectively (for instance, “If I observe $X$, can I guess $Y$?”) versus managing or causing $Y$ to change via interventions on $X$. Empirically:

• To predict $Y$, we model $\mathbb{E}\bigl[Y\mid X\bigr]$.
• To infer causality, we require identification strategies that isolate exogenous variation in $X$.

Empirical work in economics, or social science often aims to estimate partial derivatives of structural or reduced-form equations:

• $\displaystyle \frac{\partial Y}{\partial X}$: The causal derivative; tells us how $Y$ changes if we intervene on $X$.
• $\displaystyle \frac{\partial \pi}{\partial X}$: The effect of $X$ on payoff, partially mediated by changes in $Y$.

Without proper identification (e.g., randomization, instrumental variables, difference-in-differences, or other quasi-experimental designs), we risk conflating association ($\hat{f}$ that predicts $Y$) with causation ($\hat{\beta}$ that truly captures how $X$ shifts $Y$).

To illustrate these concepts, consider the following directed acyclic graph (DAG):

library(ggdag)
library(dagitty)
library(ggplot2)


# Define the DAG structure with custom coordinates
dag <- dagitty('
dag {
  X0 [pos="0,1"]
  X [pos="1,2"]
  Y [pos="1,1"]
  II [pos="1,0"]

  X0 -> Y
  X0 -> II
  X -> Y
  Y -> II
}
')

# Convert to ggdag format with manual layout
dag_plot <- ggdag(dag) +
    theme_void() +
    geom_text(aes(x = 0.5, y = 1.2, label = "Causation"), size = 4) +
    geom_text(aes(x = 0.3, y = 0.5, label = "Prediction"), size = 4) 

# Display the DAG
dag_plot

References

Kleinberg, Jon, Jens Ludwig, Sendhil Mullainathan, and Ziad Obermeyer. 2015. “Prediction Policy Problems.” American Economic Review 105 (5): 491–95.

Mullainathan, Sendhil, and Jann Spiess. 2017. “Machine Learning: An Applied Econometric Approach.” Journal of Economic Perspectives 31 (2): 87–106.