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) 

dag_plot

<img src=“20-prediction_estimation_files/figure-html/unnamed-chunk-2-1.png” alt=“Flow chart illustrating relationships between variables. Node”X0” points to “Y” with the label “Causation” and to “II” with the label “Prediction.” Node “X” points to “Y,” and “Y” points to “II.” The chart shows directional influences and predictions among the variables.” width=“90%” />

Figure 2.2: Flow Chart

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.