TL;DR: Not all measurement error is fatal in panel data. In a Two-Way Fixed Effects (TWFE) model:

• Constant and Time-Specific additive errors are harmless (absorbed by FEs).
• Unit-Specific additive errors create bias when interacting a continuous regressor with a post-treatment dummy (Continuous DiD).
• The Fix: You can eliminate this bias by including Unit × Post fixed effects or by using post-period centering (FWL residualization).

How to Ensure the Consistency of Estimators When Dealing with Systematic Errors in Continuous Variable Systems

1. Motivation

Standard econometrics training teaches us that classical measurement error (random noise) in a regressor typically biases OLS estimates toward zero (attenuation bias). However, in panel data settings—specifically Two-Way Fixed Effects (TWFE) or Continuous-Treatment Difference-in-Differences (DiD) models—the mechanism is more nuanced.

Unlike random noise, systematic measurement error can behave differently depending on its structure:

1. Some systematic errors are harmlessly absorbed by fixed effects.
2. Others “leak” into interaction terms (like $A^*_{it} \times POST_t$) and generate significant bias in the parameter of interest.

This post explores the mechanics of additive systematic error, demonstrates why unit-specific errors are particularly dangerous in continuous DiD designs, and provides two computationally simple solutions.

2. Baseline Setup: Continuous Treatment + Two-Way FE

Consider a standard panel setup with units $i$ and time $t$.

• Outcome: $y_{it}$
• True Continuous Treatment: $A^*_{it}$
• Post-Treatment Indicator: $POST_t$
• Controls: $X_{it}$
• Fixed Effects: Unit $\mu_i$ and Time $\lambda_t$

A standard continuous-treatment DiD specification is:

Interpretation:

• $\beta_1$: The baseline association between the treatment and outcome.
• $\beta_2$: The DiD estimand—how the effect of $A^*_{it}$ changes after the event ($POST_t = 1$).

The Problem: We do not observe the true $A^*{it}$. Instead, we observe a noisy proxy $\widehat A{it}$:

Does using $\widehat A_{it}$ destroy our ability to estimate $\beta_2$ consistently? It depends entirely on the structure of the error.

3. Three Types of Systematic Additive Error

We analyze three common structures of systematic error:

1. Constant: $\widehat A_{it} = A^*_{it} + \theta$
2. Unit-Specific: $\widehat A_{it} = A^*_{it} + \theta_i$
3. Time-Specific: $\widehat A_{it} = A^*_{it} + \theta_t$

The bias depends on how these errors interact with the fixed effects and the interaction term $A^*_{it} \times POST_t$.

3.1 Case 1: Constant Error (Harmless)

Suppose the measurement error is a constant shift:

Substituting this into the interaction term:

Plugging this back into the main equation (Eq 1):

Notice that the term $-\beta_2\theta POST_t$ varies only by time. It is perfectly collinear with the time fixed effects. We can define a re-centered time FE, $\tilde\lambda_t$:

Verdict: Consistent. The error is absorbed by the intercept and time FEs. OLS estimates for $\beta_1$ and $\beta_2$ are unbiased.

3.2 Case 2: Unit-Specific Error (Harmful)

This is the most dangerous scenario. Suppose the measurement error varies by unit (e.g., firm-specific reporting bias):

The interaction term becomes:

Substituting into the model:

Here we see the problem:

1. Main Effect ($\beta_1$): Safe. The error $\theta_i$ is absorbed by the unit fixed effect $\tilde\mu_i$.
2. Interaction ($\beta_2$): Biased. The composite error term $\xi_{it}$ contains $-\beta_2\theta_i POST_t$. This is correlated with the regressor $\widehat A_{it}POST_t$ (which contains $\theta_i POST_t$).

Verdict: Inconsistent. Standard TWFE fails to isolate $\beta_2$ because the measurement error creates a time-varying wedge that is specific to each unit’s post-period.

3.3 Case 3: Time-Specific Error (Harmless)

Suppose the error varies only by time:

The error components generated are $-\beta_1\theta_t$ and $-\beta_2\theta_t POST_t$. Both terms depend only on $t$.

Verdict: Consistent. These terms are fully absorbed by the time fixed effects $\lambda_t$.

4. The Fix for Unit-Specific Error

In Case 2, the bias comes from the term $-\beta_2\theta_i POST_t$. This term lives in the subspace defined by the interaction of unit dummies and the post dummy.

To fix this, we simply need to control for that subspace.

Strategy M1: Add Unit × Post Fixed Effects

We can explicitly add a dummy for each unit in the post period.

The spurious error term is a linear combination of these dummies, so the OLS estimator absorbs it into the $\phi_j$ coefficients, leaving $\beta_2$ clean.

Implementation in Stata:

* Generate the interaction of interest
gen a_post = a_hat * post

* Run regression including unit-specific post dummies
* c.id#1.post creates the necessary dummy interactions
reghdfe y a_hat a_post X*, absorb(id year c.id#1.post) vce(cluster id)

Implementation in R (fixest):

library(fixest)

feols(
  y ~ a_hat + I(a_hat * post) + X1 + X2 |
  id + year + id:post,  # id:post absorbs the error
  data = panel_df,
  cluster = "id"
)

Strategy M2: Post-Period Centering (FWL Logic)

If you prefer not to add a large number of dummy variables, you can use the Frisch–Waugh–Lovell (FWL) theorem. The equivalent “trick” is to residualize the data against Unit × Post.

This essentially means centering your variables within the post-period for each unit. Let $\bar{Z}_{i,\text{post}}$ be the mean of variable $Z$ for unit $i$ during the post-period. The transformation is:

Estimating the model on this transformed data yields estimates numerically identical to Strategy M1.

5. Summary and Key Takeaways

The table below summarizes the impact of different systematic error structures on Continuous DiD estimates:

Practical Implication: If you suspect your continuous treatment variable has unit-level systematic bias (e.g., a “sticky” measurement bias that differs by firm but is constant over time), you must include id × post fixed effects. This is not just adding flexibility; it is a necessary correction to ensure consistency.