Simpson's Paradox in PPML? Negative estimates for individual Sections vs. Positive pooled estimates (Sign reversal across multiple groups)

[Jianli Ding]

Hello Statalist,

I am estimating the effect of industrial agglomeration (Location Quotient, spec_lq) on new firm entry (new_entrants) using ppmlhdfe (Stata 17).
I am observing a consistent sign reversal (Simpson's Paradox) between my separate Section-level regressions and my pooled group regressions. I am looking for advice on the statistical mechanisms behind this reversal and how to properly handle/report it.

1. Data Structure & Industry Hierarchy

• Unit of Analysis: Fine-grained spatial grids $\times$ 2-digit Industry $\times$ Year.
• Hierarchy: The data follows a standard classification system where 2-digit Divisions are nested within broad 1-digit Sections. (e.g., Section I contains divisions 63, 64, and 65).
• Grouping Strategy: I constructed aggregated groups (e.g., "High-Tech Services") by selecting specific 2-digit divisions. Note: For Section I, all its constituent 2-digit divisions are classified as High-Tech.

2. The Empirical Puzzle & Model Specification

To diagnose this, I use the following PPML specification controlling for Grid, Sub-industry, and Year fixed effects.

Stata Code:

* Controlling for Grid, Sub-industry (hydm2), and Year FEs; Clustering at Grid level
ppmlhdfe new_entrants $final_ivs if reg_group == "Your_Group_Filter", absorb(grid_id hydm2 year) vce(cluster grid_id)

The Conflict:

• Result A (Section-Level Regressions): When I run the code above separately for any single Producer Service Section (e.g., restricting sample to Section I alone, or Section M alone), the coefficient for is consistently negative or insignificant.spec_lq
• Result B (Pooled Group Regressions): When I pool these codes into the "High-Tech Service" group (using the exact same FE structure: ), the coefficient flips to positive and highly significant.absorb(grid_id hydm2 year)

Robustness: This sign reversal also occurs for the "General Producer Service" group (flipping from negative/null to positive when pooled).

3. Linearity Check

I tested for non-linearity by adding a quadratic term (c.spec_lq##c.spec_lq) to the pooled High-Tech model. The squared term is negative but statistically insignificant, suggesting the positive linear effect dominates in the pooled specification and the result is not driven by a simple inverted-U shape.

4. Robustness Check (Interaction Model)

To investigate slope heterogeneity, I ran a pooled interaction model on the full dataset (ppmlhdfe ... c.spec_lq##i.Group_ID). Using lincom, the calculated slope for the "High-Tech Service" group remains positive and significant (+0.156).

5. My Questions

1. Mechanisms: What are the likely econometric reasons for this sign reversal in a PPML HDFE setting? Is the positive pooled estimate driven by the change in the fixed-effect structure (absorbing average grid quality vs. industry-specific grid quality)?
2. Validity: Given that all constituent sections show negative congestion effects individually, is the positive pooled coefficient a valid measure of "Cluster Sorting Benefits," or is it simply an aggregation bias?

Outputs: Regression Comparison

Model (1) restricts sample to Section I (Info/Software) only.

Model (2) restricts sample to Section M (Science/R&D) only.

Model (3) is the pooled "High-Tech Service" group.

--------------------------------------------------- (1) (2) (3) Section_I Section_M Pooled_HT (Subset) (Subset) (Aggregated) --------------------------------------------------- spec_lq -0.114** -0.073 0.156*** --------------------------------------------------- Grid FE Yes Yes Yes Ind FE (2-digit) Yes Yes Yes Year FE Yes Yes Yes --------------------------------------------------- * Standard errors clustered by Grid
Thank you for your advice.