R&D and patent counts (count data)

This illustrates the count-data family: the number of patents a firm files (non-negative integer) as a function of R&D and size. Figures are illustrative.

Summary: because count data is often overdispersed, we compare Poisson and Negative Binomial and select the appropriate model.

---

Step 1 — Ideation

• Question: how much does more R&D raise the patent count?

Step 2 — Literature Review

Economics of innovation, the knowledge production function; the patents count outcome.

Step 3 — Data Collection

Variable	Symbol	Measurement	Source
Patent count	patents	count/year	IP database
R&D spending	lnrd	log R&D	financials
Size	lnsize	log assets/labor	financials

Step 4 — Modeling

Choose the Count data family → Poisson, test for overdispersion; if present ⇒ Negative Binomial:

$$E[patents_i \mid X_i] = \exp(\beta_0 + \beta_1 lnrd_i + \beta_2 lnsize_i)$$

Illustrative results (format — not real results):

	Poisson	NegBin
lnrd (IRR)	1.35***	1.31***
lnsize (IRR)	1.12**	1.10**
Overdispersion $\alpha$	—	0.42 (≠0) ⇒ choose NegBin

Sample interpretation: a 1% rise in R&D is associated with a higher expected patent count (IRR > 1); the test $\alpha \ne 0$ ⇒ NegBin fits better than Poisson.

Stata

* ===== Count Models — Patents & R&D =====
* Poisson with robust SE
poisson patents lnrd lnsize, vce(robust)
estimates store pois

* Negative Binomial with robust SE
nbreg patents lnrd lnsize, vce(robust)
estimates store nb

* Compare AIC/BIC
estimates stats pois nb

* IRR for interpretation
nbreg patents lnrd lnsize, vce(robust) irr

* Overdispersion: the LR test of alpha at the bottom
* of nbreg output. p < 0.05 ⇒ NegBin preferred

R

# ===== Count Models — Patents & R&D =====
# Poisson
pois <- glm(patents ~ lnrd + lnsize,
            data   = df,
            family = poisson())

summary(pois)

# Negative Binomial
library(MASS)
nb <- glm.nb(patents ~ lnrd + lnsize, data = df)

summary(nb)

# IRR
exp(coef(nb))

# Compare AIC
AIC(pois, nb)
# Lower AIC ⇒ better fit

# Overdispersion test
library(AER)
dispersiontest(pois)

Python

# ===== Count Models — Patents & R&D =====
import statsmodels.api as sm
import numpy as np

# Prepare data
X = sm.add_constant(df[["lnrd", "lnsize"]])
y = df["patents"]

# Poisson
pois = sm.GLM(y, X, family=sm.families.Poisson()).fit(cov_type="HC1")
print("=== Poisson ===")
print(pois.summary())
print("IRR:", np.exp(pois.params).round(4))

# Negative Binomial
nb = sm.GLM(y, X, family=sm.families.NegativeBinomial()).fit()
print("\n=== Negative Binomial ===")
print(nb.summary())
print("IRR:", np.exp(nb.params).round(4))

# Compare AIC
print(f"\nAIC Poisson: {pois.aic:.1f}")
print(f"AIC NegBin:  {nb.aic:.1f}")

# Overdispersion check
print(f"Dispersion (Poisson): {pois.pearson_chi2 / pois.df_resid:.2f}")
# Value >> 1 indicates overdispersion ⇒ prefer NegBin

Step 5 — Reporting

Export a report + replication code; if there are excess zeros (many firms with 0 patents) ⇒ consider ZINB.

Video tutorial

Video Tutorial: Guide to running count models (Poisson/NegBin) in EcoLab

See also

• Poisson · Negative Binomial · ZINB · Catalog