RainbowStats command reference

GROUPED_CROSS_SECTION

Compare a pooled county regression with state-aware models, test whether state membership matters, and inspect whether the apparent fit comes from differences between states or relationships within states.

Syntax

GROUPED_CROSS_SECTION(crossSection)

The argument must be a CrossSectionOperator, normally produced by CROSS_SECTION. County FIPS row keys are resolved through USCountyRegistry, and the first two FIPS digits identify each state.

CountyModel=CROSS_SECTION(CountyData,20221231)
Grouped=GROUPED_CROSS_SECTION(CountyModel)
SLIDESHOW(Grouped)

Pass the original cross section. Do not first apply STATE_FIXED_EFFECTS. The grouped operator creates its own state indicators; passing an already-expanded model duplicates the dummy variables and can produce a singular regression matrix.

What the command estimates

Model	Purpose	Interpretation
Pooled OLS	Ignores state membership.	Describes the combined cross section as if all counties came from one population.
State indicators only	Uses state membership without the substantive predictors.	Measures the raw separation attributable to differences in state means.
State fixed effects	Adds states − 1 indicators to the predictors.	Allows different state intercepts while imposing common predictor slopes.
Within-state regression	Demeans the outcome and predictors inside each state.	Shows explanatory power after state means have been removed. Its slopes should match the fixed-effects slopes.
State interactions	Interacts each predictor with each non-reference state.	Tests whether predictor slopes vary by state as well as intercepts.

ANOVA and model tests

The output distinguishes two related but different questions:

One-way state ANOVA: do the unadjusted state means of the outcome differ?
Nested-model F-tests: do state intercepts improve the regression after controlling for the predictors, and do state-specific slopes improve it further?

The distribution chart places every state's density histogram in one visual. Because density—not county count—is plotted, states with different numbers of counties remain comparable. Separation between the histograms provides an immediate visual explanation of a small ANOVA p-value.

The fixed-effects and interaction regressions center predictors at their overall sample means. Thus Constant (IA at predictor means) is the fitted Iowa outcome for a county with typical predictor values. A coefficient such as State FE: WA is the estimated Washington–Iowa difference at those same predictor values.

Slides produced

Grouped cross-section audit
State summary and variable means
Pooled and grouped model comparison
One-way ANOVA and nested-model tests
Outcome distributions by state
State fixed-effects regression
Within-state regression
State-specific slopes and interaction test
Fixed-effects actual versus predicted
Within-state actual versus predicted
Sorted fixed-effects residuals and labeled outliers

Extract individual results

Request	Returned result
`summary` or `audit`	Data audit and state summary
`comparison` or `r2`	Comparison of pooled and grouped models
`anova`	One-way ANOVA and nested-model F-tests
`distribution`	Overlaid state density histograms
`fixed_effects` or `fe`	State fixed-effects regression
`within`	Within-state regression
`interactions` or `slopes`	State-specific slopes and joint interaction test
`actual_vs_fitted` or `fit`	Fixed-effects actual-versus-predicted chart
`within_fit`	Within-state actual-versus-predicted chart
`residuals` or `outliers`	Sorted fixed-effects residual chart

ANOVA=EXTRACT(Grouped,anova)
Distribution=EXTRACT(Grouped,distribution)
FixedEffects=EXTRACT(Grouped,fixed_effects)
WithinState=EXTRACT(Grouped,within)
Outliers=EXTRACT(Grouped,outliers)
SLIDESHOW(ANOVA,Distribution,FixedEffects,WithinState,Outliers)

Complete CDC county example

WAMental=LOOP_US_COUNTY(CDC(MHLTH_$$_AGEADJPRV),"WA")
IAMental=LOOP_US_COUNTY(CDC(MHLTH_$$_AGEADJPRV),"IA")
Mental=LIST(WAMental,IAMental)

WAIsolation=LOOP_US_COUNTY(CDC(ISOLATION_$$_AGEADJPRV),"WA")
IAIsolation=LOOP_US_COUNTY(CDC(ISOLATION_$$_AGEADJPRV),"IA")
Isolation=LIST(WAIsolation,IAIsolation)

WAFood=LOOP_US_COUNTY(CDC(FOODINSECU_$$_AGEADJPRV),"WA")
IAFood=LOOP_US_COUNTY(CDC(FOODINSECU_$$_AGEADJPRV),"IA")
Food=LIST(WAFood,IAFood)

WAIncome=LOOP_US_COUNTY(SET_NAME(MHIWA$$A052NCEN%MHIUS00000A052NCEN,"Median Income Relative to U.S. - $$"),"WA")
IAIncome=LOOP_US_COUNTY(SET_NAME(MHIIA$$A052NCEN%MHIUS00000A052NCEN,"Median Income Relative to U.S. - $$"),"IA")
Income=LIST(WAIncome,IAIncome)

CountyData=LIST(Mental,Isolation,Food,Income)
CountyModel=CROSS_SECTION(CountyData,20221231)
Grouped=GROUPED_CROSS_SECTION(CountyModel)
SLIDESHOW(Grouped)

How to read the results

A large increase from pooled to fixed-effects R² indicates that state membership contains important information omitted from the pooled model.
A significant pooled-versus-fixed-effects F-test supports adding different state intercepts.
A weak interaction F-test supports the simpler common-slope fixed-effects specification.
A low within-state R² means that much of the apparent pooled fit is between states rather than among counties within the same state.
Residual outliers remain substantively important even when state effects substantially improve overall fit.

Data handling and cautions

Rows missing the outcome or any predictor are removed as complete rows. Keys, labels, state membership, and every variable remain aligned.
The alphabetically first included state is the reference state.
At least two states and enough complete observations to estimate the expanded models are required.
This is a grouped analysis of one cross section, not a multi-period panel fixed-effects estimator.
The ANOVA is unadjusted; the fixed-effects regression is conditional on the included predictors.
The reported relationships are descriptive associations. State indicators can absorb omitted geographic differences, but they do not identify which omitted mechanism is responsible.
Standard errors are conventional OLS standard errors; they are not automatically heteroskedasticity-robust or state-clustered.