Panel data and causal inference

Hausman Test: Fixed vs Random Effects

The Hausman test compares fixed and random effects estimates. If they differ systematically, the entity effects are correlated with the regressors and only the fixed effects estimator is consistent.

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Updated 13 October 2026 · By Bibin Xavier

Data 480 × 8

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Example data 480 rows, 8 variables. Data stays in this browser tab.

Firmtext · 60 levels	Yearnumeric	Industrytext · 3 levels	ROAnumeric	Leveragenumeric	Log_Sizenumeric	RnD_Intensitynumeric	GDP_Growthnumeric
F01	2016	Manufacturing	11.28	0.67	8.45	0.08	1.63
F01	2017	Manufacturing	14.97	0.56	7.91	0.12	1.8
F01	2018	Manufacturing	12.24	0.63	7.78	0.1	2.39
F01	2019	Manufacturing	11.1	0.78	7.44	0.08	1.93
F01	2020	Manufacturing	11.04	0.75	7.65	0.09	2.13
F01	2021	Manufacturing	11.33	0.68	7.88	0.09	1.18
F01	2022	Manufacturing	13.01	0.66	7.84	0.11	1.28
F01	2023	Manufacturing	13.76	0.69	8.1	0.1	1.9

Variables and options

Entity (panel) identifier

Time variable

Dependent variable

Regressors 4 selected · min 1

Standard errors

Results update automatically when you change an option.

Results

480 rows · example data

Entities / observations60 / 480

Time periods per entity8balanced panel

Hausman test< .001fixed effects preferred

ρ (share of variance from entity effects).042

Table 1

Panel regression estimates

Variable	Pooled OLS	Fixed effects	Random effects	Two-way FE	First differences
Leverage	-3.4302*** (0.6655)	-5.9077*** (0.6979)	-3.4921*** (0.6664)	-5.9668*** (0.7027)	-6.7783*** (0.7118)
Log_Size	2.3195*** (0.1332)	0.7130*** (0.2126)	2.2820*** (0.1358)	0.6164** (0.2285)	0.8298*** (0.2328)
RnD_Intensity	35.1346*** (4.2753)	16.2880*** (4.6237)	34.7101*** (4.2868)	15.4763** (4.6829)	19.1418*** (4.6011)
GDP_Growth	0.4235*** (0.1189)	0.4327*** (0.1081)	0.4280*** (0.1169)	0.2014 (0.3335)	0.4490** (0.1548)
R² (within)	—	.206	—	.218	—
R² (overall)	.747	—	.695	—	.229

Note. Standard errors in parentheses (conventional). * p < .05, ** p < .01, *** p < .001. Random effects: Swamy–Arora, σ_u = 0.280, σ_e = 1.339. Constant terms omitted.

Table 2

Model selection tests

Test	Comparison	Statistic	p	Preferred model
F test for entity effects	Pooled OLS vs fixed effects	F(59, 416) = 2.824	< .001	Fixed effects
Breusch-Pagan LM	Pooled OLS vs random effects	χ²(1) = 0.636	.425	Pooled OLS
Hausman	Random vs fixed effects	χ²(4) = 89.361	< .001	Fixed effects

Note. The Hausman test uses conventional covariance matrices; a negative statistic is set to zero. When it rejects, entity effects are correlated with the regressors and only fixed effects are consistent.

Table 3

Hausman test details

Variable	b (fixed)	B (random)	b − B	sqrt(diag(V_b − V_B))
Leverage	-5.9077	-3.4921	-2.4156	0.2073
Log_Size	0.7130	2.2820	-1.5690	0.1637
RnD_Intensity	16.2880	34.7101	-18.4220	1.7326
GDP_Growth	0.4327	0.4280	0.0047	—

Note. Variances in the last column use the standard errors selected above.

Interpretation

The tests point to the fixed effects model (Hausman p < .001). In that model, significant regressors are Leverage (-5.908), Log_Size (0.713), RnD_Intensity (16.288), GDP_Growth (0.433).

A Hausman test favoured fixed effects, χ²(4) = 89.36, p < .001.

How to cite this tool

Using these results in a thesis, journal article, report or class assignment? Please cite this free tool so others can find it. It keeps the site free.

APA 7

Xavier, B. (2026). Hausman Test: Fixed vs Random Effects [Web application]. FreeStatisticalTools.com. https://freestatisticaltools.com/hausman-test-fixed-vs-random-effects/

BibTeX

@software{xavier_hausmantest_2026,
  author = {Xavier, Bibin},
  title = {Hausman Test: Fixed vs Random Effects},
  publisher = {FreeStatisticalTools.com},
  year = {2026},
  url = {https://freestatisticaltools.com/hausman-test-fixed-vs-random-effects/},
  note = {Web application}
}

Suggested sentence for your methods section: “The analysis was carried out with the free Hausman Test: Fixed vs Random Effects tool on FreeStatisticalTools.com (Xavier, 2026).”

What is the Hausman test?

The Hausman specification test compares the fixed effects and random effects estimates of a panel model. Under the null hypothesis both are consistent and random effects is efficient; a significant difference means only fixed effects is consistent.

When to use the Hausman test

Run it whenever you must justify fixed or random effects in a panel study.

Assumptions

Random effects must be efficient under the null for the classic test; with heteroskedasticity, prefer clustered inference and a robust (Mundlak) test.

How to use this tool

Enter your data. Paste columns from Excel, Google Sheets or SPSS with variable names in the first row, or upload a CSV or Excel file. The example data shows the expected layout.
Choose the variables. Select the entity (panel) identifier, time variable, dependent variable and regressors.
Set the options. Fill in standard errors.
Read the results. Results update automatically. Read the plain-English interpretation, then use Copy table to paste APA-style tables into Word and Copy for the results sentence.

How to report the Hausman test in APA style

“The Hausman test rejected random effects, χ²(4) = 31.2, p < .001, so fixed effects were used.”

Frequently asked questions

How do I interpret the Hausman test?

A p-value below .05 rejects random effects in favour of fixed effects; a larger p-value means random effects can be used.

What if the Hausman statistic is negative?

This can happen in finite samples when the variance difference is not positive definite; the statistic is then set to zero and random effects is not rejected.

Is the Hausman test valid with clustered errors?

The classic test assumes random effects is fully efficient. With heteroskedastic or serially correlated errors, a cluster-robust Mundlak test is preferred.

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