Solved – interpretation of Hausman test results

datasethausmanhypothesis testingpanel data

I used Hausman test in R in order to decide whether I should use fixed effects or random effects model. This is the result I got:

Hausman Test

data: Deviation ~ Concentration
chisq = 1.721, df = 1, p-value = 0.1896
alternative hypothesis: one model is inconsistent

I would appreciate some help in interpreting this result (I have not studied Statistics ever, and I am yet facing this challenge), and which model should I use – fe or re?

thank you!!

Best Answer

When the $p$-value is low, commonly less than $0.05$, the $H_0$ must go!

In short the Hausman test (sometimes also called Durbin--Wu--Hausman test) in R assumes $H_{0}$ is that the preferred model is random effects, i.e. no significant correlation vs. the alternative, $H_{a}$, the fixed effects, i.e. whether the errors ($\mu_i$) are correlated with the regressors, see see Section 4.3 in Baltagi (2005).

Running ?plm::phtest in R would give you further details and refrences.

Baltagi (2005), Econometric Analysis Of Panel Data

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Solved – Panel regression: what to do when Hausman test fails and want to keep time invariant regressors

The following not-yet-published paper is, in my opinion, an excellent introduction and answer to the problem you bring up:

http://polmeth.wustl.edu/media/Paper/FixedversusRandom_1_2.pdf

To summarize, you can still proceed with the random effects approach, but you must first modify the model to account for the fact that the within-cluster and between-cluster effects differ (i.e., what the Hausman test indicates). You can do this by adding the cluster means of your predictor as a separate predictor in the model, and then optionally also applying within-cluster centering to the original predictor. The details of this procedure and the resulting interpretations are discussed at some length in the paper linked above.

Solved – Hausman test with or without covariates

The choice between FE and RE models depends on the focus of the statistical inference. The FE model is an appropriate specification if we are focusing on a specific set of $N$ individuals (say, $N$ firms or $N$ OECD countries, or $N$ American states) and our inference is restricted to the behavior of this set of individuals. The RE model is an appropriate specifiction if we are drawing $N$ individuals randomly from a large population and are trying to make inferences about that population (see Baltagi, Econometric Analysis of Panel Data, 2008, §§2.2-3). The Hausman test can't say anything about your focus.

The Hausman test is asymptotically equivalent to a standard Wald test for the omission of $\tilde{\mathbf{X}}$, a matrix of deviations from individual means (see Baltagi, 2008, §4.3). In other words, given the model $$y_{it}=\mathbf{x}_{it}\boldsymbol{\beta}+\mu_i+u_{it}\tag{1}$$ one can split $\mathbf{x}_{it}$: $$y_{it}=(\bar{\mathbf{x}}_i+\tilde{\mathbf{x}}_{it})'\boldsymbol{\beta}+\mu_i+u_{it}\tag{2}$$ where $\bar{\mathbf{x}}_i$ is the vector of individual time-invariant means for the $i$th individual and $\tilde{\mathbf{x}}_{it}=\mathbf{x}_{it}-\bar{\mathbf{x}}_i$. Further, one can give separate parameters $\boldsymbol{\beta}_1$ to the individual means and $\boldsymbol{\beta}_2$ to the deviation variables: $$y_{it}=\bar{\mathbf{x}}_i'\boldsymbol{\beta}_1+\tilde{\mathbf{x}}_{it}'\boldsymbol{\beta}_2+\mu_i+u_{it}\tag{3}$$ $\boldsymbol{\beta}_1$ is a between regression coefficient, while $\boldsymbol{\beta}_1$ is the within (FE) regression coefficient. The Hausman test is based on $\hat{\boldsymbol{\beta}}_{RE}-\hat{\boldsymbol{\beta}}_{FE}$, but can equivalently be based on $\hat{\boldsymbol{\beta}}_1-\hat{\boldsymbol{\beta}}_2$ (see Baltagi, 2008, §4.3).

As to correlation, some variables in $\mathbf{X}$ may be correlated with $\boldsymbol{\mu}$, but $\tilde{\mathbf{x}}_{it}$ is orthogonal to $\mathbf{1}\mu_i$ ($\mathbf{1}$ is a vector of ones) for all $i$. Thus:

under a FE framework, the time-invariant terms $\bar{\mathbf{x}}_i'\boldsymbol{\beta}_1$ and $\mu_i$ are swept out and one gets unbiased and consistent estimates for $\boldsymbol{\beta}_2$;
under a RE framework,
- if one estimates models $(1)$ or $(2)$ the implicit assumption $\hat{\boldsymbol{\beta}}_1=\hat{\boldsymbol{\beta}}_2$ doesn't hold and the Hausman test fails;
- if one estimates model $(3)$, then $\hat{\boldsymbol{\beta}}_2$ is unbiased and consistent (it is exactly identical to $\hat{\boldsymbol{\beta}}_{FE}$) and one can ignore or suppress $\bar{\mathbf{x}}_i'\boldsymbol{\beta}_1$, or use $\bar{\mathbf{x}}_i'\boldsymbol{\beta}_1+\mu_i$ to model the random intercept (see, e.g., Snijders and Bosker, Multilevel Analysis, 2012, chap. 4).

In brief, you can estimate a RE model which passes the Hausman test by just splitting your $\mathbf{X}$ matrix into its individual time-invariant means $\bar{\mathbf{X}}$ and the within-individual time-varying deviations $\tilde{\mathbf{X}}$ (see here for a simple example, additional details and references). I'd say that such a coherent approach would be better than an eventual and questionable 'mixture' of consistent and inconsistent estimates.

Best Answer

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Solved – Hausman test with or without covariates

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