# Abstract

This paper develops a test for intercept homogeneity in fixed-effects one-way error component models assuming slope homogeneity. We show that the proposed test works equally well when intercepts are assumed to be either fixed (non-stochastic) or random. Moreover, this test can also be used to test for random effect vs. fixed effect although in the restrictive sense. The test is shown to be robust to cross-sectional dependence; for both *weak* and *strong* dependence. The proposed test is shown to have a standard *χ*^{2} limiting distribution and is free from nuisance parameters under the null hypothesis. Monte Carlo simulations also show that the proposed test delivers more accurate finite sample sizes than existing tests for various combinations of *N* and *T*. Simulation study shows that F-test is either over-sized or under-sized depending on the pattern of cross-sectional dependence. The performance of Hausman test (1978), on the other hand, is quite unstable across various *DGP*s; and empirical size varies from 0% to the nominal sizes depending on the structure of error variance-covariance matrix. The power of the proposed test outperforms the other two tests. It is worthwhile to mention that the power of our proposed test increases with *T* in contrast to that of Hausman test which is known to have no power as *T*→∞. An empirical illustration to examine the Kuznets’ U curve hypothesis with balanced panel data of Indian states is also provided. This empirical illustration points out the efficacy and the necessity of our robust test.

# Acknowledgments

The authors are grateful to Aman Ullah for his careful reading and constructive suggestions. The authors would also like to thank an anonymous referee for his/her insightful and constructive suggestions.

## Appendix

**Proof of Theorem 1**

Here we will use maximum eigenvalue norm.

**Result 1:***H*_{0A} and *H*_{0B}.

**Proof:**

from Assumption 1, we claim that *plim*(*X*′*MX*)/*NT*=*R*, i.e we claim that the second order moment of *MX* exists. Now note that, ||*X*′*MX*||=||*X*′*MM*′*X*||≤||*M*|| ||*X*′*X*||=||*X*′*X*||. Hence the matrix R is a finite matrix.

Note that, (*X*′*MX*)^{−1}*X*′MD̅=0, where *D*̅=[11111…1]′=1_{NT}. This is because *D*̅ is a linear combination of the columns of *D* and hence belongs to the column space of *D* and we know that MD=0.

Hence,

Now we take *plim* on both sides and we have

Now *limE*(*X*′*M*ϵ/*NT*)=0 by Assumption 2. Again, by Assumption 5,

Hence *lim*(*V*(*X*′*M*ϵ/*NT*))→0 as *T*→∞, both under weak and strong dependence. For weak dependence, *lim*(*V*(*X*′*M*ϵ/*NT*))→0 as either *T* and / or *N*→∞.

So we have

**Result 2:***H*_{1A} and *H*_{1B}.

**Proof:**

Here (*X*′*MX*)^{−1}*X*′MD=0, since MD=0 and *plim*(*X*′*MX*)^{−1}*X*′*M*ϵ=0, by similar arguments as stated in the previous proof both under weak and strong dependence. For weak dependence, *lim*(*V*(*X*′*M*ϵ/*NT*))→0 as either *T* and/or *N*→∞.

Hence

**Result 3:***H*_{0A} and *H*_{0B}.

**Proof:**

Now it can be easily seen from the form of *M*̅=*I*_{NT}–*D*̅(*D*̅′*D*̅)^{−1}*D*̅′ that *M*̅*D*̅=0.

Now, *W*=*plim*[(*X*′*M*̅*X*)/*NT*] is a finite matrix.

Now *limE*(*X*′*M*̅ε/*NT*)=0, by Assumption 2. Again, by Assumption 5,

Hence *lim*(*V*(*X*′*M̅*ϵ/*NT*))→0 as *T*→∞, both under weak and strong dependence. For weak dependence, *lim*(*V*(*X*′*M*̅ϵ/*NT*))→0 as either *T* and / or *N*→∞.

Hence *plim*(*X*′*M*̅ϵ/*NT*)=0. So we have

**Result 4:***H*_{1A} and *H*_{1B}.

**Proof:**

Now *plim*(*X*′*M*̅ϵ/*NT*)=0 as we have already seen in Result 1. But

**Proof of Theorem 2**

We will apply conditional Liapounov CLT. In both the cases, conditioning is on {*X*_{t}}. Define a matrix based on fourth moments and cross moments whose dimension is *N*^{2}×*N*^{2}. It is given by *O*(*λ*_{max}(*V*_{F}))=*O*(*N*^{2}), whereas, for weak dependence, in general, *O*(*λ*_{max}(*V*_{F}))≥*O*(*N*).

Thus,

where

For strong dependence, if the eigenvector corresponding to the largest eigenvalue of Ω belongs to the row-space of

and hence,

For weak dependence,

since *λ*_{min}(Ω)=*O*(1).

Therefore,

as *λ*_{max}(*V*_{F})=*O*(*N*).

**Proof of Theorem 3**

To prove this theorem, let us first concentrate on

Let us denote, for notational simplicity, *N*×*k*+1 matrix of all regressors including the intercept variable. Similarly, *ψ* is the *k*+1×1 vector of regression parameter including the intercept. First let us concentrate on the term ||*e*_{t}–ε_{t}||^{2}.

where,

Hence, for weak dependence,

and, for strong dependence,

Thus, if

Now let us concentrate on *r*≤2. Note that

where *C*_{r} is independent of *T*. For *r*=2, we have already shown in Theorem 2 that

Hence *X*}, in the sense that,

Therefore,

uniformly in *l*(*l*′*l*=1)

which implies ||*V*^{−1/2}*V*̅*V*^{−1/2}–*I*||_{e}→0 in probability.

Again,

Also,

Hence by Theorem 2,

Hence, MD and EMD have same limiting distribution.

**Proof of Theorem 4**

Let

as in the notation of equation (6).

Hence,

Thus, as in Theorem 3,

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## Supplemental Material:

The online version of this article (DOI: 10.1515/jem-2015-0004) offers supplementary material, available to authorized users.

**Published Online:**2016-4-19

**Published in Print:**2017-1-1

©2017 Walter de Gruyter GmbH, Berlin/Boston