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Intercept Homogeneity Test for Fixed Effect Models under Cross-sectional Dependence: Some Insights

Gopal K. Basak and Samarjit Das

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 χ2 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 DGPs; 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.

JEL Classification: C33; C12

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:β^FE is consistent under both H0A and H0B.

Proof:

β^FE=(XMX)1XMY=(XMX)1XM(D¯θ+Xβ+ϵ)=β+(XMX)1XMD¯θ+(XMX)1XMϵ.

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

Note that, (XMX)−1X′MD̅=0, where D̅=[11111…1]′=1NT. This is because D̅ 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,

β^FE=β+(XMX)1XMϵ=β+[(XMX)/NT]1(XMϵ/NT).

Now we take plim on both sides and we have

plimβ^FE=β+plim[(XMX)/NT]1plim(XMϵ/NT)=β+[plim(XMX)/NT]1plim(XMϵ/NT)=β+R1plim(XMϵ/NT).

Now limE(XMϵ/NT)=0 by Assumption 2. Again, by Assumption 5,

||V(XMϵ/NT)||=(1/N2T2)||E[XMITΩMX]||1/N2T2E||XMITΩMX||1/N2T2||Ω||E||XX||.

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

So we have plimβ^FE=β under both the Null.

Result 2:β^FE is consistent under both H1A and H1B.

Proof:

β^FE=(XMX)1XMY=(XMX)1XM(Dμ+Xβ+ϵ)=β+(XMX)1XMDμ+(XMX)1XMϵ.

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

Hence plimβ^FE=β under the alternative hypotheses.

Result 3:β^ols is consistent under both H0A and H0B.

Proof:

β^ols=(XM¯X)1XM¯Y=(XM¯X)1XM¯(D¯θ+Xβ+ϵ)=β+(XM¯X)1XM¯D¯θ+(XM¯X)1XM¯ϵ.

Now it can be easily seen from the form of M̅=INTD̅(D̅′D̅)−1D̅′ that M̅D̅=0.

Now, β^ols=β+(XM¯X)1XM¯ϵ. It is also easy to see from Assumption 1 that W=plim[(XM̅X)/NT] is a finite matrix.

plimβ^ols=β+plim(XM¯X)1XM¯ϵ=β+plim(XM¯X/NT)1(XM¯ϵ/NT)=β+W1plim(XM¯ϵ/NT).

Now limE(XM̅ε/NT)=0, by Assumption 2. Again, by Assumption 5,

||V(XM¯ϵ/NT)||=1/N2T2||EX(XM¯ITΩM¯X)||1/N2T2EX||XM¯ITΩM¯X||1/N2T2||Ω||EX||XX||.

Hence lim(V(Xϵ/NT))→0 as T→∞, both under weak and strong dependence. For weak dependence, lim(V(XM̅ϵ/NT))→0 as either T and / or N→∞.

Hence plim(XM̅ϵ/NT)=0. So we have plimβ^ols=β under both the Null.

Result 4:β^ols is inconsistent under both H1A and H1B.

Proof:

β^ols=(XM¯X)1XM¯Y=(XM¯X)1XM¯(Dμ+Xβ+ϵ)=β+(XM¯X)1XM¯Dμ+(XM¯X)1XM¯ϵ=β+(XM¯X/NT)1(XM¯Dμ/NT)+(XM¯X/NT)1(XM¯ϵ/NT).

plimβ^ols=β+plim[(XM¯X/NT)1(XM¯Dμ/NT)]+plim[(XM¯X/NT)1(XM¯ϵ/NT)]=β+W1plim(XM¯Dμ/NT)+W1plim(XM¯ϵ/NT).

Now plim(XM̅ϵ/NT)=0 as we have already seen in Result 1. But plim(XM¯Dμ/NT)=plim1N(X¯iX¯¯)(μiμ¯)0, hence the proof.

Proof of Theorem 2

We will apply conditional Liapounov CLT. In both the cases, conditioning is on {Xt}. Define a matrix based on fourth moments and cross moments whose dimension is N2×N2. It is given by VF=E((εtεt)(εtεt)). Note, trace(VF)λmax(VF)i,jΩij2λmax(Ω2)=λmax2(Ω). Hence for strong dependence O(λmax(VF))=O(N2), whereas, for weak dependence, in general, O(λmax(VF))≥O(N).

Thus,

EX(ldtεtεtdtl)2(ldtdtl)2λmax(VF)

where

(ldtdtl)2=O(1NT2)2,=O(1N2T3).

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

(ldtΩdtl)2={(ldtdtl)O(λmax(Ω)}2,=O(λmax(Ω)NT)2,

and hence,

EX(ldtεtεtdtl)2(ldtΩdtl)2=O(1T).

For weak dependence,

(ldtΩdtl)2{(ldtdtl)(λmin(Ω)}2,=O(1NT)2,

since λmin(Ω)=O(1).

Therefore,

EX(ldtεtεtdtl)2(ldtΩdtl)2=O(NT),

as λmax(VF)=O(N).

Proof of Theorem 3

To prove this theorem, let us first concentrate on EX||dt(etϵt)||2.

EX||dt(etϵt)||2{||dtdt||EX||etϵt||2}.

Let us denote, for notational simplicity, X˜t as a 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 ||et–εt||2.

||etϵt||2=[{X˜t(ψ^ψ)}{X˜t(ψ^ψ)}],={(ψ^ψ)(X˜tX˜t)(ψ^ψ)},{(ψ^ψ)(ψ^ψ)}||(X˜tX˜t)||e,=||(ψ^ψ)||2||(X˜tX˜t)||e,

where, ψ^ is the estimator of intercept and the β^FE. Note that ||dtdt||=Op(1NT2),||(X˜tX˜t)||e=Op(N). For strong dependence, ||(ψ^ψ)||2=Op(1T); and that for weak dependence is, Op(1NT), by Theorem 1.

Hence, for weak dependence,

EX||dt(etϵt)||2Op(1NT2)

and, for strong dependence,

EX||dt(etϵt)||2Op(1T2).

Thus, if dtεtεtdt converges so is dtetetdt and to the same matrix.

Now let us concentrate on EX|l(dtϵtϵtdt)llVllVl|r for r≤2. Note that EX|l(dtϵtϵtdt)llVllVl|r=1(lVl)rEX|(ldt(ϵtϵtΩ)dtl)|r. By Burkholder inequality,

EX|l(dtϵtϵtdt)llVllVl|rCr(lVl)rEX({ldt(ϵtϵtΩ)dtl}2)r/2,Cr(lVl)r(EX{ldt(ϵtϵtΩ)dtl}2)r/2,   asr/21,=Cr(lVl)r(Var{ldtϵtϵtdtl})r/2,Cr(lVl)r(EX{ldtϵtϵtdtl}2)r/2,

where Cr is independent of T. For r=2, we have already shown in Theorem 2 that

Cr(lVl)r(EX{ldtϵtϵtdtl}2)r/2={O(1T),for strong dependenceO(NT),for weak dependence.

Hence dtϵtϵtdtV in probability, conditionally on {X}, in the sense that,

l(dtϵtϵtdt)llVllVl0   in probability.

Therefore,

l(dtetetdt)llVllVl=lV¯llVl10   in probability,

uniformly in l(ll=1)

which implies ||V−1/2V̅V−1/2I||e→0 in probability.

Again,

||V||=||dtΩdt||||dtdt||||Ω||{Op(1NT) for weak dependence,Op(1T) for strong dependence.

Also,

EX||V¯||=EX||dtetetdt||||dtdt||EX||etet||||dtdt||{EX{||(ψ^ψ)||2}||(X˜tX˜t)||e+||Ω||}{Op(1NT) for weak dependence,Op(1T) for strong dependence.

Hence by Theorem 2,

(β^FEβ^ols)(V1V¯1)(β^FEβ^ols)=(β^FEβ^ols)V1/2(IV1/2V¯1V1/2)V1/2(β^FEβ^ols)IV1/2V¯1V1/2e(β^FEβ^ols)V1(β^FEβ^ols)=V1/2V¯V1/2Ie×Op(1)=op(1)   for both weak and strong dependence.

Hence, MD and EMD have same limiting distribution.

Proof of Theorem 4

Let δμ=plim[(XM¯X/NT)1(XM¯Dμ/NT)]=W1plim1N(X¯iX¯¯)(μiμ¯) and δ^μ=(XM¯X)1XM¯Dμ. Under Alternative, by Result 3 of Theorem 1, β^FE=β+(XMX)1XMϵ, and by Result 4 of Theorem 1, β^ols=β+δ^μ+(XM¯X)1XM¯ϵ. Therefore, under Alternative,

β^FE[β^olsδ^μ]=t=1Tdtϵ.t,

as in the notation of equation (6).

Hence, (β^FEβ^ols+δ^μ)V1(β^FEβ^ols+δ^μ) is asymptotically χk2 as in Theorem 2.

Thus, as in Theorem 3, (β^FEβ^ols+δ^μ)V¯1(β^FEβ^ols+δ^μ) is asymptotically χk2. Therefore, (β^FEβ^ols)V¯1(β^FEβ^ols) is asymptotically non-central χk2 with the non-centrality parameter as δμV1δμ.

References

Aghion, P., E. Caroli, and Cecilia G-Penalosa. 1999. “Inequality and Economic Growth: The Perspective of the New Growth Theories.” Journal of Economic Literature 37: 1615–1660.10.1257/jel.37.4.1615Search in Google Scholar

Andrews, D. K. 2005. “Cross-section Regression with Common Shocks.” Econometrica 73 (5): 1551–1585.10.1111/j.1468-0262.2005.00629.xSearch in Google Scholar

Atkinson, A. B. 1970. “On the Measurement of Inequality.” Journal of Economic Theory 2: 244–263.10.1016/0022-0531(70)90039-6Search in Google Scholar

Balestra, P., and M. Nerlove. 1966. “Pooling Cross-Section and Time Series Data in the Estimation of a Dynamic Model: The Demand for Natural Gas.” Econometrica 34: 585–612.10.2307/1909771Search in Google Scholar

Baltagi, B. H. 1981. “Pooling: An Experimental Study of Alternative Testing and Estimation Procedures in a Two-way Error Component Model.” Journal of Econometrics 17: 21–49.10.1016/0304-4076(81)90057-9Search in Google Scholar

Baltagi, B. H. 2008. Econometric Analysis of Panel Data, 4th ed, Vol. 17, 21–49. Hoboken, NJ, USA: John Wiley & Sons.Search in Google Scholar

Baltagi, B. H., and A. Pirotte. 2010. “Panel Data Inference Under Spatial Dependence.” Economic Modelling 27: 1368–1381.10.1016/j.econmod.2010.07.004Search in Google Scholar

Baltagi, B. H., J. Hidalgo, and Q. Li. 1996. “A Nonparametric Test for Poolability Using Panel Data.” Journal of Econometrics 75: 345–367.10.1016/0304-4076(95)01779-8Search in Google Scholar

Bun, M. J. G. 2004. “Testing Poolability in a System of Dynamic Regressions with Nonspherical Disturbances.” Empirical Economics 29: 89–106.10.1007/s00181-003-0191-3Search in Google Scholar

Chudik, A., H. Pesaran, and E. Tosetti. 2011. “Weak and Strong Cross Section Dependence and Estimation of Large Panels.” The Econometrics Journal 14: 45–90.10.1111/j.1368-423X.2010.00330.xSearch in Google Scholar

Datt, G. 1998. Poverty in India and Indian States: An Update: FCND Discussion Paper No.47.Search in Google Scholar

Driscoll, J. C., and A. C. Kraay. 1998. “Consistent Covariance Matrix Estimation with Spatially Dependent Panel Data.” Review of Economics and Statistics 80: 549–560.10.1162/003465398557825Search in Google Scholar

Fama, E., and J. MacBeth. 1973. “Risk, Return, and Equilibrium: Empirical Tests.” Journal of Political Economy 81 (3): 607636.10.1086/260061Search in Google Scholar

Gregoire T. G. 1992. “The Error Rate of Tests for Regression Coincidence and Parallelism Under Unknown Multiplicative Heteroscedasticity.” Biometrical Journal 34: 193–208.10.1002/bimj.4710340209Search in Google Scholar

Hausman. J. A. 1978. “Specification Tests in Econometrics.” Econometrica 46: 1251–1272.10.2307/1913827Search in Google Scholar

Hobijn, B., and P. H. Franses. 2000. “Asymptotically Perfect and Relative Convergence of Productivity.” Journal of Applied Econometrics 15: 59–81.10.1002/(SICI)1099-1255(200001/02)15:1<59::AID-JAE544>3.0.CO;2-1Search in Google Scholar

Hsiao, C. 2003. Analysis of Panel Data, 2nd ed. Cambridge: Cambridge University Press.10.1017/CBO9780511754203Search in Google Scholar

Kumbhakar, S. C., and C. A. K. Lovell. 2000. Stochastic Frontier Analysis. Cambridge: Cambridge University Press.10.1017/CBO9781139174411Search in Google Scholar

Kuznets. 1955. “Economic Growth and Income Inequality.” American Economic Review 45: 1–28.Search in Google Scholar

Lipton, M., and M. Ravallion. 1995. “Poverty and Policy.” In Handbook of Development Economics, edited by J. Berhman and T. N. Srinivasan, Vol. 3B, 2551–2657. Amsterdam: North Holland.10.1016/S1573-4471(95)30018-XSearch in Google Scholar

Mundlak, Y. 1978. “On the Pooling of Time Series and Cross Section Data.” Econometrica 46: 69–85.10.2307/1913646Search in Google Scholar

Persson, T., and G. Tabellini. 1994. “Is Inequality Harmful for Growth?” American Economic Review 84: 601–621.Search in Google Scholar

Rodgers, J., W. A. Nicewander, and L. Toothaker. 1984. “Linearly Independent, Orthogonal, and Uncorrelated Variables.” The American Statistician 38 (2): 133–134.Search in Google Scholar

Scariano, S. M., and J. M. Davenport. 1987. “The Effects of Violations of Independence Assumptions in the One-Way ANOVA.” The American Statistician 41: 123–129.10.1080/00031305.1987.10475459Search in Google Scholar

Schmidt, P., and R. C. Sickles. 1984. “Production Frontiers and Panel Data.” Journal of Business and Economic Statistics 2: 367–376.10.1080/07350015.1984.10509410Search in Google Scholar

Talwar, P. P., and J. E. Gentle. 1978. “A Robust Test for the Coincidence of Regressions.” Journal of Statistical Computation and Simulation 7: 107–113.10.1080/00949657808810217Search in Google Scholar

Vogelsang. T. J. 2012. “Heteroskedasticity, Autocorrelation, and Spatial Correlation Robust Inference in Linear Panel Models with Fixed-effects.” Journal of Econometrics 166: 303–319.10.1016/j.jeconom.2011.10.001Search in Google Scholar

Wooldridge. J. M. 2010. Econometric Analysis of Cross Section and Panel Data. 2nd ed. Cambridge, MA: MIT Press.Search in Google Scholar

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

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