Power of Some Tests of Heteroscedasticity: Application to Cobb-Douglas and Exponential Production Function

Transcription

1 Internatonal Journal of Statstcs and Applcatons 07, 7(6): 3-35 DOI: 0.593/j.statstcs Power of Some Tests of Heteroscedastcty: Applcaton to Cobb-Douglas and Exponental Producton Functon Iyabode Favour Oyenuga,*, Benjamn Agboola Oyejola Department of Mathematcs and Statstcs, The Polytechnc, Ibadan, Ngera Department of Statstcs, Unversty of Ilorn, Ngera Abstract There are several methods for detectng presence of heteroscedastcty n a sngle equaton econometrcs non-lnear model. Ths paper compares the power of Breusch-Pagan, Whte, Glejser, Park and Goldfeld-Quandt n testng for homoscedastcty when usng Cobb-Douglas and Exponental producton functon non-lnear models. The two nonlnear models were transformed to an ntrnscally lnear model through the natural logarthms. The sample szes of 0 and 30, 50 and 00, 50 and 00 for small, medum and large sample szes, respectvely and 0,000 replcatons. The levels of heteroscedastcty ntroduced were 0., 0.5 and 0.9 for mld, moderate and severe heteroscedastcty respectvely. The tests were carred out at 0.0 and 0.05 levels of sgnfcance. The results showed that as the sample sze ncreases, for every level of heteroscedastcty, the power of the Glejser and Park ncreases n detectng heteroscedastcty compare to other tests. The asymptotc behavor of the power of the test showed that Glejser and Park tests are more consstent and effcent. Keywords Intrnscally nonlnear model, Cobb-Douglas, Exponental, Power of the test. Introducton There are certan crcumstances n whch the assumpton of constant error varance or homoscedastcty n the lnear model s not tenable. In testng for heteroscedastcty, there s need to have a relatvely large samples n order to observe the problems of such errors and evaluate ther behavors. Researchers have observed that heteroscedastcty s usually found n cross sectonal data. For example, one expects to observe greater varaton n the profts of large frms than n those of small ones. The varance of profts mght also depend on product dversfcaton, research and development expendture and ndustry characterstcs and may therefore vary across frms of smlar szes. When analyzng famly spendng patterns, we fnd that there s greater varaton n expendture on certan commodty groups among hgh-ncome famles than low ones due to the greater dscreton allowed by hgher ncome [Pras and Houthakker (995)]. Goldfeld-Quandt (965) observed that the power of ther test depend on (a) the number of centrally omtted observatons c and (b) the nature of the sample values of the varable used as the deflator. They also observed that, for very large valves of c, the power of ther test s rather weak. Cook and Wesberg (983), n ther smulaton study, * Correspondng author: yaboadebayo00@yahoo.com (Iyabode Favour Oyenuga) Publshed onlne at Copyrght 07 Scentfc & Academc Publshng. All Rghts Reserved nvestgated the χ approxmaton of ther test. They showed that ther test s superor to Ancombe (96) and Bckel (978) because ther χ approxmaton of the samplng dstrbuton s poor. Kyung (000) presented a paper on Robustfyng Glejser test of Heteroscedastcty where they proposed a smple modfcaton to Glejser tests to make t more robust to dsturbances wth asymmetrc densty wth flat tal. Ths problem had earler been studed by Koenker and Bassett (98) and Newey and Powell (987). The later n partcular focused ther attenton on the test proposed by Glejser (969) whch test whether the regresson of the absolute values of resduals are correlated wth some other varables. Newey and Powell (987) presented a more elaborate test whch s more powerful than the squared resdual based test. Glejser test revsted by Machhado and Slva (000) showed that the proposed test statstcs are asymptotcally vald even when the dsturbances are not symmetrcally dstrbuted. Xu (006) presented new nonparametrc tests for heteroscedastcty n nonlnear and nonparametrc regresson models. Monte Carlo smulatons showed that, when the parametrc tests msspecfy the form of heteroscedastcty, the non parametrc tests have much hgher power aganst heteroscedastcty than the parametrc tests. Thus, the non parametrc tests are most useful when the form of heteroscedastcty cannot be specfed wth certanty or when the form s too complcated to be specfed parametrcally.

2 3 Iyabode Favour Oyenuga et al.: Power of Some Tests of Heteroscedastcty: Applcaton to Cobb-Douglas and Exponental Producton Functon. Methodology The Nonlnear Models Investgated. The Cobb-Douglass producton functon, developed by Charles Cobb and Paul Douglass (98) s a functon showng the relatonshp between the output and the combnaton of factors, or nputs, used to obtan t. It s of the form: 3 Y = θ K θ L θ (.) The Cobb-Douglass Producton Functon wth multplcatve error term can be represented as θ θ3 u Y= θ K Le (.) θ3 where Y s a dependent varable, θ s the ntercept, θ and are the regresson coeffcents, K and L a re ndependent varables and u s the error term. The model n (.) can be transformed to lnear model by takng the natural logarthms of both sdes of the equaton to obtan a regresson model of the followng form: ln Y = ln θ + θ ln K + θ ln L + u (.3) ( ) ( ) ( ) ( ) 3. The Exponental Producton Functon can be expressed n the form: K 3L Y = θ e θ e θ (.4) The Exponental Producton Functon wth multplcatve error term can be expressed as θk θ3l Y = θ e e u (.5) where Y s a dependent varable, θ s the ntercept, θ and are the regresson coeffcents, K and L are ndependent varables and u s the error term. By takng the natural logarthms of both sdes of (.5), we have ln Y = ln θ + Kθ + Lθ + ln u (.6) 3. Smulaton ( ) ( ) 3. A sample nfected wth heteroscedastcty usng unform dstrbuton to generate data for Captal (K), Labour (L) and Output (Y). The study used fxed values for θ =0., θ =0.5 and θ 3 =0.3 for the two models.. The set of parameter estmates obtaned were used to compute the resduals whch represented the dependent varable for the auxlary regresson. The error structure data were drawn from a normal dstrbuton wth mean, zero and varance, σ.. The sample szes for the smulaton were 0 and 30, 50 and 00, 50 and 00 for small, medum and large sample szes respectvely. Each experment was replcated n 0,000 tmes. v. The levels of heteroscedastcty,, ntroduced were 0., 0.5 and 0.9 for mld, moderate and severe heteroscedastcty respectvely. v. The Harvey (976) model of multplcatve heteroscedastcty error structure was assumed wth general formulaton ( ) σ = σ E Y σ and ( X X ) k k ( ) = + + σ β β = σ exp, q Where are both unknown real constants, whch determne the levels or degree of heteroscedastcty. The followng fve tests were consdered n ths paper namely; Breusch-Pagan (979), Whte (980), Glejser (969), Park (966) and Goldfeld-Quandt (965). 4. Results and Dscusson Table shows the power of the tests for Cobb-Douglas Model at α = 0.0. There s no result for sample sze 0 at all levels of heteroscedastcty ( ) for all the tests due to nsuffcent sample sze. The result revealed that at every level of heteroscedastcty, as the sample sze ncreases, the power of the test for Glejser and Park test mproves and remans powerful. It can also be seen that at = 0.5, the power of the test for Whte also mproved. When the sample sze s equal to 50, the power of all tests mproved at both =0.5 and 0.9. Goldfeld-Quandt test s not a relable test at all levels of heterogenety and sample szes. Table shows the power of the tests for Cobb-Douglas Model at α = The result shows that at every level of heteroscedastcty, as the sample sze ncreases, Glejser and Park tests are powerful. It can also be seen that at = 0. when n =50 and 50 all the powers of test mproved except Goldfeld-Quandt test, lkewse, at =0.9 when n=50. Table 3 shows the power of the tests for Exponental Model at α = 0.0. The result shows that at every level of heteroscedastcty, as the sample sze ncreases, the power of the test for Glejser and Park test s very hgh. The power of the test for Breusch-Pagan test also mproves as sample sze ncreases when = 0. expect when n=00. The power of all the tests mproved when =0. at n=30, 00 and 50 also when =0.9 at n=30 and 50. Goldfeld-Quandt test s stll not a relable test at all levels of heterogenety and sample szes.

3 Internatonal Journal of Statstcs and Applcatons 07, 7(6): Table. Power of the Tests for Cobb-Douglas Model at α = BREUSCH-PAGAN WHITE GLEJSER PARK GOLDFELD QUANDT BREUSCH-PAGAN WHITE GLEJSER PARK GOLDFELD QUANDT BREUSCH-PAGAN WHITE GLEJSER PARK GOLDFELD QUANDT Table. Power of the Tests for Cobb-Douglas Model at α = BREUSCH-PAGAN WHITE GLEJSER PARK GOLDFELD QUANDT BREUSCH-PAGAN WHITE GLEJSER PARK GOLDFELD QUANDT BREUSCH-PAGAN WHITE GLEJSER PARK GOLDFELD QUANDT

4 34 Iyabode Favour Oyenuga et al.: Power of Some Tests of Heteroscedastcty: Applcaton to Cobb-Douglas and Exponental Producton Functon Table 3. Power of the Tests for Exponental Model at α = BREUSCH-PAGAN WHITE PARK GOLDFELD QUANDT BREUSCH-PAGAN WHITE PARK GOLDFELD QUANDT BREUSCH-PAGAN WHITE PARK GOLDFELD QUANDT Table 4. Power of the Tests for Exponental Model at α = BREUSCH-PAGAN WHITE PARK GOLDFELD QUANDT BREUSCH-PAGAN WHITE PARK GOLDFELD QUANDT BREUSCH-PAGAN WHITE PARK GOLDFELD QUANDT

5 Internatonal Journal of Statstcs and Applcatons 07, 7(6): Table 4 shows the power of the tests for Exponental Model at α = The result shows that at every level of heteroscedastcty, as the sample sze ncreases, the power of the test for Glejser and Park test s powerful. The power of all the tests mproves at =0.5 when n=00 and 50, so also, at =0.9 when n=30 and 50 expect for Goldfeld-Quandt test. 5. Conclusons The results obtaned from the analyss conform to the asymptotc property as sample szes ncreases, the power of the two tests namely, Glejser and Park, s more powerful at every level of heteroscedastcty for the two models. It can therefore be concluded from ths study that Glejser and Park tests should be used for detectng the heteroscedastcty when nonlnear models are used n analyzng data. REFERENCES [] Ancombe, F. (96) Examnaton of Resduals Proc. 4 th Berkeley Symp., -36. [] Barro, R and Sala--Martn, X (004) Economc Growth. Cambrdge MIT Press. [3] Bckel, P. (978) Usng Resduals Robustly: Tests for Heteroscedastcty Nonlnearty Ann. Statstcs, 6, [4] Blaug, M. (985) Economc Theory n Retrospect, Fourth Edton, Cambrdge Unversty Press, Cambrdrge. [5] Breusch, T.S., and Pagan, A.R (979) A Smple Test for Heteroscedastcty and Random Coeffcent Varaton, Econometrca, 47, [6] Carmelo, G and Subhash, C. S (008) Jacknfe tests for Heteroscedastcty n the General Lnear Model, Australan and New Zealand Journal of Statstcs, 30(), [7] Cook, D. R. and Wesberg, S. (983) Dagonostcs for Heteroscedastcty n Regresson, Bometrka, 70, -0. [8] Draper, N. R. and Smth, H. (98) Appled Regresson Analyss, Second Edton, Wley, New York. [0] Goldberger, A.S. (96) Best Lnear Unbased Predcton n the Generalsed Lnear Regresson Model, Journal of Amercan Statstcal Assocaton, 57, [] Goldfeld, S. M., and Quandt, R.E. (965) Some Test of Heteroscedastcty Journal of the Amercan Statstcs Assocaton, 64, [] Harvey, A. C. (976) Estmatng Regresson Models wth Multplcatve Heteroscedastcty, Econometrca, 44, [3] Johnston, J. (984) Econometrc Methods, Thrd Edton, MacGraw-Hll, New York. [4] Koenker, R. and Basset, G. (98) Robust Tests for Heteroscedastcty Based on Regresson Quantles, Econometrca, 50 (), [5] Kyung, S.I. (000) Robustfyng Glejser Test of Heteroscedastcty, Journal of Econometrcs, 97, [6] Lloyd, P. J. (969) Elementary Geometrc/Arthmetrc Seres and Early Producton Theory, Journal of Poltcal Economy, 77, -34. [7] Machado, J. A. F. and Slva, J. M. C. (000) Glejser s Test Revsted, Journal of Econometrcs, 97, [8] Neter, J., Wasserman, W and Kutner, M. H (985) Appled Lnear Regresson Models, Irwn, Homewood, Illnos. [9] Newey, W. K. and Powell, J. L (987) Asymmetrc Least Squares Estmaton and Testng, Econometrca, 55, [0] Park, R. E. (966) Estmaton wth Heteroscedastc Error Term, Econometrca, 34(4), 888. [] Peter, K. (99) A gude to Econometrcs, Thrd Edton, Blackwell Publshers, Oxford. [] Pras, S. J. and Houthakker, H. S (995) The Analyss of Famly Budgets, Cambrdge Unversty Press. [3] Whte, H. (980) A Heteroscedastcty-Consstent Covarance Matrx Estmator and A Drect Test for Heteroscedastcty, Econometrca, 48(4), [4] Xu, Z. (006) Testng Heteroscedastcty n Nonlnear and Nonparametrc Regressons wth an Applcaton to Interest Rate Volatlty, School of Economcs, Shangha Unversty of Fnance and Economcs. [9] Glejser, H. (969) A New Test of Heteroscedastcty, Journal of Amercan Statstcal Assocaton, 64,