Nonparametric Test for Translog Specification of Production Function in Japanese Manufacturing Industry

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1 Noparametric Test for Traslog Specificatio of Productio Fuctio i Japaese Maufacturig Idustry Y. Koishi a ad Y.Nishiyama b a Graduate School of Ecoomics, Nagoya Uiversity b Graduate School of Evirometal Studies, Nagoya Uiversity,Huro-cho Chikusa Nagoya, Japa Abstract: Begiig with a series of pioeerig work by Cobb ad Douglas, estimatio of productio fuctio has bee oe of the mai issues i empirical ecoomics ad ecoometrics i various aspects. The fuctioal form they used is called Cobb-Douglas productio fuctio. It is later geeralized to so-called traslog productio fuctio which has a more flexible form to describe the relatio betwee the output ad iput levels. It is also aalytically coveiet for obtaiig factor demad fuctios or cost fuctio so that it has bee widely used i both macroecoomic ad microecoomic empirical studies. However, it is well kow that statistical aalysis leads to icorrect coclusios i geeral if the specified parametric model is wrog. This paper tests for the traslog specificatio of productio fuctio for Japaese maufacturig idustry. We apply the oparametric misspecificatio test by Hog ad White [995]. It is a cosistet test havig otrivial power agaist local alteratives. We test the ull hypothesis of traslog specificatio usig cross sectio data of Japaese maufacturig firms listed i Tokyo Stock Exchage market. We also test the Cobb-Douglas specificatio by the same method. Keywords: Traslog productio fuctio; Noparametric specificatio test; Japaese maufacturig idutry. INTRODUCTION Productio techology of a firm or a ecoomy is characterized by its productio fuctio (or cost fuctio alteratively) so that if oe would like to ivestigate ecoomic aspects associated with producer s behaviour, we eed to study the productio fuctio. Begiig with a series of pioeerig work by Cobb ad Douglas [98, 948] ad Douglas [97, 934], estimatio of productio fuctio has bee oe of the mai issues i empirical ecoomics ad ecoometrics. The productio fuctio they cosider has the form, Y = AK α L β (.) where Y, K, L idicate the output level, capital ad labour iputs respectively ad A, α, β are parameters determiig the productio techology. Ifα + β =, this productio techology is said to be costat returs to scale. Takig the logarithm, we obtai logy = log A + α log K + β log L. (.) (.) or equivaletly (.) is called the Cobb-Douglas productio fuctio. Christese, Jorgeso ad Lau [973] cosider a extesio of the Cobb-Douglas productio fuctio to the followig more geeral ad flexible fuctioal form. logy = α + αk logk + αl log L + αkk LL (.3) ( logk ) + α ( log L) + α logk logl They call it the trascedetal logarithmic (traslog) productio fuctio. Productio fuctio estimatio itself could be of direct iterest, but it is ofte the case that we are more iterested i some other ecoomic quatities associated with productio fuctios. For example, researchers studyig ecoomic growth may like to quatitatively determie the techological progress, while labour ecoomists may be iterested i effects of huma capital to productivity. Public ecoomists may wish to measure the margial effect of social capital stock. Either Cobb-Douglas or traslog fuctio has bee employed for these purposes ad used to estimate the productio fuctio. Solow [957] looks at the costat term of the Cobb-Douglas productio fuctio to calculate the total factor productivity (TFP). Recet developmets o this are i Romer [986], Lucas [988], Makiw, Romer ad Weil [99] ad Behabib ad Spiegel [994] amog others. KL 597

2 They have iflueced the subsequet research i empirical macro ecoomic studies. Temple [999] gives a detailed survey about the ew growth theory ad the empirical work based o it. There is a cosiderable umber of studies focusig o the compariso of productivity amog coutries or idustries usig maily the traslog productio fuctio, e.g. Jorgeso ad Nishimizu [978]. There are some research o productivity i Japa or its compariso with other coutries as i e.g. Jorgeso, Kuroda ad Nishimizu [987]. There is also a vast umber of studies o the effects of huma capital ad social capital stock. To the best of our kowledge, Nerlove [963] is the first empirical study which uses cross sectio data of idividual firms to ivestigate productio techoology. He estimates Cobb-Douglas cost fuctio usig 45 observatios o America electric geeratig compaies to aalyze U.S. electric power idustry, while Christese ad Greee [976] ad others exted it to employ the traslog form. To examie productio efficiecy, scale ecoomy, ad techological chages, productio frotier approach is developed by Farrell [957]. Empirical applicatios based o it are foud i Aiger ad Chu [968] ad others. The early studies assume that the frotier is determiistic. Aiger et al. [977] exted it to a stochastic frotier model, which is maily estimated usig pael data. See Battese ad Coelli [988], Kumbhakar [987, 988, 99] amog others. Various extesios allowig for time- ad firm-specific effects are cosidered i Corwell et al. [99], ad Kumbhakar [99, 99]. For this kid of aalysis, Cobb-Douglas ad traslog models are used. Parametric models such as (.) ad (.3) have bee widely used i a lot of empirical studies as see i the above. However, it is well kow that if the employed parametric model is icorrect i fact, statistical ifereces based o it is wrog i geeral. I the curret cotext, for example, if we specify the productio fuctio as the traslog, but it is icorrect i fact, the TFP calculated from the estimates will be differet from the true value. We, therefore, would like to test for the fuctioal specificatio (.) ad (.3) i view that they are very commoly used. There are a umber of specificatio tests we ca use. A classical method is Ramsey s RESET test. It assumes that the regressio fuctio icludes higher order powers of the ull regressio fuctio uder the alterative. There have bee developped some oparametric specificatio tests which eed ot specify the alterative fuctioal form such as Bieres [98], Bieres ad Ploberger [987], Hog ad White [995] ad Hitomi [] amog others. They are compared i Hitomi [] uder various alteratives i small sample by Mote Carlo, ad it is show that Hog ad White test has a relatively better power amog well-established alteratives i small sample. We are cocered with the productio fuctio of Japaese maufacturig firms. We test the ull of (.) or (.3) agaist the oparametric alterative usig the aual fiacial report of Japaese maufacturig firms listed i Tokyo Stock Exchage market, divisio oe, for the years from 965 to. We observe that both fuctioal forms are reected after aroud 98, while they are ot reected before the. The followig sectio shows prelimiary results of iferece o the productio fuctio based o Cobb-Douglas ad Traslog specificatio ad gives some commets. Sectio 3 reviews some oparametric specificatio methods, while Sectio 4 gives results of the Hog ad White oparametric specificatio test for the empirical data, while cocludig remarks are i Sectio 5.. PRELIMINARY STATISTICAL ANALYSIS. Estimatio We implemeted cross sectioal OLS regressio as a prelimiary study based o (.) ad (.3) for each year of It is because we are ot sure if the parameters or more geerally productio techology icludig its fuctioal form is uchaged across time α β Figure. Coefficiet estimates of Cobb-Douglas fuctio. Figures shows the coefficiet estimates of Cobb-Douglas specificatio (.). The horizotal axis is the caledar year, ad circles ad squares idicate estimates of α ad β respectively for 983 Year

3 each year. We fid the followig features: [] the parameters appear to have chaged a lot durig the period of [] The parameter associated with labour iput has bee mostly icreasig sice the middle of 97 s, while that associated with capital has bee decreasig i the same period. [3] Labour ad capital productivity show a amazig mirror image after late 97 s eve though we did ot assume the costat returs to scale restrictio α + β =. We observe the followigs for coefficiet estimates of traslog i Figures ad 3: [] all the parameters look more or less time varyig, especially α K ad α L. [] It appears α ad α chage with mirror image, but ot K L so clear as i Cobb-Douglas estimates. [3] ad α co-moves, symmetrically with LL α. KL α KK. Figure. Coefficiet estimates of traslog fuctio Figure 3. Coefficiet estimates of traslog fuctio. We thik that these fidigs are quite iterestig, but we do ot discuss these here aymore, because estimatio of productio ad drawig implicatios from it are ot our primary obect. 983 Year 983 Year αk αl αkk αll αkl RESET test The above observatios have a importat implicatio i our curret aalysis. They suggest that the productio techology seems to have bee chagig throughout the time, so that it is atural to suppose that ot oly the parameters but also the fuctioal form could be differet across time. Thus, it will be iappropriate to take a stadard pael model i textbooks to aalyze this data because it caot hadle fuctioal form chages across time. Therefore, i the followig, we treat the data as a sequece of cross sectio data. A classical specificatio test of regressio fuctio is the Ramsey s RESET test. Suppose y is a scalar radom variable, x is a d vector of radom variables, ad we are iterested i the regressio fuctio E ( y x). It tries to test if the regressio fuctio is liear i x or ot. Specifically, the test is agaist H : y = x' β + ε H : y = x' β + α( x' β ) +... P + α ( x' β ) + ε P for some P. If the true regressio fuctio admits a polyomial approximatio as i H, this test will have power. The results of RESET test are i Table (a). The secod colum shows test results of the Cobb-Douglas ull, while the secod oe idicates those of the traslog ull. O, X ad XX mea respectively that the ull hypothesis is ot reected, reected at 5% size ad reected at % size. We fid that both Cobb-Douglas ad traslog specificatios are mostly appropriate i the old days before aroud the ed of 97 s, but they are reected after 98. These results say that ot oly the parameters of productio fuctio but also its fuctioal form is chagig over time. Whichever specificatio we take, a structural chage appears to have take place aroud looks to be the time of the structural chage. I Table (a), we fid a logical icosistecy that traslog is reected at % size ispite that Cobb-Douglas is ot i 979. It looks uusual because Cobb-Douglas is ested i traslog model. It could possibly be because of the small sample size, but this ca happe i RESET test i fact because the alteratives correspodig to Cobb-Douglas ull ad traslog ull are differet so that the test with Cobb-Douglas ull may ot have a sufficiet power. 599

4 3. NONPARAMETRIC TESTS FOR FUNCTIONAL FORM OF REGRESSION FUNCTIONS I regressio aalysis, we caot make a correct iferece if we use a wrog fuctioal form of the regressio. We would like to test if a parametric regressio fuctio employed is correctly specified. Oe such test is the RESET test i the previous sectio. But its power is ot so high if the specified alterative hypothesis is icorrect i fact. Godfrey et. al. (988) poit it out by some Mote Carlo studies. Specifically, this is a test for the ull hypothesis of E ( ε x' β ) =, ot that of E( ε x) =, (3.) which is of our iterest. Sice 98 s there have bee developed some tests which directly check (3.). Mostly orthogoality coditios, that disturbaces have zero mea coditioally o the regressors, is tested without employig a well-specified alterative. Bieres [98] first take this approach to propose a test of the ull H : P[ E( y x) = m( x; β )] (3.) agaist = H : P[ E( y x) = m( x; β )] (3.3) < give a parametric fuctioal form of the regressio such as m ( x; β ) = x' β. He exploits the fact that H is true if ad oly if E[{ y m( x; β )}exp{ it' Φ( x)}] = for ay measurable fuctio Φ (.). Thus his test statistic is { y m( x ; ˆ)}exp{ β it' Φ( x )} dt = give a iid sample ( y, x ), =,..., ad a cosistet estimate βˆ uder H. This test is called a coditioal momet (CM) test. However, this test statistic ivolves itractable ull distributio. This idea is exteded i a series of articles by Bieres ad his co-authors to various data geeratig processes. It is possible to costruct fuctioal form tests for (3.) agaist (3.3) based o a direct compariso betwee residuals from parametric ad oparametric estimates for the regressio fuctio. Oe such approach take by Hog ad White [995] is the followig. For simplicity, deotig m( x) E( y x), suppose we would like to test the ull of m ( x) = x' β. It is easily see that E { m( x) x' β}( y x' β ) = if ad oly if H holds. Lettig βˆ ad ˆm (.) be the OLS estimate of β ad a series estimate of m(.) respectively, they propose to test the ull based o the sample aalogue, M { mˆ ( xi ) x ' ˆ}( β y x ' ˆ) β. = Their test statistic is M T = ( p ) ( p ) ˆ σ where ˆσ is a cosistet estimate for the variace of error term uder H, ad p as, is the umber of orthogoal basis fuctios i the estimatio of ˆm (.). They prove d T N (,) as uder H. A similar test is proposed by De Jog ad Bieres [994]. Hitomi [] fids CM test does ot have much power i small samples uder certai alteratives because some momet coditios are ot effectively used. He also fids that some cosistet misspecificatio test statistics icludig CM, Hog ad White ad some others have a commo structure of weighted squared sum of ormalized correlatio coefficiets betwee residuals ad orthoormal base fuctios. Related articles iclude Eubak ad Spiegelma [99], Wooldridge [99], Yatchew [99], Gozalo [993] ad Hardle ad Mamme [993]. There is a paper which compares oparametric ad semiparametric models as Fa ad Li [996]. 4. RESULTS Amog the alterative cadidates, we take Hog ad White [995] test (abbreviated to HW test hereafter) followig Hitomi s [] simulatio study. Table (b) shows the results of HW test. O, X ad XX mea the same as those for RESET test results i Table (a). The results from the two tests are ot terribly differet. Cobb-Douglas is ot reected before 978, but is reected after 979. The boarder is clear. We fid the traslog used to be a model we ca employ before about , but ot ay more after 984. The boarder appears to be ust about the secod oil shock. We may be able to explai these test results that Japaese firms try to adust the structural chage caused by the two shocks ad chage their techology suitably. We do ot try to discuss i detail why this happes i this paper. 6

5 Table. Results of (a) RESET test ad (b) HWtest. (a) RESET test (b) HW test Year C-D Traslog C-D Traslog 965 O O O O 966 O O O O 967 O O O O 968 X O O O 969 X O O O 97 XX X O O 97 O O O O 97 O O O O 973 O O O O 974 O O O O 975 O O O O 976 O O O O 977 X O O O 978 X X O O 979 X XX XX X 98 XX XX XX X 98 XX XX XX O 98 XX XX XX O 983 XX XX XX O 984 XX XX XX XX 985 XX XX XX XX 986 XX XX XX XX 987 XX XX XX XX 988 XX XX XX XX 989 XX XX XX XX 99 XX XX XX XX 99 XX XX XX XX 99 XX XX XX XX 993 XX XX XX XX 994 XX XX XX XX 995 XX XX XX XX 996 XX XX XX XX 997 XX XX XX XX 998 XX XX XX XX 999 XX XX XX XX XX XX XX XX XX XX XX XX O : ull ot reected X : ull reected at 5% size XX : ull reected at % size Oe iterestig feature is that the logically icosistet coclusio i the RESET for 979 disappeared i HW test. I HW ad other oparametric specificatio tests, the alterative is the same for both Cobb-Douglas ad traslog ulls so that this kid of icosistecy foud i RESET does ot ormally happe. These results of RESET ad HW tests war us agaist usig Cobb-Douglas or traslog specificatio i empirical studies especially for years after 98. We also kow from them that it may ot be a suitable way to estimate macro productio fuctio usig time series data, which typically assumes that ot oly parameters of the productio fuctio but also its fuctioal form does ot chage over time. Cobb-Douglas productio fuctio was empirically developed i 95 s, whe this model fit the data well. However, the productio techology seems to have bee improved ad that of old day form does ot apply ay more. To deeply ivestigate why ad how it happeed, ad how we ca aalyze it with what kid of model are left for the future research. We believe this is a iterestig fidig for both ecoomists ad ecoometricias. 5. CONCLUDING REMARKS We tested if Cobb-Douglas ad traslog specificatio of productio fuctio is correct or ot for Japaese maufacturig idustry i the period of We foud that they are roughly correct before 97 s, but icorrect after 98. It wars that estimated macro or micro productio fuctios from time series or pael data based o either fuctioal form with fixed coefficiets across time could be distorted i fact ad statistical ifereces based o it may be icorrect. We fid it serious because most of well-established research o growth theory or huma capital effects uses either of them. Our aalysis however icludes some problems. Firstly, we treated the data as a sequece of cross sectio data, where we eed to assume there exists the productio fuctio of maufacturig idustry ad we ca make a iferece o it usig cross sectio data of differet maufacturig firms. Ivestigatig uder which coditios it is possible ad if they are satisfied are still ope questios. Secodly, we limited our ull hypothesis to traslog fuctio up to secod order because it is most widely used, but it is possible to iclude higher order terms. Their iclusio could chage the results. Thirdly, we employed a very simple cross sectio model, estimated ad tested it for differet years. Doubtlessly it is ot the best model. It would be possible to costruct a pael model which could hadle time-varyig fuctioal form ad/or parameters. We will be able to draw a better iferece from it. Research o this directio is curretly uder way. Other future research possibility will be, for example, to icorporate productio frotier aalysis. Research for o-maufacturig idustry is also curretly uder way. 6. REFERENCES Aiger, D.J., ad S.F. Chu, O estimatig the 6

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