AN EXACT TEST FOR THE CHOICE OF THE COMBINATION OF FIRST DIFFERENCES AND PERCENTAGE CHANGES IN LINEAR MODELS

Transcription

1 AN EXACT TEST FOR THE CHOICE OF THE COMBINATION OF FIRST DIFFERENCES AND PERCENTAGE CHANGES IN LINEAR MODELS Wai Cheung Ip Deparmen of Applied Mahemaics The Hong Kong Polyechnic Universiy Hung Hom Kowloon Hong Kong Absrac Economeric models are ofen formulaed in erms of he firs difference or he percenage change, generalized by he Box-Cox difference ransformaion. The choice of suiable funcional forms has relied heavily upon esablished saisical procedures which adop primarily he likelihood approach and confine o a single ransformaion parameer only. We have derived an exac es for he parameer vecor of ransformaion in linear models. By uilizing Taylor series approximaions his reduces o a choice beween wo regression equaions. The es saisic which has an exac F-disribuion can be easily calculaed from hese wo regressions by leas squares algorihm. Mone Carlo resuls demonsrae ha our proposed es is more capable han he likelihood approach in capuring he correc size ye is as powerful as he laer. I is herefore a simple and ready saisical procedure for assessing he suiable choice of he combinaion of firs differences and percenage changes in economic forecas models. JEL Classificaion: C Key Words: funcional form, Box-Cox Transformaion, Taylor series, leas squares

2 . Inroducion The Box-Cox ransformaion of a posiive series { y } is given by (. ( y y =, ln y, 0, =,..., = 0 T where y is an economic variable a ime period and is a Box-Cox ransformaion parameer. Typically many economeric applicaions examine no he level of he variable, bu is change per ime period. Examples include he S. Louis equaion due o Anderson and Carlson (970, he money demand model by Hafer and Hein (980 and he consumer index of Colclough and Lange (98. A discree approximaion o he ime derivaive of y ( is given by Layson and Seaks (984 as (. ( y = y y, =,..., T, where y = y - y - denoes he firs difference. The usefulness of (. is ha = yields he firs difference while =0 gives he percenage change. This is ermed he Box-Cox difference ransformaion by Seaks and Vines (990. Boh he firs difference and he percenage change of an economic variable may herefore be generalized by y (. Thus a regression model which uilizes boh he firs difference and he percenage change can be represened by (.3 ( ( ( q y = α x + L+ α x + z β + L + z β + ε, p p q q where i (i=,.., q are Box-Cox difference ransformaion parameers, x j (j=,,..., p are observaions on he p independen variables which are no ransformed, is defined as in ( y (., ( i z i, (i=,, q are similarly defined, α j (j=,..., p and β i (i=,..., q are regression coefficiens, and ε is he error erm. We may regard (.3 as a general model in which he

3 ransformed parameers permi more general forms han firs differences or percenage changes hough =0 or has a sraighforward and simple inerpreaion. Unforunaely model selecion is a very difficul ask in pracice. As Dhrymes e al. (97, p94 have poined ou ha economic heory ypically provides lile guidance as o he proper funcional form appropriae o he specificaion of he economic relaionships. The choice of suiable funcional forms in economeric applicaions hus relies heavily upon esablished saisical procedures. In his paper, we shall develop an Andrews (97 ype procedure for esing he choice of he se of Box-Cox difference ransformaion parameers in (.3, allowing differen ransformaion parameers for differen variables. Is saisical mehodology follows he line of Milliken and Graybill (970 and is based on he F-disribuion. Secion reviews exising single ransformaion parameer procedures. In secion 3 we shall develop he esing procedure which reduces o a choice beween wo regression equaions. The alernaive regression, hough of no economic ineres, is easy o undersand. Secion 4 oulines he seps how he es saisic can be compued using a sandard saisical package. In secion 5 he new procedure is compared wih he likelihood raio approach by Mone Carlo models. Secion 6 presens he conclusions of his paper.. Single Transformaion Parameer Procedures Exising procedures include he likelihood raio (LR es and varians of he Lagrange muliplier (LM ess. They assume he same ransformaion parameer be used on boh he dependen and independen variables. The procedure of Layson and Seaks (984 is primarily a LR es. I requires a search for an opimal soluion of he unresriced model which in pracice can be compleed by a sequence of regressions on ransformed daa wih various values of he ransformaion parameer over a grid. Though he firs difference of he dependen variable and percenage change of he regressor are used in he povery and growh model of Thornon e al. (978 Layson and Seaks (984 have only applied heir procedure o es he laer choice bu no he use of he firs difference of he dependen variable. The search could be very expensive if no impossible when differen parameers of ransformaion are allowed. Those of Coulson

4 and Robins (985 and of Park (99 are differen varians of LM ess, consruced from raher complicaed auxiliary regressions in which he regressors involve eiher he ime derivaives of he ransformed variables or he derivaives of he unresriced log-likelihood funcion wih respec o he individual variables. The regressions are arificially consruced o generae ess for misspecificaion and heir relaions o he hypohesis o be esed are no obvious. The compuaion involved is by no means simple or sraighforward. Furher, he LR and LM procedures are asympoic and herefore are only approximae in small and moderae sample siuaions. The es saisic of he proposed procedure can be calculaed easily by ordinary leas squares esimaion of wo regression equaions, hus avoiding he relaively more laborious search over a grid of values. 3. The Tes Procedure Using marix noaion we may express (.3 as (3. ( ( l y = Xa + Z b + e where l =( 3... q, y ( is he Tx vecor of ransformed dependen variables, X is ( ( ( 3 q he Txp marix of original independen variables, Z l = ( z z... z 3 ( q is he T (q- marix of ransformed independen variables, a=(α α... α p is a p and b=(β β 3... β q is a (q- vecors of coefficiens, and e is he T vecor of independenly and normally disribued disurbances wih zero mean and consan variance σ. Thus he model selecion problem reduces o esing he hypohesis abou he ransformaion parameer vecor l =(... q in model (3.. 3

5 Expanding y ( in Taylor series abou a hypoheical value (0 yields ( (3. ( (0 (0 y y + ( w( (0 where w( = y ( / evaluaed a =. I can be readily shown ha w( has - h elemen (0 (0 (0 (3.3 w ( = (ln y - (0 (0 ( y. Similarly, we have Z ( l (0 ( l (0 Z + V( l ( l l (0 (0 (0 (0 (0 where l ( is a hypoheical value of l, = 3 q (3.4 V (0 (0 (0 (0 ( l = [ v ( v ( v ( ] 3 3 q q and v i ( v i ( (0 i (0 i ( i = z / i being evaluaed a i = (0 i, i=,, q. The individual elemens of are given similarly o (3.3. Subsiuing (3. and (3.4 ino (3. and wriing (0 (0 (0 0 = ( q, we shall obain (3.5 (0 (0 ( ( l (0 (0 y = Xa + Z b + w( M V ( l b]( l l + e [ 0 which, on leing g =l -l 0 and U(l 0, b=[-w( (0 MV( l (0 b], may be rewrien as 4

6 (0 ( ( l (3.6 y = Xa + Z b + U l, b g + e. (0 ( 0 Box (980 calls he new explanaory variable U a consruced variable. The esing of he null hypohesis H 0 : l=l 0 agains he alernaive H : l l 0 has become a choice beween (3.6 and he null model (3.7 (0 ( ( l y = Xa + Z b + e. (0 Equivalenly, we are esing H 0 : g=0 agains H : g 0 in model (3.6. To eliminae he dependence of U on e in (3.6 he former will be replaced by is leas squares esimae from he null model. Le â and bˆ be he leas squares esimaes of a and b in (3.7, he fied values (0 ( yˆ (0 ( of y can be compued from (3.7 afer replacing he regression coefficiens by heir leas squares esimaes, and he fied values of y (0 ( can hus be deermined recursively by (3.8 yˆ yˆ yˆ (0 ( (0 ( (0 ( T = yˆ = yˆ = yˆ (0 ( (0 ( (0 ( T + y + yˆ + yˆ (0 ( 0 (0 ( (0 ( T. By he inverse of he Box-Cox formula (. he fied values of y are given by (3.9 yˆ ( = (0 yˆ ( (0 e ˆ (0 y +, (0 /, (0 (0 0 = 0, =,.., T 5

7 Subsiuing (0 ( yˆ and ln yˆ ˆ ino (3.3 we shall ge ( (0 (0 w and hence w ˆ (, he leas (0 squares esimae of w (, and finally (3.0 ˆ (0 (0 U = [ wˆ ( MV( l ˆ] b upon subsiuing bˆ in U(l 0, b. Replacing U in (3.6 by Û gives (3. y (0 ( = Xa + Z (0 ( l b + Uˆ g + e which saisfies he sandard condiions of ordinary leas squares. The use of Û o replace he unobservable U has been pu forward by Milliken and Graybill (970 for wo obvious reasons, namely, Û can be compued from he daa and is independen of he disurbance. The same acic has also been used in Andrews (97. Le S 0 and S be he residual sums of squares by leas squares on he regression equaions (3.7 and (3. respecively. I follows from he resuls of Milliken and Graybill (970 ha he quaniy (3. ( S 0 S/ q F = S /( T p q + will follow a F-disribuion wih q and T-p-q+ degrees of freedom. Hence F is a es of he hypohesis g =0 or l =l 0. A deailed developmen of he above resuls can be found in Milliken and Graybill (970 who have shown ha F has a F-disribuion when H 0 : g = 0 is rue. The es is exac in he sense ha an exac significance will be obained from which exac confidence limis may be calculaed when H 0 is rue. However, lile is known abou F when g 0. See Ward and Dick (95. Andrews (97 has poined ou ha he precision in (3. may affec he efficiency of he es bu i will no affec he exacness of he disribuion of he es saisic. 6

8 The proposed procedure is herefore more capable of capuring he correc size of he es han any oher asympoic ones. For more discussion of oher advanages of he proposed procedure over asympoic ones; see Andrews ( Compuing Procedures This secion briefly oulines procedures for compuing he es saisic F. The procedures described here assume he use of a saisical package. The seps are as follows:. Transform y and Z as in (. o ge y ( (0 and (0 ( Z l and o form he regression equaion (3.7.. Esimae equaion (3.7 by leas squares o obain coefficien esimaes â and bˆ, and he error variance esimae s 0, say. (0 (0 ( ( l 3. Compue y aˆ bˆ = X + Z, from which calculae yˆ ( (0 using he recursive relaions (3.8 and yˆ by he inverse Box-Cox formula (3.9. ˆ (0 4. Compue w ( by (3.3 using compued values from sep 3 and form he vecor (0 (0 w ˆ (. Compue V( l using a formula similar o (3.3. Augmen - w ˆ ( and (0 V( l bˆ o form Û given in (3.0. (0 5. Form he regression equaion (3. and esimae i by leas squares o obain error variance esimae s, say. 6. Finally calculae he es saisic F in (3. by puing S 0 =(T-p-q+ s and S =(T-p- q+ s. When he package such as RATS used has buil-in esing procedure for he regression coefficien vecor, Seps 5 and 6 would be combined o one of esing g=0 in he regression equaion (3.. The null model is only used o compue he fied values Û of he alernaive model. 0 7

9 5. The Mone Carlo To evaluae our proposed F-es and o compare is performance wih ha of he LR procedure programmes in RATS are wrien o generae daa for models wih combinaions of firs differences and percenage changes. Sample sizes of 0, 30, 40, 60 and 00 are seleced so ha boh he small sample and asympoic properies can be sudied. Each experimen involves 000 replicaions. The four models sudied are: M00: % y = % z + ε % z ~ U(0, 0.07, ε ~ N(0, 0.05, z 0 =.0, y 0 = 5.0 M0: % y = z + ε z ~ U(-0.5,.5, ε ~ N(0, 0.05, z 0 = 5.0, y 0 = 5.0 M0: y = % z + ε % z ~ U(0, 0.07, ε ~ N(0,, z 0 = 0, y 0 = 00 M: y = 0 + z + ε z ~ U(0,, ε ~ N(0,, z 0 = 0, y 0 = 00 The populaion R are.593,.57,.553 and.57 respecively. M00 and M have been used by Seaks and Vines (990. The models are esed under each of he null hypoheses: H 0 : ( =0, =0, H 0 : ( =0, =, H 0 : ( =, =0 and H 0 : ( =, = in urn, a %, 5% and 0% significance levels. The empirical significance level is recorded, when he null hypohesis is rue his is he Type I error and when he null is false his will be he power of he relevan es. A sligh modificaion of Layson and Seak s (984 resul gives he log likelihood funcion under normaliy: 8

10 T l = [ln (π + ln σ ] σ T T ( ε + = = ln y. The RATS procedure MAXIMIZE is used o compue he maximum likelihood esimae lˆ of l and o calculae he LR es saisic χ = ˆ. [ l( l( 0] ~ χq While our F procedure ook few seconds he LR procedure ook several minues o few hours o complee 000 replicaions even hough he rue parameer values were used o sar wih. When values oher han he parameer values were used as iniial values he laer procedure ofen locaed he opimal soluion afer several hundred ieraions. Table a Size/Power of Tes a % Significance Level Nominal True Model: =0, =0 Hypohesis = 0, = 0 = 0, = =, = 0, = = Tes F LR F LR F LR F LR Sample Size

11 Table b Size/Power of Tes a 5% Significance Level Nominal True Model: =0, =0 Hypohesis = 0, = 0 = 0, = =, = 0, = = Tes F LR F LR F LR F LR Sample Size Table c Size/Power of Tes a 0% Significance Level Nominal True Model: =0, =0 Hypohesis = 0, = 0 = 0, = =, = 0, = = Tes F LR F LR F LR F LR Sample Size

12 Tables a-c give he size and power of he ess for model M00. The F-es appears generally o capure he correc size while he LR es oversaes he Type I error a he % level in small samples and undersaes i a he 0% level for all sample sizes. I does no appear o give he correc size even for sample sizes as large as 00. The wo procedures perform equally well under H 0 : ( =0, = and H 0 : ( =, =0. They boh lead o correc decision by rejecing he incorrec nulls H 0 : ( =0, = when T 60 and H 0 : ( =, =0 when T 40 in all cases. When esing under H 0 : ( =, = F appears less powerful han LR in rejecing he incorrec null. Table a Size/Power of Tes a % Significance Level Nominal True Model: =0, = Hypohesis = 0, = 0 = 0, = =, = 0, = = Tes F LR F LR F LR F LR Sample Size

13 Table b Size/Power of Tes a 5% Significance Level Nominal True Model: =0, = Hypohesis = 0, = 0 = 0, = =, = 0, = = Tes F LR F LR F LR F LR Sample Size Table c Size/Power of Tes a 0% Significance Level Nominal True Model: =0, = Hypohesis = 0, = 0 = 0, = =, = 0, = = Tes F LR F LR F LR F LR Sample Size

14 The size and power for model M0 are given in Tables a-c. The F procedure is seen again o sae he correc nominal size. On he oher hand, LR ends o oversae he size a he % level in small samples and oversae i a he 5 and 0% levels for all sample sizes considered. The Type I error of LR by no means appear o converge o he nominal one even when he sample size is 00. When esing under incorrec hypoheses, boh es procedures seem o perform equally well in rejecing he incorrec null. In some cases, a sample size of 40 is sufficienly large o lead o rejecion of wrong nulls in 00% of he imes. Table 3a Size/Power of Tes a % Significance Level Nominal True Model: =, =0 Hypohesis =, = 0 =, = Tes F LR F LR Sample Size

15 Table 3b Size/Power of Tes a 5% Significance Level Nominal True Model: =, =0 Hypohesis =, = 0 =, = Tes F LR F LR Sample Size Table 3c Size/Power of Tes a 0% Significance Level Nominal True Model: =, =0 Hypohesis =, = 0 =, = Tes F LR F LR Sample Size

16 For model M0 he wo ess are equally powerful when in fac he null were false. When he null is H 0 : ( =0, =0 or H 0 : ( =0, = boh of hem rejec he incorrec null in all cases and herefore resuls are no presened here. As visible from Tables 3a-3c hey do no appear o ouperform he oher under he incorrec null H 0 : ( =, =. They only differ in saing he size under he rue null. F generally gives correc sizes a all levels while LR oversaes or undersaes i in mos cases. From Tables 4a-4c i can be seen ha F yields Type I errors ha are very close o he nominal ones. In conras LR only produces close o saed sizes in large samples. For example, is ype I error is a leas wo imes he nominal value when T 40 a he % level. When he null is H 0 : ( =, =0 boh ess do no appear differenly in heir abiliy of rejecing he null. When he null is H 0 : ( =0, =0 or H 0 : ( =0, =, boh approaches rejec he wrong null in all of he cases. Table 4a Size/Power of Tes a % Significance Level Nominal True Model: =, = Hypohesis =, = 0 =, = Tes F LR F LR Sample Size

17 Table 4b Size/Power of Tes a 5% Significance Level Nominal True Model: =, = Hypohesis =, = 0 =, = Tes F LR F LR Sample Size Table 4c Size/Power of Tes a 0% Significance Level Nominal True Model: =, = Hypohesis =, = 0 =, = Tes F LR F LR Sample Size

18 6. Conclusion We have derived an exac es for he parameer vecor of ransformaion in linear models. By uilizing Taylor series approximaions his reduces o a choice beween wo regression equaions. The foregoing analysis need no specifically assume he ransformaion parameers o be 0 or. Our proposed es is hus applicable o any oher ransformaion parameer values hough inerpreaion is sraighforward when hey are equal o or 0. The es saisic which has an exac F-disribuion can be easily calculaed from hese wo regression equaions by leas squares algorihm. Mone Carlo resuls have demonsraed ha our proposed procedure is generally more capable han he likelihood approach in saing he correc size of he es, ye i is equally powerful o he laer in rejecing false null hypoheses. I is herefore a simple and ready alernaive o he likelihood raio es for assessing he suiable choice of he combinaion of firs differences and percenage changes in linear models, hereby allowing more flexible and appropriae economic relaions be formulaed and heir validiy be esed. Acknowledgmen This research was suppored by a gran from The Hong Kong Polyechnic Universiy Research Commiee. References Andersen, L.C. and K.M. Carlson (970: A monearis model for economic sabilizaion, Review of he S. Louis Federal Reserve Bank, 5, 7-5. Andrews, D.F. (97: A noe on he selecion of daa ransformaions, Biomerika, 58, Colclough, W.G. and Lange, M.D. (98: Empirical evidence of causaliy from consumer o wholesale prices, Journal of Economerics, 9, Coulson, N.E. and R.P. Robins (985: A commen on he esing of funcional form in firs difference models, Review of Economics and Saisics, 67, Box, G.E.P. (980: Sampling and Bayes inference in scienific modelling and robusness (wih Discussion, Journal of he Royal Saisical Sociey A, 43,

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