Estimation of a Dynamic Demand Function for Gasoline with Different Schemes of Parameter Variation

November 21, 1975

#770

ESTIMATION OF A DYNAMIC DEMAND FUNCTION FOR GASOLINE WITH DIFFERENT SCHEMES OF PARAMETER VARIATION

by

J.S. Mehta, G.V.L. Narasimham and P.A.V.B. Swamy

NOTE: International Finance Discussion Papers are preliminary materials circulated to stimulate discussion and critical comment. References in publications to International Finance Discussion Papers (other than an acknowledgement by a writer that he has had access to unpublished material) should be cleared with the author or authors,

Estimation of A Dynamic Demand Function for Gasoline

with Different Schemes of Parameter Variation

by * J.S. Mehta, G.V.L, Narasimham and P.A.V.B. Swamy

Temple University, Department of Commerce and Federal Reserve System

1, INTRODUCTION

This paper attempts to study the demand for motor gasoline by private individuals in the U.S, The study has a three-fold objective: (i) to formulate a demand function for gasoline whose consumption is technologicaily related to the stock of automobiles owned by individuals, (ii) to specify a dynamic equation which captures the effect of the adjustment of these stocks over time on the consumption of gasoline and (iii) to empirically verify the stability of the postulated model across geographical regions in the U.S.

The interesting studies by Houthakker, Verleger and Sheehan (1974), Verleger and Sheehan (1973) and by Ramsey, Rasche and Allen (1974) are allied to the work in this paper. Our study differs from these studies in three important respects. First, we consider the hypothesis that the gasoline purchased on the market by consumers are inputs into the production of automobile services which are the arguments of a household utility function. This hypothesis provides the microeconomic foundations of our aggregate demand function. Second, we treat the aggregation problem explicitly. Third, the parameters of the aggregate demand function are allowed to vary freely across geographical regions.

In Section 2 we present the theoretical formulation of the dynamic model for gasoline. In Section 3 the results of the estimation of the gasoline model from aggregate time serie: data are presented. In Section 4 we take advantage of recently developed tools to estimate the coefficients of the dynamic model from a time series of cross-sections. Some concluding remarks are given in

Section 5.

*/ . The views expressed herein are solely those of the authors and do not necessarily represent the views of the Federal Reserve System.

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-3-

then the gross investment in automobiles in the period t is given by I, (t,t) = S5,7-6; 0534 04° In terms of these variables a putty-clay production function can be stated as

Ss (2) Mee = 9544, 0-85 ,)854 1 °F, G, (tot). 1, (tot), t)

where AP is the output of automobile services, ds is the fixed proportion

of gasoline to automobiles of vintage v < t?, 8., is the output-gasoline

it ratio and F. is the ex ante production function with three inputs: gasoline, gross investment in automobiles and households’ time. We assume that

d

_ s = Ait ~ Ait A

it’ It follows from the results of Pollak and Wachter (1975) that the demand for the commodity Aye is

d @) Ae 85 O-85 NM AGSi ed = Fie Bier Pie)

where Pit is a vector of goods prices and Xe is the labor and nonlabor income of the jth household in the period t. Changes in the jth households’ technology as well as taste changes will take the form of shifts in the demand function fee:

The implication of eq. (3) is that the incremental demand for gasoline due to the replacement of automobiles and net increases in the stock of automobiles is determined in the manner suggested by the household production function model.

For the sake of simplicity we assume that the functions in (1) and (3) are linear. The quantity 6, 0-65.) is not likely to vary violently from one period to the next. It may be safely assumed for gasoline that

0,.G-55,) = 0; G-4,) over our sample period. We also assume that the function

2 ‘the assumption of constancy of i Nerlove (1966, p. 587).

4 can be justified as in Balestra and

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3. Estimation from Aggregate Time Series Data

The preceding analysis is microeconomic in nature. Since data on individual households are not available, an explicit treatment of the

aggregation problem is in order.

Assumption 1: (a) Us, = TEHEGy where the T, are regarded as the

individual-invariant time effects which are not accounted for by the included explanatory variables in eq. (4):

(b) The vectors (455%. 9%42°VG3)' with different i subscripts are the realizations of a 4-dimensional variable with the mean vector

(a,Y,,Y>5»¥~)' and a finite symmetric variance-covariance matrix, The 17°2’°°3

.. ar { variable Use is independent of CPASE EAE LALED ‘

(c) -l <y,, <1 with probability Mi.

(d) The variables Xie and Poit are predetermined from the standpoint

h

of the it household such that sup |X, | and sup Ip. | are uniformly, it <t<T git

1<t<T 1 bounded almost surely for all i. G._,G. »X. and p_.. are known constants. io” i,-1’° io gio (e) fu, ,} is a stationary process such that E Ui, = 0 V (i,t). Now aggregate eq.(4) across individuals and divide through by the number

of individuals; this gives

n t 1 "t _ "t (5) — I G., =aty, — f= G, +yY,n, ._ it ‘3 n git nm j=1 it 1 n i=l it-l 2 t i=l t 1 ; ot +— £ e..+T ne i=l it ¢t

where n, is the total number of individuals in a population in the period t,

a, =o + bio? Yee = Yo + €5,0 = 1,2,3) and

(6) Pit * Sin * FirSieea * baaXit * SasPoit eit

"7° QOUBTIVA JUBYSUOD

pue uvow Q YIM painqtzystp AT[eotTquept pue AyTIUepuedeput oie +1 eyL 32 uot ydumssy

T=? 3 3 T=? 3, way ety . y ty 38g og = Ma pue My ge 2 gg 2 Mgttg gy — = *9 wa Wy "MA WW 104M (ietrr’zft = ata wae, + eat gly ae 49 aa) sowoseq (Ss) ‘ba onzy st (F)T uot idunssy usyM

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AYIPTLVA 94} LOZ SuUOTATpuod YUueTOTFINS Jo Jas Be 9Ze[NUIOF ATISea UBD 9UO

T=Tt 3 o« U T=t 2, o« U

u 3T

aA o= C's x —) sea utr pueg= (a g spat 7 u

tu

voy, ‘Cures <tuy urw = u 30 (3) TT uoTydunssy

When we estimated eq. (7) using quarterly data for the U.S. for the period

19631 - 1973IV we obtained the following results.‘

A (8) G_ = 26.85 + 0.65 G+ 5.56 X,-0.57 p, +4 . (0.12) ** (1.73) * Co.ss) Bt t

R’ = 0.99,

Figures given in parentheses are the standard errors. The value of Durbin's (1970) h-statistic is -~1.4 which is greater than the critical value of -1.96 at the 0.05 level of significance for N(0,1). Hence the null hypothesis that there is no

first-order serial correlation among the t,'s cannot be rejected. The coefficient

t estimates have the right signs, although the price coefficient is not precisely determined. In assessing the parameter estimates in (8), attention should be given to their economic import, that is, to their interpretation in terms of the income and price responses of the classical theory of consumer demand. It should be

remembered that the demand function used here refers to the new demand for gasoline.

Therefore, the average income and own-price elasticities are given by

_ aG*,X | age, P (9) ny = Gr a and nog = Ope

ko ~y XY. Ge D * where Gr G, ¥)S,_) and X, G* and Py are the sample means of Xe GF and Pot

respectively. These definitions are borrowed from Balestra and Nerlove (1966, p. 592.) The income and own-price elasticities have been obtained from our parameter

estimates in (8) by evaluating, at the sample means, the formulas in (9). These are:

(10). ny = 1.22 and Nog =~0.38

This value of Ng is implausible because we do not believe that the gasoline is a

luxury good for majority of people. Since we do not regard the coefficient of Giy

The source and nature of data are explained in the Appendix of the paper.

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a

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to the analysis of time series of cross-sections.

4. Estimation from Temporal Cross-Section Data

We might get improved estimates if we use diaggregated data. Furthermore, the richness of disaggregated data base allows a varied menu of alternative parameter variations. Fortunately, we could get the temporal cross-section data for the U.S. which consisted of quarterly observations drawn from the forty-eight continental states plus the District of Columbia for the period 19631 - 19731V.>_ In this section we consider the problem of estimating eq. (4) from these data. We shall begin with the case in which the intercept varies in both the time and cross-section dimensions but not the slope coefficients. In the time series analysis of cross-section data, if the inter-individual and inter-temporal parameter variation takes the form of mere shifts in the regression intercept, then the specification of an equation with a shifting intercept and fixed slopes is adequate. We then go on to consider a ‘model in which all parameters vary in a systematic or random fashion.

We may rewrite eq. (4) as

Q1) Sige * ij * YagrSaje-a * Vagakaje * VagsPoaje * Mage

where i stands for the gth household, j stands for the jth state and t represents

time. pti : se, = + Vv, +e i Assumption 3 (a) Usit T, Vt + fait where the term Ty, reflects the effects that

are specific to the quarter but common to all individuals and states and Vee is

common to ali households within the jth state at a given time.

. : tows . . . (b) For given j the vectors (a5; Vagy Vag2° aga? with different i subscripts

are the realizations of a 4-dimensional variable with mean vector (95515 19%j 29% 55)"

be state se ieee tee eee Sthe data base and sources are described in detail in the Appendix.

4q (Z1) “be

ejewtxoidde ued om aniz st (z)¢ uotzdumssy uoyy G7 Ar ae | Ceper'2"] =f)

T=? af o + U

—) I®a wit pue

f = (tf, T

u JO UNUTUTW oY} ST U aI0YyM Q = +e TsP 20, we eu v o= ("9 3 -—)qa wy i pue f ueat3 roy (x) te uoT dunssy 30

ry

aft, , tftB elt i +ft, Zft

UE, , vf sft

he MOLT, | EL ft,

pue (¢‘Z‘T = 4)

ie)

*2 potaed ay. ut 9383s uaf 942 UT SpToyasnoy Jo requmu sy st +, @.194mM

T=t 36

5

fay TY T=t 40 u ef . 301 Ist ve zf, ; raft T=T +e rf c acr_ ‘Ts? +o

—*rh X J = 9 3 — +7) DOD 2 — (2D tT - T tT - = T uf +E, +f +E, uTeIqgO OM

STeNpTATpuT Fo Lequmu ay Aq Yy3no1y2 SUIPTATp puke T IdA0 (11) *be sutuums up

"(a6 CT) A 0 = {Tog yey? Yons ssadoid AreuotzeIs e& ST ¢ Thy (9)

oft3

*sjueysuod uMOUY are d pue

ft eI Slt <Olt . 4 X “9 DS sonqtea TeTzTur ey, ‘Cf pue t [[e z0z ATaans +SoWTe papunoq ATwxrozFtun 3018 L>T>1 L>2>T aie | d| dns pue \** Ty dns ey2 yons 93e35 ua! 242 UT pToyssnoy ua! ay2

JO uTodpueys ayi worz poutwrezopaad are +fT8 pue aT, seTqetiea ayy (p)

"(f*t) At AatTTqeqoad yatm T> TTT (9)

+ FFE ZT, TET, (ft “0) JO JUspusdoput

(Tt CC ST haar eTqetzen ay, °" "Vy xTI1eEW SOUBTIVAOD-9OUBTIEA STIZOMUIAS OYTUTF B pue

~ OI -

(14) G., =a, +

jt 9 7

nN.

1 jt where G.. = r J jt isl

4.1. Error Components Model

j1°jt-1 * %52

- ll -

jt * VpsPgit *

V5 Gt Ls eccoms t= 12500057)

n, -1i =,it Nie j-1 ijt and 1 Jt Mt i=l Pgijt

In the special case in which the slope coefficients are independent of j eq. (14)

can be written as

C15) G5, = Ot VG * VOX + VRP

where a, =O + u., u; is a time-invariant state effect, T,

J

j

gjt

+4

j

+ T

t

+

G = 1,2,...,m; t = 1,2,...,T)

is a state-invariant period

effect and the Vat represent the combined effect on Git of the neglected variables.

For our data m = 49 and T = 43.

If we arrange the observations on each variable first by state, then according

to period, we may depict the observations as

(16)

yray.t Zy. + G@1,) + Q.@ +v

where y is a mTxl vector of observations on Gays 1 denotes a "summer" vector, all of

whose elements are 1, the subscripts on 1 indicates its order, Z is a mTx3 matrix of

observations on G,

jt-1’ x t

te (Ty >TorseesTp)ts v is a mTxl vector of observations on Vit and @ denotes the

Kronecker product.

We now make the following assumption.

Assumption 4:

(a) The rank of Qar?2) is 43

‘(ofetz = zd = "2 (Rem) =o (zd = FE peeez & 1) RD = TD © Tk oroym

Ya 4 2°z = "2% (pet)

tn + rz = *% (gt)

N =| +

I>

N

N t

= °% (gt)

M+ 9X = ' (est)

*(£L6T) e24aW pue AuweMS UT paqT1oSep UOTIeUZOZSUeIR TeuoZoyz1O ay} SZITTIN 9M asodind sty} 10x *sqza1ed quepuedeput (Tenbaun)

INOF BUTMOTTOF YX OUT SUOTIeATESGO JW JO aTdues ay. AT[Tds 03 JUeTUaAUOD ST IT MON

(5 @ fx) 0 + Co @") 2» + CTT ®@ 'y) = ty (aa)

WIOF SY JO XT1JeW DULTILAOD-20ULTILA PU Q I019N9A UPOU YITM paynqts3sIp

A[TewLoOU st A + (i eo") + CT @n = M 10299A 9y2 p UOT dumssy Jepun *(2961) TITH pue (¢€/61) e2YyaW pue Awems Aq 1eaTS apeu aie y uoTIdumssy yo suoTeOTTdUT yy

*snouasoxe o18 3684 pue +f (3)

peangtaystp AT[ew10U ere A pue 1TH (3)

to 70>0 pue t > "0 >. (1-)- epir(Po-t) + TTP] = "yep > fo | (r-w)-

‘[rGo-1) + “attay = £y osoya ('s ® *u) 90 = ,AAq pue 0 = Ag (p)

: > “9 > Q pue “2

t> oo> y-{I-L)- «ttr(f0-1) + Ili? 9) = “y azoyn “pro = ,11g pue Q = 2g (9) feo > ny > Q pue ’ z Pp _— UW Wit d > > los ,-(t-4)- ‘Pr(lo-q) + Ti tto] = Ty osoym by 2 tig pue o = Ag (q)

- @I-

- 13 - Q, = (C,@c,), 0; = (1/7, ci)" is an orthogonal matrix of order T, and 0, =

G,,/ 7m, C3) ' is an orthogonal matrix of order m.

Swamy and Mehta (1973) have shown that

(19) EW, = 0, EW, = 0 (2 = 2,3,4) “ -2_ 2 (20a) E Wy = 9); 2 ~~ ot = t = me (20b) E WoW> = 95 re m m-1, - - 2 q = t sven (20c) E WoW, =o, Ly, T= 4-1, - 2 ae (20d) E WAW4 oq Ty and (21) E WH, = 0 (2 = 2,3,4) and E Wes = 0 for 2 # 2' = 2,3,4. The variances 2 2 2 2 . 2.2 2... | F119 0550% and o4 are functions of Tyre T res P,(% = 1,2,3,4), m and T.

Some Point Estimates ee ee MATES

Alternative estimators of x developed by Swamy (1971°, Ch. III) are

A a =: = as | =): = =\= = t ' ! t vre7e 7Iy> TI7tyS hs 71 (22) Y= fm Z525 + T Laan + m'T 2424) Cm LY + T ZY + m'T Z4¥4) rN =.s .-lsos, - (23) Yo = 522) 23Y, 24) $= (212.97) ais (24) ¥, = @325) 275 (25) % = Giz yt ni Y4 4°4 4X4 71 717 = Fe (26) of 22 + 7a", coke Z4v4 ) Y24 ry) x2 a2 a2 2 4 2 % Z 212, 1 2y7,. 2 3 4 3° “4X4

17 1Z

(27) 7 = (3 + 44 34 42 AZ

3 4

*(6961) uTessny pue OOBTTBM Bes ‘azTUTFZ zou o1B (S- F)auy jo ae dt OYdMASe oYyA JO sodUBTICA ey? *XT@ATIadsaz

£

adsWs pue ia‘ ,ws (P81)-(48T)°sbe = ATdt3 {nw you op am FIT

) = (eX oava (ee)

IN

ray) <o U

= ("2 oava (zg)

<o t

= CDoava (1s)

(2) oava (os)

wo) Catz. time®zizisezhziu) = Doava (62)

pore (82)-(%Z) sa1oqewtyse ay Fo SOOTIZEU (DAVY) SoUeTIBAOD-sdUeTIeA 2T303dUKS

povenrase SY} ‘SOOTIZEW DDUCTIVAOD-d9DUBTIVA @ITULF savy (A- FECT) guy pue (Z-"*D) up ‘Rb: :

Luafte (A- ¥1) yup Dy (Gb _y (A- 2) uup30 SUOTINGIIISIp BuTAZIWIT sy. souts *sz9y3O ay} ueYy Satdwes [Tews UT uoTze3T19 SSaUupesetqun s.UeTIeA WNWTUTU 942 02 BuTpiosse yUeTOTFZe Sow (10299A r0}OWeIed uMoUyUN 942 UT) ATWIOFTUN YoU St (gz)-($z) ‘(ZZ) ut sxojewt3se aya zo Aue ‘Q = hh ueYyM 2BYy2 UMOYS eAeYy (7Z6T) eLory pue (III pue II ‘syd ‘TZ61) Awemg ‘uotinqtaz3stp Suraqwty owes ay? sary pue 2UezSTsUOD arp (8Z)-(SZ) sxozEMT Se eY4r 2eYr SMOYUS (STH-sOy ‘dd ‘Iz6T) TTEUL pue (242-722 ‘dd ‘p961) sze8x1eqptoy Fo yey 02 TeTTezed QuauM3ze Ue ‘XAT quenbesuo) “SeoueqINISTp 942 JO SenTeA BuTpeaedons ay2 pue snosueroduejUOD ay2 Jo quepuedaput st atqetizea ArozeueTdxe ue se savedde YSTYM aTqetaeaA uspusdeap passe]

943 JO onTeA yoee (pgt)-(qgr) “s be: ut aey2 uMoys sey (gp ‘d ‘pz6T) AweMs [e- (1-4) (1-w) 1 /(°2"2-") , (VA"7-"DH = ¥ : Vv Vv pue (p-1)/O2*2-*D) FR 2-"D = £0 y-wy CF22-*2) (7422-22) = Zo oxoyn v7 v7 v7 Vv cv

v £

wv, zy. ay a

v

pa Ae a —— eran — ———) = A

vx,” Exe, * Zz Gor * Ee, * B27.) (82) vz *kiz hz | Mate Safe 2a%e

< 15 -

as A Z1Z, 212 A sey A gty (34) EAVC(F,,) = +S , 3 “4 , 232, 212, 212, vt (35) EAVC(Y,,,) = (4 4+38+t45 . Py) o %4

Using the temporal cross-section data, referred to above, we have evaluated the quantities in (22)-(35). The results are listed in Table 1. The sample means

are given below:

49 43 1 £5 £ G,, = 384.6068, 2107 i=l i=l J 49 43 1 © & G,, | = 381.7850, | Zi07 j=1 t-1 Jt} 49 43 1 2 £ X,, = 30,9802, 2107 j=1 t=1 J and 49 43 1 © © p.., = 102.5186.

107 j=l t=1 85

1 ATWUBITFTIUBTS ST YOTYM 2° LOZ 9JVUTISS BAT IISOd eB pue 2 LO} 9}eWTISA SATIeSOU B

Ajdwt (9¢) ut sa .ewt ise ay} eased STy} ul °* Ao = Yo ue “oto us 2 «A nq am ar a ar A a an a

4 ‘ % — = 79 ueya “(pe‘2°t = 10 = “dO sonqea Tiotid @ 043 Aotdwa om g] .° (e260

A 1 on ; BIYyoW pue Auems aes ‘(9¢) worz 2° pue 29°29 SjusUodUOD adUeTIeA Jo SezeWTISa 942

SATIOP OF JapzIO UT vy pue fgetgely JO SONTeA sy} UO papasu ST UOTeMIOZUT TIOTad y

. . _v . == ete = © vol'zet = 22 PUB ¥Z9"OSPT = 7D ‘ShL'IE = 70 (92)

are

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UT USATS SOOTIZEM DY JO SjUsUIaTS [TBUOSeTp 9Y2 FO 001 aaenbs 943 Butye3 Aq paqnduos

SLOIIO PIEPUBIS 94} 91IB ONTBA pazeUTIse Ydea MOT9q SesoyUared UT SeANBTz aul

(72£0°0) (s$z0°0) (6100°0) poze

TEz°z- Sc0°I- L6L0°0- - $@ZT°0- 8L66°0 A (2) (1Z90°0) (6922°0) (v9T0°0) ber

861 °0- LSL°0 vl£82°0- 96Z9°¢ €819°0 A (9) (9070°0) (0920°0) (6100°0) ber

8fL°7Z- £v0°T- TSOT‘O- 9ZET°0- 1Z66°0 A (¢) (8990°0) (7682 °0) (Z610°0) be

S8zZ°0- LSz°0 9$7S°0- €S9S°T ZZ1S°0 A (py) , (szZ¢°0) (6882 °T) (szot"o) os

082 °0- £90°T 8L0S°0- 908¢°9 10zS "0 A (g) (91S0°0) (£920°0) (0z00°0) Z_

6£5°0- 600°Z- 6S00°0- O9TT‘*0- LZ00°T A (Zz)

(ZZ90°0 ) (O€9T’ 0) (L110°0) 682°0- 892°0 TOzE’O- ZE86 0 L60L°0 & (1) pesn 33, ay £4 a ir BTNUIO4

pezeurtzsy ATIUeNd

(St) ‘bg ut 3utaeadde sieqoweieg Jo sazeUtisg

T 9TqeL

- OT -

-17-

different from zero. On the other hand, if we employ the a priori values

_ _ -l _ = -771y\71 2 _ nT 2 2 _ mT 2.2. P) = 3 = -(m-1) and Po = Py = (T-1) ~, then 94 ‘(m-1)_ Fy9z = (T-1) 0994 =

Tm %y and the estimates of variance components are (37) 3 = 0.723, &- = 28.916 and G2 = 174.296 .

In the absence of acceptable a priori information about the p,'s it is hard to assess

a

the estimates of variance components. The a priori values , =0 which imply a negative estimate for “¥ are not acceptable to us.

In cases where the p,'s are truly 0 one can estimate the variance components by the "fitting-of-constants" method exposited by Fuller and Battese (1974). We do not follow this method for two reasons. (1) Since the individuals in our study are geographical regions with arbitrarily drawn boundaries, we would hardly expect the error components Hprsesst (Or Viger Vme) to be mutually independent, see Nerlove (1965, p. 160). The assumption that P) = P32 = 0 is of dubious merit or

validity. (2) The fitting-of-constants estimators for 0. and a are based on the

following residuals,

ir> t

(CIC, (69) L,){w-Z[Z" (cic, Q) 1,)Z] “lo (CIC, @ I,)u}

ta> u

(1, ® Ci, ){w-212', @cie, )z] oral (1, @ C}C, Jw)

where C, and Cc, are as defined in (18). The elements of fi (or tT) are not mutually independent even when the Py's are 0

so that each value of the lagged dependent variable is not independent of the

contemporaneous and the succeeding values of uy (or T,)- Consequently, the

generalized least-squares estimator of Y¥ with the variance-covariance matrix

estimated by the fitting-of-constants method is not consistent.

“I 9TqeL JO MOL YAXTS dy} UT USATS saqeWTysSe oy? UTeIGO em ‘(7ZZ) UT Se wey. [ood om zy ‘peatood oq ued Ady. 19YIOYM 1899 30u ST IT ‘JQuaLezztTp AcOA 918 [ BTGeL FO SMOX YAINOF pUe PITY. 9Yy. UT UdATS YUSTOTZJO09 QWOSUT BY} FO SON[BA pazeWTIsa sy. SdUTS “AITOTYSe[S aWOoUT ay TOF ONTeA

MOT ® ATdut [ 9TqGeL JO MOL YIINOZ 9YR UT UdATB (Pi) SdVUTISS BIUCTIVAOD 9YUL ‘anTeA eTqtsnetdut ue ‘T oaoge suni [ 91qeL JO MOL PITY 9YyQ UT UBATS AJTOTISB[O9 BWOOUT poyeUTIS® SY_ “[ 9TQeL Jo MOI PITY oy. pue (g) -bo WOLZ US9S 9q UBD SB S}[NSII [VOTIUSPT 9ATB You op Asay. asnedeq s$jas eIep OM} asay. usemMjeq SaTouederostp Jueredde ore or0yy_ °(DgT) UT eIeEp FO yeYW Se oes 9Y3 you st (2) ut e3ep JO ddaN0s 9Y43 Inq ‘potied owes sy TOF SUOTIBALESgO SaTIOS UT? 93e80133e yotdep (Dg1) pue (Z) UT SuoTJenba ay YyI0g ‘stoyrenb YUssezzZIp 105 SUOTZBATISGO 9383S FO SUBS 9YR 03 [eUOTIIOdOId BIe YDTYM SUOTJBAIBSGO Fo YSTSUOD to

XTIZeU sy. pue &K IODA OY? VeEYI (ZZ-1Z “dd *Iz61T) Awemg uT uMoYSs uVeq Sey 2] "sn 03 a[qezdadze YoU 91e [ ATQeL JO SMOI YYUSADS puB YIFTF ‘puoosss oy uT poqussaid saqewtzse sy_, *(GgI) ‘ba UT SUOTJZeAIASGO 9Y2 03 WStem ase, A[npun ue soaat3 co JO ON[TeA MOT 942 (EZ) pue (97) UT UsATS OB[NUWIOFZ ay2 UT 3eYI ST STY} TOF uosesr oy ‘“UBTS YDIITOOUT Ue BABY [T 9IQeL FO SMO YyWUaAeS pue YIZFTF OY UT UBATS OWOSUT JO JUSTOTJFO0S SY FO SoyeUITAS® 9YL “T 9TqeL JO MOL pUOddS 94} UT USATS sazeWTyse Jo ssousnotands ay} sazeoTpUuT STYL ‘0 PIEMOZ PASeTG 9q T[IM DOULTIVA IOIIO By pue soTaid pue sUODUT FO SUSTITFFE09 9YR JO SozeWTISS BuTI[NSaI ay. pue [ p1eMOY paseTq 9q [TIM eTqeTzeA YUopuedep paszZeT

O42 JO YUSTOTFFOOS 9YI JO 9ABWTISSA BuTAINSeI 9YyI e

Z uo eR ssoi3aI om J] *yoezZI0d A[T1e9U ST C7 JO UUMTOD ISITF 9YQ pue ck uaemzeq UOTIe[eII02 243 (gps “d ‘eTZET) ELeppeW UT pauoTIUSW SUOSBAI SNOTAGO IOJ *S8IeIS YUaIEFJIP 10F SuOTeAIESQO SOTIOS BWI JO SUBOW 9Yy 02 [eUOTIIOdoOId oe YSTYM SUOTJeALASGO FO YSTSuOD (qQgT) ut Co XTIZeW 942 pue eK 102D9A yr ‘(gc-z¢e ‘dd ‘* TZ61) Awemg ut pe dtsodxe sy |

-

- 8I-

- 19 -

The Likelihood Approach

The likelihood function of y when the transformed observations in (18a)-(18d)

and the initial conditions are taken as data, is

a) . 7) -@uery

(38) L(Fogo04 8 |data) = (20) M2 (9270/2 (2) 2 qty 2 (ary 1 (aT @-8) +8, (F-F00))}?

exp{- 3l 5 °y

v90e , (E45) 1232, 9-75)

+° 22+ ‘+ +2 22 =2 o of 2 2

a2 & 1213 (7 4 ) + 393 «4 (-¥3)'252, (7-73 of o* 3 3 A2. _ A =.= 4 + 4% + OY) "242, 0-Y, , 2 2 ] % %4 Ce ee ee Ge AWA AWA 717 'y ' 1% A ar ee co “Xe 259, -2Y, %2 %% % "2 % 4 v2 = n-4, v3 = T-4 and 7a (m-1) (T-1)-3, see Swamy and Mehta (1973).

If, = p= -(m-1}7* and Py = Py = -(T-1)"1, then of = 0 and the likelihood

function gets modified as

_(m-1) _(T-1) _(m-1) (T-1) 2 2 (39) LCfogoa|data) = (297/292) (02)

nN

1 2 exp{- 5l we > a)

*sz[NSer [edTSUssUOU adNpord YYSTW poy pooYyTTeyT, wnuTxew ayi Fo uotzeottdde TedTueY.aM YeYI No qutod (9zz-pzz ‘dd ‘oz61) sutyuer pue xog

ze lew *h-b*ziz, CLD +1} eo : 2-0) CLD 2iz, 1-2 +t} = Cerep|) 7 (eep|) "7 dns /(exep|D "4 - cerep|Dwo -

st A 30 poourtTeytt

SATLETOL ,,[BUOTITPUOd,, SY Wey UMOYS Oq UBD FT poyzou TToyr SUTMOTTOY ‘UOT eN{IS Iajoweredizy[Nu B FO YO UOTIeWIOFUT POOYTTOEXTT B3utzeanbs Jo sAem 9Y} FO JUNOIIL Sut3uei optM SOW eB UdATS SALy (OL6T) 3301ds pue YosTteTsqrey *szejowered aouestnu ay} ere D Fo SqUaUaTe 9Yy pue YsaTEqUT Ino FO siaqowered oy} aie id Jo sqUuowe Ta

aul ‘BIep SY} WOLF BuTWod uoTyewrozuT Jo AIT{[eIOI 9YZ SUTeIUOD YSTYM UOTIOUNF POOYTTOXTT 942 FO BsSanod aToym ey} ST 4] gt aTqeL UT sejeUT se ayy Aq passaidxa

A[ [NJ JOU ST SUTEJUOD UOTJOUNZ POOYTTOXTT OY? YOTYM UOTPeUWLIOFUT OYA * 19 AOMO} *UOTJOUNF POOYT[OXT[ 842 UT peuTeqUOd ST Slojouered ay. Jnoqe sn [[9} 0} SBYy BIep 9Y}

BY [Te 3901109 ST Tepow pounsse oy FT yey sXkes ,atdtoutad pooytTeyxtt. UL

vo ¥ {[ = g — + 4 + (P4-4y%7%7,(PE-2) Pon v _- wv cv £5 £5 ee Sees (4-4) *7%7, F4-D ¥ oka v - v sv

- 0% -

- 21 -

‘<I>

A (7-7,)'Z!Z, (7-7,) -[(m-1) (T-1)-1]/2 += -4° 4 4 ~- +4 }

A V4%%q

It can be seen that (40) allows in the exponent for the loss of precision due to the estimation of all the nuisance parameters. If we replace the exponents -(m-2)/2, -(T-2)/2, -[(m-1) (T-1)-1]/2 in L, {| data) by -(m-1)/2, -(T-1)/2 and -[(m-1) (T-1)]/2 respectively, we obtain the posterior distribution of Y corresponding to an improper prior distribution employed by Swamy and Mehta (1973). If we can plot (40) we have summarized almost all the information contained in the data about Y. Unfortunately, there are practical difficulties in plotting a three dimensional function. The function in (40) contains available information about ¥ in such a way that information about an element of 7 is inextricably mixed up with the other elements of y. We cannot integrate L, (¥|data) with respect to the elements of Y to obtain 'marginal likelihoods", see Box and Tiao (1970, p. 189). Maddala's (1971b, p. 945) definition of the relative average likelihoods does not make any sense. His use of improper priors to integrate out nuisance parameters from a likelihood function clearly violates the "likelihood principle". Bayesians who employ improper prior distributions are in fact proper in their handling of improper integrals, treating them as limits of any reasonable sequence of proper integrals, see Dempster (1973). Improper distributions are devices for approximating a careful exact assessment that would be too costly to carry out. Though the Kalbfleisch and Sprott definition of conditional likelihood is interesting, it is important to note that the conditional likelihoods do not necessarily satisfy the strong "likelihood principle" that observations in different sample spaces leading to proportional likelihood functions yield identical conclusions.

With some loss of information we can plot second-order likelihood functions

defined by Kalbfleisch and Sprott (1970, p. 189). For specified Y, and Y3, the

* [epou yeot3910943 ino

YIIM JOTTJUOD UT ST STYL ‘eIeP Ino YITM oTQTsneTd ATOITUTJop are WOSTITZJI09

adtad ay} LOZ SONTVA SATITSOd pue JYUSTITFF9OS SUlOIUT BY TOF SONTBA SATIEZOU

2eYy. Uses oq UBD AT «‘“poeTsreA ST Ty se A[Tpoyreu aeZueyd sanoquod aseyy Jo UOTIBIOT

pue odeys oy} yyog °Z SANSTY UT uMOYS 1B (ezep TrEneZajuy jo sinojzu0s

au. JO S}OTYg Th z0 ON[TeA aNnI oY. BuTpreZer AQUTeIIedUN Fo o3ueI sy peonper

eyep ano snyL ‘T > ty > 0 TeAsequt srotid ay2 uey. zeqzoys ST yoTyA £°0-S°0

st y yO SON[TeA Jo asueI aTqtTsnetd Jsow oYL Endy uT sazuRYyd 0} AATITSUSSUT

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°9398 Sr01la uoTyeoTFTIeds

‘ejyep ay} FO SUOTIEIIWTT se yons sioyoezy Auew Butatoaut ATQtssod ‘juewaspnf

@ATJOOLQnS eB ST BOUSPTAS BY} JO SUOTIEITTAWT BY SAaAeTTEq BUO IAaYyIOYM ‘Seep

ey} FO uoTeJeIdIe4UT 9aATIDaLQoO ue SATB 02 ST pOoyTTeyxT[ B8utsn ut esodaind ano

‘LT aTqel UT sezeUTyse ay? ueYyy eIep Jo AIewWMS 19}}9q YOnW aptAoId SpooyTI{exTT

Jap1o-puosss ay Ey pue ae jo Soentea Jo ited patztoeds Aue 105 Yh JO on[eA eB

Jo AYTTTqtsne[d sat IeTII ayy FO aunseew ev SaAT3 (e ep Bela T ayy Aytquenb ay}

‘le[notized uy ‘Bzep ay Aq sanyeA Jeqewered snoTIeA 0} UsATS 41oddns JO aaizap oy}

Ajtquenb 0} SUOT}OUNF poOoyTTeyXT{ IepLo-puosas asaya asn Aew am ‘(e2ep Tad yy

Aq uaaTt3 ST ae pue a JO UOTJOUNJZ POOYT[eYXT][ Iaepio-puodes sayz Th petztoeds e 103

‘AT IBTIWIS *(e1ep Bey Ty yy Aq UueaTts st Th JO UOTJOUNF PoOOYTT[eYTT IEep1o0-puoses

~ 22 -

cr(y, |¥, = 3.6296, Ys = -.2874, data)

CR(Y,|¥, = --1223, y, = -.0797, data)

3

Y °S5 60 “66 1

cR(7, 17, = .0637, V5 = -.0290, data)

Ki. ra

“S$ +60 "65 Figure 1: Plots of the second-order likelihood function of v1

for different specified values of Y9 and Yz[(CR(Y,1¥+¥,+data) ].

a

“[(erep « Talfye 2Auo] Ts go sonten potztoeds

queLessIp Loz eA pue cA JO UOTIOUNF POOYTTOYTT LepLo-puoses Jo sOTd +2 sxnsTy

09 On or ° oe

> ee

(erep ‘9°90 = TAlEar@ayuo

(erep ‘g'o = TAlFar@ajuo

(erep ‘seo = TAlfAsCAyuy

9:9

Qe]

- 23 -

4.2 Random Coefficients Model

In this subsection we propose to test explicitly the geographic stability of the slope coefficients of eq.(14). Finding that we cannot accept the hypothesis of geographic stability, we go on to specify and estimate a random coefficients model which permits random variation in all the coefficients across geographic units. Estimates in Table 1 and in Figures 1. and 2 which ignore variations in slope coefficients across states are subject to the type of specification error considered by Zeliner (1962a). Hence they are not acceptable to us.

For given j, let y; be a 43 x 1 vector of observations on G z; be a

jt’

43 x 3 matrix of observations on G and p

jt-1 Xie gti’ Yj be a 3 x 1 vector of slope coefficients and ‘5 be a 43 x 1 vector of disturbances. For all the time series observations on the jth state together, eq.(14) can be written as

41 81,4, +2. %, +v. j = 1,2,...,49 (41) Yp tp 4 * 4p tY; J )

In eq. (14) we set % = 0. Alternatively, we could subsume % into Vit and assume,

as in Zellner (1962b), that Vit

assumption the computational formulae require a matrix inversion of rank 49 x 43,

and Vit for j # i> are correlated. If we make this

an infeasible computation. We, therefore, assume that v.

jt and Vint for j # j' are

independent.

For individual states the least squares estimates of a and Vj are obtained

by using only that state's data. The income and price elasticities have been obtained from these parameter estimates by evaluating, at the corresponding sample mean point, the elasticity formulae in (9). All estimates of income coefficients

are positive and are significantly different from zero on the convention adopted

posse, sotqetzea A1OzBUBTdxe ay. Aq UsyxBI SONTeA Jo BUTASTSUOD xXTIQeEW ¢ X | B ST 1-£,

potied suo posse, eTqetzea Yuepuedap ay Aq usyeI SeNTeA FO I0ZDOA [T X J & ST I-f% o10yM

C 1-0-0

fe fact fafa - 2) f

CC - > Kd + w(Td-)tT = 7% (sv) se(Tp)*bo 02T1M ued OM MON £E,, QOUBTIVA OVTUTZ pue UvOW O19Z YIM poyNqrtsystp AT TedTJUept pue A[Quepusdseput oie Lf, creel fy pue [ > Cy > [- o1aymM

3¢

C + d= ""a (24)

24 Uf, f

Aq pozereues ore saduRqinistp oyy ¢

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LI LOZ Sp9O°T + ONTBA yestatz2 ay. Spasoxe 3T3STIeIS-Y poynduos sy, “pauTJep Jou st

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UNI SOTITOTISETS SWOSUT ‘“AaNTeA ayNTOSqe UT OM} UeY J9ZeeI3 OTeI-2 eB AToweU ‘oTZY

- 72 -

- 25 -

one period and bj = (Ciyreeebgq)'-

Eq.(43) forms a regression model with nonlinear constraints. We utilized Marquardt's (1963) nonlinear estimation technique to estimate eq.(43) from time series on each of forty-nine states. The following are the results of our nonlinear least squares (NL-LS) estimation.

All of the income coefficients are estimated to be positive. Only two of these are not significantly different from zero. All price coefficients should be negative. But 16 of the 49 are estimated to be positive; all of these 16 are not significantly different from zero. The remaining 33 have the correct negative sign, but only 4 of them are significantly different from zero. The estimate of the coefficient of the lagged dependent variable is less than 1 in absolute value for all states but it turned out to be negative for 24 states. Only 9 of these negative estimates are significantly different from zero. Negative sign for the coefficient of the lagged dependent variable is in conflict with the theoretical model. Of the remaining 25 estimates which are positive, 17 are significantly different from zero.

Thirty seven of the 49 estimates of the p;'s are significantly different from zero.

The nonlinear least squares (NL-LS) and linear least squares (L-LS) results might be compared to determine whether one of the two sets of results has a clear advantage in terms of significance of estimates and correct signs. Both the NL-LS and L-LS estimates of 15 p5's are significantly different from zero. The NL-LS estimates of 21 p5's are significantly different from zero(i.e., the estimates are largerin absolute value than twice their standard errors) whereas the L-LS estimates

of the same 21 p;'s are not significantly different from zero on "h" test. The L-LS

estimates of 2 p;'s are significantly different from zero on h test but the NL-LS

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- 9Z -

m a a Aw 4 4 (46) 1 J (B, - 8)'x!87) x, (8. - 8) (m-1)4 j=l ? 124. J 0°95 jj

Gppreniom arels ofelot, 5

isyF with (m-1)4 and (T-4)m degrees of freedom where T is the minimum of

T Th For our temporal cross-section data the value of the statistic in (46)

yr°’ is 454.86. This value is well above the 5 percent significance point 1.17 of F(192,1862) leading to a rejection of the hypothesis in (45).

We now estimate 49-state equations in (43) without forcing the coefficient

vector to be the same for all states. Suppose that the finite set of vectors

(a, 73) "yssee (Eyqr¥4q) is equivalent to a random sample from a hyperpopulation.

This assumption is usually made in applying asymptotic distribution theory to samples ‘from finite populations, see Hajek (1960). The justification of this assumption in

terms of "exchangeability" is given in Lindley (1971,pp.38-40).

Assumption 5: (a) Let 8; = (a, 5 ty, Then Byes 9B are independently and identically distributed

with mean vector 8 and positive , finite variance-covariance matrix 4;

(b) 8; is independent of vj, and 25. for every j,j' = 1,2,...,m.

(c) The P; and 955 are fixed parameters.

Now we follow the procedure outlined in Swamy (1974 , p.157) to estimate B

- and A with our data. The following results were obtained.

‘ou

(47) “Uh, {i+ s; jOghes x.y ‘\

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- 82 -

- 29 -

Houthakker et al.(1974) obtained the estimates of personal disposable income

by states by multiplying each state's personal income by the ratio of U.S. disposable

to U.S. personal income. This procedure does not give accurate measures of

personal disposable income for different states. Finally, Houthakker et al. deflated

the gasoline prices by a consummer price index. For reasons mentioned in Pollak

and Wachter’ (1975) this should not be done.

For given j the correlations between the regressors and the Sit in eq. (43)

may not be as strong as those between the regressors and the disturbances of the

aggregate equation in (7). If this is true the. inconsistencies of the estimates in A

(47) will not be as great as those of the estimates in (8). Further, since B in

A (47) is a matrix weighted average of the B

‘s, it may happen that the inconsistency A a B.

of 8 is smaller than that of any individual B.. The estimates of each B, might

Rome

be further improved by taking explicit account of the possibility of "simultaneous equation" complications. In any case, lack of inconsistency is not necessarily a very important desirable property of an estimator in small sample situations.

S. Conclusions and Discussion

We have found that the aggregate estimates in (10) and the estimates in Table 1 and in Figures 1 and 2 are based on certain assumptions which are not valid for our data. In particular, Assumption 1(b) and the assumption that the slope coefficients of eq.(14) are the same for all states are unrealistic. The estimates of policyrelevant elasticities presented in (51) are based on a general set of assumptions which are considerably weaker than those adopted either in sections 3 and 4.1 of

the present study or in other studies on the demand for gasoline.

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- 0f -

Appendix

Sources of Aggregate Data

G. : percapita consumption of gasoline by individuals in gatlons obtained from,

DATA RESOURCES, INC., LEXINGTON, MASSACHUSETTS. Data is aggregated from individual

State Data for the continental United States. xy : per capita personal income in 1967 dollars obtained from BUREAU OF ECONOMIC ANALYSIS , U.S. DEPARTMENT OF COMMERCE, WASHINGTON, D.C.

P price indices for motor gasoline with 1967 = 100 obtained from, DATA RESOURCES

gt’ | INC., LEXINGTON, MASSACHUSETTS. Computed as a weighted average of price indices be

individual States in the Continental United States Sources of Temporal

Cross-Section data.

G per capita consumption of gasoline by individuals in different states in gallons

jt obtained from DATA RESOURCES INC., LEXINGTON, MASSACHUSETTS.

Xie? per capita personal income by states in 1967 dollars obtained from DATA “RESOURCES INC., LEXINGTON, MASSACHUSETTS

Potj: price indices for motor gasoline with 1967 = 100 obtained from DATA RESOURCES, INC., LEXINGTON, MASSACHUSETTS.

a . ey

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Teot3it[Oq FO TeUINOL ,,“OWTL, FO UOTIBIOTTY 942 LOZ SUOTIBOTT AW] SJT pue uoTIDUNY UoTIONporg PTOYesNoy sy? FO soURAeTOY SUL, :(SL6T) JOIYOEM “T°W PUB “WY *yeTTOd

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QUT pue UOTIDeg-SsSoID BUTTOOY 03 Yovorddy pooyTTeyxT] oul, +: (4TL6T) “gcc-Tpg'dd ‘6g “edtazoulouosy ,,‘e2eq SeTIEg sUT] pue UO0TI9eS $s01) BUTTOOg UT STEPpoW szUsUOdWOD soUeTIeA FO OS OUL, *(PILET) “S‘D ‘eTeppen “WIS “rerudrepertua ‘MaTAeYy WV “Sotastaeag ueTsekeg :(1Z61) “A’G ‘A@TPUTT ‘OUI S TTBH-99TWUeIg :koS4O MON ‘SJFTID poomatSugq “SotrpoWOUODY OF UOTINporqUy Uy :(Z96T) “YT “UTOTY "gOZ-SLT*dd‘ze ‘q setaeg ‘X39TIOS JTeotastae3g Tedoy Oya FO [eUINOP ,,‘SLaqoweIeg JO A9quMN 981e] BUTATOAUT STEPOW OF

SPOyOW POOYTTOXTT FO uoTzecTTddy,, :(OL6T) ‘via ‘3304dS pue ‘a'r ‘YOSTeTsqTeX

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