demand systems with nonstationary prices - columbia university in

DEMAND SYSTEMS WITH NONSTATIONARY PRICES

Arthur Lewbel and Serena Ng

AbstractmdashRelative prices are nonstationary and standard root-T inferenceis invalid for demand systems But demand systems are nonlinear func-tions of relative prices and standard methods for dealing with nonstation-arity in linear models cannot be used Demand system residuals are alsofrequently found to be highly persistent further complicating estimationand inference We propose a variant of the translog demand system theNTLOG and an associated estimator that can be applied in the presenceof nonstationary prices with possibly nonstationary errors The errors inthe NTLOG can be interpreted as random utility parameters The estimateshave classical root-T limiting distributions We also propose an explana-tion for the observed nonstationarity of aggregate demand errors based onaggregation of consumers with heterogeneous preferences in a slowlychanging population Estimates using US data are provided

I Introduction

IN most industrialized economies real per capita incometrends upward and inflation rates are positive More

precisely the tendency for prices and standards of living torise makes the time series of prices and real income non-stationary Less obviously relative prices are also nonsta-tionary (see for example Ng 1995 and Lewbel 1996b)This is recognized informally in the observation that theprices of some goods such as higher education and medicalcare have been rising significantly faster than the averagerate of inflation for many years More broadly debates overwhich price measure to use to index social security or toassess monetary policy are based on the fact that differentmeasures diverge over time which can only occur if thereare differences in growth rates of the prices of differentgoods

Almost every empirical demand system study suffersfrom a severe econometric flaw namely failure to cope withthis nonstationarity of prices1 The usual techniques forhandling nonstationary regressors such as cointegration orlinear error correction models cannot be applied to demandsystem estimation because any nontrivial demand systemthat is consistent with utility maximization must be nonlin-ear in relative prices (see section IV below) But very fewestimators exist for nonlinear structural models of any formcontaining nonstationary data The problem is further exac-erbated by the facts that demands are multiple-equationsystems with nonlinear cross-equation restrictions mandatedby utility maximization and that demand systems withdimensions large enough to be empirically interesting in-volve a large number of parameters relative to the number ofavailable time periods T These problems affect demand

systems estimated using individual- household- panel-cohort- or aggregate-level data because all depend uponutility-maximizing agents facing nonstationary relativeprices2

Because of these many difficulties existing demand sys-tem studies either ignore the problem entirely or deal withnonstationarity using linear model cointegration methods3

Even if one could overcome the problems of nonlinearityand high dimension cointegration methods might still notbe appropriate because the errors in demand systems (par-ticularly those estimated with aggregate data) tend to behighly autocorrelated4 As is well known standard asymp-totic theory provides a poor guide to finite-sample inferencewhen the errors are highly persistent In cases when a unitroot in the residuals cannot be rejected the regressions arespurious and the parameter estimates are inconsistent

In this paper we provide a solution to the problem ofestimating utility-derived demand systems with nonstation-ary prices The methodology also takes care of possiblenonstationarity of the errors The key is a new functionalform that by interacting budget shares with prices producesa model that is both consistent with utility maximizationand when differenced enables nonlinear estimation of thedemand parameters by instrumental variables Classicalroot-T consistency and asymptotic normality of the esti-mates then follows from Hansenrsquos (1992) theory for thegeneralized method of moments (GMM) The model whichwe call NTLOG (nonstationary translog) is a variant ofJorgenson Lau and Stokerrsquos (1982) translog demand sys-tem Unlike the translog system in which the errors areappended to budget shares the error terms in the aggregateNTLOG model equal the average values of utility-functionparameters that vary across consumers Thus persistence inthe error of the aggregate model can be attributed to pref-erences in a slowly changing heterogeneous population

Our NTLOG model and the associated estimator providea solution to the generic empirical problem of demandsystem estimation with nonstationary relative prices andpossibly nonstationary errors We apply the model to aggre-gate data and focus on T 3 asymptotics However wealso also show how the NTLOG could be applied using dataat the level of individual households which also suffer from

Received for publication June 18 2003 Revision accepted for publica-tion November 2 2004

Boston College and University of Michigan respectivelyThis research was supported in part by the National Science Foundation

through grant SES-9905010 We would like to thank Jean-Marc Robin andCliff Attfield for helpful comments and discussions All errors are ourown

1 See for example Stock (1994) and Watson (1994) for a review ofeconometric issues relating to nonstationary variables

2 Other issues including the lack of variation in prices arise withestimation using cross-section data

3 For example Attfield (1997 2004) applies linear cointegration tech-niques to Deaton and Muellbauerrsquos (1980) almost ideal model replacingthe true nonlinear (quadratic) price deflator terms with an approximatelinear index Ogaki (1992) employs a two-good demand systems alongwith a functional form that restricts cross price effects to obtain a linearmodel for cointegration Adda and Robin (1996) provide conditions forunbiased multiple-cross-section demand system estimates with nonsta-tionary prices but they also assume a linear model

4 See eg Berndt and Savin (1995) Stoker (1986) Lewbel (19911996a) and Pollak and Wales (1992)

The Review of Economics and Statistics August 2005 87(3) 479ndash494copy 2005 by the President and Fellows of Harvard College and the Massachusetts Institute of Technology

the same nonstationarity of relative prices In addition theNTLOG model allows for individual-specific fixed effectsthat are consistent with utility maximization and if suffi-cient cross-sectional price variation is present can be usedto obtain consistent demand system estimates with fixed-Tpanels and fixed effects

The plan of this paper is as follows In the next sectionwe use US demand data to provide additional evidence ofnonstationarity of demand system regressors We also showthat nonstationarity in the residuals is not due simply tomissing variables In section III we propose a possibleexplanation for these nonstationary errors by showing thaterror persistence could arise as the result of aggregationacross utility-maximizing individuals with heterogeneouspreferences in a slowly changing population We later pro-vide empirical evidence that this explanation is at leastplausible using household-level data on food demand fromthe Michigan PSID surveys Sections IV and V of the papergive the derivation of the NTLOG functional form SectionVI provides our estimator of this NTLOG model and theassociated empirical results and section VII concludes

II Nonstationary Demands Prices and Incomes

We begin in this section with an exploratory empiricalanalysis of quarterly seasonally adjusted data for the UnitedStates documenting nonstationarity of regressors and a highdegree of persistence in demand model residuals We also

provide evidence that residual nonstationarity is not due toomitted variables We use aggregate data because that iswhere nonstationarity problems are most obvious and se-vere and because price effects can be accurately estimatedusing long time series with a great deal of relative pricevariation

Let pit be the price of good or service (or group of goodsand services) i at time t i 1 N t 1 T Let Mt

be per capita expenditures on nondurable goods and ser-vices at time t Wit be the fraction of Mt spent on group i attime t and rit ln(pitMt) By homogeneity demands arefunctions of rt the vector of elements rit Figures 1 and 2present graphs of log prices and rit for four groups ofnondurable goods and services food (good 1) energy (good2) clothing (good 3) and all other nondurable goods andservices (good 4) Even after taking logarithms the graphsclearly show the drifts and trends of nonstationary behaviorSimilar results are obtained when deflating by an overallprice index like the CPI instead of Mt Figure 3 shows thecorresponding graph for aggregate budget shares Wit indi-cating that at least some of these shares may also appearnonstationary Budget shares must by construction lie be-tween 0 and 1 and so cannot remain nonstationary foreverbut as long as the magnitudes of changes from year to yearare small (relative to the range 0 to 1) shares can closelyapproximate a nonstationary process for decades as may bethe case for some shares in these US data

Results of formal tests of nonstationarity using theDFGLS test of Elliot Rothenberg and Stock (1996) and the

FIGURE 1mdashLOG PRICES

THE REVIEW OF ECONOMICS AND STATISTICS480

MZGLS test of Ng and Perron (2001) are summarized intable 15 For a system of four goods we consider log prices

log normalized prices total expenditure budget shares andsome cross terms for a total of 35 variables over the period

5 The DFGLS and the MZGLS tests estimate the trend parameters moreefficiently and are more powerful than the Dickey-Fuller (DF) test and themodified Phillips-Perron MZ tests of Perron and Ng (1996) The MZ tests

have better size properties than the Phillips-Perron Z tests Results arereported for MZ and MZGLS which are improved versions of Z

FIGURE 2mdashLOG NORMALIZED PRICES (R)

FIGURE 3mdashSHARES

DEMAND SYSTEMS WITH NONSTATIONARY PRICES 481

1954ndash1998 Neither test can reject the null hypothesis of aunit root around a linear trend6 However when the firstdifference of each series is tested for a unit root the testsreject nonstationarity in 33 of the 35 series being testedPrices rt are faced by all consumers so estimates of demandequations at any level of aggregation or disaggregation willneed to deal with nonstationarity of prices

Though our later estimates will permit budget shares tobe stationary we assume for now that budget shares andlogged scale prices are I (1) as indicated by the tests in table1 In that case demand equations linear in N

Wit a0i alit birt eit (1)

could be consistently estimated by standard least squaresmethods if the errors eit in equation (1) were stationary

Stationary errors for these equations would require that Wit

and rt be cointegrated for each commodity group i To testfor cointegration we include a deterministic time trend inequation (1) for linear trends are found to be significant infor example Banks Blundell and Lewbel (1997) Tests forthe null hypothesis of no cointegration in table 2 indicatethat these errors are not stationary (and remain nonstation-ary even when a quadratic trend is included as a regressor)We use two variants of the residuals-based cointegration testdeveloped in Phillips and Ouliaris (1991) The 5 criticalvalue with four regressors and a linear trend is 449 for theDickey-Fuller (DF) test and 377 for the modified Phillips-Perron test (MZ) For three of the four consumption groupsthe evidence of no cointegration is overwhelming In thecase of clothing the DF test is 4885 and rejects a unitroot in eit but the MZ test is 31804 and does not rejectthe null hypothesis of no cointegration

We might not expect cointegration in equation (1) be-cause utility maximization and in particular Slutsky sym-

6 The lag lengths of the augmented autoregressions are selected by theMAIC developed in Ng and Perron (2001) The BIC (not reported) leadsto the same conclusion that the levels of the series are not stationary

TABLE 1mdashTESTS FOR NONSTATIONARITY OF PRICES AND TOTAL EXPENDITURE

Series

Levels First Differences

DFGLS MZGLS Lags DFGLS MZGLS Lags

logp1 1541 5506 3 3153 17191 2logp2 1268 4901 3 3813 25618 2logp3 1107 5574 4 2613 11675 4logp4 1442 4650 3 2044 8031 2

log1 2384 11371 1 2509 11856 3log2 1842 9818 4 4525 32102 2log3 0975 2408 2 1568 5173 4log4 0924 2141 2 4580 33948 2

log(p1p2) 1624 7065 4 5813 56045 2log(p1p3) 1370 4183 2 4620 36045 3log(p1p4) 1200 3713 3 3944 25392 3log(p2p3) 1182 2903 1 3757 24921 4log(p2p4) 0732 2091 1 6030 62858 4log(p3p4) 0773 2028 4 3594 17646 4

logM 1284 3937 3 1958 7532 4W1 1707 5887 1 4321 30873 4W2 0953 2237 0 6644 68654 2W3 1701 6441 0 5944 58423 3W4 0058 02149 0 2551 8446 4

w1logp1 0756 5204 3 2060 8524 2w1logp2 0664 3189 3 3647 26940 4w1logp3 0179 085 2 2339 10237 4w1logp4 0878 5478 3 3726 21141 2w2logp1 0858 1662 1 4522 33241 3w2logp2 1168 4446 3 2804 12830 3w2logp3 0994 4832 4 3158 12997 4w2logp4 0739 1057 1 2258 7651 4w3logp1 0458 1236 1 4564 28905 3w3logp2 07751 2652 2 4172 29847 2w3logp3 0169 813 2 4752 35906 3w3logp4 0880 2968 2 4012 19959 3w4logp1 0946 2344 3 2441 9365 3w4logp2 1042 2278 3 2927 15130 4w4logp3 1046 3110 3 2743 10189 4w4logp4 1175 3110 4 0947 1518 4

The 5 critical values for DFGLST and MZGLS (which include a constant and a linear time trend) are 29 and 191 respectively The critical values for DFGLS and MZGLS (which include a constant)are 19 and 81 respectively


metry would impose implausibly strong cross-equation re-strictions on the coefficients (see section IV below) A muchmore reasonable class of demand equations is

Wit a0i alit birt cig xt eit (2)

where g(xt) is some function that is common to all of thedemand equations and xt is a vector of observed or unob-served variables (which could include t and rt) that affectdemand In particular one of the most frequently employeddemand systems in empirical work Deaton and Muellbau-errsquos (1980) almost ideal demand system (AIDS) is a specialcase of equation (2) in which g is a constrained quadratic int and rt We tested for cointegration in the approximateAIDS model which uses Stonersquos price index (P) to deflatetotal expenditure as is common practice in this literature(see for example Deaton amp Muellbauer 1980) Thisamounts to using gxt yenj1

N wjtrjt in equation (2) Theresults are given in the second panel of table 2 The critical

values for the two tests with five regressors are 474 and425 respectively Once again there is strong evidence fornoncointegration in three of the four cases with clothingbeing the possible exception

Having equation (2) hold for every group i implies thatfor i 1

Wit ai a1it birt ciW1t eit (3)

where eit eit cie1tc1 and the other tilde parametersare similarly defined Therefore if the errors eit in equation(2) were stationary then the errors eit in equation (3) wouldalso be stationary This means that a necessary condition forthe AIDS model or for any other demand equation in theform of equation (2) to have well-behaved (that is station-ary) errors is that Wit rt and Wjt must be cointegrated foreach group i j However the test statistics in table 3indicate that Wit rt and Wjt (for any j i) are not cointe-grated and hence any model in the form of equation (2)including the exact AIDS model will yield inconsistentparameter estimates

More generally the test results based on equation (3)show that failure of cointegration is not due to any singlemissing variable or regressor This is because utility maxi-mization would require that any omitted variable appear inthe demand equations for all goods i Cointegration ofequation (2) with any variable or function g(xt) would implycointegration of equation (3) which is rejected

An even more general class of demand systems is

Wit a0i a1it birt c1igxt c2ig2xt eit (4)

for arbitrary functions g(xt) and g2(xt) Examples are theapproximate quadratic AIDS (QUAIDS) model of BlundellPashardes and Weber (1993) and the exact integrableQUAIDS model of Banks Blundell and Lewbel (1997)The third panel of table 2 reports cointegration tests forequation (4) taking g to be log deflated income andg2 g2 corresponding to the approximate QUAIDSmodel7 In all cases MZa is less than the approximatecritical value of 475 The approximate critical value forthe DF is 505 Again clothing is the only good for whichthere is some support for cointegration

Similar to equation (3) having equation (4) hold forevery group i implies that for i 1 and 2

Wit ai a1it birt c1iW1t c2iW2t eit (5)

where eit is linear in eit e1t and e2t Therefore if the errorseit in equation (4) were stationary then the errors eit inequation (5) would also be stationary so a necessary con-dition for any demand equation in the form of equation (5)to have stationary errors is that Wit rt Wjt and Wkt are

7 Exact critical values have not been tabulated for systems of such highdimensions Ng (1993) finds that an approximate guide is to raise thecritical value of the DF by 035 and of MZ by 5 for each added regressor

TABLE 2mdashTESTS FOR THE NULL HYPOTHESIS OF NO COINTEGRATION

Equation (1) Wit a0i a1it j1

N bijrjt eit

Good DF M Z Lags

1 2835 11512 02 2446 13545 13 4885 31804 04 2665 10673 0

CV 449 377

Equation (2) Wit a0i a1it birt cilogMtPt eit

Good DF M Z Lags

1 3139 18098 02 3226 20042 13 5210 30300 04 3732 14835 0

CV 474 425

Equation (4) Wit a0i a1i t birt

c1ilogMtPt c2ilogMtPt 2

eit

Good DF M Z Lags

1 3568 22125 02 2608 13831 13 5426 43275 04 2242 9785 0

CV 504 475

Equation (14) Wit a0it a1i t j1N bijrjt

j1N cjzijt eit

Good DF M Z Lags

1 4356 28923 02 4430 35425 13 6029 50875 04 4344 32421 0

CV 65 675


cointegrated for each group i j k As with all the othermodels tested the test statistics in the second panel of table3 indicate that Wit rt Wjt and Wkt (for any ordering of thegoods) are not cointegrated

Analogously to the discussion regarding equation (3)failure of cointegration of equation (5) implies that nonsta-tionarity of the demand system errors could not be due toany two missing regressors Other evidence of nonstation-arity is provided by Ng (1995) Lewbel (1996a) and Attfield(1997) We will later give one more example of demandsthat are linear in variables based on the translog systemand show that it too appears to have nonstationary errors

The test statistics used in this section are based onasymptotic theory assuming that T is extremely large andalso that T is large relative to the number of regressorswhich is not the case here The small-sample distortions insome of these tests could therefore be substantial Never-theless the evidence of trends or drifts in relative pricesaggregate total expenditures and aggregate demand systemerrors seems strong even if exact p-values for many of thesetests might be in doubt

III Aggregation and Nonstationary Errors

In this section we propose one possible explanation forthe empirically observed high autocorrelation and possiblenonstationarity of aggregate demand system errors Weshow that this persistence could be caused by aggregationacross a slowly changing population of consumers withheterogeneous preferences Our NTLOG model does notdepend on the validity of this explanation and in fact can beapplied to deal with nonstationary prices even if the demandsystem errors are stationary but it is useful to understandwhy demand errors could be persistent

Blundell Pashardes and Weber (1993) suggest that ag-gregation over consumers with time-varying individual-specific effects can lead to omitted variations in the aggre-gate demand system Here we show that even if theindividuals have specific effects that are time-invariantaggregating over an evolving population with heteroge-neous preferences will induce omitted variations (that isaggregate errors) Moreover because the population evolvesslowly over time these omitted effects are likely to behighly persistent

To see how aggregation across consumers could causepersistence in aggregate demand system errors let ahi be afixed effect of consumer h for good i This fixed effect canbe interpreted as a taste parameter that is a parameter inconsumer hrsquos utility function Let t be the set of all consumersin the economy in time t and Ht 13ht 1 be the enumer-ation of t Note that t t1 t

t1 where t

isthe set of consumers who enter the economy in period t and t1

is the set of consumers that leave the economy in periodt 1 Then ait (1Ht)yenht

ahi is the simple average of ahi

across the consumers We can write

ait Ht1

Htai t1

htahi ht1

ahi

Ht

tai t1 it

where t is the relative size of the population between thetwo periods and it is the average difference between thepreferences of the consumers that dropped out and thosethat were added in time t The dynamic properties of ait thusdepend on t and it Consider first the latter Taste param-eters ahi depend in part on age family size and otherdemographic characteristics All these variables changeslowly over time Also to the extent that taste parametersvary across households and cohorts the average taste pa-rameter of those who drop out will generally differ from theaverage taste parameter of those who enter the sample inany given period Both considerations suggest that it

should exhibit random variationsNow t depends on the number of consumers in two

consecutive periods and does not depend on i Because theset of consumers in an economy changes slowly over timethe large majority of consumers in t are also in t1 If it


Equation (3) Wit a0i a1i birt ciWjt eit j i

Good i Good j DF M Z Lags

2 1 2450 13525 13 1 4883 32419 04 1 2317 9802 0

1 2 2881 11895 03 2 5456 37449 04 2 4180 20909 0

1 3 2825 12175 02 3 3012 17663 14 3 3524 23658 0

1 4 2487 10667 02 4 3724 21577 13 4 5479 42895 0

CV 474 425

Equation (5) Wit a0i a1i birt cijWjt cikWkt eit j k i

Good i Goods j k DFGLS MZGLS Lags

1 23 2851 12200 024 2411 10487 034 2127 7509 1

2 13 2997 17252 114 3722 22032 134 3558 22159 1

3 14 5404 41440 012 5438 37798 024 5050 38951 0

4 12 3825 19744 013 2889 1844 123 3660 21983 0

CV 504 475


is uncorrelated with ai t1 then this implies that ait is ahighly persistent near-unit-root process More generally it

can be correlated with ai t1 which could increase ordecrease the persistence in ait In postwar quarterly data t

ranged from a low of 09921 to a high of 09982 with astandard deviation of 00009 so empirically t is very close(but not exactly equal) to 1

Whether ait is a near-unit-root process or not depends onboth the evolution of the population t1 and the distribu-tion of the demand system errors ahi in each time periodThese are not directly observed but we will later provideempirical evidence that substantial persistence in ait is atleast plausible based on an analysis of food demand at thehousehold level using PSID data

The above argument for persistence in the average fixedeffect assumes that each household receives the sameweight of 1Ht but the argument also holds when ait isdefined as an unequally weighted average Let ht be theweight applied to household h at time t Then for ait yenht ihtaiht it can be shown that

ait tai t1 it ht

ht 1

Htahi

t ht1

h t1 1

Ht1ahi

In addition to heterogeneous preferences time variations inweights (the last two terms) will also introduce randomnessinto ait If the weights ht are budget shares then thechanges in the income distribution between periods will bethe additional source of randomness In consequence onewould still expect ait to be an autoregressive process with aroot very close to unity

More generally a fixed effect can be the sum of anaggregate component which is unaffected by aggregationover households (for example common trends in tastes) anda household-specific component Then the aggregate fixedeffect it is

it ai0 ai1t ait (6)

The implications of a slowly increasing but heterogeneouspopulation for the aggregate fixed effect are threefold Firsta model which approximates it by a deterministic trendfunction ai0 ai1t will have omitted the random variationsait Second given the size of t in the data the aggregatefixed effect is likely to be well approximated by a randomwalk with drift The magnitude of t also implies that evenif we were to observe it unit root tests would have verylow power in rejecting the null hypothesis of nonstationar-ity Third when demand system errors have autoregressiveroots so close to the unit circle the distribution of theestimated parameters will not be well approximated by thenormal distribution even asymptotically and hence standardinference will be inaccurate (this is in addition to the

problems stemming from nonstationary prices) Persistencearising from time aggregation of fixed effects is consistentwith the empirical evidence of nonstationarity and non-cointegration documented in the previous section and withthe high degree of serial correlation found in the errors ofestimated demand systems cited in the introduction We willlater present evidence from the PSID to show that persis-tence can indeed arise from aggregation

IV A Linear Form for Translog Demands

In the time series literature nonstationarity is readilyhandled in the context of linear models The difficulty fordemand systems is that in linear models the Slutsky sym-metry implied by utility maximization results in extremelyrestrictive and implausible constraints on cross-price elas-ticities Linear models are also resoundingly rejected em-pirically

To illustrate the problem suppose the demands of anindividual household were given by the general linearmodel it ai yenj1

N bij ln pjt ci ln mt for goods i 1 N where mt is the consumerrsquos total expenditures ongoods and services in time t and wit is the fraction of mt

spent on good i in time t To be consistent with utilitymaximization this demand model must satisfy homogeneityand Slutsky symmetry Homogeneity requires ci yenj1

N bij

for i 1 N which is not overly restrictive Howeverit can be directly verified that Slutsky symmetry requireseither that ci 0 for all goods i implying homotheticdemands (budget shares independent of the total expendi-ture level) or that ai 0 and bij ij for some scalars1 N so that all cross price elasticities are forced to beproportional to own price elasticities Virtually all empiricaldemand studies reject these restrictions Of course resultsof empirical tests of symmetry and homogeneity will de-pend in part on the precision with which the associatedparameters are estimated Estimates of Slutsky matrix termsare often very imprecise

Similar restrictions arise in linear models expressed interms of quantities rather than budget shares as observed byDeaton (1975) who raised these objections in the context ofthe Stone-Geary linear expenditure system Phlips (1974)describes similar restrictions regarding the Rotterdammodel (see Barten 1967 and Theil 1971) which is a lineardemand system based on time differencing of quantities andprices Deaton and Muellbauer (1980) provide further dis-cussion of these points (they attribute the Rotterdam modelobjection to unpublished results by McFadden)

To see how we construct a model that is linear invariables while overcoming these constraints consider thetranslog indirect utility function of Christensen Jorgensonand Lau (1975)

U ptmt i1

N i 1

2j1

N

bij lnpjt

mt ln

pit

mt (7)


The function U here is the indirect utility function for thehousehold Without loss of generality assume yeni1

N i 1and bij bji Define ci yenj1

N bij with yeni1N ci 0 to make

the translog exactly aggregable see Muellbauer (1975)Jorgenson Lau and Stoker (1982) and Lewbel (1987) ByRoyrsquos identity the resulting translog budget shares are

wit

i j1

N

bij ln pjt ci ln mt

1 j1

N

cj ln pjt

(8)

Unlike the severe restrictions on elasticities implied bylinear models equation (8) satisfies Slutsky symmetry andhomogeneity without constraints on own price cross priceor total expenditure elasticities at a point This feature ofunrestricted elasticities at a point is known as Diewert(1974) flexibility and was one of the motivations for thederivation of both the popular translog and the almost idealdemand model Diewert and Wales (1987) show that im-posing negative definiteness on the translog does limit itsflexibility at some points (see also Moschini 1999) but theresulting constraints on elasticities are minimal compared tothe above-described constraints required of linear models

Now observe that equation (8) can be rewritten as

wit i j1

N

bij ln pjtmt j1

N

cjwit ln pjt (9)

Equation (9) is a model for a single household but can bereadily extended to a panel of households by adding appro-priate household subscripts h The relevant point for esti-mation is that equation (9) is linear in the variables ln(pjt mt)and wit ln pjt for j 1 N Hence if some or all of thesevariables (in particular log prices) are nonstationary themodel is at least in principle amenable to estimation usingtime series methods which we will make precise in sectionVI Furthermore i could be random implying that if wewere to estimate equation (9) in the cross section the errorscould be interpreted as random utility function parametersThe next section provides details for our particular estima-tion method in the context of an aggregate version of thismodel

V The Nonstationary Translog Demand System

A convenient implication of the linearity of equation (9)is that it facilitates aggregation across households (forestimation with household-level data the aggregation stepbelow can be ignored) Let mht be consumer (or household)hrsquos total expenditures on goods and services in time t whit bethe fraction of mht spent on goods i in time t and rhit ln(pit mht) Also for each good i let hi denote the value of the

parameter i for household h so the vector of utilityfunction parameters (h1 hN) embody preference het-erogeneity The household level translog budget shares fromequation (8) are

whit

hi j1

N

bij ln pjt ci ln mht

1 j1

N

cj ln pjt

(10)

Let Mt 1Ht yenht mht Wit yenhtwhit mht yenht mht andt yenht mht ln mhtyenht ln Mt Then

ait hthi mht

ht mht

cit it cit (11)

Notice that it is the average fixed effect for good i usingexpenditure shares as weights It then follows that theaggregate budget shares are given by

Wit

ait j1

N

bij ln pjt ci ln Mt

1 j1

N

cj ln pjt

ait j1

N

bijrjt

1 j1

N

cj ln pjt

(12)

because rit ln pitMt Models like this aggregate translogwould usually be estimated as in Jorgenson Lau and Stoker(1982) that is by replacing ait with a linear combination oftrend or demographic variables and appending an additiveerror to equation (12)

We propose to estimate the aggregate analog of equation(9) instead Define

zijt Wit ln pjt (13)

and let eit ait cit Substituting equations (6) and (13)into (12) then gives

Wit ai0 ai1t j1

N

bijrjt j1

N

cjzijt eit (14)

Separate from any considerations of nonstationarity or ag-gregation one advantage of defining the model this way isthat the errors eit are by definition equal to ait cit and socan be directly interpreted as preference heterogeneity


(taste) parameters as in McElroy (1987) and Brown andWalker (1989) More importantly equation (14) is linear inrt and zit Nonstationarity in the variables and the errors cannow be dealt with as described in the next section

We call the system of equations (14) for all goods i thenonstationary translog demand system (NTLOG) for it isbased on demands derived from translog utility functionsand some or all of its component variables may be nonsta-tionary If the system (14) were cointegrated for every goodi then the demand equations could be estimated using anerror correction model This would require that eit be sta-tionary Lewbel (1991) found using both UK and USdata that t varies very little over time with little or notrend or drift Thus eit is stationary if ait is stationaryHowever the analysis in section III suggests that eit is likelyto be nonstationary (or nearly so) because there are likelyrandom variations in preferences and in the income distri-bution over time Therefore tests of equation (14) shouldfind no cointegration In addition even in our small systemwith few goods 10 regressors in each of the equations nowhave to be tested for cointegration and the power ofcointegration tests is known to decrease with increasingnumber of regressors Based on approximate critical valuesof 65 and 675 the fourth panel of table 2 shows thatthe variables in the system (14) do not appear to be cointe-grated

Thus both theory and empirical tests are consistent witheit being an integrated or nearly integrated process In thetime series literature it is recognized that imposing a unitroot on nearly integrated processes can be desirable whenthe limiting distributions of estimators and test statistics arenot well approximated by the normal distribution In thepresent context the unit root restriction can be justifiedgiven both the test results and the magnitude of vt Consis-tent with unit roots the first differences of Wit rjt and zijt forall goods i and j all appear to be I(0) and thus standard toolsfor inference can be applied

The above NTLOG model is designed for estimation withaggregate data but it or some similar variant of equation (9)could be applied to cohort- or household-level data Atdisaggregate levels errors and income may well be station-ary But relative prices which are faced by all householdswill still be nonstationary so the NTLOG will be usefulwith disaggregate data also8

A key feature of equation (13) is that it is linear invariables These variables include zijt which is the productof the nonstationary log prices and budget shares Althoughthe budget shares appear nonstationary in the data they arebounded between 0 and 1 The nonstationarity in the cross-product term can thus be expected to be weaker Evidentlythe variables zijt all appear stationary when differenced9 Incontrast Deaton and Muellbauerrsquos (1980) almost ideal de-mand model was designed to be nearly linear but missesthat ideal because of the presence of a quadratic pricedeflator which includes terms like rjt

2 First differences of rjt2

terms are not close to stationary Thus even in first-differenced form the correct limiting distribution for theAIDS model may still be nonstandard Recognizing prob-lems of high autocorrelation in levels Deaton and Muell-bauer reported estimates from differencing the AIDS modelbut assumed a standard limiting distribution for the result

It is also of some interest to compare the nonstationarytranslog with the Rotterdam model (see for example Bar-ten 1967 and Theil 1971) The Rotterdam model consistedof regressing differenced quantities on differenced pricesand income The Rotterdam model has the virtue of makingthe regressors stationary Its shortcoming is that it is notconsistent with utility maximization without imposing ex-treme restrictions on its coefficients as described in theprevious section Unlike the Rotterdam model the NTLOGis derived from a utility function that has flexible demandsFurthermore the error terms of the Rotterdam model likethe errors in the ordinary aggregate Translog and AIDSmodels are appended to demands with no economic inter-pretation In contrast the error terms of the NTLOG aredirectly derived from heterogeneity in taste parameters andvariations in the income distribution

VI Estimation and Results

Equation (14) manages full linearity but at the cost ofhaving some of the regressors (the zijt) depend on Wit andhence those regressors could be correlated with the errorseit This issue must be dealt with upon estimation Assumewe have a vector of stationary instrumental variables st thatare uncorrelated with the stationary difference eit eit eit1 Then

E stWit ai1 j1

N

bij rjt j1

N

cj zijt 0 (15)8 A limitation on using NTLOG for disaggregate data is that the translog

is a rank-two demand system with budget shares linear in log incomewhereas empirical evidence on household-level data suggests demands arequadratic and of rank three See Howe Pollak and Wales (1979) Gorman(1981) Lewbel (1991) Blundell Pashardes and Weber (1993) andBanks Blundell and Lewbel (1997) Although demands for individualhouseholds appear to be of rank three there is evidence that aggregatedemands may be adequately modeled as rank two Lewbel (1991) showsthat rank-three curvature arises primarily from households at the extremesof the income distribution and that excluding a small percentage ofhouseholds in these tails results in demands that are empirically of ranktwo If the contribution of these few extreme households to the aggregateis small then the aggregate will appear to be of rank two Also the range

of observed aggregate (per capita) income is small relative to the range ofincomes that exists across households The effect of these rank-threehouseholds in the aggregate is therefore small In our empirical applica-tion later we find that the rank-two NTLOG is satisfactory for aggregatedata Nonetheless rank-three extensions of the NTLOG could be con-structed and might be desirable for future applications using disaggregatedata

9 Ogaki and Reinhart (1998) encountered a similar problem and alsoargued that first-differencing is likely to make nonstationarity in a ratioterm empirically unimportant


The set of equations (15) for all goods i can be stacked toyield a collection of moment conditions for the parameterswhich can be estimated using the standard GMM Theinstruments and the differenced variables in the equations(15) are all stationary so the coefficients in this GMM willhave the standard root-T normal limiting distribution Be-cause these are demand equations and the errors arise frompreference heterogeneity suitable instruments will be vari-ables that affect the supply side of the economy

To check sensitivity to the choice of instruments weconsider two sets of instruments The first simply uses differ-ences in the lags of the variables in the system Wi t 2 i 1 N 1 ln pi t2 i 1 N ln Mt2 zi j t2j 1 2 the lag of the differenced log population aconstant and a time trend These instruments deal with thedependence of z on endogenous budget shares but fail tocontrol for classical simultaneity of demand with supply

The second set of instruments which should be suitablefor both these problems consist of supply variables likethose used for example by Jorgenson Lau and Stoker(1982) These instruments are the deflator for civilian com-pensation of government employees government purchasesand its deflator imports of goods and services wages andsalaries unit labor costs and participation rate governmenttransfers to individuals unemployment and populationThese are also differenced to stationarity Also included area constant and a time trend The first set of instruments has13 variables and the second has 14 yielding a total of 39and 42 moment conditions respectively It is not feasible touse both sets of instruments simultaneously because doingso will result in too many moment conditions relative to thesample size

A nonstandard feature of our application of GMM is thefollowing The adding-up constraint means that the condi-tion yeni1

N eit 0 must be satisfied This imposes strongcross-equation restrictions on the dynamic structure of theerrors if the eit terms are serially correlated See forexample Berndt and Savin (1975) and Moschini and Moro(1994) We first estimate the parameters with the White-Huber correction for heteroskedasticity and then test forserial correlation in the residuals First differencing appearsto be sufficient to render eit approximately white noise andthe Box-Ljung statistic with six lags cannot reject the nullhypothesis of no serial correlation at the 5 level forequations (1) and (2) or at the 10 level for equation (3)We also tried quasi-differencing the first-differenced data toestimate a common AR(1) parameter for the differencedresiduals corresponding to Berndt and Savinrsquos (1975) errorspecification after differencing The autocorrelation param-eter estimate is numerically small and insignificant so thoseresults are not reported10

For a system of N consumption groups only N 1equations need to be estimated given the adding-up con-straint After imposing the symmetry condition bij bji thehomogeneity condition ci yenj1

N bij and the exact aggrega-tion condition yeni1

N ci 0 we still have 12 parameters in amodel with four goods We first obtain unrestricted esti-mates of all parameters and then restrict those bij i j thatare statistically insignificant to 0 to improve precision of theestimates These results are reported in table 4 Overall the2 test for overidentifying restrictions cannot reject theorthogonality conditions

A Testing the Model

We consider two additional tests of the empirical ade-quacy of the NTLOG model The first is a test for stabilityof the coefficients (that are not statistically different from 0in the full sample) For both sets of instruments the sup LMtest of Andrews (1993) is maximized at 02 where Tis the breakpoint for a sample of size T The test statistic is1638 and 1143 for the two sets of instruments respec-tively and the 5 critical value for seven parameters is2107 Thus we cannot reject the null hypothesis of param-eter constancy

The second is a general test for any omitted factorsanalogous to our earlier use of equation (3) to test for theexistence of any function g in equation (2) Suppose the

10 If we had seen stronger evidence of serial correlation then a moreflexible treatment of autocorrelation could have been used as in Moschiniand Moro (1994)

TABLE 4mdashRESTRICTED AND UNRESTRICTED ESTIMATES OF THE

PARAMETERS BY GMM

INST1 INST2

Unrestricted Restricted Unrestricted Restricted

b11 01186 00838 01665 00988Se 01077 00886 01086 00883

b12 00020 mdash 00062 mdashSe 00563 mdash 00482 mdash


b22 01661 01206 01408 01048Se 00487 00410 00509 00390


b24 01672 01835 01474 01768Se 00623 00532 00621 00466

b33 00617 00715 01121 01387Se 00361 00392 00407 00415


b44 04851 03153 05064 04075Se 01214 01230 01475 01350

a11 00009 00008 00008 00007Se 00003 00002 00003 00002

a12 00001 mdash 00001 mdashSe 00002 mdash 00002 mdash

a13 00007 00005 00010 00008Se 00002 00002 00002 00002

2 25095 32360 25924 30323Df 27 32 30 35


nonstationary translog omits some variable or some func-tion of variables gt which could be price-related because offlexible regularity income related due to rank consider-ations or some other source of misspecification such asomitted dynamic or demographic effects Then

Wit digt ai0 ai1t j1

N

bijrjt j1

N

cjzijt eit (16)

For example digt could be a component of it or equation(16) could arise from the aggregation of demands of apotentially rank-three utility function

Analogously to how equation (2) implies equation (3) wehave that if equation (16) holds for any gt then

Wit diWkt ai0 ai1t

j1

N

bijrjt j1

N

cjzijt dizkjt eit (17)

Each equation (17) for i 2 N 1 is linear in theobservables and so can be estimated by differencing andGMM again using our instruments st We may therebyindirectly test for the existence of any omitted factor gt bytesting whether the coefficients di are statistically signifi-cant We may similarly test for two omitted variables byincluding two different budget shares as regressors in placeof just Wkt in equation (17) analogously to using equation(5) to test for the structure of equation (4) A total of 27variations of the model exist depending on which budgetshares are modeled and which are used as regressors Toconserve space table 5 only reports results for 12 configu-rations When good 4 (others) is added to the food equationthe t-statistic is sometimes significant at the two-tailed 5level suggesting some (though not overwhelming) evidenceof omitted variables But for both sets of instruments thet-statistics on other di are generally insignificant The J-testfor overidentifying restrictions is reported in the last columnof table 5 Compared with the J test in table 4 (25095 and25924) the difference never exceeds 7814 the criticalvalue from the 2 distribution with three degrees of free-dom Thus we cannot reject that the di are jointly 0

B Elasticities

Aggregate quantities are given by Qit MtWitpit Onecan verify from equation (12) that the corresponding aggre-gate price and income elasticities are given by

ln Qit

ln pjt

bijWit cj

1 k1

N

ck ln pkt

1ij (18)

ln Qit

ln Mt

ciWit

1 k1

N

ck ln pkt

1 (19)

where 1ij is the Kronecker delta which equals 1 if i j and0 otherwise The constants a0i are not identified whendifferencing as in equation (15) The elasticity formulasgiven in equations (18) and (19) do not make use of a0i and

TABLE 5mdashSPECIFICATION TESTS

(A) With INST1(ijk) ti tj tk 24

2

(211) 06336 03359 03005 260097(311) 07389 01753 02321 258998(411) 22732 04706 03154 232479(234) 03955 19064 03559 265953(314) 06823 01775 05276 258385(414) 21087 04602 00459 232554(341) 06355 07701 00733 269111(342) 06164 07175 01656 268183(344) 05854 06251 03532 267020(431) 22817 17406 03635 242229(432) 22068 17463 02535 242755(434) 21489 18058 00476 242617

(B) With INST2(ijk) ti tj tk 27

2

(211) 09181 05734 08393 212570(311) 14878 06800 09548 212513(411) 19244 01801 09574 202703(234) 12099 02463 03557 226511(314) 14857 06412 06393 218971(414) 18274 01964 06944 202437(341) 10192 22839 06290 248406(342) 10342 21522 04570 250925(344) 09126 22949 03867 253738(431) 20192 01576 07604 209346(432) 20149 01076 07502 209106(434) 19207 01403 05289 211045

The first column shows the variables being added to the equation for goods 1 2 and 3 respectivelyThe next three columns are the t-statistic on the variable being added

TABLE 6mdashESTIMATES OF PRICE AND INCOME ELASTICITIES FOR NTLOG

Good

Price

Income1 2 3 4

(A) Using INST1

1 (food) 06808 00638 00724 02682 08128Se 03847 00734 00722 02995 060902 (energy) 00438 01661 00724 12644 14019Se 01424 09802 00722 12932 046263 (clothing) 00438 00638 18580 00924 19305Se 01424 00734 08553 00901 092754 (others) 01326 03379 00724 04024 08004Se 01361 03798 00722 03885 01946

(B) Using INST2



so are identified We present estimates of price and incomeelasticities (evaluated at the mean) in table 6 The standarderrors are calculated using the delta method

We find that spending on energy and other goods is notprice-sensitive The income elasticities for energy and forclothing are above 1 whereas food and other goods areincome-inelastic Also according to the NTLOG estimatesa 1 increase in the price of food reduces expenditure onfood by 068 and a 1 increase in the price of clothingreduces expenditure on clothing by around 2 These elas-ticities are statistically significant and are larger than mostothers based on time series data in the literature which aregenerally estimated over a shorter sample See for exampleDenton Mountain and Spencer (1999) for a survey ofestimates Using the standard translog model JorgensonLau and Stoker (1982) found a very large price elasticityfor a combined food and clothing group In results notreported we find that estimation of the standard translogmodel with our data set over the same time period yields apositive own price elasticity for food and income elastici-ties for food and clothing that are approximately doublethose based on the NTLOG One cannot make inferenceabout the statistical significance of the standard translogestimates because the standard translog model is expressedin terms of nonstationary variables The asymptotic normal-ity of the NTLOG estimates on the other hand allows forstandard inference

The elasticities evaluated at the sample means reportedin table 6 have reasonable magnitudes and signs Aninteresting implication of nonstationarity of prices is that

elasticities may drift over time This is illustrated infigure 4 with estimates taken from INST2 The priceelasticity for energy appears to change little over timeand has historically been quite small The price elasticityfor food has fallen somewhat during the course of thepast forty years but the variations around the meanelasticity of 06 are rather small The price elasticity forclothing has increased in recent years The price elasticityfor other goods seems to have increased since the mid-sixties when these goods became a much larger share oftotal spending (see figure 1) The time series of incomeelasticities are presented in figure 5 A notable feature isthat not only has clothing become more price-sensitiveover time but its income elasticity has also gone upThese time variations in price and income elasticitiesmay reflect substantial changes in the composition ofthese categories over time

C Aggregate Fixed-Effect Estimates

Our empirical analyses provide evidence that aggregatedemand system errors are nonstationary We have suggestedthat nonstationarity of errors could be due to aggregation ofconsumersrsquo fixed effects across a slowly evolving popula-tion of consumers with heterogeneous preferences We nowprovide some empirical evidence to suggest that this expla-nation is at least plausible The ideal data for this exercisewould be consumer-level information over a long span butdetailed information on consumption by households (suchas the CEX) is generally available only in the form of short

FIGURE 4mdashOWN PRICE ELASTICITIES


panels or rotating panels that do not track the same house-holds neither of which are suited for the estimation ofindividual specific effects The best available data for ourpurpose appear to be the PSID which tracks householdsrsquofood consumption and income since 1968

We begin by estimating the food demand equation

wht ah fht t t log mht εht (20)

where fht is age and family size to control for observedsources of heterogeneity and ah is the fixed effect for eachhousehold h The regression includes year dummies bothadditively and interacted with log income to obtain thetime-varying coefficients t and t This is equivalent toestimating a separate Engel curve for each time period sothis analysis does not require measuring prices or specifyinghow prices affect food demands at the household level

We only consider male-headed households with at least10 observations over our estimation sample of 1974 to 1992and whose heads are between ages 25 and 5511 By per-forming fixed effect estimation we can obtain ah for 1308households

Given a sample we cannot observe households enteringor leaving the true population of households t so weproxy changes in the population by changes in subpopula-tions defined by age As a first check we aggregate theestimated fixed effect corresponding to the 147 households

that are in the sample all 17 years In this first example thereis by construction no change over time in the compositionof this subpopulation of households so the aggregate isconstant over time (see table 7) This first example corre-sponds to the extreme case of vt 1 and it 0 in sectionIII We then construct three estimated aggregate fixed ef-fects based on different subpopulations The first averagesthe fixed effect across those households whose heads arebetween ages 30 and 50 in each time period the secondbetween ages 30 and 40 and the third between ages 40 and

11 Food is the sum of food consumed at home plus food consumedoutside of home Food data were not collected in 1973 1987 and 1988The SEO sample was excluded from the analysis Households whoreported zero income andor consumption are dropped

FIGURE 5mdashINCOME ELASTICITIES

TABLE 7mdashESTIMATED AGGREGATE FIXED EFFECT

YearBalanced

PanelAge

30ndash50Age

30ndash40Age

40ndash50

74 0757 1915 1176 323975 0757 1755 1019 293876 0757 1510 0731 277577 0757 1286 0505 256678 0757 1083 0265 248779 0757 0806 0010 224380 0757 0555 0189 202981 0757 0364 0301 172982 0757 0171 0444 148983 0757 0082 0690 131284 0757 0301 0874 100385 0757 0478 1092 089286 0757 0607 1270 069987 0757 0699 1427 047390 0757 0914 1894 019291 0757 0955 1999 033592 0757 1011 2180 0526AR(1) 9545 9945 1015

The first column gives the fixed effect aggregated over a fixed set of households The remainingcolumns are based on aggregation over household heads in each year that are between ages 30 and 5030 and 40 and 40 and 50 respectively The estimated individual fixed effects are from estimates of thehousehold-level food demand equation (20) using the fixed-effect estimator as implemented in Stata


50 In all three cases the sample size changes over timeboth as the size of the subpopulation changes and as house-holds drop in and out of the interviews The averagenumbers of observations used in the aggregations are 803505 and 337 respectively with standard deviations of 147243 and 146 respectively

The estimates reported in table 7 suggest strong trends inthe resulting aggregate fixed effects consistent with ourconjecture that these series can be highly persistent Thefirst-order autoregressive parameter is estimated to be nearunity in every case12

This simple exercise is subject to many caveats due todata limitations For example there is likely to be moreperiod-to-period change in the survey respondents than inthe population at large Nonetheless the results suggest thataggregation of demand equation fixed effects over a slowlyevolving heterogeneous population could be a plausiblecause of apparent nonstationarity of errors in aggregatedemand systems

VII Household-Level Data

We implemented NTLOG with aggregate rather thanhousehold-level data because that is the context in whichthe nonstationarity problem is most obvious and severe Inthis section we briefly describe how the estimator could beapplied at the household level Assume household h has thetime t indirect utility function

Uht ptmt

i1

N hi di fht ehit 1

2j1

N

bij lnphjt

mht ln

phit

mht

where fht is a vector of observed characteristics of householdh that can affect utility and change over time hi is aconstant preference parameter for each household h andgood i ehit embodies time-varying unobserved preferenceheterogeneity yen i1

N di 0 yen i1N ehit 0 and yeni1

N hi 1and we drop the law-of-one-price assumption and allowprices to vary across households If the utility function isover nondurables and services then fht could include stocksof durables yielding conditional demand functions Uht willthen be a conditional rank-two utility function which isequivalent to an unconditional rank-three model (see Lew-bel 2002)

Let zhijt whit ln phjt and rhit ln phitmht and followthe same steps used to derive equation (9) to obtain

whit hi di fht j1

N

bijrhjt j1

N

cjzhijt ehit

Instead of (or in addition to) appearing in the utility func-tion the error term ehit can embody measurement error inwhit or optimization error on the part of household h Thepreference parameter hi is a household specific fixed effectfor good i Assuming that each household is observed in atleast two time periods estimation is then GMM based onthe moment conditions

Eshtwhit di fht j1

N

bij rhjt j1

N

cj zhijt 0

The instruments sht can be lags of ln phit ln mht andpossibly fht which are assumed to be uncorrelated withehit This specification incorporates observed and unob-served constant sources of heterogeneity in preferencesacross households into the fixed-effect parameters hiThese fixed effects are differenced out so the incidental-parameters problem does not arise even when each house-hold is only observed for a small number of time periodsConsistency requires that either the number of householdsor the number of time periods go to infinity

Although this model gives consistent estimates with fewtime periods when the number of households goes to infin-ity it should be noted that in industrialized economiesfactors including price competition and antidiscriminationlaws result in limited variation in the prices faced bydifferent households for identical goods in the same timeperiod Therefore in short panel data sets (where nonsta-tionarity might not be a problem) the available price vari-ation will generally be very limited and hence price effectswill be estimated very imprecisely For example whenprices only vary by time and region (so phit is the same forall households h in a region) vastly increasing the numberof households in each region provides no increase in ob-served relative price variation At least for some goods longtime series such as are available with aggregate data areneeded to observe substantial relative price variation

If observed prices vary by region as well as time then ouraggregate model could be applied by adding a regionsubscript and using region-specific aggregates W and MThese aggregates could be constructed from panel or fromrepeated cross-section data The time trends and aggregateerrors can now be due to aggregation of fht in addition totrends in the (regional) population means of hi

VIII Conclusions

Price and income elasticities are important statisticswhich characterize consumersrsquo behavior and are fundamen-tal to the evaluation of tax policies and welfare programsDemand systems provide a conceptually coherent frame-work for estimating these elasticities Utility maximizationrequires any reasonable specification of demand systems tobe nonlinear in relative prices and relative prices them-selves are nonstationary

12 Formal tests of nonstationarity or unit roots in the data in table 7 arenot practical because the number of time periods is very short no data areavailable in some years and ordinary tests would fail to allow forestimation errors in the generation of these data


Very few techniques exist for estimation of structuralnonlinear models with nonstationary data The vast majorityof existing empirical demand system studies with eitherhousehold- or more aggregate-level data simply ignore thisproblem treating the data as if they were stationary The fewempirical studies that do consider price nonstationarityassume linearity by for example estimating an almost idealmodel while ignoring its nonlinear component which is aquadratic price index

To deal with price nonstationarity we propose a refor-mulation of the utility-derived translog model that can bewritten in a linear form (albeit with endogeneity in theregressors caused by interacting prices with budget shares)thereby avoiding the severe constraints of ordinary utility-derived linear demand models while preserving sufficientlinear structure to deal with nonstationarity Our NTLOGmodel provides a solution to the empirical problem whichexists at both the household and the aggregate level ofdemand system estimation with nonstationary relativeprices At the household data level the NTLOG also permitsconsistent estimation in the presence of preference hetero-geneity that takes the form of utility-derived household andgood-specific fixed effects

In addition to handling nonstationarity of relative pricesour NTLOG model can also cope with possible nonstation-arity of demand system errors a feature commonly found inmodels using aggregate data We show theoretically thatnonstationarity of demand system errors could arise fromaggregation across heterogeneous consumers in a slowlychanging population and we provide some empirical evi-dence for this effect based on a panel of household demandsfor food Other possible sources of nonstationarity areomitted variables omitted dynamics and aggregationacross goods as in Lewbel (1996a) We provide someempirical evidence against the omitted variables explana-tion

We estimate this NTLOG model using aggregate USdata over the sample 1954ndash1998 The model is subjected toand passes a variety of specification tests Estimates of themodel parameters and elasticities are also reported and arefound to be economically plausible Unlike other demandsystem estimates in the literature given nonstationary dataand nonstationary errors these NTLOG estimates haveroot-T asymptotically normal distributions and so allow forstandard inference

An open problem in all time series estimation of demandsystems is to reconcile the apparent nonstationary behaviorof budget shares with the fact that budget shares arebounded between 0 and 1 In this paper we first-differencethe model because the resulting GMM estimator has stan-dard root-T limiting distribution However as discussed inDavidson and Terasvirta (2002) fractional instead of firstdifferencing could be an appealing alternative when thevariables displaying strong persistence are strictly boundedIt remains to be seen whether fractional cointegration esti-

mation as in Davidson (2002) can be applied to theNTLOG and the estimates remain root-T consistent andasymptotically normal Alternatively one might constructmodels where budget share behavior changes from nonsta-tionary to stationary in the neighborhood of boundariesPersistent movements in budget shares could be a result ofchanges in demographics tastes and the composition ofgoods A further decomposition of these effects might pro-vide a better understanding of the sources of apparentnonstationarity

REFERENCES

Adda J and J Robin ldquoAggregation of Nonstationary Demand SystemsrdquoCREST-INSEE working paper no 9634 (1996)

Andrews D W K ldquoTests for Parameter Instability and Structural Changewith Un known Change Pointrdquo Econometrica 61 (1993) 821ndash856

Attfield C ldquoEstimating a Cointegrated Demand Systemrdquo EuropeanEconomic Review 41 (1997) 61ndash73ldquoStochastic Trends Demographics and Demand Systemsrdquo Uni-versity of Bristol Mimeograph (2004)

Banks J R W Blundell and A Lewbel ldquoQuadratic Engel Curves andConsumer Demandrdquo Review of Economics and Statistics 794(1997) 527ndash539

Barten A P ldquoEvidence on the Slutsky Conditions for Demand Equa-tionsrdquo Review of Economics and Statistics 49 (1967) 77ndash84

Berndt E R and N E Savin ldquoEstimation and Hypothesis Testing inSingular Equation Systems with Autoregressive DisturbancesrdquoEconometrica 435 (1975) 937ndash958

Blundell R W P Pashardes and G Weber ldquoWhat Do We Learn aboutConsumer Demand Patterns from Micro Datardquo American Eco-nomic Review 83 (1993) 570ndash597

Brown B W and M B Walker ldquoThe Random Utility Hypothesis andInference in Demand Systemsrdquo Econometrica 57 (1989) 815ndash829

Campbell J Y and N G Mankiw ldquoPermanent Income Current Incomeand Consumptionrdquo Journal of Business and Economic Statistics 8(1990) 265ndash279

Christensen L R D W Jorgenson and L J Lau ldquoTranscendentalLogarithmic Utility Functionsrdquo American Economic Review 65(1975) 367ndash383

Davidson J ldquoA Model of Fractional Cointegration and Tests for Cointe-gration Using the Bootstraprdquo Journal of Econometrics 110 (2002)187ndash212

Davidson J and T Terasvirta ldquoLong Memory and Nonlinear TimeSeriesrdquo Journal of Econometrics 110 (2002) 105ndash112

Deaton A Models and Projections of Demand in Post-war Britain(London Chapman and Hall (1975)

Deaton A S and J Muellbauer ldquoAn Almost Ideal Demand SystemrdquoAmerican Economic Review 70 (1980) 312ndash326

Denton F D Mountain and B Spencer ldquoAge Trend and Cohort Effectsin a Macro Model of Canadian Expenditure Patternsrdquo Journal ofBusiness and Economic Statistics 174 (1999) 430ndash443

Diewert W E ldquoApplications of Duality Theoryrdquo (pp 106ndash171) in MIntriligator and D A Kendrick (Eds) Frontiers of QuantitativeEconomics vol II (Amsterdam North-Holland 1974)

Diewert W E and T J Wales ldquoFlexible Function Forms and GlobalCurvature Conditionsrdquo Econometrica 55 (1987) 43ndash68

Elliott G T J Rothenberg and J H Stock ldquoEfficient Tests for anAutoregressive Unit Rootrdquo Econometrica 64 (1996) 813ndash836

Gorman W M ldquoSome Engel Curvesrdquo in A Deaton (Ed) Essays inHonour of Sir Richard Stone (Cambridge Cambridge UniversityPress 1981)

Hansen L P ldquoLarge Sample Properties of Generalized Method ofMoments Estimatorsrdquo (pp 97ndash122) in K D Hoover (Ed) TheNew Classical Macroeconomics (Aldershot UK Elgar 1992)

Howe H R A Pollak and T J Wales ldquoTheory and Time SeriesEstimation of the Quadratic Expenditure Systemrdquo Econometrica47 (1979) 1231ndash1247

Jorgenson D W L J Lau and T M Stoker ldquoThe TranscendentalLogarithmic Model of Aggregate Consumer Behaviorrdquo in R


Basmann and G Rhodes (Eds) Advances in Econometrics(Greenwich CT JAI Press 1982)

Lewbel A ldquoFractional Demand Systemsrdquo Journal of Econometrics 36(1987) 311ndash337ldquoThe Rank of Demand Systems Theory and NonparametericEstimationlsquo Econometrica 59 (1991) 711ndash730ldquoAggregation without Separability A Generalized CompositeCommodity Theoremrdquo American Economic Review 86 (1996a)524ndash543ldquoLarge Demand Systems and Nonstationary Prices Income andPrice Elasticities for Many US Goodsrdquo Brandeis University mim-eograph (1996b)ldquoRank Separability and Conditional Demandsrdquo Canadian Jour-nal of Economics 35 (2002) 410ndash413

McElroy M B ldquoAdditive General Error Models for Production Costsand Derived Demand or Share Equationsrdquo Journal of PoliticalEconomy 95 (1987) 737ndash757

Moschini G ldquoImposing Local Curvature Conditions in Flexible DemandSystemsrdquo Journal of Business and Economic Statistics 174(1999) 487ndash490

Moschini G and D Moro ldquoAutocorrelation Specification in SingularEquation Systemslsquo Economics Letters 46 (1994) 303ndash309

Muellbauer J ldquoAggregation Income Distribution and Consumer De-mandrdquo Review of Economic Studies 42 (1975) 525ndash543

Ng S ldquoEssays in Time Series and Econometricsrdquo Princeton Universitydoctoral dissertation (1993)ldquoTesting for Homogeneity in Demand Systems When the Regres-sors Are Non-stationarylsquo Journal of Applied Econometrics 10(1995) 147ndash163

Ng S and P Perron ldquoLag Length Selection and the Construction of UnitRoot Tests with Good Size and Powerrdquo Econometrica 696 (2001)1519ndash1554

Ogaki M ldquoEngelrsquos Law and Cointegrationrdquo Journal of Political Econ-omy 100 (1992) 1027ndash1046

Ogaki M and C Reinhart ldquoMeasuring Intertemporal Substitution TheRole of Durable Goodsrdquo Journal of Political Economy 1065(1998) 1078ndash1098

Perron P and S Ng ldquoUseful Modifications to Unit Root Tests withDependent Errors and Their Local Asymptotic Propertiesrdquo Reviewof Economic Studies 63 (1996) 435ndash465

Phlips L Applied Consumption Analysis (Amsterdam North Holland1974)

Phillips P C B and S Ouliaris ldquoAsymptotic Properties of ResidualBased Tests for Cointegrationrdquo Econometrica 58 (1990) 165ndash193

Pollak R A and T J Wales Demand System Specification and Estima-tion (Oxford Oxford University Press 1992)

Stock J H ldquoUnit Roots Structural Breaks and Trendsrdquo (pp 2740ndash2831)in R Engel and D McFadden (Eds) Handbook of Econometricsvol 4 (Elsevier Science 1994)

Stoker T ldquoAggregation Efficiency and Cross-Section RegressionsrdquoEconometrica 54 (1986) 171ndash188

Theil H Principles of Econometrics (New York Wiley 1971)Watson M W 1994 ldquoVector Autoregressions and Cointegrationrdquo (pp

2844ndash2910) in R Engel and D McFadden (Eds) Handbook ofEconometrics Vol 4 (Elsevier Science 1994)

APPENDIX

Data Sources

The data are from the US National Income and Product Accountsobtained via Citibase

The sample period is 1954Q1ndash1998Q4 In Citibase mnemonics M GC GCD Nominal expenditures on the four groups are

1 GCFO (food)2 GCNF GCNG GCST GCSHO (energy)3 GCNC (clothing)4 Others M food energy clothing

Price indices are obtained by dividing nominal by real expenditures inthese groups Following many other authors (such as Campbell amp Man-kiw 1990) data from before the mid-1950s are excluded to avoid theeffects of both the Korean war and measurement errors in the first fewyears of data collection

In Citibase the second set of instruments are GGE GDGE GGCGEGIMQ GW GMPT GPOP LBLCPU LHUR and LHP16 We take logs ofthe first six of these variables before first-differencing them

For table 7 household-level data from 1974 to 1992 are taken from thePanel Study of Income Dynamics excluding the SEO sample We useobservations with male household heads who are between age 25 and 55and have no missing data on age sex marital status number of childrenor income Income is defined as the sum of earned and transfer income ofthe husband wife and other family members Food is defined as foodconsumed at home and outside the home Consumption data are notavailable for 1973 1987 and 1988 A total of 17568 observations over 17years were used in the fixed-effect estimation


the same nonstationarity of relative prices In addition theNTLOG model allows for individual-specific fixed effectsthat are consistent with utility maximization and if suffi-cient cross-sectional price variation is present can be usedto obtain consistent demand system estimates with fixed-Tpanels and fixed effects

The plan of this paper is as follows In the next sectionwe use US demand data to provide additional evidence ofnonstationarity of demand system regressors We also showthat nonstationarity in the residuals is not due simply tomissing variables In section III we propose a possibleexplanation for these nonstationary errors by showing thaterror persistence could arise as the result of aggregationacross utility-maximizing individuals with heterogeneouspreferences in a slowly changing population We later pro-vide empirical evidence that this explanation is at leastplausible using household-level data on food demand fromthe Michigan PSID surveys Sections IV and V of the papergive the derivation of the NTLOG functional form SectionVI provides our estimator of this NTLOG model and theassociated empirical results and section VII concludes

II Nonstationary Demands Prices and Incomes

We begin in this section with an exploratory empiricalanalysis of quarterly seasonally adjusted data for the UnitedStates documenting nonstationarity of regressors and a highdegree of persistence in demand model residuals We also

provide evidence that residual nonstationarity is not due toomitted variables We use aggregate data because that iswhere nonstationarity problems are most obvious and se-vere and because price effects can be accurately estimatedusing long time series with a great deal of relative pricevariation

Let pit be the price of good or service (or group of goodsand services) i at time t i 1 N t 1 T Let Mt

be per capita expenditures on nondurable goods and ser-vices at time t Wit be the fraction of Mt spent on group i attime t and rit ln(pitMt) By homogeneity demands arefunctions of rt the vector of elements rit Figures 1 and 2present graphs of log prices and rit for four groups ofnondurable goods and services food (good 1) energy (good2) clothing (good 3) and all other nondurable goods andservices (good 4) Even after taking logarithms the graphsclearly show the drifts and trends of nonstationary behaviorSimilar results are obtained when deflating by an overallprice index like the CPI instead of Mt Figure 3 shows thecorresponding graph for aggregate budget shares Wit indi-cating that at least some of these shares may also appearnonstationary Budget shares must by construction lie be-tween 0 and 1 and so cannot remain nonstationary foreverbut as long as the magnitudes of changes from year to yearare small (relative to the range 0 to 1) shares can closelyapproximate a nonstationary process for decades as may bethe case for some shares in these US data

Results of formal tests of nonstationarity using theDFGLS test of Elliot Rothenberg and Stock (1996) and the

FIGURE 1mdashLOG PRICES







FIGURE 3mdashSHARES











Series




log1 2384 11371 1 2509 11856 3log2 1842 9818 4 4525 32102 2log3 0975 2408 2 1568 5173 4log4 0924 2141 2 4580 33948 2


logM 1284 3937 3 1958 7532 4W1 1707 5887 1 4321 30873 4W2 0953 2237 0 6644 68654 2W3 1701 6441 0 5944 58423 3W4 0058 02149 0 2551 8446 4






















N bijrjt eit

Good DF M Z Lags

1 2835 11512 02 2446 13545 13 4885 31804 04 2665 10673 0

CV 449 377


Good DF M Z Lags

1 3139 18098 02 3226 20042 13 5210 30300 04 3732 14835 0

CV 474 425



eit

Good DF M Z Lags

1 3568 22125 02 2608 13831 13 5426 43275 04 2242 9785 0

CV 504 475


j1N cjzijt eit

Good DF M Z Lags

1 4356 28923 02 4430 35425 13 6029 50875 04 4344 32421 0

CV 65 675









t1 where t





ait Ht1

Htai t1

htahi ht1

ahi

Ht

tai t1 it







2 1 2450 13525 13 1 4883 32419 04 1 2317 9802 0

1 2 2881 11895 03 2 5456 37449 04 2 4180 20909 0

1 3 2825 12175 02 3 3012 17663 14 3 3524 23658 0

1 4 2487 10667 02 4 3724 21577 13 4 5479 42895 0

CV 474 425



1 23 2851 12200 024 2411 10487 034 2127 7509 1

2 13 2997 17252 114 3722 22032 134 3558 22159 1

3 14 5404 41440 012 5438 37798 024 5050 38951 0

4 12 3825 19744 013 2889 1844 123 3660 21983 0

CV 504 475







ait tai t1 it ht

ht 1

Htahi

t ht1

h t1 1

Ht1ahi



it ai0 ai1t ait (6)








N bij




U ptmt i1

N i 1

2j1

N

bij lnpjt

mt ln

pit

mt (7)






wit

i j1

N

bij ln pjt ci ln mt

1 j1

N

cj ln pjt

(8)



wit i j1

N

bij ln pjtmt j1

N

cjwit ln pjt (9)





whit

hi j1

N


1 j1

N

cj ln pjt

(10)


ait hthi mht

ht mht

cit it cit (11)


Wit

ait j1

N

bij ln pjt ci ln Mt

1 j1

N

cj ln pjt

ait j1

N

bijrjt

1 j1

N

cj ln pjt

(12)





Wit ai0 ai1t j1

N

bijrjt j1

N

cjzijt eit (14)













E stWit ai1 j1

N

bij rjt j1

N














A Testing the Model





PARAMETERS BY GMM

INST1 INST2


b11 01186 00838 01665 00988Se 01077 00886 01086 00883



b22 01661 01206 01408 01048Se 00487 00410 00509 00390


b24 01672 01835 01474 01768Se 00623 00532 00621 00466

b33 00617 00715 01121 01387Se 00361 00392 00407 00415


b44 04851 03153 05064 04075Se 01214 01230 01475 01350

a11 00009 00008 00008 00007Se 00003 00002 00003 00002


a13 00007 00005 00010 00008Se 00002 00002 00002 00002

2 25095 32360 25924 30323Df 27 32 30 35




N

bijrjt j1

N

cjzijt eit (16)



Wit diWkt ai0 ai1t

j1

N

bijrjt j1

N



B Elasticities


ln Qit

ln pjt

bijWit cj

1 k1

N

ck ln pkt

1ij (18)

ln Qit

ln Mt

ciWit

1 k1

N

ck ln pkt

1 (19)




2

(211) 06336 03359 03005 260097(311) 07389 01753 02321 258998(411) 22732 04706 03154 232479(234) 03955 19064 03559 265953(314) 06823 01775 05276 258385(414) 21087 04602 00459 232554(341) 06355 07701 00733 269111(342) 06164 07175 01656 268183(344) 05854 06251 03532 267020(431) 22817 17406 03635 242229(432) 22068 17463 02535 242755(434) 21489 18058 00476 242617


2

(211) 09181 05734 08393 212570(311) 14878 06800 09548 212513(411) 19244 01801 09574 202703(234) 12099 02463 03557 226511(314) 14857 06412 06393 218971(414) 18274 01964 06944 202437(341) 10192 22839 06290 248406(342) 10342 21522 04570 250925(344) 09126 22949 03867 253738(431) 20192 01576 07604 209346(432) 20149 01076 07502 209106(434) 19207 01403 05289 211045



Good

Price

Income1 2 3 4

(A) Using INST1


(B) Using INST2





















YearBalanced

PanelAge

30ndash50Age

30ndash40Age

40ndash50

74 0757 1915 1176 323975 0757 1755 1019 293876 0757 1510 0731 277577 0757 1286 0505 256678 0757 1083 0265 248779 0757 0806 0010 224380 0757 0555 0189 202981 0757 0364 0301 172982 0757 0171 0444 148983 0757 0082 0690 131284 0757 0301 0874 100385 0757 0478 1092 089286 0757 0607 1270 069987 0757 0699 1427 047390 0757 0914 1894 019291 0757 0955 1999 033592 0757 1011 2180 0526AR(1) 9545 9945 1015








Uht ptmt

i1

N hi di fht ehit 1

2j1

N

bij lnphjt

mht ln

phit

mht





whit hi di fht j1

N

bijrhjt j1

N

cjzhijt ehit


Eshtwhit di fht j1

N

bij rhjt j1

N

cj zhijt 0




VIII Conclusions










REFERENCES










































APPENDIX

Data Sources













FIGURE 3mdashSHARES











Series




log1 2384 11371 1 2509 11856 3log2 1842 9818 4 4525 32102 2log3 0975 2408 2 1568 5173 4log4 0924 2141 2 4580 33948 2


logM 1284 3937 3 1958 7532 4W1 1707 5887 1 4321 30873 4W2 0953 2237 0 6644 68654 2W3 1701 6441 0 5944 58423 3W4 0058 02149 0 2551 8446 4






















N bijrjt eit

Good DF M Z Lags

1 2835 11512 02 2446 13545 13 4885 31804 04 2665 10673 0

CV 449 377


Good DF M Z Lags

1 3139 18098 02 3226 20042 13 5210 30300 04 3732 14835 0

CV 474 425



eit

Good DF M Z Lags

1 3568 22125 02 2608 13831 13 5426 43275 04 2242 9785 0

CV 504 475


j1N cjzijt eit

Good DF M Z Lags

1 4356 28923 02 4430 35425 13 6029 50875 04 4344 32421 0

CV 65 675









t1 where t





ait Ht1

Htai t1

htahi ht1

ahi

Ht

tai t1 it







2 1 2450 13525 13 1 4883 32419 04 1 2317 9802 0

1 2 2881 11895 03 2 5456 37449 04 2 4180 20909 0

1 3 2825 12175 02 3 3012 17663 14 3 3524 23658 0

1 4 2487 10667 02 4 3724 21577 13 4 5479 42895 0

CV 474 425



1 23 2851 12200 024 2411 10487 034 2127 7509 1

2 13 2997 17252 114 3722 22032 134 3558 22159 1

3 14 5404 41440 012 5438 37798 024 5050 38951 0

4 12 3825 19744 013 2889 1844 123 3660 21983 0

CV 504 475







ait tai t1 it ht

ht 1

Htahi

t ht1

h t1 1

Ht1ahi



it ai0 ai1t ait (6)








N bij




U ptmt i1

N i 1

2j1

N

bij lnpjt

mt ln

pit

mt (7)






wit

i j1

N

bij ln pjt ci ln mt

1 j1

N

cj ln pjt

(8)



wit i j1

N

bij ln pjtmt j1

N

cjwit ln pjt (9)





whit

hi j1

N


1 j1

N

cj ln pjt

(10)


ait hthi mht

ht mht

cit it cit (11)


Wit

ait j1

N

bij ln pjt ci ln Mt

1 j1

N

cj ln pjt

ait j1

N

bijrjt

1 j1

N

cj ln pjt

(12)





Wit ai0 ai1t j1

N

bijrjt j1

N

cjzijt eit (14)













E stWit ai1 j1

N

bij rjt j1

N














A Testing the Model





PARAMETERS BY GMM

INST1 INST2


b11 01186 00838 01665 00988Se 01077 00886 01086 00883



b22 01661 01206 01408 01048Se 00487 00410 00509 00390


b24 01672 01835 01474 01768Se 00623 00532 00621 00466

b33 00617 00715 01121 01387Se 00361 00392 00407 00415


b44 04851 03153 05064 04075Se 01214 01230 01475 01350

a11 00009 00008 00008 00007Se 00003 00002 00003 00002


a13 00007 00005 00010 00008Se 00002 00002 00002 00002

2 25095 32360 25924 30323Df 27 32 30 35




N

bijrjt j1

N

cjzijt eit (16)



Wit diWkt ai0 ai1t

j1

N

bijrjt j1

N



B Elasticities


ln Qit

ln pjt

bijWit cj

1 k1

N

ck ln pkt

1ij (18)

ln Qit

ln Mt

ciWit

1 k1

N

ck ln pkt

1 (19)




2

(211) 06336 03359 03005 260097(311) 07389 01753 02321 258998(411) 22732 04706 03154 232479(234) 03955 19064 03559 265953(314) 06823 01775 05276 258385(414) 21087 04602 00459 232554(341) 06355 07701 00733 269111(342) 06164 07175 01656 268183(344) 05854 06251 03532 267020(431) 22817 17406 03635 242229(432) 22068 17463 02535 242755(434) 21489 18058 00476 242617


2

(211) 09181 05734 08393 212570(311) 14878 06800 09548 212513(411) 19244 01801 09574 202703(234) 12099 02463 03557 226511(314) 14857 06412 06393 218971(414) 18274 01964 06944 202437(341) 10192 22839 06290 248406(342) 10342 21522 04570 250925(344) 09126 22949 03867 253738(431) 20192 01576 07604 209346(432) 20149 01076 07502 209106(434) 19207 01403 05289 211045



Good

Price

Income1 2 3 4

(A) Using INST1


(B) Using INST2





















YearBalanced

PanelAge

30ndash50Age

30ndash40Age

40ndash50

74 0757 1915 1176 323975 0757 1755 1019 293876 0757 1510 0731 277577 0757 1286 0505 256678 0757 1083 0265 248779 0757 0806 0010 224380 0757 0555 0189 202981 0757 0364 0301 172982 0757 0171 0444 148983 0757 0082 0690 131284 0757 0301 0874 100385 0757 0478 1092 089286 0757 0607 1270 069987 0757 0699 1427 047390 0757 0914 1894 019291 0757 0955 1999 033592 0757 1011 2180 0526AR(1) 9545 9945 1015








Uht ptmt

i1

N hi di fht ehit 1

2j1

N

bij lnphjt

mht ln

phit

mht





whit hi di fht j1

N

bijrhjt j1

N

cjzhijt ehit


Eshtwhit di fht j1

N

bij rhjt j1

N

cj zhijt 0




VIII Conclusions










REFERENCES










































APPENDIX

Data Sources

















Series




log1 2384 11371 1 2509 11856 3log2 1842 9818 4 4525 32102 2log3 0975 2408 2 1568 5173 4log4 0924 2141 2 4580 33948 2


logM 1284 3937 3 1958 7532 4W1 1707 5887 1 4321 30873 4W2 0953 2237 0 6644 68654 2W3 1701 6441 0 5944 58423 3W4 0058 02149 0 2551 8446 4






















N bijrjt eit

Good DF M Z Lags

1 2835 11512 02 2446 13545 13 4885 31804 04 2665 10673 0

CV 449 377


Good DF M Z Lags

1 3139 18098 02 3226 20042 13 5210 30300 04 3732 14835 0

CV 474 425



eit

Good DF M Z Lags

1 3568 22125 02 2608 13831 13 5426 43275 04 2242 9785 0

CV 504 475


j1N cjzijt eit

Good DF M Z Lags

1 4356 28923 02 4430 35425 13 6029 50875 04 4344 32421 0

CV 65 675









t1 where t





ait Ht1

Htai t1

htahi ht1

ahi

Ht

tai t1 it







2 1 2450 13525 13 1 4883 32419 04 1 2317 9802 0

1 2 2881 11895 03 2 5456 37449 04 2 4180 20909 0

1 3 2825 12175 02 3 3012 17663 14 3 3524 23658 0

1 4 2487 10667 02 4 3724 21577 13 4 5479 42895 0

CV 474 425



1 23 2851 12200 024 2411 10487 034 2127 7509 1

2 13 2997 17252 114 3722 22032 134 3558 22159 1

3 14 5404 41440 012 5438 37798 024 5050 38951 0

4 12 3825 19744 013 2889 1844 123 3660 21983 0

CV 504 475







ait tai t1 it ht

ht 1

Htahi

t ht1

h t1 1

Ht1ahi



it ai0 ai1t ait (6)








N bij




U ptmt i1

N i 1

2j1

N

bij lnpjt

mt ln

pit

mt (7)






wit

i j1

N

bij ln pjt ci ln mt

1 j1

N

cj ln pjt

(8)



wit i j1

N

bij ln pjtmt j1

N

cjwit ln pjt (9)





whit

hi j1

N


1 j1

N

cj ln pjt

(10)


ait hthi mht

ht mht

cit it cit (11)


Wit

ait j1

N

bij ln pjt ci ln Mt

1 j1

N

cj ln pjt

ait j1

N

bijrjt

1 j1

N

cj ln pjt

(12)





Wit ai0 ai1t j1

N

bijrjt j1

N

cjzijt eit (14)













E stWit ai1 j1

N

bij rjt j1

N














A Testing the Model





PARAMETERS BY GMM

INST1 INST2


b11 01186 00838 01665 00988Se 01077 00886 01086 00883



b22 01661 01206 01408 01048Se 00487 00410 00509 00390


b24 01672 01835 01474 01768Se 00623 00532 00621 00466

b33 00617 00715 01121 01387Se 00361 00392 00407 00415


b44 04851 03153 05064 04075Se 01214 01230 01475 01350

a11 00009 00008 00008 00007Se 00003 00002 00003 00002


a13 00007 00005 00010 00008Se 00002 00002 00002 00002

2 25095 32360 25924 30323Df 27 32 30 35




N

bijrjt j1

N

cjzijt eit (16)



Wit diWkt ai0 ai1t

j1

N

bijrjt j1

N



B Elasticities


ln Qit

ln pjt

bijWit cj

1 k1

N

ck ln pkt

1ij (18)

ln Qit

ln Mt

ciWit

1 k1

N

ck ln pkt

1 (19)




2

(211) 06336 03359 03005 260097(311) 07389 01753 02321 258998(411) 22732 04706 03154 232479(234) 03955 19064 03559 265953(314) 06823 01775 05276 258385(414) 21087 04602 00459 232554(341) 06355 07701 00733 269111(342) 06164 07175 01656 268183(344) 05854 06251 03532 267020(431) 22817 17406 03635 242229(432) 22068 17463 02535 242755(434) 21489 18058 00476 242617


2

(211) 09181 05734 08393 212570(311) 14878 06800 09548 212513(411) 19244 01801 09574 202703(234) 12099 02463 03557 226511(314) 14857 06412 06393 218971(414) 18274 01964 06944 202437(341) 10192 22839 06290 248406(342) 10342 21522 04570 250925(344) 09126 22949 03867 253738(431) 20192 01576 07604 209346(432) 20149 01076 07502 209106(434) 19207 01403 05289 211045



Good

Price

Income1 2 3 4

(A) Using INST1


(B) Using INST2





















YearBalanced

PanelAge

30ndash50Age

30ndash40Age

40ndash50

74 0757 1915 1176 323975 0757 1755 1019 293876 0757 1510 0731 277577 0757 1286 0505 256678 0757 1083 0265 248779 0757 0806 0010 224380 0757 0555 0189 202981 0757 0364 0301 172982 0757 0171 0444 148983 0757 0082 0690 131284 0757 0301 0874 100385 0757 0478 1092 089286 0757 0607 1270 069987 0757 0699 1427 047390 0757 0914 1894 019291 0757 0955 1999 033592 0757 1011 2180 0526AR(1) 9545 9945 1015








Uht ptmt

i1

N hi di fht ehit 1

2j1

N

bij lnphjt

mht ln

phit

mht





whit hi di fht j1

N

bijrhjt j1

N

cjzhijt ehit


Eshtwhit di fht j1

N

bij rhjt j1

N

cj zhijt 0




VIII Conclusions










REFERENCES










































APPENDIX

Data Sources


























N bijrjt eit

Good DF M Z Lags

1 2835 11512 02 2446 13545 13 4885 31804 04 2665 10673 0

CV 449 377


Good DF M Z Lags

1 3139 18098 02 3226 20042 13 5210 30300 04 3732 14835 0

CV 474 425



eit

Good DF M Z Lags

1 3568 22125 02 2608 13831 13 5426 43275 04 2242 9785 0

CV 504 475


j1N cjzijt eit

Good DF M Z Lags

1 4356 28923 02 4430 35425 13 6029 50875 04 4344 32421 0

CV 65 675









t1 where t





ait Ht1

Htai t1

htahi ht1

ahi

Ht

tai t1 it







2 1 2450 13525 13 1 4883 32419 04 1 2317 9802 0

1 2 2881 11895 03 2 5456 37449 04 2 4180 20909 0

1 3 2825 12175 02 3 3012 17663 14 3 3524 23658 0

1 4 2487 10667 02 4 3724 21577 13 4 5479 42895 0

CV 474 425



1 23 2851 12200 024 2411 10487 034 2127 7509 1

2 13 2997 17252 114 3722 22032 134 3558 22159 1

3 14 5404 41440 012 5438 37798 024 5050 38951 0

4 12 3825 19744 013 2889 1844 123 3660 21983 0

CV 504 475







ait tai t1 it ht

ht 1

Htahi

t ht1

h t1 1

Ht1ahi



it ai0 ai1t ait (6)








N bij




U ptmt i1

N i 1

2j1

N

bij lnpjt

mt ln

pit

mt (7)






wit

i j1

N

bij ln pjt ci ln mt

1 j1

N

cj ln pjt

(8)



wit i j1

N

bij ln pjtmt j1

N

cjwit ln pjt (9)





whit

hi j1

N


1 j1

N

cj ln pjt

(10)


ait hthi mht

ht mht

cit it cit (11)


Wit

ait j1

N

bij ln pjt ci ln Mt

1 j1

N

cj ln pjt

ait j1

N

bijrjt

1 j1

N

cj ln pjt

(12)





Wit ai0 ai1t j1

N

bijrjt j1

N

cjzijt eit (14)













E stWit ai1 j1

N

bij rjt j1

N














A Testing the Model





PARAMETERS BY GMM

INST1 INST2


b11 01186 00838 01665 00988Se 01077 00886 01086 00883



b22 01661 01206 01408 01048Se 00487 00410 00509 00390


b24 01672 01835 01474 01768Se 00623 00532 00621 00466

b33 00617 00715 01121 01387Se 00361 00392 00407 00415


b44 04851 03153 05064 04075Se 01214 01230 01475 01350

a11 00009 00008 00008 00007Se 00003 00002 00003 00002


a13 00007 00005 00010 00008Se 00002 00002 00002 00002

2 25095 32360 25924 30323Df 27 32 30 35




N

bijrjt j1

N

cjzijt eit (16)



Wit diWkt ai0 ai1t

j1

N

bijrjt j1

N



B Elasticities


ln Qit

ln pjt

bijWit cj

1 k1

N

ck ln pkt

1ij (18)

ln Qit

ln Mt

ciWit

1 k1

N

ck ln pkt

1 (19)




2

(211) 06336 03359 03005 260097(311) 07389 01753 02321 258998(411) 22732 04706 03154 232479(234) 03955 19064 03559 265953(314) 06823 01775 05276 258385(414) 21087 04602 00459 232554(341) 06355 07701 00733 269111(342) 06164 07175 01656 268183(344) 05854 06251 03532 267020(431) 22817 17406 03635 242229(432) 22068 17463 02535 242755(434) 21489 18058 00476 242617


2

(211) 09181 05734 08393 212570(311) 14878 06800 09548 212513(411) 19244 01801 09574 202703(234) 12099 02463 03557 226511(314) 14857 06412 06393 218971(414) 18274 01964 06944 202437(341) 10192 22839 06290 248406(342) 10342 21522 04570 250925(344) 09126 22949 03867 253738(431) 20192 01576 07604 209346(432) 20149 01076 07502 209106(434) 19207 01403 05289 211045



Good

Price

Income1 2 3 4

(A) Using INST1


(B) Using INST2





















YearBalanced

PanelAge

30ndash50Age

30ndash40Age

40ndash50

74 0757 1915 1176 323975 0757 1755 1019 293876 0757 1510 0731 277577 0757 1286 0505 256678 0757 1083 0265 248779 0757 0806 0010 224380 0757 0555 0189 202981 0757 0364 0301 172982 0757 0171 0444 148983 0757 0082 0690 131284 0757 0301 0874 100385 0757 0478 1092 089286 0757 0607 1270 069987 0757 0699 1427 047390 0757 0914 1894 019291 0757 0955 1999 033592 0757 1011 2180 0526AR(1) 9545 9945 1015








Uht ptmt

i1

N hi di fht ehit 1

2j1

N

bij lnphjt

mht ln

phit

mht





whit hi di fht j1

N

bijrhjt j1

N

cjzhijt ehit


Eshtwhit di fht j1

N

bij rhjt j1

N

cj zhijt 0




VIII Conclusions










REFERENCES










































APPENDIX

Data Sources















t1 where t





ait Ht1

Htai t1

htahi ht1

ahi

Ht

tai t1 it







2 1 2450 13525 13 1 4883 32419 04 1 2317 9802 0

1 2 2881 11895 03 2 5456 37449 04 2 4180 20909 0

1 3 2825 12175 02 3 3012 17663 14 3 3524 23658 0

1 4 2487 10667 02 4 3724 21577 13 4 5479 42895 0

CV 474 425



1 23 2851 12200 024 2411 10487 034 2127 7509 1

2 13 2997 17252 114 3722 22032 134 3558 22159 1

3 14 5404 41440 012 5438 37798 024 5050 38951 0

4 12 3825 19744 013 2889 1844 123 3660 21983 0

CV 504 475







ait tai t1 it ht

ht 1

Htahi

t ht1

h t1 1

Ht1ahi



it ai0 ai1t ait (6)








N bij




U ptmt i1

N i 1

2j1

N

bij lnpjt

mt ln

pit

mt (7)






wit

i j1

N

bij ln pjt ci ln mt

1 j1

N

cj ln pjt

(8)



wit i j1

N

bij ln pjtmt j1

N

cjwit ln pjt (9)





whit

hi j1

N


1 j1

N

cj ln pjt

(10)


ait hthi mht

ht mht

cit it cit (11)


Wit

ait j1

N

bij ln pjt ci ln Mt

1 j1

N

cj ln pjt

ait j1

N

bijrjt

1 j1

N

cj ln pjt

(12)





Wit ai0 ai1t j1

N

bijrjt j1

N

cjzijt eit (14)













E stWit ai1 j1

N

bij rjt j1

N














A Testing the Model





PARAMETERS BY GMM

INST1 INST2


b11 01186 00838 01665 00988Se 01077 00886 01086 00883



b22 01661 01206 01408 01048Se 00487 00410 00509 00390


b24 01672 01835 01474 01768Se 00623 00532 00621 00466

b33 00617 00715 01121 01387Se 00361 00392 00407 00415


b44 04851 03153 05064 04075Se 01214 01230 01475 01350

a11 00009 00008 00008 00007Se 00003 00002 00003 00002


a13 00007 00005 00010 00008Se 00002 00002 00002 00002

2 25095 32360 25924 30323Df 27 32 30 35




N

bijrjt j1

N

cjzijt eit (16)



Wit diWkt ai0 ai1t

j1

N

bijrjt j1

N



B Elasticities


ln Qit

ln pjt

bijWit cj

1 k1

N

ck ln pkt

1ij (18)

ln Qit

ln Mt

ciWit

1 k1

N

ck ln pkt

1 (19)




2

(211) 06336 03359 03005 260097(311) 07389 01753 02321 258998(411) 22732 04706 03154 232479(234) 03955 19064 03559 265953(314) 06823 01775 05276 258385(414) 21087 04602 00459 232554(341) 06355 07701 00733 269111(342) 06164 07175 01656 268183(344) 05854 06251 03532 267020(431) 22817 17406 03635 242229(432) 22068 17463 02535 242755(434) 21489 18058 00476 242617


2

(211) 09181 05734 08393 212570(311) 14878 06800 09548 212513(411) 19244 01801 09574 202703(234) 12099 02463 03557 226511(314) 14857 06412 06393 218971(414) 18274 01964 06944 202437(341) 10192 22839 06290 248406(342) 10342 21522 04570 250925(344) 09126 22949 03867 253738(431) 20192 01576 07604 209346(432) 20149 01076 07502 209106(434) 19207 01403 05289 211045



Good

Price

Income1 2 3 4

(A) Using INST1


(B) Using INST2





















YearBalanced

PanelAge

30ndash50Age

30ndash40Age

40ndash50

74 0757 1915 1176 323975 0757 1755 1019 293876 0757 1510 0731 277577 0757 1286 0505 256678 0757 1083 0265 248779 0757 0806 0010 224380 0757 0555 0189 202981 0757 0364 0301 172982 0757 0171 0444 148983 0757 0082 0690 131284 0757 0301 0874 100385 0757 0478 1092 089286 0757 0607 1270 069987 0757 0699 1427 047390 0757 0914 1894 019291 0757 0955 1999 033592 0757 1011 2180 0526AR(1) 9545 9945 1015








Uht ptmt

i1

N hi di fht ehit 1

2j1

N

bij lnphjt

mht ln

phit

mht





whit hi di fht j1

N

bijrhjt j1

N

cjzhijt ehit


Eshtwhit di fht j1

N

bij rhjt j1

N

cj zhijt 0




VIII Conclusions










REFERENCES










































APPENDIX

Data Sources













ait tai t1 it ht

ht 1

Htahi

t ht1

h t1 1

Ht1ahi



it ai0 ai1t ait (6)








N bij




U ptmt i1

N i 1

2j1

N

bij lnpjt

mt ln

pit

mt (7)






wit

i j1

N

bij ln pjt ci ln mt

1 j1

N

cj ln pjt

(8)



wit i j1

N

bij ln pjtmt j1

N

cjwit ln pjt (9)





whit

hi j1

N


1 j1

N

cj ln pjt

(10)


ait hthi mht

ht mht

cit it cit (11)


Wit

ait j1

N

bij ln pjt ci ln Mt

1 j1

N

cj ln pjt

ait j1

N

bijrjt

1 j1

N

cj ln pjt

(12)





Wit ai0 ai1t j1

N

bijrjt j1

N

cjzijt eit (14)













E stWit ai1 j1

N

bij rjt j1

N














A Testing the Model





PARAMETERS BY GMM

INST1 INST2


b11 01186 00838 01665 00988Se 01077 00886 01086 00883



b22 01661 01206 01408 01048Se 00487 00410 00509 00390


b24 01672 01835 01474 01768Se 00623 00532 00621 00466

b33 00617 00715 01121 01387Se 00361 00392 00407 00415


b44 04851 03153 05064 04075Se 01214 01230 01475 01350

a11 00009 00008 00008 00007Se 00003 00002 00003 00002


a13 00007 00005 00010 00008Se 00002 00002 00002 00002

2 25095 32360 25924 30323Df 27 32 30 35




N

bijrjt j1

N

cjzijt eit (16)



Wit diWkt ai0 ai1t

j1

N

bijrjt j1

N



B Elasticities


ln Qit

ln pjt

bijWit cj

1 k1

N

ck ln pkt

1ij (18)

ln Qit

ln Mt

ciWit

1 k1

N

ck ln pkt

1 (19)




2

(211) 06336 03359 03005 260097(311) 07389 01753 02321 258998(411) 22732 04706 03154 232479(234) 03955 19064 03559 265953(314) 06823 01775 05276 258385(414) 21087 04602 00459 232554(341) 06355 07701 00733 269111(342) 06164 07175 01656 268183(344) 05854 06251 03532 267020(431) 22817 17406 03635 242229(432) 22068 17463 02535 242755(434) 21489 18058 00476 242617


2

(211) 09181 05734 08393 212570(311) 14878 06800 09548 212513(411) 19244 01801 09574 202703(234) 12099 02463 03557 226511(314) 14857 06412 06393 218971(414) 18274 01964 06944 202437(341) 10192 22839 06290 248406(342) 10342 21522 04570 250925(344) 09126 22949 03867 253738(431) 20192 01576 07604 209346(432) 20149 01076 07502 209106(434) 19207 01403 05289 211045



Good

Price

Income1 2 3 4

(A) Using INST1


(B) Using INST2





















YearBalanced

PanelAge

30ndash50Age

30ndash40Age

40ndash50

74 0757 1915 1176 323975 0757 1755 1019 293876 0757 1510 0731 277577 0757 1286 0505 256678 0757 1083 0265 248779 0757 0806 0010 224380 0757 0555 0189 202981 0757 0364 0301 172982 0757 0171 0444 148983 0757 0082 0690 131284 0757 0301 0874 100385 0757 0478 1092 089286 0757 0607 1270 069987 0757 0699 1427 047390 0757 0914 1894 019291 0757 0955 1999 033592 0757 1011 2180 0526AR(1) 9545 9945 1015








Uht ptmt

i1

N hi di fht ehit 1

2j1

N

bij lnphjt

mht ln

phit

mht





whit hi di fht j1

N

bijrhjt j1

N

cjzhijt ehit


Eshtwhit di fht j1

N

bij rhjt j1

N

cj zhijt 0




VIII Conclusions










REFERENCES










































APPENDIX

Data Sources












wit

i j1

N

bij ln pjt ci ln mt

1 j1

N

cj ln pjt

(8)



wit i j1

N

bij ln pjtmt j1

N

cjwit ln pjt (9)





whit

hi j1

N


1 j1

N

cj ln pjt

(10)


ait hthi mht

ht mht

cit it cit (11)


Wit

ait j1

N

bij ln pjt ci ln Mt

1 j1

N

cj ln pjt

ait j1

N

bijrjt

1 j1

N

cj ln pjt

(12)





Wit ai0 ai1t j1

N

bijrjt j1

N

cjzijt eit (14)













E stWit ai1 j1

N

bij rjt j1

N














A Testing the Model





PARAMETERS BY GMM

INST1 INST2


b11 01186 00838 01665 00988Se 01077 00886 01086 00883



b22 01661 01206 01408 01048Se 00487 00410 00509 00390


b24 01672 01835 01474 01768Se 00623 00532 00621 00466

b33 00617 00715 01121 01387Se 00361 00392 00407 00415


b44 04851 03153 05064 04075Se 01214 01230 01475 01350

a11 00009 00008 00008 00007Se 00003 00002 00003 00002


a13 00007 00005 00010 00008Se 00002 00002 00002 00002

2 25095 32360 25924 30323Df 27 32 30 35




N

bijrjt j1

N

cjzijt eit (16)



Wit diWkt ai0 ai1t

j1

N

bijrjt j1

N



B Elasticities


ln Qit

ln pjt

bijWit cj

1 k1

N

ck ln pkt

1ij (18)

ln Qit

ln Mt

ciWit

1 k1

N

ck ln pkt

1 (19)




2

(211) 06336 03359 03005 260097(311) 07389 01753 02321 258998(411) 22732 04706 03154 232479(234) 03955 19064 03559 265953(314) 06823 01775 05276 258385(414) 21087 04602 00459 232554(341) 06355 07701 00733 269111(342) 06164 07175 01656 268183(344) 05854 06251 03532 267020(431) 22817 17406 03635 242229(432) 22068 17463 02535 242755(434) 21489 18058 00476 242617


2

(211) 09181 05734 08393 212570(311) 14878 06800 09548 212513(411) 19244 01801 09574 202703(234) 12099 02463 03557 226511(314) 14857 06412 06393 218971(414) 18274 01964 06944 202437(341) 10192 22839 06290 248406(342) 10342 21522 04570 250925(344) 09126 22949 03867 253738(431) 20192 01576 07604 209346(432) 20149 01076 07502 209106(434) 19207 01403 05289 211045



Good

Price

Income1 2 3 4

(A) Using INST1


(B) Using INST2





















YearBalanced

PanelAge

30ndash50Age

30ndash40Age

40ndash50

74 0757 1915 1176 323975 0757 1755 1019 293876 0757 1510 0731 277577 0757 1286 0505 256678 0757 1083 0265 248779 0757 0806 0010 224380 0757 0555 0189 202981 0757 0364 0301 172982 0757 0171 0444 148983 0757 0082 0690 131284 0757 0301 0874 100385 0757 0478 1092 089286 0757 0607 1270 069987 0757 0699 1427 047390 0757 0914 1894 019291 0757 0955 1999 033592 0757 1011 2180 0526AR(1) 9545 9945 1015








Uht ptmt

i1

N hi di fht ehit 1

2j1

N

bij lnphjt

mht ln

phit

mht





whit hi di fht j1

N

bijrhjt j1

N

cjzhijt ehit


Eshtwhit di fht j1

N

bij rhjt j1

N

cj zhijt 0




VIII Conclusions










REFERENCES










































APPENDIX

Data Sources


















E stWit ai1 j1

N

bij rjt j1

N














A Testing the Model





PARAMETERS BY GMM

INST1 INST2


b11 01186 00838 01665 00988Se 01077 00886 01086 00883



b22 01661 01206 01408 01048Se 00487 00410 00509 00390


b24 01672 01835 01474 01768Se 00623 00532 00621 00466

b33 00617 00715 01121 01387Se 00361 00392 00407 00415


b44 04851 03153 05064 04075Se 01214 01230 01475 01350

a11 00009 00008 00008 00007Se 00003 00002 00003 00002


a13 00007 00005 00010 00008Se 00002 00002 00002 00002

2 25095 32360 25924 30323Df 27 32 30 35




N

bijrjt j1

N

cjzijt eit (16)



Wit diWkt ai0 ai1t

j1

N

bijrjt j1

N



B Elasticities


ln Qit

ln pjt

bijWit cj

1 k1

N

ck ln pkt

1ij (18)

ln Qit

ln Mt

ciWit

1 k1

N

ck ln pkt

1 (19)




2

(211) 06336 03359 03005 260097(311) 07389 01753 02321 258998(411) 22732 04706 03154 232479(234) 03955 19064 03559 265953(314) 06823 01775 05276 258385(414) 21087 04602 00459 232554(341) 06355 07701 00733 269111(342) 06164 07175 01656 268183(344) 05854 06251 03532 267020(431) 22817 17406 03635 242229(432) 22068 17463 02535 242755(434) 21489 18058 00476 242617


2

(211) 09181 05734 08393 212570(311) 14878 06800 09548 212513(411) 19244 01801 09574 202703(234) 12099 02463 03557 226511(314) 14857 06412 06393 218971(414) 18274 01964 06944 202437(341) 10192 22839 06290 248406(342) 10342 21522 04570 250925(344) 09126 22949 03867 253738(431) 20192 01576 07604 209346(432) 20149 01076 07502 209106(434) 19207 01403 05289 211045



Good

Price

Income1 2 3 4

(A) Using INST1


(B) Using INST2





















YearBalanced

PanelAge

30ndash50Age

30ndash40Age

40ndash50

74 0757 1915 1176 323975 0757 1755 1019 293876 0757 1510 0731 277577 0757 1286 0505 256678 0757 1083 0265 248779 0757 0806 0010 224380 0757 0555 0189 202981 0757 0364 0301 172982 0757 0171 0444 148983 0757 0082 0690 131284 0757 0301 0874 100385 0757 0478 1092 089286 0757 0607 1270 069987 0757 0699 1427 047390 0757 0914 1894 019291 0757 0955 1999 033592 0757 1011 2180 0526AR(1) 9545 9945 1015








Uht ptmt

i1

N hi di fht ehit 1

2j1

N

bij lnphjt

mht ln

phit

mht





whit hi di fht j1

N

bijrhjt j1

N

cjzhijt ehit


Eshtwhit di fht j1

N

bij rhjt j1

N

cj zhijt 0




VIII Conclusions










REFERENCES










































APPENDIX

Data Sources
















A Testing the Model





PARAMETERS BY GMM

INST1 INST2


b11 01186 00838 01665 00988Se 01077 00886 01086 00883



b22 01661 01206 01408 01048Se 00487 00410 00509 00390


b24 01672 01835 01474 01768Se 00623 00532 00621 00466

b33 00617 00715 01121 01387Se 00361 00392 00407 00415


b44 04851 03153 05064 04075Se 01214 01230 01475 01350

a11 00009 00008 00008 00007Se 00003 00002 00003 00002


a13 00007 00005 00010 00008Se 00002 00002 00002 00002

2 25095 32360 25924 30323Df 27 32 30 35




N

bijrjt j1

N

cjzijt eit (16)



Wit diWkt ai0 ai1t

j1

N

bijrjt j1

N



B Elasticities


ln Qit

ln pjt

bijWit cj

1 k1

N

ck ln pkt

1ij (18)

ln Qit

ln Mt

ciWit

1 k1

N

ck ln pkt

1 (19)




2

(211) 06336 03359 03005 260097(311) 07389 01753 02321 258998(411) 22732 04706 03154 232479(234) 03955 19064 03559 265953(314) 06823 01775 05276 258385(414) 21087 04602 00459 232554(341) 06355 07701 00733 269111(342) 06164 07175 01656 268183(344) 05854 06251 03532 267020(431) 22817 17406 03635 242229(432) 22068 17463 02535 242755(434) 21489 18058 00476 242617


2

(211) 09181 05734 08393 212570(311) 14878 06800 09548 212513(411) 19244 01801 09574 202703(234) 12099 02463 03557 226511(314) 14857 06412 06393 218971(414) 18274 01964 06944 202437(341) 10192 22839 06290 248406(342) 10342 21522 04570 250925(344) 09126 22949 03867 253738(431) 20192 01576 07604 209346(432) 20149 01076 07502 209106(434) 19207 01403 05289 211045



Good

Price

Income1 2 3 4

(A) Using INST1


(B) Using INST2





















YearBalanced

PanelAge

30ndash50Age

30ndash40Age

40ndash50

74 0757 1915 1176 323975 0757 1755 1019 293876 0757 1510 0731 277577 0757 1286 0505 256678 0757 1083 0265 248779 0757 0806 0010 224380 0757 0555 0189 202981 0757 0364 0301 172982 0757 0171 0444 148983 0757 0082 0690 131284 0757 0301 0874 100385 0757 0478 1092 089286 0757 0607 1270 069987 0757 0699 1427 047390 0757 0914 1894 019291 0757 0955 1999 033592 0757 1011 2180 0526AR(1) 9545 9945 1015








Uht ptmt

i1

N hi di fht ehit 1

2j1

N

bij lnphjt

mht ln

phit

mht





whit hi di fht j1

N

bijrhjt j1

N

cjzhijt ehit


Eshtwhit di fht j1

N

bij rhjt j1

N

cj zhijt 0




VIII Conclusions










REFERENCES










































APPENDIX

Data Sources










N

bijrjt j1

N

cjzijt eit (16)



Wit diWkt ai0 ai1t

j1

N

bijrjt j1

N



B Elasticities


ln Qit

ln pjt

bijWit cj

1 k1

N

ck ln pkt

1ij (18)

ln Qit

ln Mt

ciWit

1 k1

N

ck ln pkt

1 (19)




2

(211) 06336 03359 03005 260097(311) 07389 01753 02321 258998(411) 22732 04706 03154 232479(234) 03955 19064 03559 265953(314) 06823 01775 05276 258385(414) 21087 04602 00459 232554(341) 06355 07701 00733 269111(342) 06164 07175 01656 268183(344) 05854 06251 03532 267020(431) 22817 17406 03635 242229(432) 22068 17463 02535 242755(434) 21489 18058 00476 242617


2

(211) 09181 05734 08393 212570(311) 14878 06800 09548 212513(411) 19244 01801 09574 202703(234) 12099 02463 03557 226511(314) 14857 06412 06393 218971(414) 18274 01964 06944 202437(341) 10192 22839 06290 248406(342) 10342 21522 04570 250925(344) 09126 22949 03867 253738(431) 20192 01576 07604 209346(432) 20149 01076 07502 209106(434) 19207 01403 05289 211045



Good

Price

Income1 2 3 4

(A) Using INST1


(B) Using INST2





















YearBalanced

PanelAge

30ndash50Age

30ndash40Age

40ndash50

74 0757 1915 1176 323975 0757 1755 1019 293876 0757 1510 0731 277577 0757 1286 0505 256678 0757 1083 0265 248779 0757 0806 0010 224380 0757 0555 0189 202981 0757 0364 0301 172982 0757 0171 0444 148983 0757 0082 0690 131284 0757 0301 0874 100385 0757 0478 1092 089286 0757 0607 1270 069987 0757 0699 1427 047390 0757 0914 1894 019291 0757 0955 1999 033592 0757 1011 2180 0526AR(1) 9545 9945 1015








Uht ptmt

i1

N hi di fht ehit 1

2j1

N

bij lnphjt

mht ln

phit

mht





whit hi di fht j1

N

bijrhjt j1

N

cjzhijt ehit


Eshtwhit di fht j1

N

bij rhjt j1

N

cj zhijt 0




VIII Conclusions










REFERENCES










































APPENDIX

Data Sources


























YearBalanced

PanelAge

30ndash50Age

30ndash40Age

40ndash50

74 0757 1915 1176 323975 0757 1755 1019 293876 0757 1510 0731 277577 0757 1286 0505 256678 0757 1083 0265 248779 0757 0806 0010 224380 0757 0555 0189 202981 0757 0364 0301 172982 0757 0171 0444 148983 0757 0082 0690 131284 0757 0301 0874 100385 0757 0478 1092 089286 0757 0607 1270 069987 0757 0699 1427 047390 0757 0914 1894 019291 0757 0955 1999 033592 0757 1011 2180 0526AR(1) 9545 9945 1015








Uht ptmt

i1

N hi di fht ehit 1

2j1

N

bij lnphjt

mht ln

phit

mht





whit hi di fht j1

N

bijrhjt j1

N

cjzhijt ehit


Eshtwhit di fht j1

N

bij rhjt j1

N

cj zhijt 0




VIII Conclusions










REFERENCES










































APPENDIX

Data Sources













Uht ptmt

i1

N hi di fht ehit 1

2j1

N

bij lnphjt

mht ln

phit

mht





whit hi di fht j1

N

bijrhjt j1

N

cjzhijt ehit


Eshtwhit di fht j1

N

bij rhjt j1

N

cj zhijt 0




VIII Conclusions










REFERENCES










































APPENDIX

Data Sources














REFERENCES










































APPENDIX

Data Sources


























APPENDIX

Data Sources








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