on estimation of almost ideal demand system using moving blocks

Empir EconDOI 10.1007/s00181-013-0782-6

On estimation of almost ideal demand system usingmoving blocks bootstrap and pairs bootstrap methods

Ken-ichi Mizobuchi · Hisashi Tanizaki

Received: 5 August 2011 / Accepted: 5 November 2013© Springer-Verlag Berlin Heidelberg 2013

Abstract This paper applies a bootstrap method to the Almost Ideal Demand System(AIDS) proposed by Deaton and Muellbauer (Am Econ Rev 70:312–326, 1980), wherethe moving blocks bootstrap (MBB) and pairs bootstrap (PB) methods are adoptedtaking into account serially correlated error terms and limited dependent variables(note that the dependent variables in the AIDS model lie on the interval between zeroand one). We aim to obtain the empirical distribution of the expenditure and priceelasticities. Note that, the expenditure and price elasticities are obtained using theparameter estimates included in the AIDS model. In the past, a few studies reportboth the elasticity estimates and their standard errors obtained from the Delta method,but most of studies show only the elasticity estimates (i.e., statistical tests have notbeen done in most of the past studies). Applying MBB and PB methods to the AIDSmodel and using Japanese monthly household expenditure data from January, 1975 toDecember, 2012, we show in this paper that a few elasticities are statistically insignif-icant. We also compare the standard errors based on the bootstrap method with thosebased on the Delta method. We obtain the results that the differences between the Deltamethod and the bootstrap method are not negligible. In addition, the validity of thelinear approximated AIDS (LA–AIDS) model which is commonly used in empiricalstudies is examined. In consequence, we find that the LA–AIDS model shows a poorperformance, compared with the AIDS model, because the LA–AIDS model yieldsinconsistency on the elasticity estimates.

K. MizobuchiDepartment of Economics, Matsuyama University, Ehime 790-8578, Japane-mail: [email protected]

H. Tanizaki (B)Graduate School of Economics, Osaka University, Osaka 560-0043, Japane-mail: [email protected]

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K. Mizobuchi, H. Tanizaki

Keywords AIDS model · LA–AIDS model · Expenditure elasticity · Own priceelasticity · Moving blocks bootstrap · Pairs bootstrap

JEL Classification C36 · C63 · D12

1 Introduction

The Almost Ideal Demand System (hereafter, AIDS), proposed by Deaton and Muell-bauer (1980), provides the second-order approximation to an arbitrary demand sys-tem and satisfies perfect aggregation conditions over consumers. In addition, usingthe AIDS model we can easily test the constraints (i.e., homogeneity, symmetry,and negativity) in classical demand theory. The AIDS model has been extensivelyapplied in various empirical studies on demand analysis, e.g., in agricultural eco-nomics, demographic economics, development economics, environmental economics,health economics, industrial organization, monetary economics, urban economics, andso on (e.g., see Rossi 1988; Tiffin and Aguiar 1995; Filippini 1995; Oladosu 2003;Hashimoto 2004).

In demand analysis, researchers are usually interested in expenditure and priceelasticities. Past studies, however, give only point estimates of the elasticities, which arebased on the parameter estimates in the AIDS model. Therefore, statistical information(i.e., the standard error, the p-value, the confidence interval and etc) on the elasticityestimates has not been provided in the past empirical studies. There are a few studieswhich show the standard error approximated by the Delta method. Our purpose in thispaper is to obtain the distribution of the elasticity estimator by applying the bootstrapmethod into the AIDS model.

The following bootstrap procedure is heuristic: (i) estimate the AIDS model usingthe original data to obtain the parameter estimates and the residuals, (ii) given theparameter estimates in Step (i), construct pseudo data of the dependent variable byresampling the residuals obtained in Step (i), (iii) estimate the AIDS model using thepseudo data to obtain the parameter estimates, and (iv) repeat (ii) and (iii) n times.Thus, n sets of parameter estimates are obtained. This bootstrap method is called theresidual bootstrap, which is originally proposed by Efron (1979). Also, see MacKinnon(2006).

In the case of the AIDS model, however, the residual bootstrap method is notsuitable, because the dependent variables in the AIDS model line on the intervalbetween zero and one. The pseudo data constructed in Step (ii) do not guarantee togenerate the limited data. Therefore, in this paper we consider applying the PairsBootstrap (hereafter, PB) method discussed in Freedman (1981), Freedman (1984)and Freedman and Peters (1984). Also, see Efron and Tibshirani (1993, pp. 113–115)and MacKinnon (2006). In the PB method, the dependent and independent variabledata at time t are taken as one vector. T vectors are available when the sample sizeis T , and the bootstrap data at time t is resampled from T vectors, i.e., the originallyobserved data are resampled T times. Then, the AIDS model is estimated using theresampled data consisting of T vectors. Repeating this procedure n times, we obtainn parameter estimates. We take n = 10, 000 in this paper.

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On Estimation of AIDS using MBB and PB Methods

In addition to the PB procedure above, taking into account serial correlation inthe error terms, we utilize the Moving Blocks Bootstrap (hereafter, MBB) method inwhich the order of the original data vectors is kept in a certain period (i.e., some blocksof the original data vectors are resampled). For example, see Künsch (1989), Liu andSingh (1992), and Fitzenberger (1997) for the MBB method, which is also discussedin Sect. 4. In a field of the AIDS model, the MBB is utilized in Tiffin and Balcombe(2005). Thus, the empirical distribution of elasticity estimates based on the parameterestimates is obtained. From the n elasticity estimates, we can compute the standarderror, confidence interval, p-value, etc. Freedman and Peters (1984) give us a pioneerstudy which applies the bootstrap method to the econometric model and computes thestandard errors of the parameter estimates. They show that the bootstrap procedureis clearly better than the other conventional procedures such as the Delta method infinite sample. Green et al. (1987) estimate the linear expenditure system (LES) modelby combining the bootstrap method with the full information maximum likelihood(FIML) estimator, and compares standard errors of the elasticities for both the boot-strap and traditional Delta methods. As a result, the bootstrap method is preferred tothe Delta method in the sense of the standard error criterion. Furthermore, Krinskyand Robb (1991) suggest the simulation procedure under normality assumption fordisturbance terms, and compare the Delta method, the simulation procedure and thebootstrap method. They estimate the trans-log cost function by the seemingly unre-lated regression (hereafter, SUR), and compute the standard errors of the elasticitiesusing the three methods. Unlike Green et al. (1987) and Krinsky and Robb (1991)conclude that there are no major differences between the three methods in a senseof the standard error criterion. However, they also mention that the conclusion maychange, depending on some other factors such as the functional forms, the estimationmethod, data sets, etc.

There are no studies in which the bootstrap method is compared with the Deltamethod using the AIDS model. Tiffin and Balcombe (2005) apply the MBB, which isdiscussed in Sect. 4, to the linear approximated AIDS (hereafter, LA–AIDS) model,where the bootstrap method is combined with the fully modified SUR (i.e., FM-SUR) method to reduce the estimated bias and the size distortion in the small sample.However, as it will be shown later, the dependent variables in the AIDS model areshare variables, and Tiffin and Balcombe (2005) do not consider the range of the sharevariables which takes the interval between zero and one. Therefore, the standard errorsof the elasticity estimates might be unreliable.1 Besides Tiffin and Balcombe (2005),a lot of the empirical studies on the AIDS model do not take into account the range ofthe share variables. Some of the empirical studies on the AIDS model use the FIMLmethod, where a normal distribution is assumed for the disturbance term (e.g., seeRossi 1988; Oladosu 2003). In this case, the range of a dependent variable should befrom −∞ to ∞. However, the dependent variables (i.e., share variables) in the AIDSmodel should lie on the interval between zero and one. Similarly, in the case wherethe Bayesian estimation method is adopted, we have the exactly same problem (e.g.,see Tiffin and Balcombe 2005; Xiao et al. 2007).

1 Note that, Tiffin and Balcombe (2005) do not show the standard errors of the elasticity estimates.

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This paper extends Tiffin and Balcombe (2005) applying MBB and PB methods totake into account the serially correlated errors and the range of dependent variables,and aims to obtain the distribution of the elasticities which are computed from theparameter estimates in the AIDS model. Moreover, the empirical analysis is shownusing Japanese monthly household expenditure data, and we compare the standarderrors of elasticity estimates for the bootstrap method and the Delta method. Ourresults show that large difference between the two methods is observed.

The LA–AIDS model is applied to a lot of the empirical studies on demand analy-sis, but it is not based on any theoretical foundation in the sense that the approx-imation of price index is not theoretically justified. Note that, the approximationof price index in the LA–AIDS model is called the Stone price index. Pashardes(1993) and Buse (1994) point out that the parameter estimates are biased if the Stoneprice index does not work well. Through our empirical analysis, we compare LA–AIDS and AIDS models, and examine whether the LA–AIDS model works well ornot.

This paper is organized as follows. Section 2 provides the model specification, inwhich AIDS and LA–AIDS models are discussed. Section 3 presents some commentsabout the past empirical studies in the AIDS model. In Sect. 4, MBB and PB methodsare applied to the AIDS model, and Sect. 5 examines an empirical application usingJapanese household expenditure data from January, 1975 to December, 2012. We alsoconsider testing the structural change during the bubble economy periods in Japan.Section 6 presents concluding remarks.

2 AIDS and LA–AIDS Models

The AIDS model (see Deaton and Muellbauer 1980) is derived from a PIGLOG (i.e.,Price Independence Generalized LOGarithmic) type of the indirect cost function.This cost function has a lot of parameters and it can be interpreted as the second-order approximation to any cost function. Shephard’s lemma gives us a set of demandequations which can be considered as an approximation of any demand system. Let wi,t

be the share of the i th good, i.e., wi,t ≡ Xi,t/Xt , where Xi,t indicates the expenditureof the i th good at time t and Xt represents Xt = ∑M

i=1 Xi,t . Let M be the number ofgoods, pj,t be the price of the i th good and Pt be the price index. The AIDS model isexpressed as:

wi,t = αi + βi lnXt

Pt+

M∑

j=1

γi j ln p j,t + ui,t , t = 1, 2, . . . , T,

i = 1, 2, . . . , M,

(1)

where ui,t indicates the error term in the i th equation, and the variance–covariancestructure is usually assumed to be as follows:

E[ui,t u j,s] ={

σi j , if t = s,0, otherwise.

In the case where the error term is serially correlated, the assumption of E[ui,t u j,s] = 0for t �= s is violated. In Sect. 4, we discuss the bootstrap method in the case where the

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error term is serially correlated, which is called the MBB method, and the bootstrapmethod in the case where the domain of the dependent variables is not from −∞ to∞, which is called the PB method.

Equation (1) is called the AIDS model when ln Pt is defined as follows:

ln Pt = α0 +M∑

k=1

αk ln pk,t + 1

2

M∑

k=1

M∑

j=1

γk j ln pk,t ln p j,t . (2)

Instead of Eq. (2), however, most of the empirical studies utilize the following:

ln Pt =M∑

k=1

wk,t ln pk,t , (3)

where Pt is called the Stone price index and it is known as an approximation of Pt in(2). Hereafter, in this paper (2) is called the original price index to distinguish fromthe Stone price index. The AIDS model which utilizes (3) is known as the LA–AIDSmodel.

Deaton and Muellbauer (1980) mention that the Stone price index (3) is close to theoriginal price index (2). This LA–AIDS model is widely used because of simplicity ofthe estimation procedure. However, it is well known that the LA–AIDS model yieldsthe biased parameter estimates.2 Therefore, in this paper we compare (2) and (3) in(1).

Constraint conditions From the demand theory, there are some constraints in theAIDS model (1), i.e.,

(i) Additivity∑M

i=1 αi = 1,∑M

i=1 βi = 0 and∑M

i=1 γi j = 0,(ii) Homogeneity

∑Mj=1 γi j = 0,

(iii) Symmetry γi j = γ j i .(iv) Negativity The Slutsky matrix (or equivalently, the substitution matrix) is

negative semidefinite.

For both AIDS and LA–AIDS models, practically we estimate the first M − 1equations. Substituting two additivity constraints αM = 1 − ∑M−1

k=1 αk and γM j =−∑M−1

k=1 γk j into (2), we can rewrite the price index as:

ln Pt = α0 +M−1∑

k=1

αk ln pk,t + αM ln pM,t

+1

2

M−1∑

k=1

M∑

j=1

γk j ln pk,t ln p j,t + 1

2

M∑

j=1

γM j ln pM,t ln p j,t

2 Pashardes (1993) shows that the LA–AIDS model results in an omitted variable problem, which is asource of the biased parameter estimates. This problem is discussed in Buse (1994, 1998). Through someMonte Carlo experiments, Buse (1994) finds that not only SUR but also 3SLS (three stage least squares)yield inconsistent parameter estimates because of the omitted variable problem.

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= α0 +M−1∑

k=1

αk ln pk,t +(

1 −M−1∑

k=1

αk

)

ln pM,t

+1

2

M−1∑

k=1

M∑

j=1

γk j ln pk,t ln p j,t + 1

2

M∑

j=1

(

−M−1∑

k=1

γk j

)

ln pM,t ln p j,t . (4)

Moreover, substituting (4) into (1), the AIDS model is estimated as follows:

wi,t = (αi − βiα0) + βi ln Xt +M−1∑

j=1

(γi j − βiα j ) lnp j,t

pM,t

+⎛

⎝M∑

j=1

γi j − βi

⎞

⎠ ln pM,t − 1

2βi

M−1∑

k=1

M∑

j=1

γk j lnpk,t

pM,tln p j,t + ui,t , (5)

for i = 1, 2, . . . , M −1, where the two additivity constraints αM = 1−∑M−1k=1 αk and

γM j = −∑M−1k=1 γk j are included. Moreover, for the identification problem, α0 = 0 is

set in this paper. Thus, the AIDS model leads to the nonlinear demand system, becauseit includes (γi j −βiα j ) and βiγk j . Thus, for the AIDS model, the first M −1 equationsin (5) are simultaneously estimated as the nonlinear demand system.

As for the LA–AIDS model, the price index data is constructed by Eq. (3), whichdoes not depend on parameters αi , βi and γi j . Therefore, simply we can estimate thefirst M − 1 equations in (1) as the linear demand system.

Elasticity Let ηEi,t be the expenditure elasticity of the i th good at time t and ηP

i j,t bethe price elasticity of the j th good in the i th equation at time t . According to Alstonet al. (1994), based on (1) and (2), the expenditure and price elasticities of the AIDSmodel are obtained as:

ηEi,t = 1 + βi

wi,t, (6)

ηPi j,t = −δi j + γi j

wi,t− βiα j

wi,t− βi

wi,t

M∑

k=1

γk j ln pk,t , (7)

where δi j = 1 if i = j and δi j = 0 otherwise. Given wi,t , ln(Xt/Pt ) and ln pi,t , theparameters αi , βi and γi j can be estimated. From the parameter estimates, ηE

i,t and

ηPi j,t are computed based on Eqs. (6) and (7).

According to Green and Alston (1990), when we use the Stone price index Eqs. (3)in (1), i.e., the LA–AIDS model, the expenditure and price elasticities are given bythe following forms:

ηEi,t = 1 + βi

wi,t

(1 −

M∑

j=1

w j,t (ηEj,t − 1) ln p j,t

), (8)

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ηPi j,t = −δi j + γi j

wi,t− βi

wi,t

(w j,t +

M∑

k=1

wk,t (ηPk j,t + δk j ) ln pk,t

), (9)

where ηEi,t and ηP

i j,t are solved as simultaneous equations.

3 Problems in the AIDS model

A number of papers have discussed the relationship between the AIDS and LA–AIDS.To avoid the criticisms of the LA–AIDS model which uses the Stone price index (e.g.,see Buse 1994, 1998; Moschini 1995; Feenstra and Reinsdorf 2000, for the criticisms),the recent empirical studies tend to shift gradually from the LA–AIDS model to theAIDS model. However, the AIDS model still has some problems, which are discussedbelow.

First, an assumption of the normally distributed error is imposed in some studies.The dependent variable wi,t in the AIDS model Eq. (1) lies on the interval between zeroand one. A lot of studies use the FIML method which assumes a normal distributionon the disturbance term ui,t (e.g., see Rossi 1988; Oladosu 2003). When a disturbanceterm is assumed to be normal, a dependent variable should take the range from −∞ to∞. Moreover, there are some studies which utilize the Bayesian estimation method,where the normal distribution is assumed for the error ui,t (e.g., see Tiffin and Aguiar1995; Xiao et al. 2007).

Second, in past studies there is no statistical inference on the expenditure andprice elasticity estimates. In the empirical studies on demand analysis, usually, peo-ple are interested in the estimates of elasticities ηE

i,t and ηPi j,t in (6–9), not the

estimates of parameters αi , βi , and γi j in the AIDS model (1). Most of the pastempirical studies show only point estimates of the elasticities, but do not presentother statistical information, such as the standard error (SE), the p-value, the con-fidence interval and so on. There are a few studies which show SE (e.g., seePashardes 1993), where the SE is usually based on the Delta method. Freedman andPeters (1984) conclude that the SE based on the Delta method is under-estimatedin general and that the SE which utilizes the bootstrap procedure might be moreplausible.

Third, there is an endogeneity problem in the AIDS model (1). Both the totalexpenditures, denoted by Xt = ∑M

i=1 Xi,t , and the expenditure share of the i th good,denoted by wi,t = Xi,t/Xt , are expressed as a function of Xi,t . Therefore, thereis a correlation between the disturbance term ui,t and the explanatory variable Xt .We have the biased estimates of αi , βi , and γi j in the AIDS model (1) when weestimate the AIDS model by SUR. A lot of studies utilize SUR or Bayesian estimationmethod ignoring the endogeneity problem. Recently, however, taking into accountendogeneity, the studies which apply the Generalized Method of Moments (hereafter,GMM) to the AIDS model gradually increase.

Thus, the AIDS model involves some serious problems in the empirical studies.For the first and second problems, we consider adopting the bootstrap procedure forestimation, where any distribution is not assumed for the error term. Using the boot-

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strap sample based on the originally observed data, the expenditure and price elasticityestimates can be computed. Moreover, the standard errors (SE), the p-values, and theconfidence intervals of the elasticities can be obtained. Freedman and Peters (1984),Green et al. (1987), and Krinsky and Robb (1991) have suggested the bootstrap proce-dure to obtain the standard errors of the parameter estimates. In the case of the AIDSmodel, the dependent variable wi,t has to lie on the interval between zero and one.Therefore, in this paper we utilize the pairs bootstrap, rather than the residual boot-strap. Green et al. (1987) and Krinsky and Robb (1991) estimate the LES model and thetranslog cost function, respectively. Since the dependent variables in the LES modelare not the share variables, Green et al. (1987) and Krinsky and Robb (1991) do nothave the truncation problem. In the next section, we discuss the bootstrap procedureand the truncation problem in more detail in order to obtain the empirical distributionof the elasticity estimates.

Moreover, as a solution of the third problem (i.e., the endogeneity problem), weutilize the 3SLS estimation, which yields consistent parameter estimates in the AIDSmodel. For the empirical study in Sect. 5, we use (ln p1,t , ln p2,t , . . . , ln pM,t ), theircross-terms and the constant term as the instrumental variables in 3SLS. Note that inSect. 5 the nonlinear 3SLS method or the nonlinear SUR method are utilized for theAIDS model while the linear 3SLS method or the linear SUR method are adoptedfor the LA–AIDS model. However, simply we call 3SLS or SUR in both AIDS andLA–AIDS cases.

4 On MBB and PB methods

In this paper, the bootstrap method is applied to obtain empirical distributions forexpenditure and price elasticity estimators. Freedman and Peters (1984) applied thebootstrap procedure to several econometric models. For the demand models whichutilize the bootstrap procedure, Green et al. (1987) discussed the LES model, andKrinsky and Robb (1991) investigated the translog cost function. Taking account ofserially correlated errors, in this paper we adopt the MBB method for estimation (e.g.,see Künsch 1989; Liu and Singh 1992; Fitzenberger 1997 for the MBB method).

Supposing that all the data are I (1) processes and that there is a co-integrationrelationship in (1), Tiffin and Balcombe (2005) apply the bootstrap procedure to theLA–AIDS model and they show the testing procedure on symmetry and homogeneity,where the FM-SUR method is applied for estimation. However, their analysis is basedon the misleading assumption, i.e., wi,t = Xi,t/Xt ∼ I (1), where they do not takeinto account the constraint that the share wi,t lies on the interval between zero andone.

In this paper, we apply the bootstrap method to the AIDS model. Taking into accountthe constraint that wi,t lies on the interval between zero and one for all i = 1, 2, . . . , Mand t = 1, 2, . . . , T , the PB method is adopted (e.g., see Efron and Tibshirani 1993;MacKinnon 2006, for the PB method). In the PB method, both dependent and inde-pendent data observed at time t are always chosen in pairs.

Furthermore, to avoid the endogenous problem in the AIDS model, the 3SLS isapplied for estimation. Thus, we apply 3SLS, MBB, and PB to the AIDS model.

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The MBB and PB method implemented in this paper is as follows. Let us definethe observed data vector at time t as yt ≡ (wt , pt , Xt ) for t = 1, 2, . . . , T , wherewt = (w1,t , w2,t . . . wM,t ) and pt = (p1,t , p2,t ,..., pM,t ), i.e., wt and pt are 1 × Mvectors and accordingly yt is a 1 × (2M + 1) vector. When the error terms are seriallycorrelated, it is known that the MBB method is an efficient tool (e.g., see Künsch 1989;Liu and Singh 1992; Fitzenberger 1997). Suppose that T and b are positive integers.Let b be the number of the date vectors included in one block. Then, we can constructT − b + 1 blocks of the observed data vectors as follows:

Yt,b =

⎛

⎜⎜⎜⎝

yt

yt+1...

yt+b−1

⎞

⎟⎟⎟⎠

, for t = 1, 2, ..., T − b + 1,

where Yt,b denotes a b × (2M + 1) matrix. Let m∗ be the maximum integer of T/b.We take m = m∗ when the remainder of T/b is zero and m = m∗ + 1 otherwise.

The MBB and PB method resamples m blocks randomly with replacement out ofT − b + 1 overlapping blocks, i.e., {Y1,b, Y2,b, ..., YT −b+1,b}. Let I1, I2, ..., Im berandom numbers generated from the discrete uniform distribution on {0, 1, ..., T −b}.The MBB pseudo data y∗

t , t = 1, 2,...,T , is the result of arranging the elements ofthe m resampled blocks {YI1+1,b, YI2+1,b,...,YIm+1,b∗}, which can be represented asfollows:

The 1st Block(i.e., YI1+1,b)

y∗1 = yI1+1,

y∗2 = yI1+2,

...,

y∗b = yI1+b,

The 2nd Block(i.e., YI2+1,b)

y∗b+1 = yI2+1,

y∗b+2 = yI2+2,

...,

y∗2b = yI2+b,

...

The mth Block(i.e., YIm+1,b∗)

y∗(m−1)b+1 = yIm−1+1,

y∗(m−1)b+2 = yIm−1+2,

...,

y∗(m−1)b+b∗ = yIm−1+b∗ ,

where the integer b∗ is chosen from T = (m − 1)b + b∗ for b∗ ≤ b. That is, YIm+1,b∗indicates the first b∗ row vectors out of YIm+1,b, taking into account the end block.Thus, y∗

t = (w∗t , p∗

t , X∗t ) represents the t th row vector of the MBB and PB sample.

Replacing yt by y∗t in (1–3), we estimate (αi , βi , γi j ) by 3SLS, where (2) for AIDS

or (3) for LA–AIDS is taken. Because Tiffin and Balcombe (2005) do not considerthis problem, there might be a possibility of overestimating the standard errors of theparameter estimates.

The bootstrap procedure shown above is repeated n times. From the n sets ofthe parameter estimates, n elasticities are computed by (6) and (7). Based on the nelasticity estimates, we have the empirical distributions of the expenditure and priceelasticities. Thus, the standard errors, the p-values and the confidence intervals areobtained from the empirical distributions. In the empirical analysis of the next section,we set n = 10, 000, M = 10, and T = 456, 180, 276.

Thus, in this paper, MBB and PB are combined to solve the serially correlated errorsand the limited dependent variables. In the PB method, both dependent and indepen-

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dent data observed at time t are always selected in pairs. Therefore, the PB methoddoes not require the assumption that the error terms are homoscedastic. However, wehave to note that we cannot use the lagged variables for the instrumental variables,because the order of the originally observed data are resampled.

5 Empirical Analysis

5.1 Data

In this section, applying the bootstrap technique discussed in the previous section, weestimate the AIDS model using Japanese monthly household expenditure data fromJanuary, 1975 to December, 2012. The household expenditure data are classified intoten categories which are reported in National Survey of Family Income and Expenditure(Statistical Survey Department, Statistics Bureau, Ministry of Internal Affairs andCommunications), and they are as follows:

• Food,• Housing,• Fuel, light and water charges (hereafter, Fuel),• Furniture and household utensils (hereafter, Furni),• Clothes and footwear (hereafter, Clothes),• Medical care (hereafter, Medical),• Transport and communication (hereafter, Trans),• Education (hereafter, Edu),• Culture and recreation (hereafter, Culture), and• Other consumption expenditures (hereafter, Other).

All the expenditure data are seasonally adjusted by X12 using Eviews 7.2. The pricedata are taken from the Consumption Price Index (CPI) corresponding to each expen-diture.

• Food,• Housing,• Fuel, light and water charges,• Furniture and household utensils,• Clothes and footwear,• Medical care,• Transportation and communication,• Education,• Reading and recreation, and• Miscellaneous.

All the price data are seasonally adjusted by X12 using Eviews 7.2, and they aredivided by seasonally adjusted General CPI data before the logarithmic transformation.Thus, we use relative prices, where the base year is 2010. Total expenditure Xt isconstructed by summing up all the seasonally adjusted expenditure data divided bythe corresponding seasonally adjusted price data.

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Table 1 The number ofsignificant parameters

The number of parameters to beestimated is shown in theparentheses

EstimationPeriod

3SLS Bootstrap+3SLS

αi βi γi j αi βi γi j

75–12 6 (9) 6 (9) 48 (90) 5 (10) 4 (10) 25 (100)

75–89 3 (9) 3 (9) 21 (90) 3 (10) 3 (10) 19 (100)

90–12 4 (9) 4 (9) 30 (90) 3 (10) 2 (10) 15 (100)

5.2 Estimation Results

As we mentioned in Sect. 3, the AIDS model has an endogeneity problem, and accord-ingly we cannot obtain consistent parameter estimates using the classical estimationmethods such as SUR. Therefore, the AIDS model (1) is estimated using 3SLS, wherethe log relative prices (ln p1,t , ln p2,t , ..., ln pM,t ), their cross-terms and the constantterm are used for the instrumental variables. Note that 3SLS is available in Eviews7.2. The first M − 1 equations are simultaneously estimated (the Other consumptionexpenditure equation, i.e., the M th equation, is omitted from estimation because ofthe additivity constraints), where M = 10 is taken. This paper uses the MBB andPB method as shown in Sect. 4, where in this paper b = 12 is taken for the lengthof block (b = 12 implies that one block consists of one year data). We generate nbootstrap samples of y∗

t , and estimate αi , βi and γi j for each bootstrap sample, wheren = 10, 000 is set.

The demand structure of Japanese households has been possibly changed in thebubble economy period, i.e., the last half of 1980s. Therefore, we divide samplesinto two regimes. The first regime is from January, 1975 to December, 1989, and thesecond regime is from January, 1990 to December, 2012. Remember that the Nikkei225 stock price was the highest value on December 29, 1989 and that the bubbleeconomy has burst since then.3 Table 1 shows the number of significant parameter

3 Actually we have tested the structural change between 1989 and 1990 as follows. Let θ1 and θ2 beparameter vectors in the first and second regimes. θ1 and θ2 which consist of {αi ,βi ,γi j } for i = 1, 2, ..., M−1 and j = 1, 2, ..., M . From the n bootstrap samples of y∗

t , t = 1, 2, ..., T , we can compute n estimatesof the parameter vector. Let θ̂1 and θ̂2 be the sample averages obtained from the n estimates of parametervectors, where the subscript denotes the first and the second regimes. ̂1 and ̂2 represent the samplevariance-covariance matrices, which are also computed from the n estimates of parameter vectors in thefirst and second regimes. Under the null hypothesis H0 : θ1 = θ2, asymptotically we can test the structuralchange as follows:

(θ̂1 − θ̂2)′(̂1 + ̂2)−1(θ̂1 − θ̂2) −→ χ2(108).

Note that, we have 120 parameters in (1) but there are 12 additivity constraints. Therefore, the degree offreedom is given by 120 − 12 = 108.We have obtained 415.9 for the above test statistic, and the critical values of 5 and 1 % are given byχ2

0.05(108) = 133.3 and χ20.01(108) = 145.1, respectively. Thus, we have rejected the null hypothesis

which there is no structural change between 1989 and 1990.Moreover, we have examined the tests of structural change for the individual parameters. As a result, thenull hypotheses are rejected for 70 parameters out of 108 parameters at 5 % significant level. Therefore, wecan conclude that the structural change has occurred between 1989 and 1990.

123


estimates at 5 % significance level for each estimation period: (i) full sample, whichis denoted by 75–12 in Table 1, where the estimation period is from January, 1975to December, 2012 and accordingly the sample size is T = 456, (ii) the first regime,denoted by 75–89, where the estimation period is from January, 1975 to December,1989 and the sample size is T = 180. and (iii) the second regime, denoted by 90–12, where the estimation period is from January, 1990 to December, 2012 and thesample size is T = 276. In Table 1, 3SLS indicates the 3SLS estimation resultsusing the originally observed data, while Bootstrap+3SLS represents 3SLS with theMBB and PB method, which has been discussed in Sect. 4. In the case of Boot-strap+3SLS, each value indicates that the number of parameter estimates in which thep-value is less than 0.025 or larger than 0.975. Note in this paper that the p-value isobtained dividing the number of positive coefficient estimates by the number of boot-strap samples, which corresponds to the empirical probability which is larger thanzero. Comparing 3SLS with Bootstrap+3SLS, 3SLS is larger than Bootstrap+3SLSin ratio for all the cases, which implies that the standard errors of coefficients esti-mated by 3SLS are underestimated. The numbers of significant parameters in 75–12 are larger than those in 75–89 and 90–12 for both 3SLS and Bootstrap+3SLS.Therefore, we can also conclude that there is a structural change between 1989 and1990.

5.2.1 Disturbance analysis

We check whether the assumption on the uncorrelated disturbance over time t isplausible. Figure 1 shows the residual plots estimated by 3SLS for the first M − 1equations in the AIDS model given by Eqs. (1) and (2), where we consider the structuralchange at the beginning of 1990 and accordingly the estimation period is divided intotwo regimes (M = 10 is taken in this paper). The vertical line between December,1989 and January, 1990 represents the time period when the bubble economy burst.Moreover, Fig. 1 shows the Durbin–Watson (DW ) statistics for each regime. The DWin the left-hand side corresponds to the first regime and the DW in the right-hand sideindicates the second regime.

Utilizing the TSP source code in the website: http://www.stanford.edu/~clint/bench/dwcrit.htm, the DW lower and upper bounds for the 5 % critical value, denotedby dl and du, are given by (dl, du) = (1.62837, 1.88692) for (T, k) = (180, 12) and(dl, du) = (1.72043, 1.88595) for (T, k) = (276, 12), where T denotes the samplesize and k indicates the number of regressors including the intercept.

In this paper, we take k = M +2 = 12, T = 180 in the first regime and T = 276 inthe second regime. The serial correlation in the error term is found in the first regimeof Food, Fuel and Culture and in the second regime of Food, Fuel, Trans, Edu, andCulture (see × in the superscript of DW in Fig. 1). In the first regime of Clothes,Trans, and Edu and in the second regime of Housing, Furni, and Clothes, we cannotjudge whether there is a serial correlation in the error term (see in the superscriptof DW in Fig. 1). Four cases out of 18 result in no serial correlation.

Although DW is a rough measure of serial correlation because the AIDS model isnonlinear in parameters, it might be plausible to utilize the MBB method, taking intoaccount serial correlation. In addition, from Fig. 1, we can observe heteroscedastic

123

http://www.stanford.edu/~clint/bench/dwcrit.htm

http://www.stanford.edu/~clint/bench/dwcrit.htm


Fig. 1 Residual Plots—3SLS. × in the superscript of DW implies that the error terms are serially correlatedat 5 % significance level, while triangle indicates that we cannot judge whether there is a serial correlation

errors for some of 18 cases. The PB method is useful for the heteroscedastic error termsbecause of resampling in pairs. The use of MBB and PB solves both serial correlationand heteroscedasticity problems.

5.2.2 Testing homogeneity, symmetry, and negativity conditions

We consider testing the homogeneity, symmetry, and negativity conditions, whichhave been discussed in Sect. 2.

Homogeneity First, Table 2(a) shows the testing results on homogeneity by the Waldtest statistics, where 3SLS is used for estimation. The null hypothesis is writtenas:

123


Table 2 Test of homogeneity

(a) 3SLS (Wald test statistic) (b) Bootstrap+3SLS (p-values)

Estimation period 75–12 75–89 90–12 75–12 75–89 90–12

Food 1.920 0.209 2.157 0.1722 0.2973 0.1268

(.1658) (.6473) (.1419)

Housing 0.244 2.548 0.307 0.2492 0.0323 0.5116

(.6215) (.1105) (.5796)

Fuel 0.823 3.310 0.918 0.6656 0.0259 0.2173

(.3644) (.0689) (.3381)

Furni 8.584* 1.390 1.241 0.9833* 0.8894 0.9182

(.0034) (.2384) (.2653)

Clothes 0.999 3.741 0.086 0.2745 0.9815* 0.5410

(.3176) (.0531) (.7693)

Medical 40.80* 0.069 2.148 0.0005* 0.3867 0.8027

(.0000) (.7933) (.1427)

Trans 3.502 0.099 0.118 0.1680 0.4816 0.7519

(.0613) (.7526) (.7310)

Edu 8.993* 4.388* 3.357 0.9951* 0.9914* 0.7317

(.0027) (.0362) (.0669)

Culture 1.146 0.438 0.128 0.7037 0.2104 0.3785

(.2843) (.5083) (.7195)

Other – – – 0.8835 0.3690 0.2756

(i) In (a), * indicates that the null hypothesis is rejected at the 5 % significance level by Wald test. Thep-values are in the parentheses. The 5 % critical value is χ2

0.05(1) = 3.841(ii) In (b), * indicates that the null hypothesis is rejected by the both-sided test at the 5 % significance level

H0 :M∑

j=1

γi j = 0, i = 1, 2, ..., M.

In Table 2(a), * indicates that the null hypothesis is rejected at the 5 % significancelevel (note that the 5 % critical value is χ2

0.05(1) = 3.841) and the p-values are inthe parentheses. For the estimation period from January, 1975 to December, 2012,Food, Housing, Fuel, Clothes, Trans, and Culture expenditures satisfy the homogeneitycondition and the other three expenditures do not (see 75-12 in Table 2(a)).

For the homogeneity test in the bootstrap procedure based on 3SLS, the test statisticis given by

∑Mj=1 γ̂ ∗

i j , where γ̂ ∗i j denotes the parameter estimate based on the bootstrap

sample: y∗t = (w∗

t , p∗t , X∗

t ) for t = 1, 2, ..., T , where w∗t = (w∗

1,t , ..., w∗M,t ) and

p∗t = (p∗

1,t , ..., p∗M,t ). Using the n bootstrap samples, we can obtain the n test statistics.

If the number of the test statistics which are less than zero is less than 0.025n or greaterthan 0.975n, we reject the null hypothesis H0. In Table 2(b), the bootstrap procedurebased on 3SLS is utilized to obtain the p-values, which are defined as the upperempirical probability of Prob(

∑Mj=1 γ̂ ∗

i j > 0) in this paper. In the case of the bootstrap

123


method, Food, Housing, Fuel, Clothes, Trans, Culture, and Other expenditures satisfythe homogeneity condition, but the other three expenditures do not (see 75-12 in Table2(b)). Note that (a) is the exactly same result as (b).

Rejections of the homogeneity condition are observed in lots of empirical studies.One of the reasons why the homogeneity condition is rejected is due to the structuralchange (see Hashimoto 2004). Therefore, we have examined the same testing proce-dure for the first and second regimes in Tables 2(a) and (b). As a consequence, thenumber of the expenditures satisfying the homogeneity condition increases, i.e., (i)in the first regime (i.e., 75-89 in Tables 2(a) and (b)) the homogeneity condition issatisfied for eight expenditures out of nine expenditures in the case of 3SLS and foreight out of ten expenditures in Bootstrap+3SLS, and (ii) in the second regime (i.e.,90-12 in Tables 2a and b) it is satisfied for nine out of nine expenditures in 3SLS andten out of ten expenditures in Bootstrap+3SLS. Thus, from the fact that in a lot ofcases the homogeneity condition is satisfied when the samples are divided, we canobserve that there is surely a structural change between 1989 and 1990.

Symmetry: Second, Tables 3 and 4 show the results of testing the symmetry condition,where the null hypothesis is as follows:

H0 : γi j − γ j i = 0, for i > j and i, j = 1, 2, ..., M.

This test is examined for both 3SLS and Bootstrap+3SLS cases and for the full sample(January, 1975–December, 2012), the first regime (January, 1975–December, 1989)and the second regime (January, 1990–December, 2012) cases. In the case of Boot-strap+3SLS, the testing procedure of the symmetry condition is: (i) compute γ̂ ∗

i j − γ̂ ∗j i

for n bootstrap samples, where n = 10, 000, and (ii) test the null hypothesis using the

p-value, i.e., the upper empirical probability of Prob(γ̂ ∗

i j − γ̂ ∗j i > 0

).

Tables 3(a) and 4(a) show that 22 and 33 out of 45 cases satisfy the symmetrycondition for 3SLS (see Table 3(a)) and Bootstrap+3SLS (see Table 4(a)), respectively.Dividing the sample, 36 and 38 cases in estimation period 1975–1989 for both 3SLSand Bootstrap+3SLS (see Tables 3(b) and 4(b)), and 36 and 42 in 1990–2012 satisfythe symmetry condition for 3SLS and Bootstrap+3SLS (see Tables 3(c) and 4(c)).Thus, Bootstrap+3SLS satisfies the symmetry condition for more cases, comparedwith 3SLS. From these results, it might be concluded that like the case of homogeneitywe can observe the structural change between 1989 and 1990, because for Tables 3and 4 (a) is smaller than (b) and (c) with respect to the number of the cases where thesymmetry condition is satisfied.

Thus, taking into account the structural change, both homogeneity and symmetryconditions are satisfied in most of cases.

Negativity Finally, we conduct the test of negativity (or concavity) condition, whichshows whether the estimated parameters guarantee the condition of local maximiza-tion. For each bootstrap sample, the negativity can be checked by calculating theeigenvalues of the Slutsky matrix (or equivalently, the substitution matrix, denoted byS), where an element in the i th row and j th column is denoted by si j . The negativity

123


Table 3 Test of symmetry (Wald test statistics)—3SLS

Food Housing Fuel Furni Clothes Medical Trans Edu Culture

(a) Estimation period: 1975–2012Housing 0.207

(.6489)

Fuel 0.408 2.547

(.5230) (.1105)

Furni 1.838 33.88* 36.38*

(.1751) (.0000) (.0000)

Clothes 0.232 2.689 2.559 0.989

(.6299) (.1010) (.1097) (.3200)

Medical 24.90* 0.553 32.85* 24.84* 0.296

(.0000) (.4573) (.0000) (.0000) (.5864)

Trans 2.788 0.223 0.113 19.16* 10.85* 16.50*

(.0950) (.6369) (.7368) (.0000) (.0010) (.0000)

Edu 15.89* 5.912* 0.000 14.74* 4.961* 2.476 0.166

(.0001) (.0150) (.9839) (.0001) (.0259) (.1156) (.6834)

Culture 4.579* 4.415* 7.733* 4.213* 10.37* 20.34* 2.540 0.366

(.0324) (.0356) (.0054) (.0401) (.0013) (.0000) (.1110) (.5453)

Other 1.227 0.729 6.862* 1.650 0.008 16.99* 12.08* 15.62* 5.676*

(.2680) (.3932) (.0088) (.1990) (.9291) (.0000) (.0005) (.0001) (.0172)

(b) Estimation period: 1975–1989

Housing 5.239*

(.0221)

Fuel 0.908 2.203

(.3406) (.1378)

Furni 2.604 0.120 0.000

(.1066) (.7293) (.9841)

Clothes 7.117* 1.074 1.437 0.912

(.0076) (.3000) (.2307) (.3395)

Medical 0.003 1.307 1.802 0.022 10.32*

(.9592) (.2528) (.1795) (.8832) (.0013)

Trans 0.073 0.492 0.691 0.520 0.470 5.654*

(.7865) (.4832) (.4060) (.4707) (.4929) (.0174)

Edu 13.40* 6.078* 1.729 0.873 22.85* 0.646 5.733*

(.0003) (.0137) (.1885) (.3502) (.0000) (.4217) (.0167)

Culture 0.448 0.000 3.704 0.097 2.308 0.057 0.064 1.626

(.5034) (1.000) (.0543) (.7556) (.1287) (.8115) (.8006) (.2023)

Other 1.165 0.092 3.736 0.727 8.919* 0.063 0.533 1.107 3.521

(.2805) (.7614) (.0533) (.3939) (.0028) (.8016) (.4655) (.2928) (.0606)

123


Table 3 continued

Food Housing Fuel Furni Clothes Medical Trans Edu Culture

(c) Estimation period: 1990–2012

Housing 0.196

(.6582)

Fuel 0.426 3.229

(.5141) (.0723)

Furni 0.071 10.83* 22.14*

(.7900) (.0010) (.0000)

Clothes 0.026 0.110 0.066 4.595*

(.8730) (.7402) (.7978) (.0321)

Medical 0.907 3.216 0.041 0.300 4.721*

(.3409) (.0729) (.8391) (.5841) (.0298)

Trans 0.516 2.950 0.372 2.569 1.936 1.079

(.4724) (.0859) (.5417) (.1089) (.1641) (.2989)

Edu 6.414* 2.125 10.05* 2.481 0.086 1.896 0.086

(.0113) (.1450) (.0015) (.1152) (.7691) (.1685) (.7699)

Culture 4.042* 1.538 1.504 0.412 2.343 0.079 0.337 1.829

(.0444) (.2150) (.2201) (.5208) (.1258) (.7785) (.5616) (.1763)

Other 1.765 1.422 1.734 0.613 0.465 7.363* 0.001 4.387* 0.024

(.1840) (.2331) (.1879) (.4335) (.4951) (.0067) (.9701) (.0362) (.8773)

* Indicates that the null hypothesis is rejected at the 5 % significance level. The p-values are in the paren-theses. The 5 % critical value is χ2

0.05(1) = 3.841

condition implies that S is a negative semidefinite matrix, i.e., all the eigenvalues arenon-positive. Deaton and Muellbauer (1980) suggest to use C whose eigenvalues hasthe same signs as those of S, where the (i, j)th element of C is given by:

ci j = γi j + βiβ j lnXt

Pt− wi,tδi j + wi,tw j,t ,

where δi j is the Kronecker delta, i.e., δi j = 1 for i = j and δi j = 0 for i �= j .4 Thistest is examined for the full sample (January, 1975–December, 2012), the first regime(January, 1975–December, 1989), and the second regime (January, 1990–December,2012) in Bootstrap+3SLS. The negativity condition is satisfied in only five cases forthe full sample, in 386 cases for the first regime and in two cases for the secondregime out of 10,000 bootstrap parameter estimates. That is, we can conclude thatthe negativity condition is rejected.5 It is extremely difficult to obtain the parameterestimates which satisfy the negativity condition.

4 Because ci j depends on wi,t , ln Xt and ln Pt , we evaluate them at the sample averages of wi,t , ln Xt and

ln Pt over time t . That is, wi,t , ln Xt and ln Pt in ci j are replaced by wi , ln X and ln P , respectively.5 Moschini (1998) suggests to use the semiflexible AIDS model, which impose the negativity condition.

123


Tabl

e4

Test

ofsy

mm

etry

(p-v

alue

s)—

Boo

tstr

ap+

3SL

S

Food

Hou

sing

Fuel

Furn

iC

loth

esM

edic

alT

rans

Edu

Cul

ture

(a)

Est

imat

ion

peri

od:1

975–

2012

Hou

sing

.452

8

Fuel

.634

3.6

536

Furn

i.7

180

.998

8*.9

876*

Clo

thes

.306

5.3

054

.316

5.9

310

Med

ical

.003

1*.3

159

.038

3.0

002*

.498

4

Tra

ns.2

847

.689

5.5

945

.001

2*.0

278

.992

7*

Edu

.999

4*.9

552

.429

4.9

786*

.978

3*.8

655

.680

9

Cul

ture

.865

6.1

182

.055

4.7

422

.989

2*.9

894*

.837

0.3

685

Oth

er.7

809

.722

1.8

963

.880

0.6

564

.952

0.9

739

.012

8*.8

423

(b)

Est

imat

ion

peri

od:1

975–

1989

Hou

sing

.001

8*

Fuel

.251

3.1

025

Furn

i.4

779

.625

1.8

472

Clo

thes

.991

4*.9

083

.953

4.5

039

Med

ical

.400

3.8

163

.555

3.5

950

.010

7*

Tra

ns.4

848

.201

1.7

408

.692

5.3

971

.048

1

Edu

.999

9*.9

946*

.418

3.2

838

.990

3*.9

680

.011

8*

Cul

ture

.631

0.5

616

.488

6.2

903

.295

4.7

991

.306

7.8

832

Oth

er.2

060

.428

4.8

342

.147

5.1

904

.243

3.6

991

.047

8.9

680

123


Tabl

e4

cont

inue

d

Food

Hou

sing

Fuel

Furn

iC

loth

esM

edic

alT

rans

Edu

Cul

ture

(c)

Est

imat

ion

peri

od:1

990–

2012

Hou

sing

.821

1

Fuel

.334

2.8

337

Furn

i.4

728

.988

6*1.

0000

*

Clo

thes

.448

8.2

471

.513

1.9

414

Med

ical

.721

0.7

836

.359

3.1

285

.978

5*

Tra

ns.8

288

.911

1.7

088

.095

7.2

115

.905

3

Edu

.871

4.6

518

.947

4.4

521

.370

2.1

097

.370

4

Cul

ture

.756

3.0

617

.251

7.4

825

.858

5.3

805

.388

5.1

998

Oth

er.1

808

.228

8.8

779

.552

6.2

367

.070

9.5

137

.175

5.4

332

Bas

edon

the

p-va

lue,

*in

dica

tes

that

the

null

hypo

thes

isis

reje

cted

byth

ebo

th-s

ided

test

atth

e5

%si

gnifi

canc

ele

vel

123


5.2.3 Expenditure elasticity (6) and own price elasticity (7)

Tables 5 and 6 show the expenditure and own price elasticities based on Eqs. (6)and (7) in the case of (a) the full sample (January, 1975–December, 2012), (b) thefirst regime (January, 1975–December, 1989), and (c) the second regime (January,1990–December, 2012).6 As discussed in Sect. 4, using the bootstrap procedure wecan obtain the expenditure and price elasticities, the standard error, the p-value andthe 2.5 and 97.5 percent points (i.e., 95 % confidence interval), which are denoted byElas, SE, p-value, 2.5 and 97.5 %, respectively. For comparison, in Tables 5 and 6, wealso show each elasticity (Elas) and its standard error (SE) in the case where 3SLS isutilized. SE in 3SLS is given by the Delta method.7

From Bootstrap+3SLS in Table 5 (i.e., the expenditure elasticities), 7 or 8 out of 10expenditures are statistically different from zero for each estimation period. A good iselastic, which is known as a luxury good, when the expenditure elasticity is larger than1, and a good is inelastic, which is known as a necessity good, when the expenditureelasticity is smaller than 1. In Table 5(a), Furni, Clothes, Trans, Edu, Culture, andOther are luxury goods, and Food, Housing, Fuel, and Medical are necessity goods.Moreover, a good is known to be inferior when the expenditure elasticity is smallerthan 0. The expenditure elasticity of Fuel (1975–1989) is negative but statisticallyinsignificant. This result is not plausible because Fuel is known more commonly asthe normal good (that is, the expenditure elasticity should be larger than 0). It is naturalto consider that the demand structure might be changed during the Japanese bubbleeconomy period.

Next, we check the results of own price elasticities. Table 6 shows the uncompen-sated price elasticity given by Eq. (7). The results in the full sample case (Table 6(a))indicate that all the own price elasticities except for Housing are statistically sig-nificant and that they are negative. Therefore, the demand principle holds in thenine expenditures. However, the elasticity of Housing is positive but statisticallyinsignificant. Taking into account the structural change, the elasticity of Housingis negative and significant in the first regime (Table 6(b)), and it is positive butinsignificant in the second regime (Table 6(c)). Therefore, it might be concludedthat for Housing the demand principle holds only in the first regime. For Food,Fuel, Furni, Clothes, and Culture, the price elasticities in the second regime arelarger in absolute value than those in the first regime. It might be plausible to

6 As it is the case of the negativity test, in Tables 5 and 6 we evaluate wi,t and ln pi,t at the sample averagesof wi,t and ln pi,t over time t . Remember that the expenditure and price elasticities are given by (6) and(7), which depend on wi,t and ln pi,t .7 The Delta method is as follows. Suppose that an elasticity η is expressed as a function η(θ) of a parametervector θ = {αi , βi , γi j } for i = 1, 2, ..., M − 1 and j = 1, 2, ..., M . Let V be the variance–covariance

matrix of θ̂ and D be the gradient of η(θ). The asymptotic standard errors, denoted by SE(η), are obtainedas follows:

SE(η) ≈ D′V D.

Here, because of the additivity constraint, the M th expenditure parameters are represented by those of theother expenditure parameters. Therefore, it is not easy to obtain the standard errors of the M th elasticities.

123


Table 5 Expenditure elasticity (6)

Bootstrap+3SLS 3SLS

Elas SE p-value 2.5 % 97.5 % Elas SE

(a) Estimation period: 1975–2012

Food 0.5129∗ 0.0819 1.0000 0.348 0.670 0.5136∗ 0.0409

Housing 0.4495 0.3439 0.9104 −0.202 1.158 0.2389 0.2070

Fuel 0.5144 0.2725 0.9539 −0.132 1.025 0.6009∗ 0.2473

Furni 1.3900∗ 0.2507 1.0000 0.880 1.885 1.3360∗ 0.2655

Clothes 1.4321∗ 0.2436 1.0000 0.972 1.947 1.3440∗ 0.1518

Medical 0.0018 0.2777 0.4636 −0.479 0.705 −0.1360 0.4199

Trans 1.0735∗ 0.2550 0.9998 0.565 1.586 1.0900∗ 0.1549

Edu 2.0579∗ 0.3046 1.0000 1.461 2.656 2.1602∗ 0.2576

Culture 1.1553∗ 0.2378 1.0000 0.723 1.631 1.0748∗ 0.1212

Other 1.4241∗ 0.1004 1.0000 1.222 1.620 – –

(b) Estimation Period: 1975–989

Food 0.4703∗ 0.1156 0.9999 0.237 0.689 0.6166∗ 0.1122

Housing 1.1616∗ 0.5425 0.9922 0.193 2.284 0.9281 0.6788

Fuel −0.0292 0.3554 0.4907 −0.774 0.627 −0.7713∗ 0.2626

Furni 1.8703∗ 0.7113 0.9980 0.537 3.323 1.4820 1.3918

Clothes 1.9840∗ 0.4640 1.0000 1.093 2.903 2.3030∗ 0.1294

Medical 1.2627∗ 0.4497 0.9963 0.370 2.135 1.9318 0.9872

Trans 0.8771 0.5218 0.9542 −0.153 1.894 0.5624 0.5219

Edu 1.5529∗ 0.6291 0.9943 0.342 2.796 1.2161∗ 0.2900

Culture 1.1141∗ 0.2684 1.0000 0.598 1.656 1.3450∗ 0.2237

Other 1.2051∗ 0.2101 1.0000 0.803 1.616 – –

(c) Estimation Period: 1990–2012

Food 0.4864∗ 0.0925 1.0000 0.294 0.661 0.6012∗ 0.0652

Housing 1.4805∗ 0.3429 1.0000 0.814 2.177 1.2122∗ 0.2662

Fuel 0.1556 0.1448 0.8600 −0.131 0.445 0.0441 0.3402

Furni 1.5768∗ 0.4536 1.0000 0.731 2.509 1.4372∗ 0.5079

Clothes 1.4682∗ 0.2506 1.0000 0.951 1.930 1.7037∗ 0.3215

Medical 0.4319 0.3351 0.8985 −0.228 1.080 0.6770 0.7043

Trans 1.6611∗ 0.3709 0.9999 0.944 2.399 1.6757∗ 0.2305

Edu 1.4989∗ 0.4353 0.9993 0.654 2.374 1.4503∗ 0.3689

Culture 1.1128∗ 0.2848 1.0000 0.589 1.712 0.8106∗ 0.1421

Other 1.1012∗ 0.1611 1.0000 0.793 1.434 – –

* Indicates that the null hypothesis is rejected by the both-sided test at the 5 % significance level. SE in3SLS is obtained by the Delta method

consider that the long run deflation in the second regime forces Japanese house-holds to be sensitive against the price increase. Here, we confirm whether the para-meter shifts of some elasticities was occurred during the bubble economy period.

123


Table 6 Own price elasticity (7)

Bootstrap+3SLS 3SLS

Elas SE p-value 2.5 % 97.5 % Elas SE

(a) Estimation period: 1975–2012

Food −1.3736∗ 0.2819 0.0000 −1.924 −0.818 −1.2872∗ 0.6529

Housing 0.9966 0.8511 0.8812 −0.706 2.676 1.1553∗ 0.3022

Fuel −0.9794∗ 0.1896 0.0000 −1.472 −0.662 −0.9210 0.4763

Furni −1.1614∗ 0.2274 0.0000 −1.584 −0.694 −1.1371∗ 0.4461

Clothes −1.2304∗ 0.2756 0.0000 −1.771 −0.672 −1.2785∗ 0.2522

Medical −1.9583∗ 0.1989 0.0000 −2.347 −1.551 −2.0126∗ 0.3137

Trans −1.8426∗ 0.3444 0.0000 −2.524 −1.184 −1.9933∗ 0.4226

Edu −0.9450∗ 0.1862 0.0002 −1.325 −0.586 −0.9997∗ 0.2588

Culture −2.0018∗ 0.5259 0.0000 −3.021 −0.927 −2.2545∗ 0.1236

Other −1.6113∗ 0.0931 0.0000 −1.810 −1.438 – –

(b) Estimation period: 1975–1989

Food −1.3538∗ 0.2855 0.0000 −1.942 −0.821 −1.3193 1.1530

Housing −3.1584∗ 1.2414 0.0021 −5.836 −0.904 −3.0177∗ 0.4495

Fuel −1.4530∗ 0.1822 0.0000 −1.780 −1.062 −1.4820∗ 0.6158

Furni −0.5367 0.8625 0.2502 −2.243 1.238 −0.1221 1.2286

Clothes −1.5787∗ 0.3944 0.0001 −2.361 −0.818 −1.5445 1.0332

Medical −1.9887∗ 0.3534 0.0000 −2.690 −1.321 −2.0135∗ 0.4538

Trans −1.5412∗ 0.5719 0.0036 −2.681 −0.433 −1.6476 1.6109

Edu −1.7457∗ 0.5139 0.0000 −2.945 −0.907 −1.6421∗ 0.3986

Culture −1.5928∗ 0.4949 0.0032 −2.472 −0.510 −1.8623∗ 0.3480

Other −1.8689∗ 0.1814 0.0000 −2.283 −1.552 – –

(c) Estimation period: 1990–2012

Food −1.6449∗ 0.3882 0.0000 −2.412 −0.883 −1.6777 1.2058

Housing 1.8830 1.1704 0.9477 −0.458 4.126 1.2647∗ 0.3059

Fuel −1.5608∗ 0.2079 0.0000 −1.985 −1.142 −1.5789∗ 0.7118

Furni −1.0359∗ 0.3692 0.0097 −1.703 −0.247 −1.1378 0.8865

Clothes −1.7745∗ 0.3429 0.0000 −2.489 −1.150 −1.7027∗ 0.4096

Medical −1.2183∗ 0.2430 0.0000 −1.715 −0.760 −1.0645∗ 0.1697

Trans −0.7129 0.7125 0.1534 −2.000 0.793 −0.9957 0.7717

Edu −1.3166∗ 0.7137 0.0022 −3.174 −0.367 −0.9849∗ 0.3408

Culture −1.7965∗ 0.7249 0.0105 −3.105 −0.270 −1.6362∗ 0.1700

Other −1.7198∗ 0.2803 0.0000 −2.357 −1.243 – –

* Indicates that the null hypothesis is rejected by the both-sided test at the 5 % significance level. SE in3SLS is obtained by the Delta method

The testing procedure of the structural change in the elasticities will be discussedlater. We conclude that the structural change in Housing price elasticity occurred in1990.

123


5.2.4 Comparison of standard errors

In Tables 5 and 6, SE in Bootstrap+3SLS is compared with SE in 3SLS. We justifyuse of the bootstrap method to obtain plausible standard errors for all the elasticityestimates. A few past studies report the standard errors of elasticities but most of paststudies do not report them. The elasticities are converted from the parameter estimatesincluded in the AIDS model Eq. (1), using Eqs. (6) and (7). In the past studies inwhich the standard errors of the elasticities are shown, the Delta method is utilized.See Footnote 7 for computation of the standard errors by the Delta method. Regard-ing the standard errors, Tables 5 and 6 show a comparison between the traditionalapproximation method (i.e., the Delta method) and the bootstrap method in the threeestimation periods (a–c). In the case of own price elasticity, the standard errors of thebootstrap method (i.e., SE in Bootstrap+3SLS) are very different from those of theDelta method (i.e., SE in 3SLS). Freedman and Peters (1984) and Green et al. (1987)show that the bootstrap method is preferred to the Delta method in a sense of accuratestandard errors and that the bootstrap method performs well even in the case of smallsample and/or small number of the bootstrap samples.8

Krinsky and Robb (1991) compare the two methods (i.e., the Delta method andthe bootstrap method) with respect to parameter estimates and conclude that the twomethods essentially have no differences in the case of small sample and large numberof the bootstrap samples (i.e., T = 24 and n = 1, 000). However, Krinsky and Robb(1991) show that the Delta method is different from the bootstrap method for thestandard errors of price elasticities, not expenditure elasticities. The results in thispaper are basically consistent with the findings in Krinsky and Robb (1991), wherewe take T = 456, 180, 276 and n = 10, 000. Green et al. (1987) compare the standarderrors of expenditure elasticities for the two methods using the LES model, and theyfind a considerable difference between the two methods.9 Another problem of usingthe Delta method is the difficulty in estimating the standard errors of M th expenditure’selasticities, while the bootstrap method can easily obtain them. From these reasons,the bootstrap method might be more attractive than the traditional Delta method.

5.2.5 Testing structural change in the elasticities

We have tested the structural change on individual parameters in Footnote 3. Here, weconsider testing whether there is a structural change on the elasticities. Let η̂1 and η̂2 bethe sample averages from the n elasticities in the first and second regimes, respectively.

8 Freedman and Peters (1984) examine the case of T = 18 and n = 100, and Green et al. (1987) study thecase of T = 36 and n = 50, where T denotes the sample size and n represents the number of the bootstrapsamples.9 Note that Green et al. (1987) are based on T = 36 and n = 50, which indicates the small sample size andthe small number of bootstrap samples. Mizobuchi and Tanizaki (2007) examine the same comparison forthe LA–AIDS model in the case of the large sample size and the large number of the bootstrap samples (i.e.,T = 370 and n = 10, 000). In consequence, for the standard errors of expenditure elasticities the Deltamethod is clearly different from the bootstrap method. On the other hands, Mizobuchi and Tanizaki (2007)obtain the results that the two methods are almost the same for the standard errors of price elasticities. Thereare no other studies which compare the Delta method and the bootstrap method in large sample case.

123


Table 7 Test of structuralchange between 1989 and 1990(t test)—Bootstrap+3SLS

Utilizing asymptotic normality,* indicates that the nullhypothesis of no structuralchange is rejected by theboth-sided test at the 5 %significance level

Expenditure Elasticity Own price elasticity

Food −0.1087 0.6041

Housing −0.4969 −2.9549∗Fuel −0.4815 0.3900

Furni 0.3479 0.5321

Clothes 0.9781 0.3747

Medical 1.4814 −1.7963

Trans −1.2246 −0.9066

Edu 0.0706 −0.4879

Culture 0.0033 0.2321

Other 0.3924 −0.4466

σ̂1 and σ̂2 represent the standard errors of η̂1 and η̂2, which are also computed fromthe n elasticities. Under the null assumption that η1 is equal to η2, as the number ofobservations increase for both the first and second regimes, we have the following:

η̂1 − η̂2√

σ̂ 21 + σ̂ 2

2

−→ N (0, 1).

Note that η̂1 and σ̂1 correspond to Elas and SE in Tables 5(b) or 6(b), while η̂2 and σ̂2correspond to Elas and SE in Tables 5(c) or 6(c).

Thus, we can test whether the structural change occurs between 1989 and 1990,using the expenditure elasticities and the standard errors in Tables 5b and c. In the sameway, using Tables 6b and 6c, we can examine the structural change of the own priceelasticities. The test statistics are shown in Table 7, and they are compared with thestandard normal distribution. From the table, the own price elasticity of Housing haschanged between 1989 and 1990, while we can not find the structural change on theother elasticities. Thus, we conclude that only in the case of Housing price elasticitythe structural change has occurred between 1989 and 1990.

5.2.6 AIDS versus LA–AIDS

In empirical studies, the LA–AIDS model which consists of Eqs. (1) and (3) is moreoften utilized, rather than the AIDS model given by Eqs. (1) and (2). In the LA–AIDSmodel the Stone price index (3) is used for ln Pt in (1), while in the AIDS model theoriginal price index (2) is utilized for ln Pt . The Stone price index Eq. (3) is known as anapproximation of Eq. (2). In the case of LA–AIDS, expenditure, and price elasticitiesare given by Eqs. (8) and (9), which are discussed in Green and Alston (1990). Now,we examine how different results we obtain, depending on the Stone price index (3) orthe original price index (2). As it will be discussed below (Fig. 2), the Stone price index(3) does not approximate the original price index (2). Table 8 shows the differencebetween AIDS and LA–AIDS models for the full sample (January, 1975–December,2012). (a) in Table 8 is exactly equivalent to (a) in Bootstrap+3SLS of Tables 5and 6.

123


(a)

(b)

Fig. 2 Price indices: ln Pt —3SLS. (a) Original price index (2). b Stone price index (3)

We compare (a) and (b) in Table 8, where (a) and (b) indicate AIDS and LA–AIDS,respectively. Both are estimated by the bootstrap method based on 3SLS. The elasticityestimates in (a) are very close to those in (b), where the differences between (a) and(b) are −0.0733 to 0.1226 for the expenditure elasticity and −0.0727 to 0.0546 for theown price elasticity. From (a) and (b) in Table 8, there is no evidence that the elasticityestimates in (b) are biased toward either the left-hand side or the right-hand side. Inaddition, SEs in (a) are very close to those in (b).

Buse (1994, 1998) and Moschini (1995) conclude that the LA-AIDS elasticity esti-mates are biased.10 In this paper, however, we can observe that using Bootstrap+3SLS,AIDS is very close to LA–AIDS with respect to the expenditure and own price elas-ticities.

5.2.7 Endogenous problem (3SLS versus SUR)

In order to check the endogenous problem on Xt , we compare the elasticities of 3SLSwith those of SUR in Table 8, where (a) should be compared with (c), and (b) shouldbe compared with (d). Generally, it is known that SUR yields biased and inconsistentestimates, but 3SLS gives us consistent estimates. As it is expected, we can find thedivergence between both elasticities. That is, the elasticity estimates in SUR are verydifferent from those in 3SLS. As for AIDS, the differences between (a) and (c) are−0.6018 to 0.3305 for the expenditure elasticities and −0.0917 to 0.4139 for theprice elasticities. For LA–AIDS, the differences between (b) and (d) are −0.5853 to0.3148 for the expenditure elasticities and −0.1277 to 0.4443 for the price elasticities.Moreover, in most of the cases, the standard errors of the elasticity estimates in SURare smaller than those in 3SLS for both expenditure and price elasticities. That is, it isobserved that the standard errors of SUR are under-estimated. Thus, we see that theendogenous problem is serious in both AIDS and LA–AIDS models.

10 They also criticize use of the Stone price index. The problem cannot be avoided even if the demandsystem is estimated by 3SLS (e.g., see Buse 1994, 1998). Moschini (1995) suggests some alternative priceindices, but concludes that they are not suitable to approximate the original price index.

123


Tabl

e8

AID

Sve

rsus

LA

–AID

S(1

975–

2012

)

Exp

endi

ture

elas

ticity

Ow

npr

ice

elas

ticity

Ela

sSE

p-va

lue

2.5

%97

.5%

Ela

sSE

p-va

lue

2.5

%97

.5%

(a)

AID

S(B

oots

trap

+3SL

S)

Food

0.51

29∗

0.08

191.

0000

0.34

80.

670

−1.3

736∗

0.28

190.

0000

−1.9

24−0

.818

Hou

sing

0.44

950.

3439

0.91

04−0

.202

1.15

80.

9966

0.85

110.

8812

−0.7

062.

676

Fuel

0.51

440.

2725

0.95

39−0

.132

1.02

5−0

.979

4∗0.

1896

0.00

00−1

.472

−0.6

62

Furn

i1.

3900

∗0.

2507

1.00

000.

880

1.88

5−1

.161

4∗0.

2274

0.00

00−1

.584

−0.6

94

Clo

thes

1.43

21∗

0.24

361.

0000

0.97

21.

947

−1.2

304∗

0.27

560.

0000

−1.7

71−0

.672

Med

ical

0.00

180.

2777

0.46

36−0

.479

0.70

5−1

.958

3∗0.

1989

0.00

00−2

.347

−1.5

51

Tra

ns1.

0735

∗0.

2550

0.99

980.

565

1.58

6−1

.842

6∗0.

3444

0.00

00−2

.524

−1.1

84

Edu

2.05

79∗

0.30

461.

0000

1.46

12.

656

−0.9

450∗

0.18

620.

0002

−1.3

25−0

.586

Cul

ture

1.15

53∗

0.23

781.

0000

0.72

31.

631

−2.0

018∗

0.52

590.

0000

−3.0

21−0

.927

Oth

er1.

4241

∗0.

1004

1.00

001.

222

1.62

0−1

.611

3∗0.

0931

0.00

00−1

.810

−1.4

38

(b)

LA

-AID

S(B

oots

trap

+3SL

S)

Food

0.53

22∗

0.07

571.

0000

0.37

90.

679

−1.3

384∗

0.29

210.

0000

−1.9

06−0

.760

Hou

sing

0.50

120.

3437

0.93

16−0

.152

1.21

50.

9987

0.85

180.

8814

−0.7

112.

679

Fuel

0.48

030.

2655

0.95

05−0

.151

0.97

4−0

.962

4∗0.

1908

0.00

00−1

.461

−0.6

46

Furn

i1.

3374

∗0.

2451

1.00

000.

830

1.81

3−1

.144

5∗0.

2276

0.00

00−1

.568

−0.6

75

Clo

thes

1.48

15∗

0.24

141.

0000

1.03

41.

995

−1.2

850∗

0.28

100.

0000

−1.8

39−0

.725

Med

ical

0.03

350.

2728

0.51

72−0

.432

0.72

2−1

.935

6∗0.

1972

0.00

00−2

.325

−1.5

36

Tra

ns0.

9509

∗0.

2684

0.99

920.

412

1.47

9−1

.769

9∗0.

3481

0.00

00−2

.462

−1.1

11

Edu

2.13

12∗

0.29

081.

0000

1.54

22.

687

−0.9

723∗

0.19

110.

0000

−1.3

73−0

.608

Cul

ture

1.13

17∗

0.24

071.

0000

0.70

21.

615

−1.9

923∗

0.52

830.

0001

−3.0

10−0

.909

Oth

er1.

4300

∗0.

0989

1.00

001.

230

1.62

1−1

.624

0∗0.

0915

0.00

00−1

.817

−1.4

51

123


Tabl

e8

cont

inue

d Exp

endi

ture

elas

ticity

Ow

npr

ice

elas

ticity

Ela

sSE

p-va

lue

2.5

%97

.5%

Ela

sSE

p-va

lue

2.5

%97

.5%

(c)

AID

S(B

oots

trap

+SU

R)

Food

0.45

66∗

0.04

771.

0000

0.36

00.

548

−1.3

012∗

0.26

710.

0000

−1.8

32−0

.780

Hou

sing

1.05

13∗

0.21

841.

0000

0.64

31.

503

0.58

270.

9399

0.73

36−1

.258

2.46

7

Fuel

0.36

98∗

0.14

030.

9867

0.04

30.

626

−0.9

574∗

0.18

950.

0000

−1.4

48−0

.641

Furn

i1.

4273

∗0.

2030

1.00

001.

048

1.84

0−1

.174

3∗0.

2259

0.00

00−1

.601

−0.7

09

Clo

thes

1.32

63∗

0.15

111.

0000

1.03

81.

628

−1.1

521∗

0.24

780.

0000

−1.6

22−0

.639

Med

ical

0.26

590.

1993

0.92

07−0

.087

0.73

4−1

.888

0∗0.

1900

0.00

00−2

.251

−1.5

10

Tra

ns1.

4865

∗0.

1931

1.00

001.

108

1.87

3−2

.101

6∗0.

3253

0.00

00−2

.743

−1.4

72

Edu

1.72

74∗

0.21

981.

0000

1.28

72.

154

−0.8

533∗

0.17

110.

0000

−1.1

73−0

.503

Cul

ture

1.08

75∗

0.15

411.

0000

0.80

11.

394

−1.9

774∗

0.53

830.

0001

−3.0

23−0

.897

Oth

er1.

2976

∗0.

0666

1.00

001.

164

1.42

8−1

.599

1∗0.

0984

0.00

00−1

.810

−1.4

12

(d)

LA

-AID

S(B

oots

trap

+SU

R)

Food

0.46

30∗

0.04

531.

0000

0.37

10.

550

−1.2

583∗

0.27

600.

0000

−1.8

08−0

.720

Hou

sing

1.08

65∗

0.21

271.

0000

0.69

01.

528

0.55

440.

9450

0.72

23−1

.299

2.45

2

Fuel

0.35

60∗

0.13

720.

9861

0.03

80.

605

−0.9

401∗

0.18

890.

0000

−1.4

26−0

.627

Furn

i1.

3170

∗0.

1921

1.00

000.

947

1.70

1−1

.138

4∗0.

2233

0.00

00−1

.553

−0.6

78

Clo

thes

1.32

33∗

0.14

901.

0000

1.03

91.

622

−1.1

573∗

0.25

010.

0000

−1.6

29−0

.639

Med

ical

0.28

480.

1970

0.93

90−0

.064

0.74

5−1

.871

8∗0.

1892

0.00

00−2

.235

−1.4

95

Tra

ns1.

4053

∗0.

1976

1.00

001.

016

1.79

9−2

.068

8∗0.

3256

0.00

00−2

.715

−1.4

42

Edu

1.81

64∗

0.21

331.

0000

1.39

32.

228

−0.8

759∗

0.17

080.

0000

−1.2

01−0

.533

Cul

ture

1.04

99∗

0.15

371.

0000

0.76

41.

355

−1.9

604∗

0.54

090.

0001

−3.0

13−0

.871

Oth

er1.

3213

∗0.

0658

1.00

001.

189

1.45

1−1

.605

6∗0.

0964

0.00

00−1

.809

−1.4

21

123


5.2.8 Original price index (2) versus Stone price index (3)

As it is easy to handle, empirical demand studies prefer the LA–AIDS model to theAIDS model. In the past, however, there is a lot of literature on accuracy of the Stoneprice index (3), compared with the original price index (2). For example, see Pashardes(1993), Buse (1994, 1998), Hahn (1994), Moschini (1995), Buse and Chan (2000),Feenstra and Reinsdorf (2000), etc. They show that the Stone price index is not a goodapproximation of the original price index, and therefore they strongly suggest to usethe original price index. In this paper, we compare the Stone index and the originalprice index, and examine whether the LA–AIDS model is valid enough. Figure 2 showsthe plots of two price indices, which show that the divergence between two indices isvery large. The original price index is more volatile than the Stone price index. We canconclude that the Stone price index is not an approximation of the original price indexand accordingly that the original price index should be adopted in the AIDS model.

6 Conclusion

In past empirical studies, AIDS and LA–AIDS models have been utilized in a field ofdemand analysis (e.g., marketing, political economics, agricultural economics, etc.),where few statistical properties on the estimated elasticities are discussed althoughpeople are interested in the expenditure and price elasticities, not the parameter esti-mates themselves in the AIDS model. That is, in a lot of empirical studies, the pointestimate on the expenditure and price elasticities is reported, but the standard error,the confidence interval, the p-values and so on are not discussed at all. Applying thebootstrap method, this paper has suggested how to implement statistical inference onthe elasticity estimates obtained from AIDS and LA–AIDS models. The differencesbetween the past studies and this paper are as follows:

(i) The dependent variables {wi,t , i = 1, 2, ..., M} in the AIDS model are sharevariables which lie on the interval between zero and one. There are few studieswhich apply the bootstrap method to the demand system having share’s dependentvariable. Tiffin and Balcombe (2005) applies the bootstrap method to the LA–AIDS model. However, they do not consider the “share” variable, and thereforetheir bootstrap samples of wi,t are not guaranteed to be in the interval [0, 1]. Inthis paper, we take this problem into consideration, using the PB method (seeSect. 4, in detail).

(ii) From the DW statistics, we have the possibility that the error terms are seriallycorrelated. Therefore, we utilize the MBB method, preserving the order of thebootstrap sample during b periods (see Sect. 4).

(iii) We have shown how to test the homogeneity, symmetry, and negativity con-straints, using the bootstrap method. Especially, it is not easy to consider testingthe negativity condition by the Delta method. Using the bootstrap method, theseconstraints are tested very easily.

(iv) There are no other comparative analysis between the bootstrap method and thetraditional Delta method, using the AIDS model. This paper uses the large samplesize (i.e., T = 456, 180, 276) and compares the bootstrap method and the Delta

123


method. As a result, the bootstrap method is quite similar to the Delta method forthe expenditure and price elasticity estimates, but the standard errors based onthe Delta method are very different from those based on the bootstrap method.

(v) As an additional problem, the AIDS model has an endogeneity, which resultsin inconsistency of the parameter estimates. Most of the past empirical studieshave not considered about the inconsistent parameter estimates. In this paper,the 3SLS estimation method is adopted to obtain the consistent estimates. Notethat, GMM is recently used to resolve the endogeneity problem. 3SLS is a sortof GMM (see Hayashi 2000).

In the empirical analysis, we estimate the AIDS model using monthly Japanesehouseholds expenditure data, where ten expenditure data from January, 1975 toDecember, 2012 are used for estimation. From the results, the elasticities of someexpenditures are insignificant. Therefore, if we do not perform the statistical test forthe expenditure and price elasticities, we may have incorrect interpretation on the esti-mation results. Moreover, during the Japanese bubble economy period, we find thestructural change on the own price elasticity for Housing.

Finally, we compare the Stone price index with the original price index to examinevalidity of using the LA–AIDS model. From Fig. 2 and Table 8, there is large diver-gence between the two indices. The divergence causes the biased elasticity estimates.Therefore, considering a recent development of computer capability, use of the AIDSmodel is more appropriate than that of the LA–AIDS model. Moreover, for inferenceon expenditure and price elasticities, the bootstrap method might be a useful tool inthe AIDS model.

Thus, AIDS is preferred to LA–AIDS, the Stone price index is not a good approxi-mation of the original price index. 3SLS, rather than SUR, should be utilized to obtainconsistent parameter estimates. Moreover, the bootstrap method should be applied forstatistical inference on expenditure and price elasticities.

Acknowledgments The authors are grateful to Professor Noriko Hashimoto (Kansai University) and twoanonymous referees for valuable comments and suggestions. K. Mizobuchi gratefully acknowledges theresearch support from Matsuyama University (i.e., sabbatical term from August, 2011 to July, 2012) andJapan Society for the Promotion of Science, Grant-in-Aid for Young Scientists (B) #22730217, with deepestappreciation, while H. Tanizaki acknowledges the research support from Japan Society for the Promotionof Science, Grant-in-Aid for Scientific Research (A) #23243038.

References

Alston JM, Foster KA, Green RD (1994) Estimating elasticities with the linear approximate almost idealdemand system: some Monte Carlo results. Rev Econ Stat 76(22):351–356

Buse A (1994) Evaluating the linearized almost ideal demand system. Am J Agr Econ 76(4):781–793Buse A (1998) Testing homogeneity in the linearized almost ideal demand system. Am J Agr Econ

80(1):208–220Buse A, Chan WH (2000) Invariance, price indices and estimation in almost ideal demand systems. Empir

Econ 25(3):519–539Deaton A, Muellbauer J (1980) An almost ideal demand system. Am Econ Rev 70:312–326Efron B (1979) Bootstrapping methods: another look at the Jackknife. Ann Stat 7:1–26Efron B, Tibshirani RJ (1993) An introduction to the bootstrap. Monographs on statistics and applied

probability, vol 57. Chapman & Hall, New York

123


Feenstra RC, Reinsdorf MB (2000) An exact price index for the almost ideal demand system. Econ Lett66:159–162

Filippini M (1995) Electricity demand by time of use: an application of the household AIDS model. EnergEcon 17(3):197–204

Fitzenberger B (1997) The moving blocks bootstrap and robust inference for linear least squares and quantileregressions. J Econom 82:235–287

Freedman DA (1981) Bootstrapping regression models. Ann Stat 9:1218–1228Freedman DA (1984) On bootstrapping stationary two-stage least-squares estimates in stationary linear

models. Ann Stat 12:827–842Freedman DA, Peters SC (1984) Bootstrapping a regression equation: some empirical results. J Am Stat

Assoc 79(385):97–106Green R, Rocke D, Hahn W (1987) Standard errors for elasticities: a comparison of bootstrap and asymptotic

standard errors. J Bus Econ Stat 5(1):145–149Green RD, Alston JM (1990) Elasticities in AIDS models. Am J Agr Econ 72(2):442–445Green RD, Alston JM (1991) Elasticities in AIDS models: a clarification and extension. Am J Agr Econ

73(4):874–875Hahn W (1994) Elasticities in AIDS models: comments. Am J Agr Econ 76:972–977Hashimoto N (2004) Changing Japanese society and behavior of household consumption: analysis with the

AIDS model. Kansai University Publishing, Osaka (in Japanese)Hayashi F (2000) Econometrics. Princeton University Press, PrincetonKrinsky I, Robb AL (1991) Three methods for calculating the statistical properties of elasticities: a com-

parison. Empir Econ 16(2):199–209Künsch HR (1989) The jackknife and the bootstrap for general stationary observations. Ann Stat 17:

1217–1241Liu RY, Singh K (1992) Moving blocks jackknife and bootstrap capture weak dependence. In: Lepage R,

Billiard L (eds) Exploring the limits of the bootstrap. Wiley, New York, pp 224–248MacKinnon JG (2006) Bootstrap methods in econometrics. Econ Record 82:S2–S18Mizobuchi K, Tanizaki H (2007) On estimation of income and price elasticities in the almost ideal demand

system using the bootstrap method. J Jpn Stat Soc 37(1):161–178 (in Japanese)Moschini G (1995) Units of measurement and the Stone index in demand system estimation. Am J Agr

Econ 77:63–68Moschini G (1998) The semiflexible almost ideal demand system. Eur Econ Rev 42:349–364Oladosu G (2003) An almost ideal demand system model of household vehicle fuel expenditure allocation

in the United State. Energ J 24(1):1–21Pashardes P (1993) Bias in estimating the almost ideal demand system with the Stone index approximation.

Econ J 103(419):908–915Rossi N (1988) Budget share demographic translation and the aggregate almost ideal demand system. Eur

Econ Rev 32(6):1301–1318Tiffin R, Aguiar M (1995) Bayesian estimation of an almost ideal demand system for fresh fruit in Portugal.

Eur Rev Agr Econ 22:469–480Tiffin R, Balcombe K (2005) Testing symmetry and homogeneity in the AIDS with cointegrated data using

fully-modified estimation and the bootstrap. J Agr Econ 56(2):253–270Xiao N, Zarnikau J, Damien P (2007) Testing functional forms in energy modeling: an application of the

Bayesian approach to U.S. electricity demand. Energy Econ 29(2):158–166

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