# Box-Cox Test: the theoretical justification and US-China empirical study

**ABSTRACT** In econometrics, the derivation of a theoretical model leads sometimes to two econometric models, which can be considered justified based on their respective approximation approaches. Hence, the decision of choosing one between the two hinges on applied econometric tools. In this paper, the authors develop a theoretical econometrics consumer maximization model to measure the flow of durables’ expenditures where depreciation is added to former classical econometrics model. The proposed model was formulated in both linear and logarithmic forms. Box-Cox tests were used to choose the most appropriate one among them. The proposed model was then applied to the historical data from the U.S. and China for a comparative study and the results discussed.

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**ABSTRACT:**For more than a half a decade, the fact that expenditure on durables can be well approximated by a random walk has remained a hidden puzzle, challenging almost any theory in which agents smooth the use of their wealth. This paper shows that once a nonparsimonious approach is used, or lower frequencies of the data are examined, the fact itself disappears; changes in expenditures on durables reveal a degree of reversion consistent with the permanent income hypothesis, although this reversion occurs at a rate significantly slower than what is suggested by a frictionless permanent income hypothesis model. Copyright 1990, the President and Fellows of Harvard College and the Massachusetts Institute of Technology.Quarterly Journal of Economics 02/1990; 105(3):727-43. · 5.92 Impact Factor - SourceAvailable from: D. A. DickeyJournal of the American Statistical Association 06/1979; 74. · 1.83 Impact Factor
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**ABSTRACT:**Consumer durables expenditures are normally assumed to be of linear form with additive error term. Hansen and Singleton (1983) derive the log linear form for consumption, but they do not employ a depreciation rate. In this article, we use a multiplicative error term to obtain a log linear AR(1) with a unit root as an approximate process that drives durables expenditure. The Box-Cox test rejects the linear form in favour of the log linear one. The Box-Jenkins model selection procedure and the augmented Dickey-Fuller test support an AR(1) in log linear form.Applied Economics Letters 01/2007; 14(9):643-646. · 0.23 Impact Factor

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* Corresponding author. Tel: 808-974-7462, Fax: 808-974-7685

E-mail addresses: tamr@hawaii.edu (T. B. Vu),

© 2010 Growing Science Ltd. All rights reserved.

doi: 10.5267/j.ijiec.2010.04.003

International Journal of Industrial Engineering Computations 2 (2011) 203–212

Contents lists available at GrowingScience

International Journal of Industrial Engineering Computations

homepage: www.GrowingScience.com/ijiec

Box-Cox Test: the theoretical justification and US-China empirical study

Tam Bang Vua* and Eric Iksoon Ima

aCollege of Business and Economics, University of Hawaii Hilo, 200 K. Kawili street, Hilo, HI, 96720, USA

A R T I C L E I N F O A B S T R A C T

Article history:

Received 1 April 2010

Received in revised form

17 July 2010

Accepted 17 July 2010

Available online 17 July 2010

Keywords:

Model specification

Approximations

Box-Cox test

US-China study

In econometrics, the derivation of a theoretical model leads sometimes to two econometric

models, which can be considered justified based on their respective approximation approaches.

Hence, the decision of choosing one between the two hinges on applied econometric tools. In

this paper, the authors develop a theoretical econometrics consumer maximization model to

measure the flow of durables’ expenditures where depreciation is added to former classical

econometrics model. The proposed model was formulated in both linear and logarithmic forms.

Box-Cox tests were used to choose the most appropriate one among them. The proposed model

was then applied to the historical data from the U.S. and China for a comparative study and the

results discussed.

© 2010 Growing Science Ltd. All rights reserved.

1. Introduction

Econometric analysis is an important tool in many fields of social sciences, business, and

engineering. The derivation of a theoretical model might lead to two econometrics models, which

can be considered justified based on their respective approximation approaches. Hence, the only way

an economist can choose one between the two is by using applied econometric tools. To demonstrate

this, the authors develop in this paper a theoretical model of consumer maximization problem. It is

shown that once the first order condition is written, two econometric models can be obtained

depending on whether a linear approximation or a log approximation is used. This justifies the use of

the Box-Cox test to distinguish the more appropriate model from the less appropriate one. The

traditional Keynesian theory posits that consumption depends on current disposable income. Keynes

(1936) shows that this relationship is a fairly stable function. He argues that a higher absolute level

of income will lead, as a rule, to a greater proportion of income being saved. However, empirical

evidence does not support his claims. Inspired by the failure of Keynes' theory, Friedman (1957)

develops a model of utility maximization that relates consumption to permanent income. He assumes

that the representative consumer knows his lifetime income with certainty, and is able to save and

borrow at will, subject to the constraint that any remaining debt is repaid at the end of his life. Under

these assumptions, the individual will divide his lifetime income equally among different periods of

his life. Hence, the consumer’s consumption in any period does not depend on his current disposable

income. Instead, it depends on his permanent income, which equals the average of his lifetime

income. Just as Friedman (1957), Modigliani (1971) also believes that the current income is not

crucial to consumption. But, he differs with Friedman by showing that the individual’s consumption

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depends on his lifetime wealth. The representative consumer is able to estimate his lifetime wealth

with certainty. Based on this estimation, the individual determines his average consumption in the

long run, and then sets current consumption to a fraction of that amount. That fraction equals the

discounted present value of his wealth. Integration of the theories of Friedman (1957) and

Modigliani (1971) leads to the “life cycle-permanent income hypothesis.” The hypothesis refutes

Keynesian argument that consumption depends on current disposable income alone.

Hall (1978) extends Friedman’s and Modigliani’s models to the case of ‘consumption under

uncertainty’. He assumes that the individual does not know his lifetime earnings with certainty.

Hence, he has to maximize the expectation of his discounted utility. The result shows that the

individual consumes the weighted average of his expected lifetime resources, which is equal to the

expectation of the present discounted value of his earnings plus his initial wealth. Hence,

consumption follows an autoregressive process of order one, an AR(1), in which the prediction of the

dependent variable in period t+1 depends on its own value in period t plus a stochastic disturbance.

Furthermore, if the explanatory variable has a unit root, then the process is non-stationary. This is

Hall’s famous random walk hypothesis. While testing the theory, he finds that the stock prices

significantly contribute to the prediction of future consumption. He argues that this happens because

some part of consumption takes time to adjust to a change in permanent income. He suggests a

modified version of the random walk hypothesis that allows a brief lag between permanent income

and consumption. The reason behind this suggestion might be the linear form of his model, instead of

the log form, which might fit US data better than its linear counterpart.

Mankiw (1982) expands Hall’s framework to deal with consumer expenditure on durable goods. He

also argues like Hall (1978) that the representative consumer maximizes the expectation of his

discounted utility subject to his budget constraint: the present value of the lifetime stock of durables

goods does not exceed initial wealth, plus the present value of lifetime labor earnings. Contrary to

Hall’s model, the durable goods depreciate over time. Hence, the stock of durable goods in any

period equals the expenditure on durable goods, plus the previous period durable goods net

depreciation. Mankiw (op. cit) assumes a quadratic utility function and an additive error term. He

argues that if Hall is correct, consumer durables expenditures follow an ARMA (1,1) process. The

moving average parameter is related to the rate of depreciation. The empirical results give Mankiw an

autoregressive process, an AR(1), instead. This implies that there is no difference between durable

and nondurable goods and services concerning the relationship between consumption expenditure and

its own lags. He is puzzled by the contradiction between his theoretical model of ARMA (1,1) and

the empirical result of AR(1) for the flow of consumer durables. It appears that depreciation rates

play no role in determining consumer expenditure on durable goods. Again, this might be due to the

linear form of his model, which he choses arbitrarily without any justification. Hansen and Singleton

(1983) were the first to derive a theoretical model in log form, supposedly for composite

consumption, but it does not involve the depreciation rate. This restricts the application of this model

basically to non-durables and services. With a utility function in constant elasticity form and without

rates of depreciation, the Euler equation for the growth rate of consumption is in multiplicative form.

From this equation, a multiplicative error term gives an econometric model in log linear form. They

conclude that consumption in log linear form follows a random walk. As they built the model mainly

to address the asset price problem, the robustness of their model was not rigorously tested with

empirical evidence. Also related to consumer durables is the Taylor series approximation of Mankiw

(1985) which obtains a model in log form relating consumer durables to consumer nondurables or

interest rates. Mankiw (op. cit) points out that log of consumer nondurables follows a random walk,

as shown theoretically by Hansen and Singleton (1983). This means that log of consumer durables

also follows a random walk. However, he does not carry out any empirical study to test his theory.

Campbell and Mankiw (1989) develop a new theory wherein they consider that the assumption of one

representative consumer in the conventional optimization problem is not appropriate for

consumption. Instead, they introduce two representative consumers: one consumes his permanent

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205

income and the other consumes his current income owing to borrowing constraints. What they get

from this assumption contrasts with the finding of Hansen and Singleton (1983), i.e., composite

consumption in log-linear form does not follow a random walk. It is not clear, however, why they

develop a theoretical model for a composite consumption function without depreciation rate.

Caballero (1990) develops a one-consumer linear model which is similar to that of Mankiw (1982).

Unlike Mankiw, he employs five annual or eight quarterly moving averages. Assuming that

consumers adjust their expenditures in durable goods slowly, he revises Mankiw’s model to allow for

significant delays in consuming the goods. He claims that consumer durables expenditures follow an

ARMA as in Mankiw’s theoretical model: although each of the five or eight moving averages is

insignificant, they together yield a significant moving average. Nevertheless, there is no econometric

theory which justifies the assertion that the combination of insignificant moving average effects in an

ARMA(1,5) or ARMA(1,8) is equivalent to a significant moving average effect in an ARMA(1,1)

model.

Ermini (1988) attributes the failure of Hall's theory to the sampling and temporal aggregation

problem (STA), because of which the econometrician observes only a sampled version of the actual

time series or its temporally aggregated version. Instead of providing a theoretical model, Ermini

(1988) uses these stylized facts in econometrics to suggest that the consumption expenditure should

follow an IMA (1,1) in linear form. Using quarterly data for the period 1947.1-1984.3 on the U.S.

nondurable and service expenditure per capita, seasonally adjusted, and in constant 1972 dollars, he

finds a positive and significant moving average. Ermini (1989) reinforces his IMA (1,1) model with

a theoretical model. He verifies it with the U.S. monthly data on nondurable and service expenditures

from 1959 to 1985 in constant 1982 dollars and finds a negative moving average. Since this is not

what Hall (1978) finds, it again brings to the fore the possible misspecification of the model. Winder

and Palm (1996) use a utility function in exponential form with nonseparable preferences between

leisure and consumption. This results in a theoretical model in linear form, which is slightly different

from that of Mankiw owing to the interaction between leisure and consumption. Assuming that the

representative consumer follows rational expectation, they find that consumer expenditure on durable

goods follows an ARIMA (1,1,1) process, which is very close to Mankiw's model. Nevertheless, they

do not explain the empirical result of an AR(1) in Mankiw (1982). Romer (2005) extends Mankiw’s

model by analyzing the stock of consumer durables and the flow of their expenditures, separately. He

concludes that, theoretically, the stock of consumer durables follows a random walk, but the flow of

their expenditures does not. His explanation is based on the involvement of the depreciation rates:

intuitively, some of the durable goods purchased in period t – 1 are still around in period t.

Therefore, to keep the expected consumption in period t at the same level, it is not necessary to buy

one unit of goods all over again. The representative consumer has to purchase only enough to replace

the fraction of the existing goods that depreciate. Thus, as of period t – 1, part of the change in

expenditure between t and t – 1 is predictable, i.e., the expenditure itself does not follow a random

walk. By doing this, Romer tries to justify Mankiw's theoretical model of ARMA(1,1). Just as

Winder and Palm (1996), he does not explain why Mankiw obtains an AR(1) empirically. Vu (2007)

shows that durables expenditures can be approximated as an AR(1) process in log form but does not

discuss the case of composite consumption. Owing partly to the controversial nature of the previous

results, the focus of recent literature has shifted to either multivariate models instead of univariate

ones, or panel data instead of time series. For example, Wachter (2006) develops a model of

consumption in relation to interest rates whereas Pounder (2009) uses a panel data set to show that

consumption depends less on permanent income in the data than on the theory. The authors’ response

to the inconclusive results of the univariate models is to develop a slightly different model for

composite consumption, where depreciation rate is also added. Once the theoretical model is derived,

it can be shown that two econometric models of linear or log form can be obtained. By testing these

two models using the model transformations developed by Box-Cox (1964), it is shown that the

results can be different for two particular countries: the U.S. and China. The disparity is then

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206

justified by pointing out the problems of using the Ljung-Box-Pierce tests for US data, as shown in

Box and Pierce (1970) and Ljung and Box (1978). Finally, the unit root test, which was introduced

by Dickey-Fuller (1979) and discussed in details in Hamilton (1994), is carried out to verify our

theoretical and econometric models.

The remainder of this paper is organized as follows: Section 1 presents the proposed theoretical and

econometrics models; Section 2 discusses the Box-Cox test; Section 3 presents a comparative

empirical study between the U.S. and China; Section 4 presents the summary and conclusions of this

work.

1.1 Theoretical and Econometric Models

A representative consumer who

()

00

0

t

=

⋅⋅

=−−+

t

ttt

Maximizes E UEeu K C

⎡

⎣

dt

ρ

∞

∫

−

=⎤

⎦

(1)

t

tttt

subject to A wKK rA

δ

(2)

t

tt

and K

is considered here. This is a combination of the Ramsey-Cass and Mankiw (1982) models. If the

depreciation rate is one, then one has to revert to the case of consumer nondurables and services as in

Hall (1978). Or, one can consider this an extended version of the one in Hansen and Singleton (1983)

and Lucas (1978), with depreciation rate added. Hence, this model is a generalization of a specific

model for consumption of either durables or nondurables and services. The definitions of the

variables are shown in Table 1.

Table 1

Variable Definitions

Notation Definition

KC

δ

⋅

≡ −+

(3)

Eou The expected lifetime utility at time zero

ut Instantaneous utility with the usual properties of u’ > 0 and u’’ <0

Kt The stock of durable goods

Ct The flow of durables expenditures

At The asset

wt The wage income

r The interest rate

ρ

The time preferences

δ

The depreciation rate

From (3), one has,

⋅

=+

t

tt

CKK

δ

.

(4)

Combining (2) and (4) yields

⋅

=−+

t

ttt

AwCrA

.

(5)

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T. B. Vu and E. I. Im / International Journal of Industrial Engineering Computations 2 (2011)

207

Thus, the current Hamiltonian is

(

0

ctt

H E u K C

=

⎡

⎣

The first order condition is

H

E u C

C

∂

)()

ttt

wC rA

λ+−+

⎤

⎦

.

(6)

()()

00

0''

c

tt

t

K

C

E u K

λ

∂∂

∂

=→==

,

(7)

c

t

H

A

∂

r

λ

λ

λλρρ

⋅

⋅

∂

=−+→−= −

(8)

and the tranversality condition implies that

()

lim'

tt

Au C e

→∞

= .

From (7), one has

()

ln

0

E e

λ=

.

Assuming that lnu’ is normally distributed with mean Eo lnu’ and variance v, as shown in Romer

(2008), one can write thus:

0

t

ρ−

(9)

t

U C

′

(10)

()

ln '/2

lnln '

2

"

u

'22

o

Euvt

o

oo

e

v

Eut

u C CvC

C

v

EE

C

λ

λ

λ

λ

θ

+

⋅⋅⋅

=

=+

⎛ ⎞

⎜ ⎟

⎝ ⎠

−= −−=−

(11)

Combining (8) and (11) yields

⋅

=+ −

2

o

C

C

v

Er

θρ

.

Hence, the first order condition for the maximization problem is as follows:

⋅

⎛⎞

=+ −= −

⎜⎟

⎝⎠

That is, θ is the agents’ consumption elasticity of marginal utility and 1

0

1

θ

"

u

,

2'

t

t

C

C

v u C

Er where

ρθ

.

(12)

φ

θ=

the agent’s consumption

elasticity of intertemporal substitution.