Download Production Smoothing in Developed Countries

Preliminary
Comments welcome
Production Smoothing in Developed Countries
Isamu Ginama
Mitsuhiro Odaki
Graduate School of Social Sciences
Hiroshima University
1-2-1 Kagamiyama, Higashi-Hiroshima
739-8525, Japan
Tel: +81 82 424 7267
Fax:+ 81 82 424 7267
E-mail: [email protected]
JEL :E23・E32
Abstract: This paper extends the method introduced by Krane and Braun(1991) and
Wang(2002) in the way that the unit root testing procedures form the basis of
constructing the bootstrap BCa confidence intervals for the ratio of the variance of
production to the variance of sales. Models to test for a unit root when there are
structural breaks in the linear time trend were converted to the equations that consist
of the deterministic and stochastic components. Bootstrap resamplings are made to
construct confidence intervals for the relative variance ratio of production and sales.
Keywords: Unit root, Structural breaks, Production smoothing, Inventory investment,
BCa confidence intervals.
1. Introduction
This paper investigates the macroeconomic data of G7 countries on the production,
sales and inventory investment to test the production smoothing hypothesis of
inventory investment. The analytical framework is an extension of the models of
Krane and Braun(1991)( Krane and Braun, hereafter ) and Wang(2002)( Wang,
hereafter ) in which they assumed that the U.S. and Taiwanese industrial data can
be converted to stationary series in the way that the industrial production and sales
1
series are regressed on the seasonal dummy variables and the linear time trend,
and estimated the variances of the production and sales as the sum of the variances
of the seasonal dummy variables and the variance of the residuals. In so doing,
Wang applied the bootstrap method to construct the confidence intervals( BCa ) of
the ratio of the variances of production and sales. This method is adopted in the
current paper. The stationarity of the residuals, however, was not formally tested in
those analyses. In applying their methodological framework to the macroeconomic
data such as the GDP, inventory investment, and aggregate sales, it is crucial to test
whether these variables are trend stationary or not. Explicit unit root tests on these
variables would involve the trend terms with or without structural breaks. In this
paper, the unit root tests take into account the possibility of endogenous structural
break(s).
G7 countries were chosen as the group of developed countries which were
characterized
by
the
production
literature(
West(1988),
counter
smoothing
Wilkinson(1989,
1991),
property
Fukuda
in
the
and
Teruyama(1988)(Fukuda and Teruyama, hereafter)).
The paper is organized as follows. The next section describes the data analyzed.
The third section represents the models of the unit root tests used in this paper. The
econometric method for constructing the BCa intervals of the ratio of the variance of
production to the variance of sales is explained in section four. Section five shows a
sketch of the proof that the presence of the deterministic seasonal dummy variables
does not affect the limiting distribution for the test parameter in the equations to
test for a unit root . The results of the unit root tests on the production and sales of
the G7 countries are represented. Estimated BCa confidence intervals and their
interpretations are shown in section six. Concluding remarks follow.
2. Data
Real quarterly time series of the GDP and inventory investment were extracted
from the database software Datastream( Thomson Inc.) 1 . The importance of
seasonal variations in the analysis of inventory fluctuations was emphasized in the
literature(Ghali(1987), Krane and Braun, and Wang, for example.). The data on
Japan, UK, Italy, and Germany are not seasonally adjusted, and the seasonal
Datastream provides the Japanese macroeconomic data for the period after 1980
which are constructed on the 1993 SNA basis. Japanese data, however, were available
for the longer period on the 1968 SNA basis on the data base software Reuters Ecowin.
Japanese data used in this paper is on the 1968 SNA basis corresponding to the longer
sample period.
1
2
variations form a part of the variability of production and sales for these countries.
The data for three other countries are seasonally adjusted due to constraint on the
availability of data that are not seasonally adjusted. Sales is defined as the GDP
minus the inventory investment as in the literature. GDP and sales are transformed
to logarithmic terms before testing for a unit root. Granger and White(2011) defines
some concepts of trend in time series data. The concept of local trends there means
trends “ only for some finite time interval “ (Definition 9 ). This definition of trends
seems to be applicable to the concept of deterministic trends in the current paper.
The sample periods to be analyzed are somewhat different across the countries
depending on whether significant results can be obtained in testing the variables for
stationarity around the deterministic trend with and without structural breaks.
As for Germany, the first differences of production and sales were tested for a unit
root in the augmented Dickey Fuller ( ADF, here after ) equations with seasonal
dummy variables after attempting to analyze for the possibility of trend stationary
results for various sample periods of German data. German data turned out to be
trend stationary having no structural break in this specification. When Germany
was referred to as the country of production counter smoothing in the literature, it
used to be West Germany. We extracted the annual West German data from
International Financial Statistics for the period from 1957 through 19912.
Canadian data were partitioned into two sample periods which are not
successive in time. The thirteen years of time interval between the two observation
periods were not included in our analysis because of the outcomes that the unit root
tests for the sample periods including this interval did not indicate any significant
results.
3.
Method
The following three types of models are applied to test the production and sales time
series for a unit root in this paper.
(i)
The ADF tests with the linear time trend, that is
Δyt = μ̂ + β̂t + 𝛼̂1 yt−1 + ∑ki=1 β̂i Δyt−i + et
(1)
̂ i s are the parameters, and et is the residual. The value of k is
, where μ,
̂ β̂, 𝛼̂1 and β′
Attempts to construct the consistent data on West Germany on the quarterly basis
were not successful due to incompatible numbers for inventory investment represented
on different volumes of the data set.
2
3
determined by the Akaike information criterion. ∆ stands for the first difference
operator.
(ii)
The Zivot and Andrews tests(1992) with a single endogenous structural
break for the break type C specification is
yt = μ̂ + θ̂DUt + β̂t + γ̂DTt + α
̂yt−1 + ∑ki=1 β̂i Δ yt−i + et
(2)
,where θ̂ and γ̂ are parameters, and k is determined in the same way as in the ADF
test3. The variables DUt and DTt are defined as follows:
Let the break point be denoted as TB , then DUt = 1 for t ≥ TB + 1, and DUt =
0 otherwise.
DTt is defined in the way that DTt = t − TB
for t ≥ TB + 1 , and DTt = 0
otherwise.
(iii)
The Lumsdaine and Papell tests(1997) with two endogenous structural
breaks for the break type BB is
yt = μ̂ + θ̂DT1t + β̂t + γ̂DT2t + α
̂yt−1 + ∑ki=1 β̂i Δ yt−i + et
(3)
DTjt is defined in the way that DTjt = t − TBj+1 for t ≥ TBj + 1 , and DTjt = 0 otherwise,
where j=1,2.
4. Bootstrapping
Following Wang, BCa confidence intervals of the variance ratio, that is, the ratio of
the variance of production to the variance of sales, are constructed by bootstrapping the
deterministic part of equation (1) , (2), and (3). Unlike his presumption that the
production and sales time series can be expressed as the sum of the deterministic terms
that consists of seasonal dummy variables and the linear time trend, and the stationary
random residuals, equations (1), (2), and (3) for the unit root tests need to be converted
to the form specified in Krane and Braun and Wang.
(i)
Conversion of equation (1)
Let α
̂ =1+ α
̂1, then equation (1) can be written as
The maximum lag was eight in Zivot and Andrews in which the annual data were
used. When the quarterly data are used in this paper, the maximum value of k was set
equal to 12.
3
4
yt = μ̂ + β̂t + α
̂yt−1 + ∑ki=1 β̂i Δyt−i + et
(1)’
which can further be written as
yt = μ̂′ + β̂′ t + OTt′
(1)′′
̂
1
μ
1
where β̂′ = β̂ (1−α) , μ̂′ = 1−α̂ − β̂̂
α ((1−α̂)2 ) , and
OTt′ =
1
̂L
1−α
(∑ki=1 β̂i Δyt−i + et )4 . Using
equation (1)’’ for the GDP and sales at the same time, bootstrap resamplings on yt and t
are made to run a series of OLS regressions with the bootstrap samples on equation (1)’’.
The variance of OT′t term gives the variance of production for GDP equation, and gives
the variance of sales for sales equation. Based on the replications of the ratio of those
variances, BCa confidence intervals can be constructed as in Efron(1987), and Efron
and Tibshirani(1998).
(ii) Conversion of equation (2)
1
1
In rewriting equation (2), the terms (1−α̂L) DUt and (1−α̂L) DTt must be defined. Let
1
1
DUCt= (1−α̂L) DUt and DTCt= (1−α̂L) DTt , then it is straightforward to demonstrate that
̂i
1−α
DUCt = (
̂
1−α
) for t=（TB + i）, i=1,2, .............. , N, and DUCt =0 for t < TB , and that
DTC(TB +i) = ∑ij=1[DTi − (j − 1)]α
̂(j−1) for i=1,2,...... , N, and DTCt = 0 for t ≤ TB , where N
stands for the number of observations. Using these terms, equation (2) can be rewritten
as
yt = μ̂′ + θ̂DUCt + β̂′ t + γ̂DTCt + OTt′′
where μ̂′ =
̂
μ
̂
1−α
− β̂α
̂(
1
̂ )2
(1−α
(2)′
1
) 、 β̂′ = β̂ (1−α̂) 、 and
OT′′t = (
1
) (∑ki=1 β̂i Δ yt−i + et ) .
̂L
1−α
Bootstrap resamplings are made on yt , DUCt , t, and DTCt to run a series of OLS
regressions with these variables on equation (2)’ for the GDP and sales series to
calculate the variances of 𝑂𝑇𝑡′′ which give the variances of production and sales from
which the BCa confidence intervals on the variance ratio are constructed.
(ii)
Conversion of equation (3)
Equation (3) can be written as
4
L stands for the lag operator.
5
yt = μ̂′ + θ̂DTC1t + β̂′ t + γ̂DTC2t + OTt′′
(3)′
,where DTCj(TBj+i) = ∑ih=1[DTji − (h − 1)]α
̂(h−1) for i=1,2,...... , N −𝑇𝐵 , and DTCjt = 0
for t ≤ TBj, where N stands for the number of observations, and j=1,2.
(iii)
The seasonal dummy variables
Equatoin (3) with the seasonal dummy variables is written as
k
yt = μ̂ + φ1 d1 + φ2 d2 + φ3 d3 + θ̂DT1t + β̂t + γ̂DT2t + α
̂yt−1 + ∑ β̂i Δ yt−i
i=1
+ et
(4)
,which can be converted to
yt = μ̂′ + φ1 dc1 + φ2 dc2 + φ3 dc3 + θ̂DTC1t + β̂′ t + γ̂DTC2t
+ OTt′′
(4)′
See Table 1 for the definitions of 𝑑𝑐𝑖 , i = 1, 2 , 3. The variance of 𝑦𝑡 (V(𝑦𝑡 )) in this
equation is given as V(𝑦𝑡 ) =
Wang, where 𝜑0 = 𝜇̂ ′ , 𝜑̅ =
1
∑3 (𝜑𝑖 − 𝜑̅)2 + 𝑉(𝑂𝑇𝑡′′ ) as in Krane and Braun,
4 𝑖=0
1 3
∑ 𝜑 , 𝑎𝑛𝑑 𝑉(𝑂𝑇𝑡′′ ) stands for the variance of 𝑂𝑇𝑡′′.
4 𝑖=0 𝑖
and
Applying equations (1)’, (2)’, (3)’, and (4)’, or an appropriate combination of these
equations to production and sales series, bootstrap resamplings were made over the
deterministic parts of the converted equations simultaneously to construct the BCa
confidence intervals for the relative variance ratio of production and sales.
Table 1 about here
5. Unit root tests
Dickey, Bell, and Miller(1986) derived the proposition that the limiting
distribution for 𝛼̂ in equations (1) and (1)’ is not affected by the presence of the
deterministic seasonal dummy variables for the case of the ADF test. We represent a
6
sketch of the proof on the similar proposition that the inclusion of the deterministic
seasonal dummy variables in the testing equation with a single endogenous break does
not affect the limiting distribution of the test statistic, and conjecture that the same
proposition holds for the case of two endogenous breaks
Simply incorporating the ordinary dummy variables into Model (A) of
Perron(1989)(Perron, hereafter ) or essentially (A) of Zivot and Andrews(1992),
𝑦𝑡 = 𝜇1 + (𝜇2 − 𝜇1 )𝐷𝑈𝑡 + 𝜃𝑡 + ∑3𝑖=1 𝛾𝑖 𝑑𝑖𝑡 + 𝛼𝑦𝑡−1 + 𝜖𝑡 , ∀𝑡 ≥ 1 ,
(5 − 1)
Where 𝐷𝑈𝑡 follows the definition of Zivot and Andrews, as 𝐷𝑈𝑡 = 1 if t ≥ 𝑇𝐵 + 1 , 0
otherwise, 𝑑𝑖𝑡 = 1 for any t expressed as t=4s+i with a nonnegative integer s, 0
otherwise, i=1,2,3, 𝑦0 = 𝑂𝑝 (1)
or O(1) , and {𝜖𝑡 } is a sequence of unobservable
random variables which are iid with E𝜖𝑡 = 0 , 𝐸𝜖𝑡2 = 𝜎 2 > 0 and finite fourth and
eighth order cumulants. We also use the regression equation
∆𝑦𝑡 = 𝜇̂ 1 + (𝜇̂ 2 − 𝜇̂ 1 )𝐷𝑈𝑡 + 𝜃̂ 𝑡 + ∑3𝑖=1 𝛾̂𝑖 𝑑𝑖𝑡 + ( 𝛼̂ − 1 )𝑦𝑡−1 + 𝑒̂𝑡 , 𝑡 = 2, … . , 𝑇,
(5 − 2)
To obtain the test statistic, and in connection with (2), let 𝑙 𝑇 , 𝑙 𝑇𝐵 , 𝜏𝑇 and 𝑙𝑖;𝑇 , i=1,2,3 be
the (T−1)-dimentional vectors composed of 1, 𝐷𝑈𝑡 , 𝑡 𝑎𝑛𝑑 𝑑𝑖𝑡 , respectively, as
𝑙 𝑇 = (1, … … , 1)′ , 𝜏𝑇 = (2,3,…. , T)′,
𝑙 𝑇𝐵 = (0, … . . ,0,1, … . . ,1)′
provided that the (𝑇𝐵 + 1)𝑡ℎ component is the first one being 1,
𝑙1;𝑇 = ( 1,0,0,0,1,0,0,0, … … … , )′ ,
𝑙2;𝑇 = (0,1,0,0,0,1, ,0,0,0, … … , )′ ,
𝑙3;𝑇 = (0,0,1,0,0,0,1,0,0,0, … … , )′ ,
with the breakpoint 𝑇𝐵 defined in the above-mentioned articles. Also let λ =
𝑙𝑖𝑚 𝑇→∞
𝑇𝐵⁄
𝑇 and further define
Y= ( 𝑦2 , … . , 𝑦𝑇 )′ ,
𝑌−1 = ( 𝑦1 , … , 𝑦𝑇−1 )′ , 𝐸̅ = (𝜖2 , … . , 𝜖 𝑇 )′ ,
2
𝑆 ̅ = ( ∑ 𝜖ℎ ,
ℎ=1
3
𝑇
′
∑ 𝜖ℎ , … , ∑ 𝜖ℎ ) ,
ℎ=1
ℎ=1
2
𝑇−1
̅ ( 𝜖1 , ∑ 𝜖ℎ , … ., ∑ 𝜖ℎ )′ ,
𝑆−1
ℎ=1
ℎ=1
𝑋1 = [ 𝑙 𝑇 , 𝑙 𝑇𝐵 , 𝜏𝑇 ], 𝑋2 = [ 𝑙1;𝑇 , 𝑙2;𝑇 , 𝑙3;𝑇 ] , 𝑋 = [ 𝑋1 , 𝑋2 ], 𝑍1 = [ 𝑌−1 , 𝑋1 ], 𝑍 = [ 𝑌−1 , 𝑋 ],
7
𝑀1𝑇 = 𝐼𝑇−1 − 𝑙 𝑇 (𝑙′𝑇 𝑙 𝑇 )−1 𝑙 ′𝑇 , 𝑀𝑋𝑖 = 𝐼𝑇−1 − 𝑋𝑖 (𝑋𝑖′ 𝑋𝑖 )−1𝑋𝑖′ ,
𝑖 = 1,2,
𝑀𝑋 = 𝐼𝑇−1 − 𝑋(𝑋 ′ 𝑋)′ 𝑋 ′ ,
𝑀𝑍1 = 𝐼𝑇−1 − 𝑍1 (𝑍1′ 𝑍1 )−1 𝑍 ′ , 𝑀𝑍 = 𝐼𝑇−1 − 𝑍(𝑍 ′ 𝑍)−1 𝑍 ′ ,
𝐷𝑇;𝑋 = 𝑑𝑖𝑎𝑔 {𝑇
1⁄
2,
𝑇
1⁄
2,
𝑇
3⁄
2,
𝑇
1⁄
2
, 𝑇
1⁄
2,
𝑇
1⁄
2 },
𝑇
0
𝐷𝑇;𝑍 = [0 𝐷 ] .
𝑇;𝑋
Letting 𝑇4 be the largest integer that is either equal to or less than
(𝑇 − 1)⁄
4 and
4
4 −1
{𝑖 + 4(𝑡 − 1)} = 𝑖𝑇4 + 4 ∑𝑇𝑡=1
∑𝑇𝑡=1
𝑡 = 𝑖 𝑇4 + 2(𝑇4 − 1)( 𝑇4 − 2)
2
2
𝑇
⁄8 + 𝑂(𝑇) , 𝑖 = 1,2,3,
= 2𝑇4 + 𝑂(𝑇) =
noting that
It is not difficult to check that
1
1−𝜆
1−𝜆
1
1−𝜆
2
2
1−𝜆
1
4
−1
−1
𝐷𝑇;𝑋
𝑋 ′ 𝑋 𝐷𝑇;𝑋
=
1
1
1
1
2
4
4
4
1−𝜆
1−𝜆
4
1−𝜆
1
4
1
4
1−𝜆
4
4
1−𝜆
1−𝜆
1−𝜆
2
1
4
1
4
1
4
1
3
1
8
1
8
8
8
1
4
0
1
0
8
1
4
0
8
0
0
0
1
4
[
+ O(𝑇
−1
]
(5-3)
)
The test statistic 𝑡𝛼 (i.e., the t-statistic of the regression coefficient ( 𝛼̂-1) in (5-2)) based
on (5-2) is now expressed as
1
1
′
′
𝑡𝛼 (𝜆) = ( 𝑇 −1 𝑌 ′ 𝑀𝑍 𝑌 )−2 (𝑌−1
𝑀𝑋 𝑌−1 )−2 {𝑌−1
𝑀𝑋 ( 𝑌 − 𝑌−1 )}
Also let 𝑊𝐴 (𝜆, 𝑟) denote a stochastic process on [0 , 1] that is the projection residuals in
𝐿2 [0 , 1 ] of a Brownian motion projected onto the subspace generated by 1, du(λ, r), r
with du(λ, r) such that du(λ, r) = 1 if r ≥ λ and 0 otherwise, as done by (9) and (10) in
Zivot and Andrews. Then:
8
Lemma1:
Suppose that 𝑦𝑡 is generated by (5-1). If the null is formulated as 𝜇1 =
𝜇2 , 𝜃 = 𝛾1 = 𝛾2 = 𝛾3 = 0 and α = 1 , and (5-2) is the regression for the unit root test,
the limiting distribution of 𝑡𝛼 (𝜆) under the null is expressed as
1
1
( ∫0 𝑊𝐴 (𝜆, 𝑟 )2 𝑑𝑟 )−1/2 (∫0 𝑊𝐴 (𝜆, 𝑟)𝑑𝑊𝐴 (𝜆, 𝑟))
Prrof: First, note that for any integers i ,j such that 4 > j > i > 0 and s=1, ….. 𝑇4 ,
∑𝑗𝑛=𝑖+1 𝜖4(𝑠
−1)+𝑛
∑𝑗𝑛=𝑖+1 𝜖4(𝑠′ −1)+𝑛
𝑗
is the sum of (j−i) pieces of 𝜖4(𝑠−1)+𝑛 and that ∑𝑛=𝑖+1 𝜖4(𝑠−1)+𝑛 and
are independent as s ≠ s ′ . It is then shown by these matters that for
i , j given above,
𝑇
4(𝑠−1)+𝑗
4
∑ℎ=1
𝑇 −3/2 ∑𝑠=1
3
𝑇4
𝑇4
𝑗
∑4(𝑠−1)+𝑖
𝜖ℎ − 𝑇 −2 ∑𝑠=1
𝜖ℎ = 𝑇 −3/2 ∑𝑠=1
(∑𝑛=𝑖+1 𝜖4(𝑠−1)+𝑛 ) = 𝑂𝑃 (𝑇 −1 ),
ℎ=1
which implies that
𝑇
4(𝑠−1)+𝑖
4
∑ℎ=1
𝑇 −3/2 ∑𝑠=1
1
−1
𝜖ℎ = (4) 𝑇 −3/2 ∑𝑇𝑡=2 ∑𝑡−1
ℎ=1 𝜖ℎ + 𝑂𝑃 (𝑇 ). 𝑖 = 1,2,3, (𝐴. 1)
Noting that
3
−3/2 ∑𝑇4 ∑4(𝑠−1)+𝑖
𝑇 −3/2 ∑𝑇𝑡=2 ∑𝑡−1
𝜖ℎ
ℎ=1 𝜖ℎ = ∑𝑖=1 𝑇
𝑠=1 ℎ=1
.
Next, note that the (T−1) observations of (5-1) under the null are written as
Y= 𝑌−1 + 𝐸̅ + 𝜇1 𝑙 𝑇 = 𝑌−1 + 𝑋𝑏 + 𝐸̅ = 𝑍(1, 𝑏′ )′ + 𝐸̅ ,
Where b is the coefficient vector difined suitably, and this expression leads to
1
′
′
𝑡𝛼 (𝜆) = ( 𝑇 −1 𝐸̅ ′ 𝑀𝑍 𝐸̅ )−1/2 ( 𝑇 −2 𝑌−1
𝑀𝑋 𝑌−1 )−2 (𝑇 −1 𝑌−1
𝑀𝑋 𝐸̅ ) ,
(𝐴. 2)
Noting that
3
3
′
′
′
𝑇 −2 𝑌−1
𝑀𝑋 𝑌−1 = 𝑇 −2 𝑌−1
𝑀𝑋1 𝑌−1 − ( 𝑇 −2 𝑌−1
𝑀𝑋1 𝑋2 ) (𝑇 −1 𝑋2′ 𝑀𝑋1 𝑋2)−1 ( 𝑇 −2 𝑋2′ 𝑀𝑋1 𝑌−1 ),
(A.3)
9
3
1
′
′
′
𝑇 −1 𝑌−1
𝑀𝑋 𝐸̅ = 𝑇 −1 𝑌−1
𝑀𝑋1 𝐸̅ − ( 𝑇 −2 𝑌−1
𝑀𝑋1 𝑋2 ) (𝑇 −1 𝑋2′ 𝑀𝑋1 𝑋2 )−1 ( 𝑇 −2 𝑋2′ 𝑀𝑋1 𝐸̅)
In view of (5-3) and
𝑦𝑡−1 = ∑𝑡−1
ℎ=1 𝜖ℎ + (𝑡 − 1)𝜇1 + 𝑦0
Owing to (5-1) under the null and noting that
𝑀𝑋1 = 𝑀1𝑇 − 𝑀1𝑇 𝜏(𝜏 ′ 𝑀1𝑇 𝜏)−1 𝜏′𝑀1𝑇 ,
It is trivial to see that
(𝑇 −1 𝑋2′ 𝑀𝑋1 𝑋2 )−1 = O(1),
(A.4)
and that
3
3
3
′
̅ ′ 𝑀𝑋 𝑋2 = 𝑇 −2 ∑𝑇𝑡=2 𝑠̅𝑡−1 (𝑑1𝑡 , 𝑑2𝑡 , 𝑑3𝑡 )
𝑇 −2 𝑌−1
𝑀𝑋1 𝑋2 = 𝑇 −2 𝑆−1
1
3
− ( 𝑇 −2 ∑
𝑇
𝑠̅𝑡−1 ) {(𝑇 − 1)−1 ∑
𝑡=2
𝑇
( 𝑑1𝑡 , 𝑑2𝑡 , 𝑑3𝑡 )}
𝑡−2
5
̅ ′ 𝑀𝑙 𝜏𝑇 ) ( 𝑇 −3𝜏𝑇′ 𝑀𝑙 𝜏𝑇 )−1 [𝑇 −2 ∑𝑇𝑡=2 𝑡(𝑑1𝑡 , 𝑑2𝑡 , 𝑑3𝑡 ) − 𝑇 −2 (∑𝑇𝑡=2 𝑡) {(𝑇 −
− (𝑇 −2 𝑆−1
𝑇
𝑇
(A.5)
1)−1 ∑𝑇𝑡=2(𝑑1𝑡 , 𝑑2𝑡 , 𝑑3𝑡 )}], ,
where 𝑠̅𝑡−1 = ∑𝑡−1
ℎ=1 𝜖𝑡 . Since
𝑇4
=
𝑇
1
4
+ 𝑂(𝑇 −1 ) and 𝑑𝑖𝑡 = 1 for any t expressed as
t=4s+i with a nonnegative integer s, 0 otherwise, by definition, we obtain
𝑇 −3/2 ∑𝑇𝑡=2 𝑠̅𝑡−1 ( 𝑑1𝑡 , 𝑑2𝑡 , 𝑑3𝑡 )
𝑇4
= 𝑇 −3/2 ∑𝑠=1
( ∑4(𝑠−1)+1
𝜖ℎ , ∑4(𝑠−1)+2
𝜖ℎ , ∑4(𝑠−1)+3
𝜖ℎ ) + 𝑂𝑃 (𝑇 −1 ),
ℎ=1
ℎ=1
ℎ=1
( 𝑇 −3/2 ∑𝑇𝑡=2 𝑠̅𝑡−1 ) {(𝑇 − 1)−1 ∑𝑇𝑡=2(𝑑1𝑡 , 𝑑2𝑡 , 𝑑3𝑡 )}
= ( 𝑇 −3/2 ∑
𝑇
𝑡−1
∑
𝑡=2
1 1 1
𝜖ℎ ) ( , , ) + 𝑂𝑃 (𝑇 −1) ,
4 4 4
ℎ=1
1 1 1
𝑇 −2 ∑𝑇𝑡=2 𝑡( 𝑑1𝑡 , 𝑑2𝑡 , 𝑑3𝑡 ) = (8 , 8 , 8) + 𝑂(𝑇 −1 )
1 1 1
𝑇 −2 ( ∑𝑇𝑡=2 𝑡){(𝑇 − 1)−1 ∑𝑇𝑡=2( 𝑑1𝑡 , 𝑑2𝑡 , 𝑑3𝑡 )} = (8 , 8 , 8) + 𝑂(𝑇 −1 ) .
Using (A.6) as well as (A.1) , (A.5) is converted into
10
(A.6)
(A.7)
′
𝑇 −3/2 𝑌−1
𝑀𝑋1 𝑋2 = 𝑂𝑃 ( 𝑇 −1 ) .
It is then derived by (A.3), (A.4) and (A.7) that
′
′
𝑇 −2 𝑌−1
𝑀𝑋 𝑌−1 = 𝑇 −2 𝑌−1
𝑀𝑋1 𝑌−1 + 𝑂𝑃 ( 𝑇 −2 ) ,
(A.8)
′
′
𝑇 −1 𝑌−1
𝑀𝑋 𝐸̅ = 𝑇 −1 𝑌−1
𝑀𝑋1 𝐸̅ + 𝑂𝑃 (𝑇 −1 ) .
It also follows from the well-known asymptotic results on the I(0) and I(1) series that
1
𝑇 −1 𝐸̅′ 𝑀𝑍 𝐸̅ = 𝑇 −1 𝐸̅ ′ 𝐸̅ + 𝑂𝑃 (𝑇 −1 ) = 𝜎 2 + 𝑂𝑃 ( 𝑇 −2 ),
1
𝑇 −1 𝐸̅′ 𝑀𝑍1 𝐸̅ = 𝑇 −1 𝐸̅ ′ 𝐸̅ + 𝑂𝑃 (𝑇 −1 ) = 𝜎 2 + 𝑂𝑃 ( 𝑇 −2 ) .
(A.9)
Putting (A.2), (A.8) and (A.9) together shows that the limiting distribution of 𝑡𝛼 (𝜆) is
equal to that of
1
1
′
′
( 𝑇 −1𝐸̅ ′ 𝑀𝑍1 𝐸̅ )−2 ( 𝑇 −2 𝑌−1
𝑀𝑋1 𝑌−1 )−2 ( 𝑇 −1 𝑌−1
𝑀𝑋1 𝐸̅ ).
We can easily find that this quantity is the one derived from the test statistic for Model
(A) of Perron or Zivot and Andrews, noting that the seasonal dummies are excluded
from the list of the regressors in construction. Thus, it is established by results or
arguments in the above-mentioned articles that the limiting distribution of the above
quantity is the one claimed by the lemma.
Tables 2, 3, and 3 continued represent the results of unit root tests on the production
and sales for G7 countries. The US sample period was divided into two parts to be able
to apply the models to test for a unit root that are used in the current paper. In each
time period, the US data turned out to be trend stationary with two structural breaks.
The confidence intervals for the variance ratio, therefore, will be constructed in each
time period for the US. The results on Germany in Table 3 are for the period after
unification of West abd East Germany. GDP and sales for this period were trend
stationary in the first differences. Table 3 continued carries the results on West
Germany using the annual data in first differences. Both GDP and sales are identified
as trend stationary series in the ADF tests. Table 3 continued represents the results on
Canadian data for two different observation periods based on which the BCa confidence
intervals will be represented in the next section.
11
Tables 2 , 3 and 3 continued
about here
6. BCa confidence intervals
Tables 4 to 12 represent the BCa intervals of the 90 percent, 95 percent, and 99 percent
confidence levels for the seven countries of Japan, US, France, UK, Italy, Germany, and
Canada. Japan, the US, France, and Italy clearly show the production counter
smoothing property of inventory investment because the widest 99 percent confidence
intervals are located in the region toward the right( greater ) side of the value of one. In
case that the 90 percent confidence interval is applied, it is possible to add the UK to the
group of countries where production is more volatile than sales. The production counter
smoothing characteristic of Germany indicated in the literature are related to the era of
West Germany. The results of testing the West German data for a unit root were
reported in Table 3 continued. The BCa confidence intervals were calculated based on
this model, and were represented in Table 10 where production is interpreted as being
more volatile than sales as in the literature5. The results in Table 9, however, show that
production is smoother than sales. There seems to be a possibility that the event of
unification of West and East Germany gave rise to a change in the relative volatility of
production and sales.
The probability that the calculation of BCa confidence intervals stops in the
middle of the prescribed number of repetitions is not negligible when a break point of
the time trend is located near the end of the sample period because the resampling
process may then involves picking up only zeros for the dummy variable that represents
a structural break. The second break point of the Canadian production series was found
to be in the third period from the end of the sample period( Table 3 ). The BCa
confidence intervals for Canada using this model are not reported due to the occasion
that a series of OLS routines was suspended within the small number of repetitions.
Due to this difficulty, Canadian data were analyzed for two other observation periods,
and the results obtained for these cases are represented in Table 3 continued. Using the
data in these time periods, Table 11 and 12 represent that the estimated BCa confidence
intervals for Canada are shown to be located on the both sides of the value of one
indicating that the test on the production smoothing theory is indeterminate. The point
Germany was one of a group of developed countries where production was more
volatile than sales in the literature.
5
12
estimate of the relative variance in Table 11 which is represented by the value of 𝜃̂ is
greater than one. The relative variance of Canada documented in the literature
corresponds to this case. Interval estimates on the relative variance, however, are
interpreted not to be conclusive. The point estimate of the relative variance for more
recent sample period used in Table 12 took the value that is less than one. These results
on Canadian data are against the implications for the numerical estimates on the
relative variances of production and sales in the literaute. Germany which used to be a
country where production was more volatile than sales before unification seems
currently to be a country where the production smoothing hypothesis empirically holds.
Tables 4 to 12 about here
The 99 percent confidence intervals which are the widest among the estimated
three intervals are located in the area to the right of one for Japan, the US, France, Italy,
and West Germany indicating that the data for these countries firmly support the
production counter smoothing property of inventory fluctuations . The characteristic of
Canadian inventory fluctuations is indeterminate with respect to whether production is
more volatile than sales or not throughout the sample periods used in the current paper.
Concluding remarks
This paper extended the method introduced by Krane and Braun, and Wang in the
way that the unit root testing procedures form the basis of constructing the bootstrap
BCa confidence intervals. Models to test for a unit root with and without the structural
break(s) in a linear time trend were converted to the equations that consist of the
deterministic and stochastic components.
The information on the confidence interval of the variance ratio makes it possible to
interpret the results of the tests on the production smoothing hypothesis of inventory
investment in terms of the relative volatility of production and sales in light of
statistical variations of the sample data. A stylized fact 6 for the hypothesis that
mentions that production is more volatile than sales explains inventory fluctuations of
developed countries better than of developing countries according to Fukuda and
Teruyama. Based on the interval estimation method in the current paper, Japan, the US,
France, the UK, and Italy seem to have more volatile production than sales. It seems to
6
Blanchard(1983), Blinder(1986), Holt et al(1960), and Kahn(1987).
13
be desirable to empirically investigate whether the alleged international duality in the
smoothing property of inventory investment remains to be a remarkable feature in an
analysis which takes into account statistical variations of the sample data. The
analytical method represented in this paper, therefore, needs to be implemented for the
data on other countries, especially for the data on a group of developing countries.
14
References
Blanchard, O. J. (1983), “ The Production and Inventory Behavior of the American
automobile Industry,” Journal of political Economy, 91, 365-400.
Blinder, A. S. (1986), “ Can the Production Smoothing Model of Inventory Behavior be
Saved ?,” Quarterly Journal of Economics, 101, 431-454.
Dickey, D. and Fuller, W.(1979), “Distribution of the Estimates for Autoregressive Time
Series with a Unit Root”, Journal of the American Statistical Association 74, 427-31.
Dickey, D., Bell, W., and Miller, R.(1986), “ Unit roots, in Time Series Models: Tests and
Implications.” American Statistician 40, 12-26.
Dolado, J., Jenkinson, T., and Simon, S.(1990), “ Cointegration and unit roots”, Journal
of Economic Surveys”, 4, 249-273.
Efron, B. (1987), “ Better Bootstrap Confidence Intervals”, Journal of the American
Statistical Association, 82, 171-200.
Efron, B. and Tibshirani, R. (1998,), An Introduction to the Bootstrap, Chapman & Hall/
CRC.
Fukuda, S. and Teruyama (1988), “ Some International Evidence on Inventory
Fluctuations,” Economics Letters, 28, 225-230.
Ghali, M. A. (1987), “ Seasonality, Aggregation and the testing of the Production
Smoothing hypothesis.” American Economic Review, 77, 464-469.
Ginama, I., Odaki, M., and Wang, H(2009)., “An International Study on Inventory
Fluctuations”, Discussion Paper series, No. 2009-2.
Ginama, I.(2010) “An Approach toward the Production Smoothing Phenomenon “(in
Japanese), Oita University Keizai Ronbunshu, vol. 62, no. 3・4.
Ginama, I. and Yamanouchi, K.(2011), “An Analysis of Inventory Fluctuations”( in
Japanese), Hiroshima University Keizaigakukenkyu, vol. 28.
Granger, C. and White, H.(2011), “Consideration of Trends in Time Series”, Journal of
Time Series Econometrics, v3, issue 1, 1-38.
Gregory, A. and Hansen, B.(1992), “ Residual-based Tests for Cointegration in Models
with Regime Shifts”, Journal of Econometrics, v70, no. 1, 99-126.
Holt, C. C., Modigliani, F. Muth, J. and Simon, H. (1960),
Planning Production,
Inventories, and Work Force, Prentice-Hall, Englewood Cliffs, NJ.
Kahn, J. A. (1987), “ Inventories and the Volatility of Production,” American Economic
Review, 77, 667-679.
Krane, S. D. and Braun, S. N. (1991), “ Production Smoothing Evidence from
Physical-Product Data,” Journal of Political Economy, 99, 558-581.
15
Lumsdaine, R. L. and Papell, D. H. (1997) “ Multiple Trend Breaks and the Unit-Root
Hypothesis”, Review of Economics and Statistics, vol. 79, No. 2, pp 79-109.
MacKinnon, J. (1996), “ Numerical Distribution Functions for Unit Root and
Cointegration Tests,” Journal of Applied Econometrics, 11, 601-618.
Perron, P.(1989), “ The Great Crash, the Oil Price Shock, and the Unit Root hypothesis”,
Econometrica 57, 1361-1401.
Wang,
H.
(2002),
“
Nominal
Data
and
the
Production
Smoothing
Hypothesis ,“ Economics Letters, 76, 245-250.
West, K. D.(1988), “ Evidence from Seven Countries on Whether Inventories Smooth
Aggregate Output ,“ NBER Working Paper No. 2664.
Wilkinson,
M.(1989)
“
Aggregate
Inventory
Behavior
in
Large
European
Economies ,“ European Economic Review, 33, 181-194.
(1991),
“
Inventory
Behavior
and
Economic
Instability
in
Japan ,“ Journal of the Japanese and International Economies, 5, 189-198.
Zivot, E., and Andrews, D. W. K.(1992), “ Further Evidence on the Great Crash, the
Oil- Price Shock, and the Unit-Root Hypothesis “, Journal of Business and
Economic Statistics, 10, no. 3. 251-270.
16
Table 1 Conversions of seasonal dummy variables
dc1t
1st quarter
2nd quarter
3rd quarter
4th quarter
dc2t
dc3t
1
̂3
α
̂2
α
̂4
1−α
̂4
1−α
̂4
1−α
̂
α
1
̂3
α
̂4
1−α
̂4
1−α
̂4
1−α
̂2
α
̂
α
1
̂4
1−α
̂4
1−α
̂4
1−α
̂3
α
̂2
α
̂
α
̂4
1−α
̂4
1−α
̂4
1−α
Notes: Let the equation to test for a unit root of the variable yt be yt = μ̂ + β̂1 d1t +
β̂2 d2t + β̂3 d + γ̂t + α
̂yt−1 + et , where dit ′s are the seasonal dummy variables that takes
the value one in the i th quarter( i=1, 2, and 3), and t, et , are the linear time trend, and
the residual, respectively. μ̂, β̂1 , β̂2 , β̂3 , γ̂, and α
̂ are the parameters in the equation. Then,
the table represents the values that the term
Dit
̂L
1−α
=𝑑𝑐𝑖𝑡 , (i=1,2,3) takes in each
quarter ( i = 1, 2, and 3 ) over the period of observations.
17
Table 2
Summary of Unit Root Tests
Japan
US
GDP Sales
Obs.
GDP
Sales
France
UK
GDP Sales
GDP Sales
(1955.1 – 1999.1) (1967.1 – 2008.1) (1978.1 – 2008.1) (1983.1 – 2008.1)
Period I
# of Breaks
2
2
2
2
0
1
1
1
N/A
C
A
A
1995.2
1990.1
Period II
2
2
Period I
Break Type
BB
BB
AA
AA
Period II
BB
BB
Period I
TB1
1972.3 1972.3
1978.1
1978.1
TB2
1992.3 1992.3
1981.3
1981.3
1990.4
Period II
TB1
1995.2
1995.2
TB2
2000.2
2000.2
Period I
Level of
Significance(τ) **
**
**
*
*
*
**
**
SA
SA
NSA
NSA
Period II
*
Data type
NSA
NSA
***
SA
SA
Notes: All of the data are in real and logarithmic terms. Notations of the break types are
the same as in the literature (Perron(1989), Zivot and Andrews(1992), and Lumsdaine
and Papell(1997)). “ * “, “ ** “, “ *** “ indicate the 10%, 5%, and 1% level of significance,
respectively. NSA and SA stand for “ not seasonally adjusted “ and
“ seasonally
adjusted “ series for quarterly data. TBi stands for the i th break point (i = 1, 2 ).
18
Table 3 Summary of Unit Root Tests
Obs.
Italy
Germany
Canada
GDP Sales
∆GDP ∆Sales
GDP Sales
(1991.1－2008.1)
(1990.2 –2008.1)
(1981.1 – 2007.1)
# of Breaks
2
Break Type
BC
2
0
BB
N/A
TB1
1991.3 1991.3
TB2
2003.1 2003.3
0
0
0
N/A
AB
CB
.
1998.2
1995.4
2007.3
2002.1
TB1
TB2
Level of
Significance(τ) **
Data type
NSA
**
NSA
***
NSA
NSA
***
***
***
SA
SA
Notes: All of the data are in real and logarithmic terms. Notations of the break types are
the same as in the literature (Perron(1989), Zivot and Andrews(1992), and Lumsdaine
and Papell(1997)). “ * “, “ ** “, “ *** “ indicate the 10%, 5%, and 1% level of significance,
respectively. NSA and SA stand for “ not seasonally adjusted “ and
“ seasonally
adjusted “ series for quarterly data. TBi stands for the i th break point (i = 1, 2 ).
19
Table 3 continued
Summary of Unit Root Tests
Germany
∆GDP
Obs.
Canada
GDP
∆Sales
(1957 ― 1991)
Canada
Sales
(1965.1 ― 1978.4)
GDP
Sales
(1992.1 ― 2008.1)
# of Breaks
0
0
0
0
1
Break Type
N/A
N/A
N/A
N/A
B
B
2003.4
2004.2
TB1
1
TB2
Level of
Significance
Data Type
***
Annual
***
Annual
*
*
**
**
SA
SA
SA
SA
Notes: All of the data are in real and logarithmic terms. Notations of the break types are
the same as in the literature (Perron(1989), Zivot and Andrews(1992), and Lumsdaine
and Papell(1997)). “ * “, “ ** “, “ *** “ indicate the 10%, 5%, and 1% level of significance,
respectively. NSA and SA stand for “ not seasonally adjusted “ and
“ seasonally
adjusted “ series for quarterly data. TBi stands for the i th break point (i = 1, 2 ).
20
Table 4 BCa Confidence Intervals for Japan ( Variance Ratio )
V(y)/V(S)
Confidence Intervals
BCa
99 %
( 1.01335 , 1.02716 )
95 %
( 1.01473 , 1.02552 )
90 %
( 1.01555 , 1.02448 )
ẑ0
-0.00334
â
0.04991
θ̂
1.01993
Notes: ẑ0 and â stand for the bias correction constant, and the acceleration constant.
θ̂ is the point estimate of the sample variance ratio. The sample period for the unit root
test is 1955.1～1999.1（The number of observations ＝177）. The number of observation
period used to construct BCa confidence intervals was set equal to 1958.2～1999.1 （The
number of observations ＝164）. The number of Bootstrap resamplings is 3,000.
21
Table 5 BCa Confidence Intervals for the US ( Variance Ratio )
V(y)/V(s)
BCa
Period
I
II
Confidence Intervals
99 %
(1.18870 , 1.68141 )
( 1.1120 , 1.5940 )
95 %
( 1.25243 , 1.60796 )
( 1.16848 , 1.52188 )
90 %
( 1.28194 , 1.57434 )
( 1.19710 , 1.48385 )
ẑ0
-0.008356
-0.29324
â
-0.002138
-0.008192
θ̂
1.420776
1.3542
Notes: ẑ0 and â stand for the bias correction constant, and the acceleration constant.
θ̂ is the point estimate of the sample variance ratio. The period I and II for the unit root
tests are 1967.1～1984.4（the number of observations ＝72）, and 1985.1 ～2008.1( the
number of observations = 93). The observation periods used to construct BCa confidence
intervals for periods I and II were set equal to 1970.2～1984.4, and 1988.2～2008.1,
respectively. ( the number of observations ＝59, and 80 ）. The number of Bootstrap
resamplings is 3,000.
22
Table 6 BCa Confidence Intervals for France ( Variance Ratio ）
V(y)/V(s)
Confidence Intervals
BCa
99 %
(1.084788,1.694079)
95 %
(1.144478,1.597168)
90 %
(1.180879,1.552616)
ẑ0
-0.1916709
â
0.012346076
θ̂
1.364265449
Notes: ẑ0 and â stand for the bias correction constant, and the acceleration constant.
θ̂ is the point estimate of the sample variance ratio. The sample period for the unit root
test is 1978.1～2008.1（the number of observations ＝121）. The observation period used
to construct BCa confidence intervals was set equal to 1981.1～2008.1 （the number of
observations ＝109）. The number of Bootstrap resamplings is 3,000.
23
Table 7 BCa Confidence Intervals for UK ( Variance Ratio )
V(y)/V(S)
Confidence Intervals
BCa
99 %
( 0.99879 , 1.00923 )
95 %
(0.99968 , 1.00733 )
90 %
( 1.00011 , 1.00647 )
ẑ0
0.06522
â
0.04700
θ̂
1.00261
Notes: ẑ0 and â stand for the bias correction constant, and the acceleration constant.
θ̂ is the point estimate of the sample variance ratio. The sample period for the unit root
test is 1983.1～2008.1（The number of observations ＝101）. The number of observation
period used to construct BCa confidence intervals was set equal to 1986.2～2008.1 （The
number of observations ＝88）. The number of Bootstrap resamplings is 3,000.
24
Table 8 BCa Confidence Intervals for Italy ( Variance Ratio )
V(y)/V(S)
Confidence Intervals
BCa
99 %
( 1.00023 , 1.00395 )
95 %
(1.00073 , 1.00361 )
90 %
( 1.00097 , 1.00344 )
ẑ0
-0.05517
â
-0.00131
θ̂
1.00227
Notes: ẑ0 and â stand for the bias correction constant, and the acceleration constant.
θ̂ is the point estimate of the sample variance ratio. The sample period for the unit root
test is 1981.1～2007.1（The number of observations ＝105）. The number of observation
period used to construct BCa confidence intervals was set equal to 1984.2～2007.1 （The
number of observations ＝92）. The number of Bootstrap resamplings is 3,000.
25
Table 9 BCa Confidence Intervals for Germany ( Variance Ratio )
V(y)/V(S)
Confidence Intervals
BCa
99 %
( 0.179115 , 0.289420 )
95 %
(0.189857 , 0.271006 )
90 %
( 0.195748 , 0.262982 )
ẑ0
0.021726
â
0.018947
θ̂
0.226948
Notes: ẑ0 and â stand for the bias correction constant, and the acceleration constant.
θ̂ is the point estimate of the sample variance ratio. The sample period for the unit root
test is 1991.1～2008.1（The number of observations ＝69）. The number of observation
period used to construct BCa confidence intervals was set equal to 1991.4～2008.1 （The
number of observations ＝66）. The number of Bootstrap resamplings is 3,000.
26
Table 10 BCa Confidence Intervals for West Germany (Variance Ratio)
V(y)/V(s)
Confidence Intervals
BCa
99 %
(1.006113, 1.001788 )
95 %
(1.148905 , 1.852629 )
90 %
( 1.233023 , 1.766839 )
𝑧̂0
-0.090361
𝑎̂
-0.065289
𝜃̂
1.522056
Notes: ẑ0 and â stand for the bias correction constant, and the acceleration constant.
θ̂ is the point estimate of the sample variance ratio. The sample period for the unit root
tests is 1957～1991（the number of observations ＝35）. The observation periods used to
construct BCa confidence intervals was set equal to 1960 ～ 1991( the number of
observations ＝32 ）. The number of bootstrap resamplings is 3,000.
27
Table 11 BCa Confidence Intervals for Canada ( Variance Ratio ）
V(y)/V(s)
Confidence Intervals
BCa
99 %
(0.823454, 1.843992)
95 %
(0.910669, 1.681195)
90 %
(0.963495, 1.592324)
ẑ0
0.010027
â
-0.041922
θ̂
1.233036
Notes: ẑ0 and â stand for the bias correction constant, and the acceleration constant.
θ̂ is the point estimate of the sample variance ratio. The sample period for the unit root
test is 1965.1～1978.4（the number of observations ＝56）. The observation period used
to construct BCa confidence intervals was identical with the sample period. The number
of Bootstrap resamplings is 3,000.
28
Table 12 BCa Confidence Intervals for Canada ( Variance Ratio ）
V(y)/V(s)
Confidence Intervals
BCa
99 %
(0.578803, 1.333072)
95 %
(0.666948, 1.267492)
90 %
(0.716487, 1.204174)
ẑ0
0.274110
â
0.002397
θ̂
0.916256
Notes: ẑ0 and â stand for the bias correction constant, and the acceleration constant.
θ̂ is the point estimate of the sample variance ratio. The sample period for the unit root
test is 1992.1～2008.1（the number of observations ＝65）. The observation period used
to construct BCa confidence intervals was set equal to 1995.2～2008.1 （the number of
observations ＝52）. The number of Bootstrap resamplings is 3,000.
29