Journal of International Fisheries▶ Article ◀

Vol.10，2011

Capturing Changes in Asymmetric Price Transmission: A Rolling Window TAR Estimation Using Bluefin Tuna Case Study
Toru NAKAJIMA†, Takahiro MATSUI＊, Yutaro SAKAI＊＊ and Nobuyuki YAGI（The University of Tokyo, Graduate School／＊Kinki University, Global COE
Postdoctoral Fellow／＊＊University of Calgary, Graduate School）
E-mail：†n_through@yahoo.co.jp【abstract】
In this paper, asymmetric price transmission (APT) from producer prices to wholesale
prices is estimated with a threshold autoregressive (TAR) rolling window regression
model using the data of bluefin tuna in Japan. Rolling window regression is amethodology that repeats regressions using dataset subsamples by shifting the startand end points with a fixed window. Hence, the rolling window regression capturespotential time-varying parameters. The findings of this study supported those of
Matsui et al. (2011). However, the rolling window regression is more robust as astatistical methodology than regressions with total sample. Furthermore, it provides
more information on the process of changes in the APT compared to estimations withsample separations based on descriptive analysis or structural change tests. TAR
estimations with sample separation can be considered to be a special case of therolling TAR without overlapping time periods.
【keywords】
rolling window, TAR model, cointegration, asymmetric price transmission

1．Introduction
The threshold autoregressive (TAR) model developed by Enders and Granger (1998)
and Enders and Siklos (2001) is a popular econometric model for estimating
asymmetric price transmission (APT) with nonstationary time series data. As
mentioned in the survey paper by Meyer and von Cramon-Taubadel (2004), pricetransmission is said to be asymmetric if the speed or the amount of adjustment of theoutput price differs after the input price increases or decreases. Many empiricalstudies on various commodities have been conducted using the TAR model. Thesestudies illustrated the existence of APT from the prices of an upstream industry to-1-Toru NAKAJIMA, Takahiro MATSUI, Yutaro SAKAI and Nobuyuki YAGI

those of a downstream industry.
Previous empirical APT studies that use the TAR model for agricultural products
include that of Abdulai (2002), who analyzed the Swiss pork market; Ghoshray (2002),
who estimated the APT for wheat export prices in major wheat-producing countries;
Hassan and Simioni (2001), who considered the APT from shipping prices to retailprices in the vegetable market in France; and Nakajima (2010), who analyzed the APTof U.S. corn from the export prices in the U.S. to the import prices in Japan.
For fishery products, Gonzales et al. (2003) analyzed the marine products market inFrance and identified downward rigidity in price transmission from the landing prices
for cod to their retail prices and upward rigidity from the import prices for salmon totheir retail prices. Jaffry (2004) analyzed the French hake value chain and concluded
that retailers responded to positive changes in auction prices more quickly than they
did to negative changes. Matsui et al. (2011) estimated the momentum-TAR, or
M-TAR, model using Japanese bluefin tuna data and showed that the pricetransmission was asymmetric, in that the consumption market prices adjusted more
rapidly when the farm-gate prices increased than when they decreased in the latter
half of the 1970s through the end of the 1980s. Then the price transmission changed
to another asymmetry in which the consumption market prices adjusted more slowly
when the farm-gate prices increased than when they decreased in the 1990s.
There are some problems, however, in these previous studies that employ the TAR
model. One problem is that the sample period was selected rather arbitrarily. Because
the values of the coefficients in the model are sensitive to the selected sample period,
it is necessary for researchers to carefully consider possible changes in the estimation
results by changing the sample period. TAR estimations should be conducted based on
the sensitivity of different sample sizes.
Another problem, which is closely related to the one mentioned above, is that
previous studies rarely focused on possible time-varying parameters in TAR
estimations. This problem is a cause for concern, because if structural changes exist
and estimated parameters vary in different subsamples, then the estimates may bebiased and the estimation results in the total sample may provide misleading
conclusions and implications. Some papers mentioned potential structural changes
based on descriptive analysis and divided the sample period into subsamples. Then,they estimated the TAR model for each subsample (Enders and Siklos (2001), Matsui

et al. (2011), etc.). However, this methodology lacks a statistical background of thesample divisions and is unable to completely eliminate the arbitrariness of-2-

Capturing Changes in Asymmetric Price Transmission: A Rolling Window TAR Estimation Using Bluefin Tuna Case Study(sub)sample selection. Other studies employed structural change tests such as thesequential Chow test (Abdulai, 2002) and Bai and Perron’s (2003) multiple structuralchange tests (Nakajima et al., 2010). Nevertheless, there is no adequacy in the
rigorous manner of estimating changes in asymmetry using OLS residuals.
The purpose of this study is to illustrate a methodology more robust than that ofprevious studies for estimation of the APT using the TAR model in a manner. Our
methodology captures possible structural changes by estimating possible time-varying
parameters in the model. In particular, in this study, rolling window regressions are
employed that run sequential regressions by shifting the start and end points of thesample with a constant interval and a fixed subsample (window) size(1).
The rest of this paper is organized as follows. In Section 2, the rolling window
methodology and the rolling TAR model are explained. In Section 3, an empiricalanalysis of the rolling TAR is conducted using bluefin tuna data, same as that
employed by Matsui et al. (2011). Finally, in Section 4, we provide the conclusions.

2．Methodology and model
2-1．Rolling window regression
As explained in Zivot and Wang (2006), rolling analysis of a time series model isoften used to assess the model’s stability over time. Although the model’s parameters
are assumed to be constant over time in the time series analysis, this assumption may
not be true, because the economic environment often changes. One of the common
techniques for assessing the constancy of a model’s parameters is to compute theparameter estimates over a rolling window of a fixed size in the sample. If theparameters are truly constant over the entire sample, then the estimates over the
rolling windows should not be very different. However, if the parameters change atsome point during the sample, then the rolling estimates should capture thisinstability.
In general, the rolling window regression is a methodology that repeats regressions
using subsamples of total data by shifting the start and end points with a fixedwindow. Consider an OLS estimation with time series data using the rolling window
as follows:y t (i ) = a i + bi xt (i ) + et (i )i = 1,  , n ,(1)where y is the dependent variable, x is the vector of independent variables, a andb are parameters, e is the error term, and i is the number of rolling regressions.
-3-Toru NAKAJIMA, Takahiro MATSUI, Yutaro SAKAI and Nobuyuki YAGI

Note that both the intercept and the slope parameter may change in each regression

i . If the total observation number is N , the window size is set to be w , and theincrement interval, or the increment of the time period, between the adjacent rolling
regressions is d , then n is written as follows: N − w
n=
(2)
 +1,
 d 
where  X  stands for the floor of X , that is, the largest integer less than or equal to

X . In most cases, the regressions are conducted by increasing the start and end

points by 1, that is, d = 1 ; hence, n = N − w + 1 . In addition, t (i ) ∈ [i, i + w − 1] for eachi.
Using data with a total sample size of 500 and setting a window size of 100,
regressions are conducted initially using periods 1–100, followed by periods 2–101,
3–102, …, and finally periods 401–500. The total number of regression results (thetotal sample size minus the window size plus one) for each parameter is 401. In thismanner, possible time-varying parameters are obtained for a given sample.
The parameters of rolling window regressions depend on the interval of each
regression and window size. The larger the interval, the more coarse must be thechange in the parameters. It is recommended that smaller intervals, such as one, bechosen by considering the functional processing ability of computers. As for window
size, the smaller the window size, the more detailed the movement of the time-varying
parameters; however, in this case, the variation in the movement is large (Su andHwang, 2009). Conversely, a large window makes it difficult to determine possible
structural changes in the subsample.
The window size depends on the choices or the aims of the research. Swanson (1998)
used 10-year (120-month) and 15-year (180-month) windows, while Su and Hwang(2009) compared the estimation results of 13, 25, 39, and 51 windows. In any case,some sensitivity tests should be conducted during the empirical analysis.
2-2．Rolling window with the TAR model
The rolling TAR model is introduced on the basis of the methodology of the rolling
window regression. The TAR model is explained in detail in Enders and Siklos (2001),
Nakajima (2010), and Matsui et al. (2011). Hence, only the essence of the model andthe difference in the rolling windows are shown here.
The first step is to regress the output price ( p 2t ) on the input price ( p1t ) to obtain
the residuals{µ̂ t } , which may be serially correlated. If
-4-

p1t

and

p 2t

are

Capturing Changes in Asymmetric Price Transmission: A Rolling Window TAR Estimation Using Bluefin Tuna Case Study

nonstationary (I(1)) variables and {µ̂ t } is the stationary (I(0)) variable, then p1t andp 2t are said to be cointegrated (Engle and Granger, 1987).
In a rolling TAR model, whether p1t (i ) and p 2t (i ) in the i th regression are
cointegrated can be tested using the following equation for each regression i :∆µt (i ) = I t (i )ρ1i µt (i )−1 + (1 − I t (i ) )ρ 2i µt (i )−1 + ∑ γ ji ∆µt (i )− j + ε t (i ) ,
Ti

(3)

j =1

1 if
I t (i ) = 0 ifµ t (i )−1 ≥ τ i,µ t (i )−1 < τ i(4)where I t (i ) is the Heaviside indicator function; ρ1i , ρ 2i (2), and γ ji are parameters;

τ i is the super-consistent estimator of the threshold that minimizes the sum of thesquared residuals (RSS) of Equation (3) by using the grid search method developed by
Chan (1993); and ε t (i ) is the white noise disturbance term. For
stationary,{µ̂ ( ) } to bet i

p1t (i ) and p 2t (i ) need to satisfy the conditions ρ1i < 0 , ρ 2i < 0 , and(1 + ρ1i )(1 + ρ 2i ) < 1 for any value of τ i (Petrucelli and Woolford, 1984). Ti is the lagorder that satisfies these conditions (hereafter referred as the Conditions) andminimizes either the Akaike information criteria (AIC) or the Bayesian information
criteria (BIC).
In the rolling TAR model, a cointegration test is performed by testing whether

ρ1i = ρ 2i = 0 in each regression. If the null hypothesis ρ1i = ρ 2i = 0 is rejected, thenp1t (i ) and p 2t (i ) are cointegrated. The F-statistics are called Φ i statistics(3). The
distribution of the Φ i statistics is different from those of the normal F-statistics. The
critical values given the lag order are shown in Enders and Siklos (2001).
If p1t (i ) and p 2t (i ) are confirmed to be cointegrated, then the APT can be tested by
testing the null hypothesis ρ1i = ρ 2i (4). If ρ1i = ρ 2i is rejected andρ1i < ρ 2i ,negative discrepancies from the equilibrium error adjust more rapidly than positive
discrepancies. This implies that a shock that decreases margins adjusts more rapidly
than a shock that increases margins. That is, the price transmission exhibits
downward rigidity, which is called positive APT. In contrast, if ρ1i = ρ 2i is rejectedand ρ1i > ρ 2i , then the positive deviations adjust toward the equilibrium error more
rapidly than the negative deviations. This implies that a shock that increases themargin adjusts more rapidly than the one that decreases the margin. This results in
upward rigidity, which is called negative APT.
Another formulation of the asymmetric adjustment process exists: instead of usingµ t (i )−1 , ∆µ t (i )−1 can be used in Equation (4) as follows:-5-Toru NAKAJIMA, Takahiro MATSUI, Yutaro SAKAI and Nobuyuki YAGI

1 if
I t (i ) = 
0 if∆µ t (i )−1 ≥ τ i,∆µ t (i )−1 < τ i(5)Equation (3) with the indicator function of Equation (5) in each regression represents
the rolling M-TAR model.
The TAR and M-TAR models correspond to two asymmetric adjustment processes,
deepness and steepness (Sichel, 1993); deepness represents a skewed time series andsteepness a skewed first-diffenrence series. However, in both models,
indicates positive APT andρ1i > ρ 2i

ρ1i < ρ 2i

indicates negative APT. In the empiricalstudies, TAR or M-TAR is selected on the basis of AIC or BIC (Enders and Granger,
1998). Furthermore, according to power tests in previous studies (Enders (2001),
Enders and Siklos (2001), etc.), the M-TAR model has greater power in detecting a
cointegration relationship than the TAR model.

3．Empirical analysis
3-1．Data
The data used in this study is the same as that used by Matsui et al. (2011), because
one of the motivations of this study is to check the validity of sample division.
Monthly prices of bluefin tuna (frozen) in both producer and wholesale markets fromJanuary 1964 to December 2006 were obtained from the “Annual Report ofDistribution Statistics on Fisheries Products” of the Ministry of Agricultural, Forestry
and Fisheries.
The price data are transformed into natural logarithmic form and used on anominal basis, as in many empirical studies of the TAR model, although a sensitivity
test is conducted using real price data, where the nominal data are deflated by the
consumer price index (CPI) of Japan. The CPI was obtained from the International
Financial Statistics of the IMF. Furthermore, we checked the seasonal variation of thedata. According to seasonal adjustments using the X12-ARIMA method, the data were
found to have limited seasonal variations. We conducted the sensitivity test using theseasonally adjusted data.
3-2．Unit root tests
The unit root tests for the stationarity of (logs of) p1t and p 2t were conductedusing the ADF and KPSS tests(5), whose results are drawn from Matsui et al. (2011)
and illustrated in Table 1. The ADF tests indicate that although the null hypothesis-6-Capturing Changes in Asymmetric Price Transmission: A Rolling Window TAR Estimation Using Bluefin Tuna Case Study

Table 1 Unit root test results
Variable

Difference

ADF test
t-statistic
p-value−2.14
0.52−16.75***
0.00
−2.51
0.32
−12.52***
0.00

KPSS test
LM statistic
0.319***
0.024
0.163**
0.035

Level
Log ofproducer priceFirst-differenced
Level
Log ofwholesale priceFirst-differenced
Notes:
1) The results are the same as those in Matsui et al. (2011).
2) The t-statistics of the ADF tests are obtained by regressions with an optimallag length, which minimizes the BIC. The p-values are calculated by
MacKinnon’s (1996) procedure.
3) Lagrange multiplier (LM) test statistics are obtained with Andrews’s
bandwidth using the Bartlett kernel. The asymptotic critical values are basedon Kwiatkowski et al. (1992).
4) The test equations in both the ADF and KPSS tests include a trend and anintercept.
5) ***, **, and * represent significance at 1%, 5%, and 10% levels, respectively.

that the time series have a unit root was not rejected in the level series, it was
rejected in the first-differenced series at the 1% significance level for both prices. In
addition, the KPSS tests indicate that the null hypothesis of no unit root was rejectedin the level series at least at the 5% level; however, it was not rejected in thefirst-differenced series for both prices. Therefore, it follows that both p1t and p 2t
are I(1) series.
3-3．Rolling TAR results
In the rolling window regression of the TAR model, the window sizes of 10 years(120 months) and 1 month interval were chosen. The window of 120 months is neither
too long nor too short. In addition, in the next subsection, rolling window regressions
using the window sizes of five years (60 months) and 15 years (180 months) are also
conducted as sensitivity tests.
Because the total sample size is 516 and the window size is 120, there are 397 TAR
model regressions using subsamples. In each regression, the super-consistent τ i isestimated using Chan’s (1993) method. The lag orders that satisfy the Conditions(6)are selected to minimize the BIC of Equation (3). In addition, in each regression, theparameters are stored only when the price series are confirmed to be cointegrated,
where Φ i statistics are greater than the critical values at the 10% significancelevel(7). Furthermore, the BIC of each regression is stored to compare the value of the
TAR model with that of the M-TAR model. Based on the result of the rolling window
regression, TAR and M-TAR models are compared in each regression using BICs.
-7-

Toru NAKAJIMA, Takahiro MATSUI, Yutaro SAKAI and Nobuyuki YAGI

Table 2 Numbers and percentages of (M-)TAR models selected by the BIC
Case A
NumberModel

Case B
Number

%

%

TAR
102
27.3
102
27.7
M-TAR
272
72.7
266
72.3
TAR + M-TAR
374
94.2
368
92.7
Number of regressions
397
397
Notes:
1) “Case A” indicates that if the BIC in one model is missing and is available in
the other, then the model having the value is chosen. Meanwhile, in “Case B”,
if the BIC in one of the models is missing, then neither model is selected.
2) “Number” of the TAR and M-TAR is the number selected by the BIC.
3) “Number” of the “TAR+M-TAR” is the sum of the “Number” in the TAR and the
M-TAR.
4) “%” of the TAR or M-TAR is the percentage of the TAR or M-TAR to“TAR+M-TAR.”
5) “%” of the “TAR+M-TAR” is the cointegration percentage of the “TAR+M-TAR”
to the number of rolling window regressions.

Table 2 illustrates the numbers and ratios of the TAR and M-TAR models selected interms of the BIC. In each regression, the model whose BIC is smaller than the other isselected. In Case A, if a BIC value in one model is missing, because at least one of theconditions previously mentioned (the cointegration conditions and the Conditions) is
not satisfied, while the other has the value, then the model having the value is chosen.
In Case B, if the BIC in one of the models is missing, then neither model is selected.
In both cases, it is concluded that the M-TAR model outperforms the TAR model interms of the BIC, which is consistent with the studies previously mentioned. In
addition, note that the ratio of “the sum of TAR and M-TAR selected” and “the numberof regressions” is larger than 92%, which indicates that most of the regressions satisfy
the Conditions and the price series are cointegrated.
Figure 1 illustrates the movements of the estimated ρ1iand ρ 2iin the rolling

M-TAR model. The vertical axis indicates the absolute values of the parameters. The
horizontal axis indicates the number of regressions ( i ) in the ascending order. Figure
1 demonstrates that the value of ρ1i

is greater or smaller than ρ 2i , where iranges from 1 to approximately 90; clearly, ρ1i < ρ 2i
185; ρ1i

is close to ρ 2i

where i ranges from 90 towhere i ranges from 185 to 265; ρ1i > ρ 2i

ranges from 265 to 330; and ρ1i

is close to ρ 2i

where i ranges from 330 to theend (397). However, it is difficult to conclude whether ρ1i
smaller than

ρ 2iwhere iis statistically greater orwhen the asymmetry changes. Therefore, significance of theasymmetry and the specific time points are then introduced.
-8-

Capturing Changes in Asymmetric Price Transmission: A Rolling Window TAR Estimation Using Bluefin Tuna Case Study

1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0

1

41

81

121

161

201

241

ρ1i
|rho1|

281

321

361

ρ 2i
|rho2|

Figure 1 Movements of ρ1i

and ρ 2i

Notes:
Horizontal axis indicates regression numbers ( i ).
1
0.8
0.6
0.4
0.2
0
'69.1
-0.2

'73.1

'77.1

'81.1

'85.1

'89.1

'93.1

'97.1

'01.1

-0.4
-0.6

ρ 2i −- |rho1|
ρ1i|rho2|

significant

Figure 2 Movements of ρ 2i − ρ1iand significance

Notes:
1) Positive and negative values indicate positive APT and negative APT,
respectively.
2) If the null hypothesis ρ1i = ρ 2i is rejected at least at the 10% level,bold circles are on the dotted line. If the null is not rejected, thecircles are on the horizontal line. The absence of bold circles indicatesno cointegration or satisfaction of the Conditions.
3) The median time point of each period of the subsample is chosen as
the time point of the regression. It follows that the APT index ranges
from January 1969 to January 2002.

-9-

Toru NAKAJIMA, Takahiro MATSUI, Yutaro SAKAI and Nobuyuki YAGIThe dotted line in Figure 2 shows the value of ρ 2i − ρ1i . In addition, the figure
shows whether ρ1i = ρ 2i is rejected at least at the 10% level. If the null hypothesis isrejected at least at the 10% level, bold circles are located on the dotted line. If the null
hypothesis is not rejected, then the circles are located on the horizontal line. The
absence of bold circles indicates no cointegration or satisfaction of the Conditions. In
addition, the time points are allocated instead of regression numbers; the median time
point for each period of the subsample is chosen as the time point of the regression,that is, the time of the i th regression t = [i + (i + w − 1)] 2 = i + (w − 1) 2

(8).

For

example, the period of the first regression is from January 1964 to December 1973,
and the median time point is January 1969. Hence, the result of the regression isassumed to represent the regression result in January 1969(9).
Based on these settings, it cannot be concluded whether ρ1ithan ρ 2i

is greater or smallerfrom the end of the 1960s to the end of 1975. However, ρ1i < ρ 2i

from1976 to mid-1984. There is no asymmetry from mid-1980s to early 1992. ρ1 > ρ 2
until mid-1996. No asymmetry was observed from late 1996 to mid-1999. From late1999 to early 2001, there is no cointegration relationship between the prices. Finally,
there is no asymmetry from late 2001 to the end (January 2002).
This result is mostly consistent with that of Matsui et al. (2011). In Matsui et al. ’s
study, no asymmetry was observed from 1964 to 1975, positive APT was observed from1976 to 1989, negative APT was observed in the 1990s, and no cointegration wasobserved in the 2000s. Although the previous study divided the sample based on a
descriptive analysis of the market structure, it did not include a statisticalbackground of the sample separation. In this sense, the result of this study has moreobjectivity or robustness and more detailed information about the changes in the APT.
3-4．Sensitivity tests
Because the estimation result may change with the characteristics of the data (suchas nominal or real and seasonally adjusted or nonadjusted) and the window size, somesensitivity tests were conducted. As shown in Table 3, the ratios of the M-TAR model
selected in terms of the BIC are much higher than those of the TAR model in all theestimation results. Hence, the subsequent discussion is based on the results of the
M-TAR model.
The results using real data, seasonally adjusted nominal data, and seasonallyadjusted real data are shown in Figure 3. Compared to the original result using
seasonally nonadjusted nominal data, the one using real data has negligible- 10 -

Capturing Changes in Asymmetric Price Transmission: A Rolling Window TAR Estimation Using Bluefin Tuna Case Study

Table 3 Percentages of (M-)TAR models selected by the BIC (for the sensitivity test)
Model

Real
A
B
21.0
21.4
79.0
78.6
93.5
91.9

SA
A
B
25.2
26.4
74.8
73.6
95.0
90.7

Real+SA
A
B
24.0
24.4
76.0
75.6
91.2
87.7

60 window
A
B
29.2
29.2
70.8
70.8
61.5
53.2

180 window
A
B
22.6
22.6
77.4
77.4
100.0 100.0

TAR
M-TAR
T+M
Unit: %
Notes:
1) “T+M,” “A,” and “B” are the same as “TAR+M-TAR,” “Case A,” and “Case B,”
respectively, in Table 2. The meaning of each notation is the same as that inTable 2.
2) “Real” is the result using real (deflated) data and “SA” is the result using
seasonally adjusted data.

difference. The APT seems to be more negative from the end of the 1960s to mid-1970s.
The result using seasonally adjusted nominal data shows less positive APT in earlier
periods, more symmetry in the early 1980s, and more positive APT in the late 1980s,
compared to the original one. Meanwhile, the result of the seasonally adjusted real
data shows less positive APT in earlier periods and more positive APT in the late1980s, compared to the original one. However, in either case, the movements of the
APT are similar to that of the original result, that is, no APT was observed in theearlier period, positive APT was observed from the mid-1970s, negative APT was
observed in the 1990s, and no APT was observed in the 2000s.
The results using the window sizes of five years (60 months) and 15 years (180months) are shown in Figure 4. Note that the time range on the horizontal axis isdifferent from the original result, because the median of the time period in each
regression differs according to the window size. The result with the 60 window showsa rather unstable movement of the APT, although the movement is similar to theoriginal one. Because the sample size is small, it is difficult to find the cointegration
relationship between the prices, particularly from the late 1990s. This is confirmed inTable 3, where the percentage of the TAR and M-TAR models showing cointegration ismuch smaller than in the other models (approximately 50% or 60%)(10).
In contrast, the result with the 180 window represents positive APT from the end of
the 1970s to mid-1980s and negative APT from the late 1980s to the end of the 1990s.
According to Table 3, the prices are cointegrated in all the results. The movement
seems rather smooth, that is, there is less fluctuation in the short term. However, thisindicates that the regressions are unable to capture possible structural changes andobtain consistent results. Comparison of these results with those of different window
sizes reveals that the results with the 120 window fluctuate moderately and capture- 11 -

Toru NAKAJIMA, Takahiro MATSUI, Yutaro SAKAI and Nobuyuki YAGI
0.8
0.6
0.4
0.2
0
69.1
-0.2

73.1

77.1

81.1

85.1

89.1

93.1

97.1

01.1

'93.1

'97.1

'01.1

'97.1

'01.1

-0.4
-0.6
-0.8

(1) Real price data
0.8
0.6
0.4
0.2
0
'69.1
-0.2

'73.1

'77.1

'81.1

'85.1

'89.1

-0.4
-0.6
-0.8

(2) Seasonally adjusted nominal price data
0.8
0.6
0.4
0.2
0
'69.1
-0.2

'73.1

'77.1

'81.1

'85.1

'89.1

'93.1

-0.4
-0.6
-0.8

(3) Seasonally adjusted real price data

Figure 3 Trend in APT (for real, seasonally adjusted, and seasonally adjusted real data)
Notes:
Notations are the same as those in Figure 2.

- 12 -

Capturing Changes in Asymmetric Price Transmission: A Rolling Window TAR Estimation Using Bluefin Tuna Case Study
1.2
1
0.8
0.6
0.4
0.2
0
-0.2'66.7

'70.7

'74.7

'78.7

'82.7

'86.7

'90.7

'94.7

'98.7

'91.7

'95.7

'02.7

-0.4
-0.6
-0.8

(1) 60 window
0.8
0.6
0.4
0.2
0
'71.7
-0.2

'75.7

'79.7

'83.7

'87.7

'99.7

-0.4
-0.6
(2) 180 window

Figure 4 Trend in APT (for the window sizes of 60 and 180)
Notes:

Notations are the same as those in Figure 2, except that the period is from
July 1966 to July 2004 for the figure using the 60 window and from July
1971 to July 1999 for the figure using the 180 window.

the movement of the APT rather well. Furthermore, the trend of the APT is similar ineach window size, although the timings of the changes differ slightly.

4．Conclusions
In this paper, the APT from producer prices to wholesale prices is estimated with- 13 -

Toru NAKAJIMA, Takahiro MATSUI, Yutaro SAKAI and Nobuyuki YAGI

the rolling window regression of the TAR model using Japanese bluefin tuna data.
The estimation results indicate positive, negative, or no APT from the 1960s to theend of 1975. Furthermore, APT was positive until mid-1984. The APT graduallychanged from positive to negative with the period of no asymmetry from themid-1980s to early 1992. The negative APT continued until mid-1996, after which
there was no asymmetry until January 2002 (the end of the rolling regression). In
addition, there was no cointegration relationship between prices from late 1999 toearly 2001.
The transition of the APT previously mentioned corresponds to the changes in themarket structure of bluefin tuna in Japan. As shown in Matsui et al. (2011), themarket structure of the sellers became oligopolistic, particularly after the promotionof vessel-based trade, which was prominent in the 1970s. In the 1990s, however, many
sellers entered the industry, increasing the number of sellers. Furthermore, thebuyers’ market power was thought to increase because of the emergence of large
retailers such as supermarkets. From the end of the 1990s, a large amount of farmed
tuna was imported and traded in Japan, which may have caused another change inthe market structure.
According to the sensitivity tests, there is little difference in the trend of the APT,even though the data are transformed from nominal to real, seasonally adjusted
nominal, and seasonally adjusted real data. Furthermore, the trend itself does notchange according to the window sizes. However, the timing of the changes in the trend
differs slightly, particularly when different window sizes are applied.
The empirical results using the rolling window TAR estimation are more convincing
than those using the conventional TAR, because the rolling window TAR estimation ismore robust since it captures the time variations of the parameters. Furthermore, it
includes more information about the changes in the APT, as demonstrated in thisstudy. That is, compared to the findings in Matsui et al. (2011), this study seems moreconvincing from the perspective of the changes in the APT. One reason for this is thatthe study captured the transition of the APT more concretely, because the movementof the APT was indexed like a set of time series data. In contrast, the sample
separation into four subsamples in Matsui et al. (2011) was based on the ex ante
information of the market structure and was not drawn ex post by an econometricanalysis.
Another advantage of this study compared to Matsui et al. (2011) is that thefindings of this study contain more information about the changes in the APT. Matsui- 14 -

Capturing Changes in Asymmetric Price Transmission: A Rolling Window TAR Estimation Using Bluefin Tuna Case Studyet al. (2011) found that the APT changed from positive in 1976–1989 to negative in the
1990s and concluded that the result “implied” a structural change in the downstream
industry, which might affect the structure of the upstream industry. This implication
should be a type of conjecture, because the opposite signs of the APT during different
time periods do not show any process of structural change. In contrast, this paper
revealed that the positive APT gradually changed to no APT and then to negative APT.
This “clearly showed” the process by which the structural change in the downstream
industry affected the structure of the upstream industry in a detailed manner. Thus,
as a statistical methodology, the rolling window regression is not only robust but also
includes more information about the process of the changes in the APT, compared toestimations with a sample separation based on descriptive analysis or structuralchange tests. From the perspective of the latter, the TAR estimation with a sample
separation can be considered to be a special case of the rolling TAR without
overlapping time periods.
This study applied the rolling window methodology to the TAR model and createdan APT index by considering the median time point of each regression as the time
point of the subsample. This methodology is applicable, with less data demand, tovarious analyses on price transmissions such as other fishery products. Moreover, therolling window methodology itself is useful and manageable when analyzingtime-varying parameters of any type of regression.
Further research on indexing the changes, which should be consistent irrespectiveof window sizes, is necessary. Allocation of precise time points to the indicator isanother related issue for future research.

Notes(1) Bermejo et al. (2011) used a recursive estimation with a different type of TAR modelin which the window size grows with a fixed start or end point.
(2) According to Engle and Granger (1987), if p1t and p 2t are cointegrated, then the
OLS regression equation of p 2t on p1t can be written as the following error
correction model (ECM):∆ p 2t = ρ ( p 2,t −1 − α − β p1,t −1 ) +

k

∑
j =1

δ 1 j ∆p 2,t − j +k∑ δ ∆p2j

1,t − j + v t

j =1

where v t is the white noise disturbance term and k is the lag length (Granger’s
representation theorem). The term p 2,t −1 − α − β p1,t −1 is called the error correction

term and ρ is known as the adjustment coefficient. Equations (3) and (4) are- 15 -

Toru NAKAJIMA, Takahiro MATSUI, Yutaro SAKAI and Nobuyuki YAGI
consistent with the ECM, and ρ1i and ρ 2i are the adjustment coefficients in the
ECM with the indicator function shown in Equation (4) (Enders and Siklos, 2001).
(3) It is possible to verify whether ρ1i = ρ 2i = 0 by the Student’s t-test instead of the F( Φ )-test. However, from the power tests using Monte Carlo simulations, Enders andSiklos (2001) showed that particularly in the M-TAR model, the power of detecting
the cointegration relationship is greater in the F ( Φ )-test than in the t-test. Previous
empirical studies that used the TAR model also employed the F ( Φ )-test, rather than
the t-test. Hence, we conducted the F ( Φ )-test.
(4) Most empirical studies employed the ECM with an asymmetric adjustment toestimate short-run parameters and simulate the impact of exogenous shocks on
endogenous variables by estimating impulse response functions. However, the test for
the APT was conducted using the TAR model, not the ECM. Furthermore, we focusedon the APT and its changes. Hence, we dedicated our analysis to the rolling TAR
regression. Analyses on short-run dynamics and impulse responses should be
conducted in future research.
(5) The KPSS test was introduced to complement unit root tests such as the ADF test;
that is, by testing both unit root and stationarity hypotheses, it is possible todistinguish series that appear to be stationary, series that appear to have a unit root,and series for which the data (or the tests) are not sufficiently informative to ensure
whether they are stationary or integrated (Kwiatkowski et al., 1992).
(6) The maxlag was set to three to reduce the computer’s burden. Most lags in the
regressions are found to be zero. Hence, the setting should not be a problem.
(7) The critical values for the TAR and M-TAR models were chosen to be 6.35 and 5.76,
respectively. These are the highest values among the values with various lag length
at the 10% level if the observation number is 100, as in Enders and Siklos (2001). In
addition, according to Enders and Siklos (2001), as the observation number increases,
the critical values decrease. Thus, if the Φ i statistics are greater than those values,
the prices are definitely cointegrated.
(8)X

denotes the nearest integer to X . In this paper, 99 / 2 = 50 .

(9) Because one of our major interests is to check when the changes in the APT occur, it isnecessary for us to allocate a specific time point to the parameters of each regression.
However, previous studies on rolling windows did not attempt to specify the time
point for each regression. Although no formal methodology exists to allocate the time
point for each regression, the one we employ here is considered to be reasonable.
Further research should be conducted on this issue.

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Capturing Changes in Asymmetric Price Transmission: A Rolling Window TAR Estimation Using Bluefin Tuna Case Study(10) These values indicate the percentages that satisfied the Conditions and where theprices were cointegrated. Meanwhile, the percentages that satisfied the Conditions
were 99.% for Case A and 93.7% for Case B. It follows that when the 60 window isused, the results have a low percentage of “T+M” primarily because no cointegration
relationship exists between the prices.

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