3.2. TVR-VAR Model
Based on the assumption that the variables have unit roots and are not cointegrated, the TVP-VAR model has a different structure from the VAR model; the estimated parameters change over time (
Primiceri 2005). To consider such parameter changes over time, we applied the TVP-VAR model on the CNY (E1), JPY (E2), Chinese LNG average import price (PL), and JCC average crude oil price. The JCC price was included in our model, which was mainly to avoid the impact of the JCC price on foreign exchange and to better understand the impact of the exchange rate on the Chinese LNG import price. The lag order of the time-varying model was determined by using the minimum AIC value obtained from the VAR model. In this study, two time-varying models for the CNY and JPY were constructed to compare the effects of these exchange rates on the Chinese LNG import price.
In the CNY TVP-VAR model, the Chinese LNG import price (PL), CNY(E1) monetary rate, and the JCC crude oil price (PJ) were set as
The model was constructed as follows:
where
denotes the first difference of the variable.
Similarly, for the JPY model, the three main variables of our interest were set as
and the model had the following form:
Here, are the time-varying constant vectors of (), are the time-varying coefficient matrices () of (), and are error term vectors of ().
The error terms
in Equations (2) and (4) were assumed to follow the variate normal distribution with an average of 0 and time-varying covariance matrices of
. The time-varying covariance matrices
were expanded by using the Cholesky decomposition:
where
are diagonal matrices of (3). Here, all the diagonal components were
In addition,
were the diagonal matrices of (
) where
Here, were the time-varying variances of structural shocks for variable , while were the parameters of the time-varying simultaneous correlations given to the variable by the structural shock of the variables where ().
Then, based on Equations (1), (2) and (5), the CNY(E1) model could be rewritten as the following equations:
Similarly, based on Equations (3), (4) and (6), the JPY (E2) model could be expressed as follows:
Here,
were the vectors corresponding to Equations (7) and (9):
is defined as below:
where
is the identity matrix of
, and
is the Kronecker product. In addition,
in Equations (8) and (10) are the normalized structural shocks.
The time-varying parameter was set by assuming the following equations:
where,
are the lower triangular components of the
matrices. The diagonal components
were converted into
(
) where
and
The time-varying parameters for the CNY and JPY models were defined as (
) and (
).
The prior distributions corresponding to (
) and (
) were set as follows:
In Equations (14) and (15), the and denote the Inverse Wishart and Inverse Gamma distributions, respectively.
In this study, the above time-varying parameter (
) where
) in the TVP-VAR model was estimated using Bayesian theory. The Markov chain Monte Carlo (MCMC) method in the framework of Bayesian Inference was used for estimating the time-varying parameters. According to
Nakajima and Watanabe (
2012),
,
.
Table 1 illustrates the sampling steps using the joint posterior probability density function
and the MCMC method. The details of the steps are explained in
Nakajima and Watanabe (
2012) and
Nakajima (
2011).
In step 1, there is a possibility that the estimated value of fixed-parameter is unstable when the estimation period is short (
Nakajima and Watanabe 2012). In this case, the prior distribution of the initial value of the time-varying parameters of the first 10 samples is drawn from the normal distribution as prior data (
Primiceri 2005). The mean and covariance matrices of the prior distribution are determined by the ordinary fixed-parameter VAR model (
Kosumi 2016). Using the obtained average estimated values (
and the estimated values of the covariance matrix (
), the following normal distribution was set:
In the MCMC method, it takes some time for the Markov chain to converge to the target distribution, so the first part of the sample sequence was discarded. The expected value was calculated using the remaining samples, and it was determined whether the chain converged (
Kosumi 2016). In this study, the convergence test was performed with the following methods.
First, we examined the convergence by plotting the sample parameters using the MCMC method. We used the plots to find out whether the fluctuation of the sample is stable (
Kosumi 2016).
Second, the CD statistic proposed by
Geweke (
1991) was used. The CD statistic was used to identify whether the averages of the first to last sub-samples are the same. If the test suggested that the sample parameters converge to samples from the posterior distribution, and if the mean difference among the first to last sub-samples extracted became zero, then we could confirm that the parameters did converge.
3.3. Impulse Response Function
The impulse response method is a way to see how the innovation given to the error term of an equation propagates to the test variables. Since the models for the CNY and JPY are constructed in the same way, we only discuss the impulse response function for the CNY.
The TVP-VAR model of Equation (1) with two lags can be rewritten as follows:
Here, is a time-varying constant vector of (), is a time-varying coefficient matrix of (), and is an error term vector of (). The initial value of was set to zero ().
The impulse response function can be obtained by the following steps. First, let the value of when innovation is not given () be . Second, according to Equation (17), let the value in period be while the next period’s value is . The value of when innovation is given () is denoted as . Hence, the value of in period is and the next period’s value is .
Next, by calculating the difference between the case without and with innovations, the effect of innovation can be expressed as
. In this case,
can be expressed as:
Equation (18) is called the impulse response function, and the cumulative response function is defined for every lag period (t = 1, 2, …).
Finally, the pass-through rate on the LNG import price is defined as (cumulative impulse response to the foreign exchange shock of the import price)/(cumulative impulse response to the own monetary shock) (
Shioji 2010). Based on the cumulative response function, the pass-through rate on the Chinese LNG import price can be expressed as:
Here, is the pass-through related to the fluctuation of the CNY on the Chinese LNG import price, and is the accumulative impulse response of the CNY fluctuation shock on the Chinese LNG import price. Finally, is the accumulative impulse response to its own shock from the CNY fluctuation. All the impulse response function estimations were performed with OxMetrics6.01.
3.4. Data
The monthly average price from China Customs (
Wind 2019) was used for the LNG import price. The monthly average price released by the Petroleum Association of Japan (
Wind 2019) was used for the JCC crude oil price. Furthermore, the nominal effective exchange rate was the exchange rate used in the study. The CNY fluctuation is the monthly average nominal effective data published by the People’s Bank of China and the data were collected from Wind Net. The JPY represents the monthly average nominal effective data released by the Bank of Japan. The sample period covered in this study was from August 2005 to September 2018. All the data used in this study is provided as a
Supplementary Material.
As US dollars are the most commonly used currency in international trades, we used US dollars as the base unit for our variables. Thus, the JCC crude oil import prices and Chinese LNG import prices were converted to dollar-denominated prices for unifying the Japanese and Chinese markets. However, because the data of the variables have different units, they were standardized by using the following formula:
Here, is the normalized value of where denotes the variable of our interest (CNY, JPY, JCC crude oil import price, and Chinese LNG import price), while and are the mean and variance of .
Figure 1 is the plots of the standardized data of our variables
calculated from Equation (20). From this figure, we can see that the CNY (
) is more volatile than the JPY (
). It is also discernible from the figure that the China LNG import price (
) seems to fluctuate along with the Japanese crude oil price (
).