The Fertitly Rate in Japan
3. Research Method
(1) Research Design
The purpose of this study is to explore the dynamic relationship between housing price, fertility rate, household expenditure and other variables. Since there could exist unit root and structure change in the time series, we use CUSUM test and unit root test to examine the stationarity of these variables before the cointegration test or vector autoregression (VAR). If all the series are stationary, we then employ VAR
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model to examine their relationship. If all or parts of series are non-stationary, we then employ the cointegration test proposed by Johansen (1998) to analyze the variables of a long-term equilibrium relationship. If there is no cointegration relationship, we then apply “vector error correction model” (VECM) for further study. Finally, we use Granger causality test (Granger, 1969) to examine the lead-lag relationship between these variables.
A. Structural Change
Many potential factors could cause structural change, including the financial crisis, the stock market crash or policy shocks to disrupt economic development.
Perron (2005) indicated that structural change will bias the unit root test statistics toward the non-rejection of unit root. It is therefore necessary to confirm whether there is structural change during the research period. In this study, we apply the Cumulative Sum of the recursive residuals (CUSUM test) to test the structural change.
If there are structural changes in the sample series, we need to reduce the research period to no structural change to avoid the error in estimation.
B. Unit Root Test
Many macro-economic time series variables have the non-stationary characteristics. Non-stationary series, if caused by random shocks, may lead time series data not to converge to the original equilibrium. If we use non-stationary variables in the regression or other statistical models, results might be spurious (Granger and Newbold, 1974). Unit root tests provide basis for assessing whether if a time series is stationary, or integrated of a particular order. Hence, we employ Augmented Dickey-Fuller (ADF) unit root test proposed by Dickey and Fuller (1981) and Phillips-Perron (PP) unit root test proposed by Phillips and Perron (1988) to examine the existence of unit root.
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C. Cointegration Test
According to Engle and Granger (1987), if time series variable is non-stationary, it could become stationary after taking d-time difference, which means the series are integrated of the d order. When two non-stationary variables are integrated of the same order, and a linear combination relationship of them is stationary, then there is cointegration relationship in these variables. However, this approach can not examine more than two variables because there may be more than one cointegration vector. In this study, we use cointegration test proposed by Johansen (1988). The Johansen cointegration approach is a maximum likelihood estimation of a fully specified error correction model, which can examine more than one cointegration vector. This method is robust to interprete the multiple long-run equilibrium relationship between variables. Johansen (1988) proposed the Trace and Maximum Eigenvalue test to determine the number of cointegration.
⋋𝑡𝑟𝑎𝑐𝑒(𝑟) = −𝑇 ∑𝑛 ln(1 −⋋𝑖)
𝑖=𝑟+1
⋋𝑚𝑎𝑥 (𝑟, 𝑟 + 1) = −𝑇𝑙𝑛(1 −⋋𝑟+1)
T is the number of observations and ⋋𝑖 is the value of characteristic roots. The null hypothesis of the Trace is 𝐻0：𝑟𝑎𝑛𝑘 ≤ 𝑟. For the null hypothesis of the Maximum Eigenvalue test is 𝐻0：𝑟𝑎𝑛𝑘 = 𝑟.
D. Vector Autoregressive Model and Vector Error Correction Model
Vector Autoregressive model (VAR) is appropriate to explore the dynamic interrelationship between all series (Sims, 1980). In each equation, each variable can be represented by the lag periods of its own and other variables. According to Engle and Granger (1987), if there is cointegration relationship between variables and there
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is long-run equilibrium, we can combine the cointegration in the error correction model for Vector Error Correction Model (VECM). VECM is an appropriate model for a set of cointegrated variables. By VECM, we can explore the short-term dynamic adjustment to estimate the relationship between variables and how they affect each other.
E. Granger Causality Test
Granger causality test was proposed by Granger (1969). The main purpose of this methodology is to examine the existence of lead-lag relationship between two variables. In other words, it can investigate the ability of one series to predict another based on its past value. If current and past value of 𝑌𝑡 is able to predict future value of 𝑍𝑡, it is said that 𝑌𝑡 does Granger cause 𝑍𝑡. Moreover, if there is an interaction between two variables, then the result indicates there is feedback relation between two variables.
(2) Data Description and Processing
A. Data employed in this study is shown in Table 1. Fertility rates of Japan are collected from the Ministry of Health, Labor, and Welfare, and housing price index is from the Ministry of Land, Infrastructure, Transport and Tourism. Other variables (household expenditure, household disposable income, consumer price index, unemployment rate and rate) are collected from Taiwan Economic Journal (TEJ) database. All the variables are quarterly data. The research was conducted from Q1 of 1996 to Q4 of 2013 with 72 quarterly data in total.
B. Data Source
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Table 1 Data Source
Variable Code Source Time Period
C. Data Analysis and Processing
The description of statistical variables includes mean, median, maximum, minimum, and standard deviation, as shown in Table 2.
Table 2 Data Analysis
Variable Mean Median Maximum Minimum Std. Dev. Obs.
18 structural change for all variables at 5% significance level during the research period.
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