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To develop a way to analyze the dynamic relationship between the economic growth and the level of inflation in Japan, we propose a Bayesian regression method to estimate the dynamic dependence of the stationary component of GDP on a stationary component of the CPI. First, we extract stationary components from original GDP and CPI time series using a set of state space models. Then, we construct a set of Bayesian regression models with a time-varying coefficient. We also analyze the dynamic relationship between the stationary components of GDP and the CPI using Japanese economic statistics from 1980 to 2005.

One of the most fundamental objectives of macroeconomic policies is to sustain high economic growth together with a fair inflation level. Thus, the question is: what level of inflation is fair? It is very difficult to answer this question, because it obviously depends on the nature and structure of an economy. That is, the answer to the question will vary from country to country, and within a specific country it will also vary over time. However, a guide to a fair inflation level can obtained from empirical studies.

To test the hypothesis that inflation has a long-run impact on output, [

These studies prompted us to analyze the relationship between real gross domestic product (GDP), which is a basic indicator of economic growth, and the consumer price index (CPI), which measures the level of fluctuations, in Japan. However, there are difficulties in undertaking such an analysis. The first problem is that GDP data are presented as a quarterly time series, while those for the CPI are monthly.

Another problem in the analysis is the dynamics in the relationship between GDP and the CPI. Regression analysis models are often used for relationship analysis with constant regression coefficients, the implication being that no structural changes occur. However, when the study period spans several decades, it is clearly unrealistic to assume constant coefficient parameters. Thus, the conventional approaches are considered inadequate for the analysis of business cycles with long-term time series. [

The first step in analyzing the dynamic relationship between GDP and the CPI is to extract the stationary components from each original time series. Then, we present a method to analyze the dynamic relationship between the stationary components of GDP and the CPI using Bayesian dynamic modeling. There are two points in the relationship between GDP and the CPI, i.e., the lead-lag relationship and the time-varying dependence between these two indicators. These are considered by introducing a lag parameter and time-varying coefficients into a set of Bayesian dynamic models.

The rest of this paper is organized as follows. In Section 2, we introduce a method for estimating a stationary component from quarterly or monthly time series data. In Section 3, we present our models and parameter estimation methods for the proposed approach. An empirical study based on the proposed approach is presented in Section 4. Section 5 concludes.

As mentioned above, the first step in analyzing the relationship between GDP and the CPI is the estimation of the stationary components of GDP and the CPI. Thus, we introduce a method for estimating the stationary components from the original GDP and CPI time series.

For quarterly GDP time series y m , we consider a set of statistical models as follows:

y m = t m y + s m y + r m y + w m y , (1)

t m y = 2 t m − 1 y − t m − 2 y + v m 1 y , (2)

s m y = − s m − 1 y − s m − 2 y − s m − 3 y + v m 2 y , (3)

r m y = ∑ j = 1 p α j r m − j y + v m 3 y ( m = 1 , 2 , … , M ) , (4)

where t m y , s m y , and r m y are the trend component, the seasonal component, and the stationary component, respectively, of the time series y m . In addition, p represents the order of an autoregressive model for the stationary components and α 1 , … , α p are the AR coefficients. w m y ~ N ( 0, σ 2 ) is the observation noise, while v m 1 y ~ N ( 0, τ 1 2 ) , v m 2 y ~ N ( 0, τ 2 2 ) , and v m 3 y ~ N ( 0, τ 3 2 ) are system noises for each component model. It is assumed that w m y , v m 1 y , v m 2 y , and v m 3 y are independent of one another.

When the model order p and the hyperparameters α 1 , … , α p , σ 2 , τ 1 2 ,

Further, to estimate a stationary component in a monthly CPI time series, we use a set of models similar to that in (1)-(4), as follows:

where q represents the order of an AR model for the stationary component and

To analyze the dynamic relationship between quarterly GDP and the monthly CPI, we propose an approach based on a set of two-mode regression models with time-varying coefficients (TMR-TVC).

We classify GDP growth into two states, the upside mode corresponding to the situation in which the stationary component of GDP continues to increase, and the downside mode corresponding to the situation in which it continues to decrease. We consider that the relationship between GDP and the CPI may differ according to the situation. Thus, we use different models for the two modes.

For the upside mode, the TMR-TVC models are given in the form of a regression model with time-varying coefficients as follows:

where

The lag

The models in (9)-(12) are essentially Bayesian linear models in which the model in (9) defines the likelihood and the models in (10)-(12) form a second-order smoothness prior for the time-varying coefficient. Thus, we can estimate the time-varying coefficient with optimal smoothness on

Similar to the upside mode, the TMR-TVC models for the downside mode are given as

with

Below, we only show the methods for estimating the hyperparameters in the TMR-TVC models for the upside mode because those for the downside mode are similar.

Now, we set

with

In the state space model comprising (17) and (18), the time-varying coefficient

Let

Given the values of

[Prediction]

[Filter-1]

[Filter-2]

Note that for each value of m, when the time point m is in an upside period, we use Filter-1, otherwise Filter-2 is applied.

Based on the results of the Kalman filter, we can obtain an estimate for

[Fixed-interval Smoothing]

Then, the posterior distribution of

Given the time series data

where

where

Moreover, for a fixed value of

Thus, an estimate

Information about the value of the lag

Thus, we can analyze the lead-lag relationship between GDP and the CPI from the distribution of the relative likelihood on

Here, we present an empirical study analyzing the relationship between real GDP and the CPI in Japan. The real GDP data were obtained from the Cabinet Office, Government of Japan, while the CPI data were obtained from the website of the Ministry of Internal Affairs and Communications, Japan.

1980Q1-2005Q4.

For simplicity of parameter estimation, we adjust the scale for the GDP time series. Specifically, letting

In addition, because

In the analysis below, we use the scale-adjusted time series

To estimate the stationary component in GDP, we compute the likelihoods for the models in (1)-(4) for

to

absolute values have continued to rise. This implies that sometimes there are positive effects of GDP on CPI and such effects become stronger.

To analyze the relationship between economic growth and inflation in Japan, we proposed a Bayesian dynamic linear modeling method and used it to analyze the dynamic relationship between quarterly GDP data and monthly CPI data in Japan. First, we extracted stationary components from GDP and CPI time series using a set of state space models. Then, we constructed a set of Bayesian regression models with a time-varying coefficient. These models are two-mode regression with time-varying coefficient (TMR-TVC) models.

It should be emphasized that there are two important parameters in the TMR-TVC models: the time lag between GDP and the CPI and the time-varying coefficient. From the value of the lag, we can determine the lead-lag relationship, while from the estimate of the time-varying coefficient, we can analyze the dynamic relationship between GDP and the CPI.

Finally, using an empirical study based on the proposed method, we analyzed the dynamic relationship between the stationary components of GDP and the CPI using Japanese economic statistics from 1980 to 2005. The empirical study produced the following results. 1) The lead-lag relationship between GDP and the CPI is complicated. 2) The CPI had a negative effect on GDP from the time of the collapse of the bubble economy in Japan, and sometimes there are positive effects of GDP on the CPI in which the effects become stronger during the expansion phase. 3) GDP has a negative effect on the CPI, and this effect becomes stronger in periods of recession.

This work is supported in part by a Grant-in-Aid for Scientific Research (C) (16K03591) from the Japan Society for the Promotion of Science. We thank Geoff Whyte, MBA, from Edanz Group (www.edanzediting.com/ac) for editing a draft of this manuscript.

Kyo, K. (2018) The Dynamic Relationship between Economic Growth and Inflation in Japan. Open Journal of Social Sciences, 6, 20-32. https://doi.org/10.4236/jss.2018.63003