Estimation of a fractionally cointegrated demand system: evidence from the Japanese expenditure data

Abstract

In this article, we investigate the exact integration of series and analyse the cointegration relationship in the Japanese demand system. We find that the series considered in this analysis follow a fractionally I(d) process, not an exact I(1) one. Further, the Japanese demand system comprises the fractional cointegration relationships. Therefore, the use of a standard cointegration analysis might have a possibility of misleading the demand system with a fractionally cointegrated I(d) process.

Keywords:

• cointegrated demand system
• Japanese expenditure data
• fractional integration
• cointegration rank test

JEL Classification:

• C32
• D12

Notes

¹ The fractional unit root test of a univariate score test for integration against the fractional alternative has been also suggested by Robinson (1994) and Tanaka (1999).

² They determine the cointegration rank of a fractionally integrated time series by solving a generalized eigenvalue problem analogous to that in Johansen (1995).

³ Recently, the AID system model has often been used to estimate the standard cointegration system (Ng, 1995; Duffy, 2002).

⁴ All series were seasonally unadjusted because the use of seasonally adjusted data may distort the estimation of long-run relationships. Therefore, we adjust for seasonality by using the dummy variables for each year over Quarters 1–3.

⁵ For example, in the standard ADF test (Dickey and Fuller, 1979), d ₀ = 1 and d ₁ = 0.

⁶ To simplify our discussion, we test the unit root under a priori known value d ₁.

⁷ Under adding-up condition, the income elasticity for fuel normally becomes to be smaller than 1.

⁸ In Hashimoto (2004), the own price elasticity also does not satisfy the negative condition in some commodities.

⁹ Ng (1995) showed that if the nonstationary variables in the demand system were not cointegrated, the tests for homogeneity would not be very meaningful because of the conflict between the model and the data. That is, the homogeneity restriction is a long-run property, and hence the demand theory should be satisfied in the long-run relationship.

¹⁰ In addition, we should note that the use of the annual data series may not yield desirable results in our analysis; using these data would pose difficulties in detecting the fractional unit root of their series in the case of Japanese expenditure data.