## Working Papers > Time Series Methods

Cointegration Tests Using Instrumental Variables with an Example of the U.K. Demand for Money

Walter Enders, Kyung So Im, and Junsoo Lee

Abstract:

Walter Enders, Kyung So Im, and Junsoo Lee

Abstract:

In this paper, we propose new cointegration tests based on stationary instrumental variables in a single equation model as well as in a system of equations. An important property of our tests is that the asymptotic distribution is standard normal or chi-square. As such, the asymptotic distribution of the IV tests does not depend on the number of the regressors, differing deterministic terms, structural changes, and even statinary covariates. Thus, our IV cointegration tests have operating advantages in the presence of nuisance parameters. Moreover, we show that including stationary covariates increases considerably the power of the tests without affecting size. We illustrate the use of the tests by examining the demand for money in the U.K.

The Flexible Fourier Form and the Dickey-Fuller Type Unit Root Tests.

Walter Enders and Junsoo Lee.

Abstract

Walter Enders and Junsoo Lee.

*Economics Letters*. ForthcomingAbstract

We suggest a new unit-root test with a Fourier function in the deterministic term in a Dickey-Fuller type regression framework. Our suggested test can complement the Fourier LM and DF-GLS unit root tests. They have good size and power properties.

A Unit Root Test Using a Fourier Series to Approximate Smooth Breaks

Walter Enders and Junsoo Lee

(Oxford Bulletin of Economics and Statistics)

(http://onlinelibrary.wiley.com/doi/10.1111/j.1468-0084.2011.00662.x/pdf)

Abstract:

Walter Enders and Junsoo Lee

(Oxford Bulletin of Economics and Statistics)

(http://onlinelibrary.wiley.com/doi/10.1111/j.1468-0084.2011.00662.x/pdf)

Abstract:

We develop a unit-root test that relies on a simple variant of Gallant’s (1981) Flexible Fourier Form. The test is based on the fact that nonlinearities of an unknown form, including structural change, can often be captured using the low frequency components of a Fourier approximation. Hence, instead of modeling the nonlinearities or selecting specific break dates, the specification problem is transformed into selecting the proper frequency components to include in the estimating equation. It is shown that the Fourier approximation does reasonably well for the types of nonlinearities and breaks often used in economic analysis. The appropriate use of the test is illustrated using several interest rate spreads.

A General Test For Time-dependence in Parameters

Ralf Becker, Walter Enders and Stan Hurn

(

Abstract:

Ralf Becker, Walter Enders and Stan Hurn

(

*Journal of Applied Econometrics*19, 2004. pp. 899–906.)Abstract:

We propose a new test based on a Fourier series to approximate the unknown form of a nonlinear time-series model. The test has good size and power properties to detect structural breaks, seasonal parameters and random coefficients. Moreover, it has reasonable power to discriminate between nonlinearity in variables and nonlinearity in parameters. We use the test to show that U.S. inflation is appropriately estimated with a time-varying intercept that jumps in the late 1960’s, peaks in the early 1980’s and then begins to decline. German income and consumption data is used to illustrate the ability of the test to suggest the form of the nonlinearity.

Modeling Inflation and Money Demand Using a Fourier Series Approximation.

Ralf Becker, Walter Enders and Stan Hurn

In C. Milas, P. Rothman, and D. van Dijk, eds.

Ralf Becker, Walter Enders and Stan Hurn

In C. Milas, P. Rothman, and D. van Dijk, eds.

*Nonlinear Time Series Analysis of Business Cycles*. (Elvesier: Amsterdam) 2006. pp. 221–44. The paper develops a test with the null of stationarity that allows for the possibility of an unknown number of structural breaks, or other nonlinearities, in the data-generating process. The test is based on the fact that the behavior of a breaking process can often be captured using a single frequency component of a Fourier approximation. Hence, instead of selecting specific break dates, the number of breaks, and the form of any nonlinearities, the specification problem is transformed into selecting a low frequency component to include in the estimating equation. Our proposed test does not exhibit any serious size distortions, and shows reasonable power. The appropriate use of the test is illustrated using real exchange rates in the post-Bretton Woods period.