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Article Excerpt
Abstract
In this article, the nominal, real, and real per capita GDP series are modeled in the UK by means of fractionally integrated techniques. A version of the tests of Robinson [1994] are used that permits the incorporation of deterministic trends with no effect on the standard (normal) limit distribution of the tests. The results show that the nominal GDP appears to be nonstationary and non-mean reverting, with its order of integration being much higher than 1. The real GDP also appears to be nonstationary and the order of integration fluctuates widely between 0.6 and 1.2. Finally, the real per capita GDP seems to be stationary, either I(0) or 1(d), with D positive but close to 0. (JEL C22)
Introduction
This article is concerned with the different ways of modeling the UK output. Traditionally, deterministic approaches based on linear (or quadratic) functions of time were employed. Using this approach, it was assumed that the series was stationary once it had been detrended. Later on, and especially after the seminal paper of Nelson and Plosser [1982], stochastic approaches were considered. In that paper, following the work and ideas of Box and Jenkins [1970], they showed that many series became stationary after taking first differences. In this context, it is said that the series contains a unit root and many test statistics have been developed for testing unit root models (for example, Dickey and Fuller [1979]; Phillips and Perron [1988]; and Kwiatkowski et al. [1992]). In the last few years, however, the unit root model has been extended to allow for a fractional degree of integration.
Suppose that ([u.sub.t], t = 0, [+ or -]1, ... ) is an I(0) process, defined for the purpose of the present paper as a covariance stationary process with spectral density function that is positive and finite at the zero frequency. It is then said that [x.sub.t] is I(d) if:
[(1 - L).sup.d] [x.sub.t] = [u.sub.t], t = 1, 2, ...
for any real value d. Clearly, if d = in (1), then [x.sub.t] = [u.sub.t], and a weakly auto-correlated [x.sub.t] is allowed for. However, if d > 0, then [x.sub.t] is said to be a long memory process, so called because of the strong association between observations widely separated in time. This type of models was introduced by Granger and Joyeux [1980], Granger [1980; 1981] and Hosking [1981] and stationary fractional models (with d < 0.50) were shown by Granger [1980] and Robinson [1978] to arise from aggregation of ARMA series with randomly varying coefficients. Parke [1999] theoretically justifies I(d) models in terms of the duration of the shocks. A complete survey of the literature based on these models can be found in Baillie [1996] and recent empirical studies based on fractional models are, among others, Hauser et al. [1998] and Gil-Alana [2001a].
This article investigates the nonstationarity in the UK output by means of fractionally integrated techniques. For this purpose, a version of the tests of Robinson [1994] is made use of that permits the incorporation of deterministic trends and different types of disturbances with no effect on the limit distribution of the tests. The outline of the paper is as follows: the next section briefly describes the version of Robinson's [1994] tests used in the paper. In the following section, the tests are applied to several series corresponding to the UK output, while the final section contains some concluding comments.
Testing of I(d) Models
Following Bhargava [1986], Schmidt and Phillips [1992], and others of parameterization of unit root models, consider the regression model:
[y.sub.t] = [beta][z.sub.t] + [x.sub.t], t = 1, 2, ... (2)
where [y.sub.t] is the time series observed; [beta] is a (kx1) vector of unknown parameters; [z.sub.t] is a (kx1) vector of deterministic regressors, and the regression errors, [x.sub.t], are of form as in (1) with I(0) [u.sub.t].
Robinson [1994] proposed a Lagrange Multiplier (LM) test of the null hypothesis:
[H.sub.0] : d = [d.sub.0], (3)...
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