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...covered time horizon, namely from 1858 to 1938, is characterized by a number of very important events, the nature of which is either economic or historical. In addition, time series stationarity is checked through a Kwiatkowski, Phillips, Schmidt, and Shin (KPSS) test. (JEL C12, C13, C22, C52)
Introduction
The investigation of time series for the presence of a unit root usually precedes the use of several econometric techniques, like ordinary least squares, time series and spectral analysis, and co-integration and error correction modeling since the stability of time series or the determination of their order for integration is essential for their application. The stability of a time series could be graphically examined with a correlogram of its level. More specifically, the rapid (slow) geometrical convergence of the graph on autocorrelation function towards zero is indicative of a stationary (non-stationary) process. The results of this methodology may, however, turn out to be quite questionable. For example, in case of a nearly integrated time series, i.e., a time series which converges to its long-run equilibrium value very slowly, its slow decay autocorrelation function may lead to the false conclusion that the considered time series is non-stationary.
Other procedures, which might be used to determine the presence of a unit root in a time series, are the ones proposed by (1) Dickey and Fuller [1979; 1981], (2) Kwiatkowski et al. [1992], and (3) Phillips and Perron [1988] who drew a unit root test using non-parametric statistical methods. (1)
Various Dickey-Fuller and Phillips-Perron test statistics are biased toward the acceptance of the unit root null in the presence of structural breaks, i.e., structural breaks reduce the power of unit root test. Therefore, Perron [1989; 1990; 1994; 1997], Zivot and Andrews [1992], Banerjee et al. [1992], Perron and Vogelsang [1992a; 1992b; 1998] have developed tests, in the context of which the significance of unit root null is tested, allowing for a break in a time series and choosing the break date either exogenously or endogenously. Moreover, Lumsdaine and Papell [1997] developed a methodology, in the context of which the unit root hypothesis is investigated allowing for two breaks in a time series with the break dates to be chosen endogenously.
The methodologies, used in the present article, were developed by (1) Kwiatkowski et al. [1992], (2) Perron and Vogelsang [1998], and (3) Lumsdaine and Papell [1997]. Covering the period between 1858 and 1938 and using annual data, the time series of Greek real GDP and real money supply (measured in natural logarithms) are tested for the presence of a unit root, allowing at most for two breaks that take place at an unknown point in time.
Methodology
In the context of a Kwiatkowski, Phillips, Schmidt, and Shin (KPSS) test, the following procedure begins with the determination of the following statistic:
[arc.[eta]](l) = [[T.sup.-2]/[[s.sup.2](l)]][T.summation over (t=1)][S.sub.t.sup.2] (1)
where: t = 1, 2,...., T, l = o([T.sup.1/2]), and T = the sample size,
[S.sub.t] = [t.summation over (i=1)][e.sub.i] (2)
[s.sup.2](l) = [T.sup.-1][T.summation over (t=1)][e.sub.t.sup.2] + 2T[l.summation over (s=1)]w(s,l)[T.summation over (t=s+1)][e.sub.t][e.sub.t-s] (3)
with:
w(s, l) = 1 - s/(l + 1) (4)
Given that l = o([T.sup.1/2]), [arc.[eta]](l)--statistic is calculated for any integer value of l within the interval [0, 9) and is denoted as [arc.[eta].sub.u](l)[[arc.[eta].sub.r](l)] when the residuals [e.sub.t] result after the regression of the examined time series on a constant (on a constant and a time trend).
On a second step, the estimated via (1) [arc.[eta].sub.[mu]](l) or [arc.[eta].sub.r](l) statistic is compared with the appropriate critical values which are provided by Kwiatkowski et al. [1992] (Table 1, page 166). More specifically, the unit root hypothesis is not rejected if the value of the test statistic, for a given value of parameter l, is greater than the critical value at a specific significance level.
Following Perron [1989; 1990], the trend shifts are modeled by two general groups of models. The first group includes the so-called 'additive outlier' models (AO), which permits sudden occurrence of a break in the trend of a time series. The second group includes the 'innovational outlier' models (IO), which allow for a break that is completed slowly over time and not within a time period (like in the case of AO models).
Three forms of breaks will be considered. The first (A) and second (B) forms are related with a positive or negative change in the mean and the slope of the examined time series respectively. The third form (C) is referred to a positive or negative change both in the mean and the slope of the time series' trend.
In the context of AO models, the investigation for a unit root in the time series {[Y.sub.t]}[.sub.t=1.sup.T] involves a three-step procedure. In the first step, the ordinary least squares (OLS) method is used to estimate one of the following equations: (2)
Form (A): [Y.sub.t] = [[mu].sup.A] + [[beta].sup.A]t + [[theta].sup.A]D[U.sub.t] + [~.Y.sub.t] (5)
Form (B): [Y.sub.t] = [[mu].sup.B] + [[beta].sup.B]t + [[gamma].sup.B]D[T*.sub.t] + [~.Y.sub.t] (6)
Form (C): [Y.sub.t] = [[mu].sup.C] + [[beta].sup.C]t + [[theta].sup.C]D[U.sub.t] + [[gamma].sup.C]D[T*.sub.t] + [~.Y.sub.t] (7)
where: D[U.sub.t] = 0(1) if t [less than or equal to] [T.sub.B](t...
NOTE: All illustrations and photos
have been removed from this article.
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