- 15 _-_

countries. These results provide additional evidence on the prevalence of departure from normality in foreign exchange auction data in Sub-Saharan Africa. However, in this case auction data could not be differentiated between the two groups of countries.

Concerning variability, it has been pointed out by Pritchett that when there is significant deparure from normality, the standard deviation may not be the appropriae indicator of uncerinty. For example the rates of changes in the auction rate and the premium in Zambia have almost similar standard deviations at 0.10 and 0.09, respectvely. However, they have two very different distributions, the auction rate has a non-symmetric and substantially left skewed kurtotic distribution, while the premium had a kurtotic but symmetric distnbution. Another point mentioned by Pritchet is that under departure from normality, there may be several but very contadictory measure of variability, none of which has a clear advantage over die others. This point is validated in our case by comparing the standard deviations of the auction rates in Nigeria and Zambia which are very smal at 0.08 and 0.10, respectively, to the ones for Uganda at 1.98 and Ghana at 9.91. This, however, -conzradicts Figure 1 which shows that exacdy the opposite happens.?

# 3.3 Unit Roots, Sructural Shifts and Stationarity Issues.

The hypothesis of the unit root (loosely speaking, this refer to a time series wi an infinite variance at the limiit) in the individual economic time series data has geeratd considerable attendon and acceptability in recent empirical and theoretic,' economic research. The seminal paper by Nelson and Plosser (1982) which found that most macroeconomic varibles have a univara time series structure with a unit root, and the more recent path-breaking work on cointegracion theory9 (Engle and Granger, 1987) has provided the impetus for a burgeoning research programn that has already started to have far-reaching implications for both theory and empincal work in economics. The most important economic implication of the unit ro hypothesis is that random shocks have permanent effects on economic variables. In our context this means that the fluctataions that chracterize the auction variables are not ranitory. Therefore, it is important to conduct careful testing of the unit D Perhaps this contradiction coud bc resolved by using the coefficent of variation rahe than the staniard deviation.

'Formally. Id y= TDt, + Z, bc an econonic series composed of a deterministic trnd TD1 and a 5ochastic component. For simplicity assune that 4 ca be **desctibed** by an autoregrcssiv_moving averagc process: A(L), = B(L) cj, whe A(LJ and B(L) arc polynomials in the lag operator fr and e, is a sequence of iLd. innovaton. The noise finction Z7 is asumed to have mea zero, the moving avenage polnomial is also assumed to have roots srictly outside the ni circl. Then Z, has a unit root if A(L) has one unit root and all ado roots stiey outside the unit circle In this case (I-L)Z, = AZ, is a stationmy procss and (1-Lby, = Ay, is staionary around a fixed man, If on the other hand A(L) has all its roots outside the unit circle, the Z is a statinr process and _y,_ is stationary aound a trend-

-Thc idca of conte_tio basically states that even though individual series may have a unit root, thcre may cxst9 various linear combinations of variables which are _statay._ Stated more formally in the context of te definigion of footnote 1, kt the n-vector y, be comnposed of (yb ... yj. whcem Ya i dcfined in footnote I above. Then y, is sai to be coinegmted if there exists at ast one n-elmnt vector 0 such that 'y, is trd sttonay. This is a mider definition of cointegrtion (Campbell and Pern, 1991), which is more suited to anablsis of eoonomic date since it permits the inion of ddernmntic components (such as tends and structural break dummics) in the cointegration model along wiith other z-stuuionary stochastic vanabls.