Market Models Table 11.1: Dickey-Fuller tests on equity indices

the first column of Table 11.1. None of the time series appear to be stationary even at 10% (the 10% critical value of DF is -2.57).

The null hypothesis that equity index prices are non-stationary cannot be rejected. This does not necessarily mean that they are generated by 7(1) processes: they may be non-stationary because they are generated by 1(2) processes, or by integrated processes of an even higher order. So the next step is to repeat the procedure but this time using Ay, instead of y„ that is, to test F£0: Ay, ~ 1) against H,: Ay, ~ 0), or equivalently

H0:j, ~/(2) against H,:j, ~ 1(1).

In efficient markets the logarithm of prices follows a random walk, but not all markets are efficient

Estimating DF regressions of the form A2y, = + pAj>, i + e, for y, being the log equity index prices, gives the DF statistics shown in the second column of the table. The null hypothesis is obviously rejected at the highest significance level and we can conclude that these equity index log prices have followed an 1) process during the period. Readers can perform similar tests using the ADF workbook and the PcGive program provided on the CD.

In efficient markets the logarithm of prices follows a random walk, but not all markets are efficient. In some energy markets prices are dominated by supply constraints (storage, transport, cartel restrictions, etc.) and by demand fluctuations (weather conditions and so on). Consider the daily data for NYMEX prompt futures prices of natural gas, from 1 January 1998 to 3 March 1999, shown in Figure 11.4.4 Natural gas storage facilities play a crucial role in balancing supply and demand. In summer months excess production is injected into storage, and in the winter months the storage gas is withdrawn to supply the extra demand. In the winter, when demand typically exceeds production, one would expect to see spot prices rise sharply during periods of extreme cold. Futures prices may also rise because depleting storage may raise future price expectations. However, the winters of 1997/98 and 1998/99 were very mild in North America and storage was filled to capacity during the autumn months of 1997 and 1998, so futures prices responded little to daily demand fluctuations.

4The futures contract represents the price for delivery of equal volumes over the entire calendar month represented by the futures contract. Many thanks to Enron for providing these data.

ADF tests simply add lagged dependent variables to the DF regression. The number of lags included should be just sufficient to remove any autocorrelation in the errors

Jan-98 Mar-98 May-98 Jul-98 Sep-98 Nov-98 Jan-99 Mar-99 Figure 11.4 NYMEX prompt futures prices for natural gas.

If the DF test is applied to the gas daily log futures prices the DF regression is

Ay, = 0.047 -0.063y, u

(2.81) (-2.87)

where y, = InF, and F, is the futures price as time t. The DF statistic of -2.87 allows one to accept the hypothesis that y, is stationary at the 5% level. But this is rare. The natural gas market is an energy market that lacks efficiency, mainly because of storage and transportation constraints. Most financial markets are much more efficient, and it is unusual to find price data that are stationary.

It is fortunate that Dickey is the first named author because subsequent refinements of the test are now referred to as augmented Dickey-Fuller (ADF) tests, rather than the other way around. ADF tests simply add lagged dependent variables to the DF regression. The number of lags included should be just sufficient to remove any autocorrelation in the errors, so that OLS will give an unbiased estimate of the coefficient of j>, , (§A.1.3). Slightly different critical values apply in the ADF(m) test,5 but otherwise the general principle of testing the significance of the coefficient on y, x is similar to the DF test: Ay, is regressed on a constant, y, x and m lags of Ay,. The hypotheses

H0: , ~ 7(1) against H,: jf ~ 7(0)

are equivalent to

H0: P = 0 against H0: p < 0

in the model

Ay, = + Bj, + Ay, i + . . . + amAy, m + ,. (11.11)

5Critical values of the ADF statistic for different values of m and different sample sizes are given in MacKinnon (1991).

The ADF test statistic is

ADF(m) = ft/(est. s.e. b), where b is the OLS estimate of p in the ADF regression (11.11).

11.1.6 Testing for the Trend in Financial Markets

If a time trend is included in the ADF regression, the test is for H0: , ~ 1) against H,: y, ~ ) + trend.

This test will distinguish between a stochastic and a deterministic trend. Different critical values apply. For example, with 250 observations the ordinary DF statistic has 5% critical value of -3.43 and the 1% critical value is -3.99. But DF tests for the presence of a deterministic trend are not as powerful as other tests, such as the Durbin-Hausmann or Schmidt-Phillips tests described below.

There are a large number of unit root tests for stochastic against deterministic trends other than the basic ADF test just described, and most of them are more powerful in many circumstances

There are a large number of unit root tests for stochastic against deterministic trends other than the basic ADF test just described, and most of them are more powerful in many circumstances. The tests proposed by Durbin and Hausmann (see Choi, 1992) are uniformly more powerful than DF tests in the presence of a deterministic trend. The maintained model of the Durbin-Hausmann test is the same as that for DF tests of H0: , ~ 1) against H yt ~ ) + trend. That is,

Ay, = + at + Pj, ! + e,. The Durbin-Hausmann test statistic for H0: p - 0 versus H,: p < 0 is

DH =

(fry - bf est. V(b)

where b denotes the OLS estimate of P and biv denotes the instrumental variables estimate of P using y, to instrument jvi- Another test, proposed by Schmidt and Phillips (1992), has a polynomial in the time trend in the maintained model, but since there is limited interest in presence of deterministic trends in financial data, the details are not described here.

Dickey-Fuller tests assume the errors in an ADF regression are i.i.d.(0, ct2), but less restrictive assumptions on the errors are possible. For example, the Phillips-Perron test allows errors to be dependent with heteroscedastic variance (Phillips and Perron, 1988). The theory of Phillips-Perron tests is quite complex and will not be covered here. But they are more useful than DF tests when the data have GARCH effects and they are available as an in-built procedure in standard econometrics packages. An empirical example of the application of Phillips-Perron tests is given in Corbae and Ouliaris (1986) to determine whether exchange rates follow a random walk.