7-16 Characteristics ofI(O),I(1)and I(2) Series differencing twice to induce stationarity p31 ould require An i(2) series contains two unit roots and so we 1and i(2) series can wander a long way from their mean value and cross this mean value rarely I(O)series should cross the mean frequently The majority of economic and financial series contain a single unit root, although some are stationary and consumer prices have been argued to have 2 unit roots c Chris Brooks2002陈磊2004
© Chris Brooks 2002 陈磊2004 7-16 Characteristics of I(0), I(1) and I(2) Series • An I(2) series contains two unit roots and so would require differencing twice to induce stationarity. p376 • I(1) and I(2) series can wander a long way from their mean value and cross this mean value rarely. • I(0) series should cross the mean frequently. • The majority of economic and financial series contain a single unit root, although some are stationary and consumer prices have been argued to have 2 unit roots
7-17 1.4 Test for a unit root 无法使用ac域或pac确定序列是否是单位根过程 The early and pioneering work on testing for a unit root in time series was done by dickey and Fuller dickey and Fuller 1979, Fuller 1976). The basic objective of the test is to test the null hypothesis that =l in y,=yutu against the one-sided alternative o <l. So we have Ho: series contains a unit root VS. H: series is stationary. We usually use the regression Ay=yy+ a test of -l is equivalent to a test of y0(since -1=y c Chris Brooks2002陈磊2004
© Chris Brooks 2002 陈磊2004 7-17 1.4 Test for a unit root • 无法使用acf或pacf确定序列是否是单位根过程。 • The early and pioneering work on testing for a unit root in time series was done by Dickey and Fuller (Dickey and Fuller 1979, Fuller 1976). The basic objective of the test is to test the null hypothesis that =1 in: yt = yt-1 + ut against the one-sided alternative <1. So we have H0 : series contains a unit root vs. H1 : series is stationary. • We usually use the regression: yt = yt-1 + ut a test of =1 is equivalent to a test of =0 (since -1=)
7-18 Different forms for the df Test Dickey Fuller tests are also known as t tests: t, us T' The null (ho and alternative(hi models in each case are i)H0:y1=y-1+a H1:y=pyn1+upφ<1 This is a test for a random walk against a stationary autoregressive process of order one(ar(l) 1i) Ho: y=ya+u t H1:y=φy21++ut,p<1 This is a test for a random walk against a stationary ar(i) with drift 0: V=v+u, 1 :y=ova++nttu, o< This is a test for a random walk against a stationary ar(i)with drift and a time trend c Chris Brooks2002陈磊2004
© Chris Brooks 2002 陈磊2004 7-18 Different forms for the DF Test • Dickey Fuller tests are also known as tests: , , . • The null (H0 ) and alternative (H1 ) models in each case are i) H0 : yt = yt-1+ut H1 : yt = yt-1+ut , <1 This is a test for a random walk against a stationary autoregressive process of order one (AR(1)) ii) H0 : yt = yt-1+ut H1 : yt = yt-1++ut , <1 This is a test for a random walk against a stationary AR(1) with drift. iii) H0 : yt = yt-1+ut H1 : yt = yt-1++ t+ut , <1 This is a test for a random walk against a stationary AR(1) with drift and a time trend
7-19 Computing the df Test statistic · We can write Ay=ut where Ay,=yr y-1, and the alternatives may be expressed as Ay,=yy+u+t +u, with FFA=0 in case i), and 1=0 in case ii)and y=o-1. In each case, the tests are based on the t-ratio on the y-i term in the estimated regression of Ay, on ytI, plus a constant in case ii)and a constant and trend in case iii). The test statistics are defined as test statistic=∧ sE(y) The test statistic does not follow the usual t-distribution under the null, since the null is one of non-stationarity, but rather follows a non-standard distribution. Critical values are derived from monte Carlo experiments in, for example, Fuller (1976). Relevant examples of the distribution are shown in table 7.l below c Chris Brooks2002陈磊2004
© Chris Brooks 2002 陈磊2004 7-19 Computing the DF Test Statistic • We can write yt =ut where yt = yt - yt-1 , and the alternatives may be expressed as yt = yt-1++ t +ut with ==0 in case i), and =0 in case ii) and = -1. In each case, the tests are based on the t-ratio on the yt-1 term in the estimated regression of yt on yt-1 , plus a constant in case ii) and a constant and trend in case iii). The test statistics are defined as test statistic = • The test statistic does not follow the usual t-distribution under the null, since the null is one of non-stationarity, but rather follows a non-standard distribution. Critical values are derived from Monte Carlo experiments in, for example, Fuller (1976). Relevant examples of the distribution are shown in table 7.1 below SE( )
7-20 Critical values for the df test Significance level% CV.for constant-2.57-2.86-343 t no trend V. for constant-3.12-3.41 3.96 and trend Table 4.1: Critical Values for DF and ADF Tests(Fuller 1976,p373) The null hypothesis of a unit root is rejected in favour of the stationary alternative in each case if the test statistic is more negative than the critical value 临界值还与样本容量有关,参见675页表A27。 c Chris Brooks2002陈磊2004
© Chris Brooks 2002 陈磊2004 7-20 Critical Values for the DF Test The null hypothesis of a unit root is rejected in favour of the stationary alternative in each case if the test statistic is more negative than the critical value. 临界值还与样本容量有关，参见675页表A2.7。 Significance level 10% 5% 1% C.V. for constant but no trend -2.57 -2.86 -3.43 C.V. for constant and trend -3.12 -3.41 -3.96 Table 4.1: Critical Values for DF and ADF Tests (Fuller, 1976, p373)