' COMPUTATION OF ASYMPTOTIC CRITICAL VALUES FOR THE H(c) and H(L) COINTEGRATION TESTS WHEN THERE ARE 2 or 3 STRUCTURAL BREAKS ' ============================================================= ' Written by David Giles, Dept. of Economics, University of Victoria. ' Last Modified: 28 June, 2011. ' The total sample size is T. The number of sub-samples due to the structural breaks is "q". So, q=2 corresponds to one structural break, and q=3 corresponds to 2 structural breaks. If q=1 there are no structural breaks. ' The break-points occur at T1, when q=2, and at T1 and T2, when q=3. ' The break-point proportions are V1=(T1/T), when q=2; and V1 and V2=(T2/T), when q=3. ' (If q=2, the value of V2 below is not used) ' r = cointegrating rank ' p = no. of variables in system ' To cross-check the results for the H(c) test with the last column of Table 5 in Johansen et al., set q=1 (V1 and V2 can be left at their current values as they will not be used) '=================================== ' EDIT THE NEXT 3 LINES, AS APPROPRIATE: scalar q=3 scalar V1=0.3 scalar V2=0.85 '================================ 'EDIT THE NEXT 2 LINES IF P-VALUES ARE REQUIRED scalar traceL=123.6 scalar traceC=114.7 '================================ vector(2) c2 vector(3) c3 vector(2) c4 vector(2) c5 vector(2) c6 scalar a scalar b ' The values of "a" and "b" depend on the number (q-1) of structural breaks. ' When q=2, we set a=0 in Table 4 of Johansen et al. (2000) & b = min[ V1 , (1-V1)] ' When q=3, we set a=min[V1, (V2-V1), (1-V2)] & b = min[remaining two V expressions ] if q=2 then c2.fill v1, 1-v1 a=0 b=@min(c2) endif if q=3 then c3.fill v1, v2-v1, 1-v2 a=@min(c3) b=@median(c3) endif if q=1 then a=0 b=0 endif ' Let pr denote (p-r) vector(10) pr vector(10) lmL vector(10) lvL vector(10) meanL vector(10) varL vector(10) lmC vector(10) lvC vector(10) meanC vector(10) varC vector(10) theta vector(10) k vector(10) crit_HL90 vector(10) crit_HL95 vector(10) crit_HL99 vector(10) crit_HC90 vector(10) crit_HC95 vector(10) crit_HC99 vector(10) pvalL vector(10) pvalC ' lm denotes the logarithm of the mean of the asymptotic distribution ' lv denotes the logarithm of the variance of the asymptotic distribution ' See Table 4 of Johansen et al. (2000). We then add L or C to the names to reflect the H(L) test or the H(c) test. ' First construct critical values for the H(L) test, and then for the H(c) test for !pr = 1 to 10 pr(!pr)=!pr lmL(!pr)=3.06+0.456*!pr+1.47*a+0.993*b-0.0269*!pr^2-0.0363*a*!pr-0.0195*b*!pr-4.21*a^2-2.35*b^2+0.000840*!pr^3+6.01*a^3-1.33*a^2*b+2.04*b^3-2.05/!pr-0.304*a/!pr+1.06*b/!pr+9.35*a^2/!pr+3.82*a*b/!pr+2.12*b^2/!pr-22.8*a^3/!pr-7.15*a*b^2/!pr-4.95*b^3/!pr+0.681/!pr^2-0.828*b/!pr^2-5.43*a^2/!pr^2+13.1*a^3/!pr^2+1.5*b^3/!pr^2 lvL(!pr)= 3.97+0.314*!pr+1.79*a+0.256*b-0.00898*!pr^2-0.0688*a*!pr-4.08*a^2+4.75*a^3-0.587*b^3-2.47/!pr+1.62*a/!pr+3.13*b/!pr-4.52*a^2/!pr-1.21*a*b/!pr-5.87*b^2/!pr+4.89*b^3/!pr+0.874/!pr^2-0.865*b/!pr^2 meanL(!pr)=exp(lmL(!pr))-(3-q)*!pr varL(!pr)=exp(lvL(!pr))-2*(3-q)*!pr ' Use the asymptotic mean and variance to obtain the shape and scale parameters of the gamma distribution to be used to approximate the asymptotic distribution of the test statistic, and hence obtain the desired quantiles under the null: theta(!pr)=(varL(!pr)/meanL(!pr)) k(!pr)=meanL(!pr)^2/varL(!pr) crit_HL90(!pr)=@qgamma(0.90, theta(!pr),k(!pr)) crit_HL95(!pr)=@qgamma(0.95, theta(!pr),k(!pr)) crit_HL99(!pr)=@qgamma(0.99, theta(!pr),k(!pr)) if traceL>0 then pvalL(!pr)=1-@cgamma(traceL, theta(!pr),k(!pr)) endif lmC(!pr)=2.80+0.501*!pr+1.43*a+0.399*b-0.0309*!pr^2-0.0600*a*!pr-5.72*a^2-1.12*a*b-1.70*b^2+0.000974*!pr^3+0.168*a^2*!pr+6.34*a^3+1.89*a*b^2+1.85*b^3-2.19/!pr-0.438*a/!pr+1.79*b/!pr+6.03*a^2/!pr+3.08*a*b/!pr-1.97*b^2/!pr-8.08*a^3/!pr-5.79*a*b^2/!pr+0.717/!pr^2-1.29*b/!pr^2-1.52*a^2/!pr^2+2.87*b^2/!pr^2-2.03*b^3/!pr^2 lvC(!pr)= 3.78+0.346*!pr+0.859*a-0.0106*!pr^2-0.0339*a*!pr-2.35*a^2+3.95*a^3-0.282*b^3-2.73/!pr+0.874*a/!pr+2.36*b/!pr-2.88*a^2/!pr-4.44*b^2/!pr+4.31*b^3/!pr+1.02/!pr^2-0.807*b/!pr^2 meanC(!pr)=exp(lmC(!pr))-(3-q)*!pr varC(!pr)=exp(lvC(!pr))-2*(3-q)*!pr ' use the asymptotic mean and variance to obtain the shape and scale parameters of the gamma distribution to be used to approximate the asymptotic distribution of the test statistic, and hence obtain the desired quantiles under the null: theta(!pr)=varC(!pr)/meanC(!pr) k(!pr)=meanC(!pr)^2/varC(!pr) crit_HC90(!pr)=@qgamma(0.90, theta(!pr),k(!pr)) crit_HC95(!pr)=@qgamma(0.95, theta(!pr),k(!pr)) crit_HC99(!pr)=@qgamma(0.99, theta(!pr),k(!pr)) if traceC>0 then pvalC(!pr)=1-@cgamma(traceC, theta(!pr),k(!pr)) endif next smpl 1 10 mtos(pr,p_minus_r) mtos(crit_HL90, crit_HL_90) mtos(crit_HL95, crit_HL_95) mtos(crit_HL99, crit_HL_99) mtos(crit_HC90, crit_HC_90) mtos(crit_HC95, crit_HC_95) mtos(crit_HC99, crit_HC_99) group crit_vals p_minus_r crit_HL_90 crit_HL_95 crit_HL_99 crit_HC_90 crit_HC_95 crit_HC_99 show crit_vals if traceL>0 then mtos(pvalL,p_val_HL) group pval_HL p_minus_r p_val_HL show pval_HL endif if traceC>0 then mtos(pvalC,p_val_HC) group pval_HC p_minus_r p_val_HC show pval_HC endif