Appendix 2: Unit Root and Time Trend Tests | RDP 8910: An Analysis of the Determinants of Imports

RDP 8910: An Analysis of the Determinants of Imports Appendix 2: Unit Root and Time Trend Tests

Tracey Horton and Jenny Wilkinson

December 1989

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This appendix contains the results of sequentially testing for three unit roots, then two unit roots then one unit root in each series used in our estimation procedure. (The results for three unit roots are not shown below because they were always rejected.) In each case test statistics are presented which test firstly for a time trend or a constant, using standard t-tests, and then test the given hypothesis regarding the order of integration, using each of the three tests Dickey-Pantula (DP), Stock-Watson (SW) and Dickey-Fuller (DF). Four AR corrections have been made in testing each variable.

The following provides an example of how to read each section in each table. (e.g. lines 1 to 8 in Table 1.) Line 1 tests the null hypothesis of no time trend; line 5 tests the null hypothesis of no constant. The presence or absence of a time trend determines whether the tests in lines 2–4 or 6–8 are valid. If a time trend is detected (that is the null is rejected in line 5) then the three tests in lines 2–4 determine whether the hypothesis being tested in the section (in this case two unit roots versus at most one) can be accepted. If no time trend is detected then one uses the tests in lines 6–8.

TABLE 1: IMPORTS AND ACTIVITY
Test	Total Imports	Endog Imports	GDP	GNE
H₀ : Two Unit Roots vs H₁: at most One Unit Root
time	0.872	0.991	1.250	1.164	(1)
DP:	−4.223^*	−4.223^*	−3.453^**	−4.028^*	(2)
SW:	−50.982^*	−51.316^*	−58.997^*	−45.7.95^*	(3)
DF:	−3.883^**	−4.152^*	−3.409	−3.846^**	(4)
constant	2.040^**	2.190^**	3.316^*	2.604^*	(5)
DP:	−4.049^*	−4.023^*	−3.217^**	−3.719^*	(6)
SW:	−50.161^*	−50.570^*	−58.140^*	−44.121^*	(7)
DF:	−3.801^*	−4.038^*	−3.217^**	−3.631^*	(8)
H₀ : One Unit Root vs H₁: No Unit Roots
time	3.611^*	4.022^*	2.464^**	3.020^*
DP:	−2.315	−2.689	−2.068	−2.174
SW:	−13.992	−15.972	−9.363	−11.632
DF:	−2.863	−2.891	−1.974	−2.792
constant	0.024	0.021	−0.667	−0.656
DP:	0.750	0.783	0.964	1.202
SW:	0.431	0.581	0.566	0.733
DF:	0.246	0.406	0.946	0.781

For example, for Total Imports, at the 1% critical value we cannot reject the null hypotheses of no time trend and no constant. We can, however, reject the presence of two unit roots with each of the three tests (lines 6–8) and therefore accept that there is at most one unit root. Moving on to the second section of the table, we accept that a time trend exists and cannot reject the presence of one unit root for all three tests.

* indicates the null hypothesis is rejected at a 1% level of significance.
** indicates the null hypothesis is rejected at a 5% level of significance.

TABLE 2: RELATIVE PRICES AND OVERTIME
Test	RP Total Imports	RP Endog Imports	RP of Exports	Overtime
H₀ : Two Unit Roots vs H₁: at most One Unit Root
time	−1.536	−1.227	−0.207	0.565
DP:	−3.524^**	−3.073	−4.048^*	−2.688
SW:	−46.004^*	−40.869^*	−48.150^*	−28.594^**
DF:	−3.081	−2.576	−3.238	−2.913
constant	−0.275	−0.306	−0.741	1.691
DP:	−3.127^**	−2.871^**	−4.091^*	−2.752
SW:	−44.409^*	−40.245^*	−48.039^*	−28.714^*
DF:	−2.677	−2.306	−3.265^**	−2.989^**
H₀ : One Unit Root vs H₁: No Unit Roots
Time	−1.428	−1.759	−2.202^**	1.662
DP:	−2.052	−3.030	−2.254	−2.140
SW:	−7.863	−8.935	−13.840	−7.734
DF:	−2.042	−2.422	−2.743	−1.528
constant	0.272	1.014	−0.403	1.453
DP:	−2.256	−2.539	−0.974	−0.562
SW:	−6.404	−6.524	−5.171	−0.847
DF:	−2.154	−2.063	−1.305	−0.343