Appendix E: Integration Tests of the Data | RDP 9607: Towards an Understanding of Australia's Co-Movement with Foreign Business Cycles

RDP 9607: Towards an Understanding of Australia's Co-Movement with Foreign Business Cycles Appendix E: Integration Tests of the Data

Nicolas de Roos and Bill Russell

November 1996

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The following tables examine the time series properties of the real GDP and real share price data. Table E1 presents augmented Dickey Fuller (Said and Dickey 1984) (ADF) tests where the null hypothesis of a unit root is tested against the alternative of stationarity. Table E2 presents Kwiatkowski et al. (1992) (KPSS) tests where the null hypothesis of stationarity is tested against the alternative of a unit root.

Domestically consumed GDP, US GDP, OECD GDP, export-markets GDP and the world real share price all appear to be integrated of order 1. For all these variables, the null hypothesis of non-stationarity cannot be rejected using the ADF tests. In addition, the null of stationarity is rejected at the 10 per cent level for all these variables using the KPSS tests, with the possible exception of domestically consumed GDP.^[31] The first differences of these series are found to be stationary under both tests.

Exports and the Australian real share price appear to be trend stationary. Using the ADF tests, the null of non-stationarity is rejected at the 10 per cent level in favour of the alternative of trend-stationarity. In addition, the KPSS tests cannot reject the null of trend-stationarity.

The time series characteristics of the US real share price are ambiguous. The ADF test rejects the null of non-stationarity in favour of trend-stationarity at the 10 per cent level. However, the KPSS test rejects the null of stationarity at the 10 per cent level. The series is characterised as trend-stationary in the paper. This interpretation is supported by graphical analysis.

Table E1: Augmented Dickey-Fuller Tests^(a)
Variable	Lags ^(b)	Φ₁	Φ₃
exports	4	6.33^*	11.03^**	3.28	0.99	−4.05^*
domestically consumed GDP	3	2.69	4.39	2.02	−0.91	−2.81
Δ domestically consumed GDP	5	8.42^**	8.41^*	−2.80^**	−3.85^**	−3.79^*
US GDP	3	3.41	3.30	2.51	−0.27	−2.44
Δ US GDP	4	10.13^**	10.25^**	−2.17^*	−4.24^**	−4.15^*
OECD GDP	5	4.79#	1.60	2.71	−0.99	−1.45
Δ OECD GDP	4	5.09^*	5.53	−1.22	−3.03^*	−3.10
export-markets GDP	2	9.36^**	1.65	4.17	−0.16	−1.74
Δ export-markets GDP	1	10.76^**	10.75^**	−1.46	−4.52^**	−4.47^**
Australian real share price	3	1.24	7.16^*	−1.44	−1.00	−3.59^*
US real share price	1	3.67	6.42^*	−1.43	−0.44	−3.46#
world real share price	3	2.15	2.14	−1.84	−0.90	−1.95
Δ world real share price	1	11.55^**	11.60^**	−4.11^**	−4.69^**	−4.66^**
Notes: (a) The likelihood ratio tests are: The ‘t-tests’ are ρ=1 for *, , and # denote significance at the 1%, 5%, and 10% levels respectively. The critical values for the likelihood ratio tests and the ‘t-tests’ are from Dickey and Fuller (1981) and Fuller (1976) respectively. The shaded box indicates the form of the model used in testing for non-stationarity. In most cases, the sample is 1980:Q1–1995:Q3. The sample is truncated when more than 3 lags of the dependent variable are included in the test. All variables are in logs. Æ indicates the change in the variable. (b) ‘Lags’ indicates the number of lags of the dependent variable included in the test to remove autocorrelation in the residuals.

Table E2: Kwiatkowski, Phillips, Schmidt and Shin Tests^(a)
		Constant		Constant and trend
Variable	Lag length^(b):	4	8	4	8
exports		1.337^**	0.795^**	0.100	0.091
domestically consumed GDP		1.300^**	0.787^**	0.164^*	0.117
Æ domestically consumed GDP		0.070	0.074	0.058	0.061
US GDP		1.324^**	0.788^**	0.201^*	0.137#
Æ US GDP		0.095	0.096	0.099	0.100
OECD GDP		1.350^**	0.798^**	0.196^*	0.130#
Æ OECD GDP		0.168	0.159	0.090	0.087
export-markets GDP		1.361^**	0.808^**	0.194^*	0.127#
Æ export-markets GDP		0.173	0.142	0.174	0.143
Australian real share price		1.189^**	0.719^*	0.085	0.074
US real share price		1.338^**	0.795^**	0.136#	0.122#
Æ US real share price		0.040	0.084	0.041	0.086
world real share price		1.242^**	0.735^*	0.201^*	0.132#
Æ world real share price		0.108	0.113	0.093	0.098
Notes: (a) The null hypothesis of stationarity is considered by testing in Y_t = ξt + r_t + ε_t, where r_t = r_t₋₁ + μ_t and . With the inclusion of a constant, the critical values at the 1%, 5%, and 10% levels of significance, are 0.739, 0.463, and 0.347 respectively. With the inclusion of a constant and trend, the critical values at the 1%, 5% and 10 % levels of significance, are 0.216, 0.146, and 0.119 respectively. *, , and # denote significance at the 1%, 5%, and 10% levels respectively. In most cases, the sample is 1980:Q1–1995:Q3. The sample is truncated when more than 3 lags of the dependent variable are included in the test. All variables are in logs. (b) The lag length refers to the value of l chosen when calculating the estimate of the error variance, used in the testing procedure.

Footnote

Using the KPSS test, the null of stationarity for domestically consumed GDP is accepted at the 10 per cent level when the lag length is 8 or greater. However, as Kwiatkowski et al. (1992) note, the power of the test is reduced as the lag length is increased. [31]