With the knowledge of the sequential procedure for the unit root test, we would now test the data for R3M, R10Y, RGDP & M1/P with the Augmented Dicky-Fuller regression models of (model ? with constant and trend), (model ? with constant, no trend), an

With the knowledge of the sequential procedure for the unit root test, we would now test the data for R3M, R10Y, RGDP & M1/P with the Augmented Dicky-Fuller regression models of (model Ê with constant and trend), (model Ë with constant, no trend), and (model Ì with no constant and no trend).

From the SHAZAM output (see appendix 1), we can work through the following test with the respective data series:

R3M (raw data) Appropriate lag length suggested by SHAZAM is 2 for the analysis:
Test Model	Test Type & Test Hypothesis	Test Result	Conclusion
Ê	F 3 test with H₀: (a ,b ,r ) = (a ,0,0) against H₁: (a ,b ,r ) ¹ (a ,0,0)	test statistic < critical value (Do not reject H₀)	Series is non-stationary, no trend with possible drift.
Ê	T-test H₀: r =0 against H₁:r ¹ 0	t^* < t_crit (Do not reject H₀)	Verified that series is non-stationary
Ê	F 2 test with H₀: (a ,r ) = (0,0) against H₁: (a ,r ) ¹ (0,0)	test statistic < critical value (Do not reject H₀)	Series is random walk (unit root, no trend) with no drift.
Ë	F 1 test with H₀: (a ,b ,r ) = (0,0,0) against H₁: (a ,b ,r ) ¹ (0,0,0)	test statistic < critical value (Do not reject H₀)	Series is non-stationary with no drift (no trend determined earlier)
Ë	T-test H₀: r =0 against H₁:r ¹ 0	t^* < t_crit (Do not reject H₀)	Verified that series is non-stationary (with restriction b =0 imposed)
Ì	T-test H₀: r =0 against H₁:r ¹ 0	t^* < t_crit (Do not reject H₀)	Verified that series is non-stationary (with restrictions a =b =0 imposed)
Therefore, we can conclude that R3M is a random walk series (non-stationary, no trend) with zero drift.
R10Y (raw data) Appropriate lag length suggested by SHAZAM is 5 for the analysis:
Test Model	Test Type & Test Hypothesis	Test Result	Conclusion
Ê	F 3 test with H₀: (a ,b ,r ) = (a ,0,0) against H₁: (a ,b ,r ) ¹ (a ,0,0)	test statistic < critical value (Do not reject H₀)	Series is non-stationary, no trend with possible drift.
Ê	T-test H₀: r =0 against H₁:r ¹ 0	t^* < t_crit (Do not reject H₀)	Verified that series is non-stationary
Ê	F 2 test with H₀: (a ,r ) = (0,0) against H₁: (a ,r ) ¹ (0,0)	test statistic < critical value (Do not reject H₀)	Series is random walk (unit root, no trend) with no drift.
Ë	F 1 test with H₀: (a ,b ,r ) = (0,0,0) against H₁: (a ,b ,r ) ¹ (0,0,0)	test statistic < critical value (Do not reject H₀)	Series is non-stationary with no drift (no trend determined earlier)
Ë	T-test H₀: r =0 against H₁:r ¹ 0	t^* < t_crit (Do not reject H₀)	Verified that series is non-stationary (with restriction b =0 imposed)
Ì	T-test H₀: r =0 against H₁:r ¹ 0	t^* < t_crit (Do not reject H₀)	Verified that series is non-stationary (with restrictions a =b =0 imposed)
Therefore, we can conclude that R10Y is a random walk series (non-stationary, no trend) with zero drift.
RGDP (raw data) Appropriate lag length suggested by SHAZAM is 2 for the analysis:
Test Model	Test Type & Test Hypothesis	Test Result	Conclusion
Ê	F 3 test with H₀: (a ,b ,r ) = (a ,0,0) against H₁: (a ,b ,r ) ¹ (a ,0,0)	test statistic < critical value (Do not reject H₀)	Series is non-stationary, no trend with possible drift.
Ê	T-test H₀: r =0 against H₁:r ¹ 0	t^* < t_crit (Do not reject H₀)	Verified that series is non-stationary
Ê	F 2 test with H₀: (a ,r ) = (0,0) against H₁: (a ,r ) ¹ (0,0)	test statistic > critical value (Reject H₀)	Series is random walk (unit root, no trend) with non-zero drift.
Ë	F 1 test with H₀: (a ,b ,r ) = (0,0,0) against H₁: (a ,b ,r ) ¹ (0,0,0)	test statistic > critical value (Reject H₀)	Series is non-stationary with non-zero drift (no trend determined earlier)
Ë	T-test H₀: r =0 against H₁:r ¹ 0	t^* < t_crit (Do not reject H₀)	Verified that series is non-stationary (with restriction b =0 imposed)
Do not need to test model Ì as one of the restriction a =0 is not true and thus the result will be biased!
Therefore, we can conclude that RGDP is a random walk series (non-stationary, no trend) with a non-zero drift.
M1/P (raw data) Appropriate lag length suggested by SHAZAM is 5 for the analysis:
Test Model	Test Type & Test Hypothesis	Test Result	Conclusion
Ê	F 3 test with H₀: (a ,b ,r ) = (a ,0,0) against H₁: (a ,b ,r ) ¹ (a ,0,0)	test statistic < critical value (Do not reject H₀)	Series is non-stationary, no trend with possible drift.
Ê	T-test H₀: r =0 against H₁:r ¹ 0	t^* < t_crit (Do not reject H₀)	Verified that series is non-stationary
Ê	F 2 test with H₀: (a ,r ) = (0,0) against H₁: (a ,r ) ¹ (0,0)	test statistic < critical value (Do not reject H₀)	Series is random walk (unit root, no trend) with no drift.
Ë	F 1 test with H₀: (a ,b ,r ) = (0,0,0) against H₁: (a ,b ,r ) ¹ (0,0,0)	test statistic < critical value (Do not reject H₀)	Series is non-stationary with no drift (no trend determined earlier)
Ë	T-test H₀: r =0 against H₁:r ¹ 0	t^* < t_crit (Do not reject H₀)	Verified that series is non-stationary (with restriction b =0 imposed)
Ì	T-test H₀: r =0 against H₁:r ¹ 0	t^* < t_crit (Do not reject H₀)	Verified that series is non-stationary (with restrictions a =b =0 imposed)
Therefore, we can conclude that M1/P is a random walk series (non-stationary, no trend) with zero drift.
R3M (First-order differenced data) Appropriate lag length suggested by SHAZAM is 8 for the analysis:
Test Model	Test Type & Test Hypothesis	Test Result	Conclusion
Ê	F 3 test with H₀: (a ,b ,r ) = (a ,0,0) against H₁: (a ,b ,r ) ¹ (a ,0,0)	test statistic > critical value (Reject H₀)	Could be either or or
Ê	T-test H₀: r =0 against H₁:r ¹ 0	t^* > t_crit (Reject H₀)	Could be either or , ie. no unit root.
Í *	T-test H₀: b =0 against H₁:b ¹ 0	t^* < t_crit (Do not reject H₀)	Series is stationary (no unit root) with no trend.
Í *	T-test H₀: a =0 against H₁:a ¹ 0	t^* < t_crit (Do not reject H₀)	Series is stationary with no trend and zero (no) drift.
Í *	T-test H₀: r =0 against H₁:r ¹ 0	t^* > t_crit (Reject H₀)	Verified that series is stationary (no unit root).
Ï *	T-test H₀: r =0 against H₁:r ¹ 0	t^* > t_crit (Reject H₀)	Verified that series is stationary (with restrictions a =b =0 imposed).
Therefore, we can conclude that the first-order differenced R3M is a stationary series (no unit root) with no trend and zero drift.
R10Y (First-order differenced data) Appropriate lag length suggested by SHAZAM is 8 for the analysis:
Test Model	Test Type & Test Hypothesis	Test Result	Conclusion
Ê	F 3 test with H₀: (a ,b ,r ) = (a ,0,0) against H₁: (a ,b ,r ) ¹ (a ,0,0)	test statistic > critical value (Reject H₀)	Could be either or or
Ê	T-test H₀: r =0 against H₁:r ¹ 0	t^* > t_crit (Reject H₀)	Could be either or , ie. no unit root.
Í *	T-test H₀: b =0 against H₁:b ¹ 0	t^* < t_crit (Do not reject H₀)	Series is stationary (no unit root) with no trend.
Í *	T-test H₀: a =0 against H₁:a ¹ 0	t^* < t_crit (Do not reject H₀)	Series is stationary with no trend and zero (no) drift.
Í *	T-test H₀: r =0 against H₁:r ¹ 0	t^* > t_crit (Reject H₀)	Verified that series is stationary (no unit root).
Ï *	T-test H₀: r =0 against H₁:r ¹ 0	t^* > t_crit (Reject H₀)	Verified that series is stationary (with restrictions a =b =0 imposed).
Therefore, we can conclude that the first-order differenced R10Y is a stationary series (no unit root) with no trend and zero drift.
RGDP (First-order differenced data) Appropriate lag length suggested by SHAZAM is 9 for the analysis:
Test Model	Test Type & Test Hypothesis	Test Result	Conclusion
Ê	F 3 test with H₀: (a ,b ,r ) = (a ,0,0) against H₁: (a ,b ,r ) ¹ (a ,0,0)	test statistic > critical value (Reject H₀)	Could be either or or
Ê	T-test H₀: r =0 against H₁:r ¹ 0	t^* > t_crit (Reject H₀)	Could be either or , ie. no unit root.
Í *	T-test H₀: b =0 against H₁:b ¹ 0	t^* < t_crit (Do not reject H₀)	Series is stationary (no unit root) with no trend.
Í *	T-test H₀: a =0 against H₁:a ¹ 0	t^* > t_crit (Reject H₀)	Series is stationary with no trend and a non-zero drift.
Í *	T-test H₀: r =0 against H₁:r ¹ 0	t^* > t_crit (Reject H₀)	Verified that series is stationary (no unit root).
Î *	T-test H₀: a =0 against H₁:a ¹ 0	t^* > t_crit (Reject H₀)	Verified that series has a drift term (with restriction b =0 imposed).
Î *	T-test H₀: r =0 against H₁:r ¹ 0	t^* > t_crit (Reject H₀)	Verified that series is stationary (with restriction b =0 imposed).
Therefore, we can conclude that the first-order differenced RGDP is a stationary series (no unit root) with no trend and a non-zero drift.
M1/P (First-order differenced data) Appropriate lag length suggested by SHAZAM is 4 for the analysis:
Test Model	Test Type & Test Hypothesis	Test Result	Conclusion
Ê	F 3 test with H₀: (a ,b ,r ) = (a ,0,0) against H₁: (a ,b ,r ) ¹ (a ,0,0)	test statistic > critical value (Reject H₀)	Could be either or or
Ê	T-test H₀: r =0 against H₁:r ¹ 0	t^* > t_crit (Reject H₀)	Could be either or , ie. no unit root.
Í *	T-test H₀: b =0 against H₁:b ¹ 0	t^* < t_crit (Do not reject H₀)	Series is stationary (no unit root) with no trend.
Í *	T-test H₀: a =0 against H₁:a ¹ 0	t^* < t_crit (Do not reject H₀)	Series is stationary with no trend and zero (no) drift.
Í *	T-test H₀: r =0 against H₁:r ¹ 0	t^* > t_crit (Reject H₀)	Verified that series is stationary (no unit root).
Ï *	T-test H₀: r =0 against H₁:r ¹ 0	t^* > t_crit (Reject H₀)	Verified that series is stationary (with restrictions a =b =0 imposed).
Therefore, we can conclude that the first-order differenced M1/P is a stationary series (no unit root) with no trend and zero drift.

*Note: Model Í is from Appendix 2, or from Appendix 3.

Model Î is from Appendix 2, or from Appendix 3.

Model Ï is from Appendix 2, or from Appendix 3.

Thus, it was illustrated that the problem of non-stationarity of the raw data set of R3M, R10Y, RGDP, M1/P can be removed by taking first-order difference of the respective data. The unit root test has been used to verify the transition of non-stationary series of the raw data to a stationary one following first-order differencing at I(1).

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