Example 5.A joint test of homogeneity between real money and real income and no deterministic linear trend in the cointegration relations.

• How many ’same restrictions’ can we impose on β?

TEST FOR EXCLUSION: LR-test, Chi-Square(r) r DGF ChiSq5 LM3R DPY RM LYR RB DS831 1 1 3.84 7.10 17.87 0.28 1.05 2.09 0.10 (0.01) (0.00) (0.60) (0.31) (0.15) (0.75) 2 2 5.99 12.36 23.17 0.92 5.63 3.10 3.52 (0.00) (0.00) (0.63) (0.06) (0.21) (0.17) 3 3 7.81 24.71 38.24 14.99 17.63 14.77 18.47 (0.00) (0.00) (0.00) (0.00) (0.00) (0.00) 4 4 9.49 32.53 46.95 22.84 22.55 22.90 24.90 (0.00) (0.00) (0.00) (0.00) (0.00) (0.00)

What to do when the test rejects!

Testing the stationarity of a single hypothesis

Testing stationarity of a known relations

Tests of stationarity around a constant mean

TEST FOR STATIONARITY: LR-test, Chi-Square(6-r) r DGF ChiSq5 LM3R DPY RM LYR RB 1 5 11.07 48.23 36.31 46.18 47.85 49.25 (0.00) (0.00) (0.00) (0.00) (0.00) 2 4 9.49 28.46 25.50 26.39 28.21 29.49 (0.00) (0.00) (0.00) (0.00) (0.00) 3 3 7.81 23.21 21.10 21.13 23.01 24.18 (0.00) (0.00) (0.00) (0.00) (0.00) 4 2 5.99 9.97 9.09 10.01 9.79 10.12 (0.01) (0.01) (0.01) (0.01) (0.01)

Tests of stationarity allowing for a shift in the mean at 1983

TEST FOR STATIONARITY: LR-test, Chi-Square(5-r) r DGF ChiSq5 LM3R DPY RM LYR RB 1 4 9.49 37.97 17.72 43.92 44.29 39.81 (0.00) (0.00) (0.00) (0.00) (0.00) 2 3 7.81 19.24 9.13 25.28 24.90 20.62 (0.00) (0.03) (0.00) (0.00) (0.00) 3 2 5.99 17.03 8.98 20.58 20.32 16.75 (0.00) (0.01) (0.00) (0.00) (0.00) 4 1 3.84 8.69 5.81 7.30 9.38 5.93 (0.00) (0.02) (0.01) (0.00) (0.01)

Testing hypotheses on a single beta relation when somecoefficients need to be estimated

Counting degrees of freedom:of the restricted vector• r-1 restrictions and 1 normalization without testing• additional restrictions change the likelihood function

Example H26: Homogeneity between interest rates and inflation

Testing hypotheses on a single beta relation

Revisiting the scenario analysis

Alternatively, the money demand relation could be a combination of two cointegration relations: