Testing non-identifying restrictions on the long-run relations.

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Testing non-identifying restrictions on the long-run relations

Specify either the number of free parameters or the number of restrictions

The LR test statistic:

How many ’same restrictions’ can we impose on β? When the test rejects!

TEST FOR EXCLUSION: LR-test, Chi-Square(r) r DGF ChiSq5 LM3R DPY RM LYR RB DS (0.01) (0.00) (0.60) (0.31) (0.15) (0.75) (0.00) (0.00) (0.63) (0.06) (0.21) (0.17) (0.00) (0.00) (0.00) (0.00) (0.00) (0.00) (0.00) (0.00) (0.00) (0.00) (0.00) (0.00)

LM3R LYR DPY RM RB DS831 TREND Beta Beta Beta Alpha1 Alpha2 Alpha3 DLM3R (-2.56) (-3.95) (1.11) DLYR (-0.26) (2.10) (-2.42) DDPY (-5.05) (2.77) (-0.84) DRM (0.92) (-1.84) (-2.14) DRB (0.81) (0.90) (2.47 )

PI matrix: LM3R LYR DPY RM RB DS831 TREND DLM3R (-4.78) (4.47) (-1.15) (2.51) (-3.35) (2.91) (0.60) DLYR (1.78) (-2.89) (-1.00) (-2.90) (2.86) (-0.46) (0.46) DDPY (-0.50) (-1.23) (-5.69) (-1.07) (0.31) (-1.56) (-2.77) DRM (-0.81) (0.08) (1.35) (-1.58) (1.36) (2.63) (2.96) DRB (0.99) (0.36) (0.60) (1.98) (-1.51) (-2.12) (-2.05)

Testing hypotheses on a single beta relation Counting degrees of freedom:of the restricted vector r-1 restrictions and 1 normalization without testing additional restrictions change the likelihood function

Testing a known β

Two complications:

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 (0.00) (0.00) (0.00) (0.00) (0.00) (0.00) (0.00) (0.00) (0.00) (0.00) (0.00) (0.00) (0.00) (0.00) (0.00) (0.01) (0.01) (0.01) (0.01) (0.01)

Some coefficients needs to be estimated

The switching algoritm

Finding the maximum likelihood estimates

The LR test:

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 (0.00) (0.00) (0.00) (0.00) (0.00) (0.00) (0.03) (0.00) (0.00) (0.00) (0.00) (0.01) (0.00) (0.00) (0.00) (0.00) (0.02) (0.01) (0.00) (0.01 )

Example H26: Homogeneity between interest rates and inflation Testing hypotheses on a single beta relation