RDP 9608: Modelling the Australian Exchange Rate, Long Bond Yield and Inflationary Expectations Appendix C: The Behavioural Model of Australian Long Bond Yields – Integration Tests and Diagnostics

Alison Tarditi

November 1996

Table C.1: Integration Tests (1979:Q4–1995:Q2)
  H0: Non-stationarity H0: Stationarity
Φ3 ττ τμ DF-GLSτ DF-GLSμ KPSSτ KPSSμ
Real 10-yr bond 10.51*** −4.37*** −4.31*** −11.24*** −11.94*** 0.08 0.43*
Real US 10-yr bond 4.38 −2.94 −2.63 −1.21 −1.46 0.12* 0.12
Return on capital 6.29** −3.53** −3.49** −3.56** −3.60*** 0.10 0.19
Cash rate 2.47 −1.96 −1.42 −5.29*** −5.90*** 0.16** 0.28
Government deficit 5.14 −3.18* −3.19** −2.64 −2.56** 0.11 0.11
Undiversifiable risk 1.36 −1.60 −1.52 −6.52*** −7.46*** 0.12* 0.13
Current account 3.54 −2.21 −2.68 −3.39** −5.54*** 0.21** 0.51**
Inflation expectations 12.03*** −4.89*** −2.80 −3.68** −2.15** 0.07 0.39*
Δ Inflation 58.8*** −10.80*** −10.84*** −1.79 −0.77 0.08 0.11

Notes: *, ** and *** denote significance at the 10%, 5% and 1% levels respectively. Φ3 refers to the likelihood ratio test of (α, β, ρ)=(α, 0, 1) in Yt = α + βt + ρYt−1 + et. The critical values are from Dickey and Fuller (1981). τ refers to the Augmented Dickey-Fuller (ADF) ‘t-tests’; ττ includes a constant and trend and τμ includes a constant only. The critical values are from Fuller (1976). DF – GLSτ and DF – GLSμ are a modified trend and constant version, respectively, of the ADF tests proposed by Elliot et al. (1992). KPSS is a test proposed by Kwiatkowski et al. (1992) which tests the null hypothesis of stationarity. A truncation lag of 8 is used for the calculation of the estimate of the error variance.

All three tests support the stationarity of the Australian real 10-year bond rate around a constant or a trend. On the other hand, evidence for the US real long bond rate is mixed; the ADF and DF-GLS tests fail to reject their null of a unit root, but the KPSS test fails to reject the null hypothesis that the US real long bond rate is stationary around either a mean or a trend. The return on domestic capital and inflationary expectations are both clearly stationary; the ratio of the Commonwealth government budget balance to GDP is mean stationary; the evidence for the undiversifiable risk term, ‘beta’, is mixed.

Table C.2: Real Long Bond Equation (3)
Dependent variable: Change in real bond
(1981:01–1995:02)
Explanatory variable
 		 Coefficient
(Standard error)
Speed of adjustment parameter −0.513***
(0.10)
Return on capital 0.164***
(0.04)
Inflation term : Inline Equation 0.256***
(0.08)
Δ Real bondt−2 0.369***
(0.12)
Δ Return on capitalt−1 0.431*
(0.23)
ΔEt (π) −1.22***
(0.22)
Δ Et–2 (π) 0.93 ***
(0.29)
Δ Government deficitt 0.89*
(0.52)
Growtht 0.40*
(0.21)
Growtht−1 −0.39*
(0.20)
Inline Equation 0.60
DW 1.76
ARCH test Inline Equation 0.882
[0.347]
AR (4) test Inline Equation 3.28
[0.512]
Jarque-Bera Normality test Inline Equation 0.96
[0.618]

Notes: *, ** and *** denote significance at the 10%, 5% and 1% respectively. Standard errors are in parentheses; probability values are in square brackets.
Figure C.1: Real Long Bond Equation (3)
Figure C.1: Real Long Bond Equation (3)
Figure C.2: Nominal Bond Equation (7)
Figure C.2: Nominal Bond Equation (7)
Table C.3: Nominal Bond Equation (7) Specification
Dependent variable: Change in nominal bond
(1981:Q1–1995:Q2)
Nominal Bondt−1 −0.241***
(0.055)
Return on Capitalt−1 0.079***
(0.029)
US Real Ratet−1 0.127*
(0.67)
Perfect Foresight 1-year-aheadMarkov Expectationst−1 0.204***
(0.042)
Inline Equation 0.268*
{0.084}
Δ Markov Expectationst−1 −0.375***
(0.136)
Δ GDPt−3 −0.304***
(0.111)
Inline Equation 0.332
DW 2.07
ARCH test Inline Equation 0.416
[0.519]
AR (4) test Inline Equation 3.0846
[0.544]
Jarque-Bera Normality test Inline Equation 1.408
[0.495]

Notes: *, ** and *** denote significance at the 10%, 5% and 1% respectively. Standard errors are in brackets (), probability values are in square brackets and the F-test for the joint significance of the US real rate dynamics are in parenthesis.

The small negative coefficient on the third lag of growth in the dynamics of equation (7) corresponds with the (roughly) three-year cycle in bond yields in Australia.