Exploring persistence in the forward rate series

Transcription

1 From the SelectedWorks of Derek Bond February, 2009 Exploring persistence in the forward rate series Derek Bond, University of Ulster Available at:

2 Exploring persistence in the Forward Rate Series Derek Bond 1 Mike Harrison 2 Ed O Brien 3 1 University of Ulster at Coleraine 2 University College Dublin 3 European Central Bank FSRG th February Copies of Presentation available at

3 The Problem Non-stationarity and nonlinearity are closely related. Perron(1989) and Harrison and Bond (1992) It is difficult to statistically distinguish between difference stationary series and nonlinear but stationary series.

4 The Aim of the Presentation To explore the use of some recent tests on non-linearity using semi-parametric estimates of the fractional integration parameter d Smith(2005) modified gph test. Perron and Qu(2009) t d (a, c 1 ; b, c 2 ) tests Shimotsu (2006) Wald and d th differencing tests Qu(2009) Wald test To test the behaviour of time series involved in the Forward Rate Unbiasness debate.

5 Integration Background {y t } t=0 is said to be integrated of order d denoted by I(d)if the series is differenced d times before it is stationary. Classical analysis d is integer. Normally either: y t = y t y t 1 or is stationary y t

6 The restriction that d is integer is relaxed. Thus: d y t = y t dy t ! d(d 1)y t j! d(d 1)..(d j +1)y t j...

7 Stationary & Non-Stationary When 0 < d < 1 it follows that long memory exists If 0 < d < 0.5 then {y t } t = 0 is stationary if 0.5 d < 1.0 then the series {y t } t = 0 is non stationary.

8 Testing for : Dolado(2002) Traditional methods use semi-parametric spectral estimates Dolado(2002) approach is based on the distribution of the t statistic on φ from the generalised ADF regression: d 0 y t = φ d 1 y t 1 + p ζ i y t i + υ t i=1 where υ t is a hypothesised white noise error. Dolado (2002) set d 0 equal to 1.

9 Testing for : Lobato(2007) Dolado(2002) test is inefficient Lobato and Velasco (2007) propose an efficient Wald test based on the model: where if d > 0.5 the normal test. y t = φ 2 z t 1 (d2) + u t z t 1 (d 2 ) = ( d 2 1 1) 1 d 2 y t

10 Estimating ˆd Background Background to Tests Overview of Random Field Models Two main approaches to estimating ˆd: parametric Maximum Likelihood Non-linear least squares Semi parametric GPH Whittle

11 Logic behind tests Background Background to Tests Overview of Random Field Models If the series is linear changing the frequency domain should not effect the estimated value of d

12 Smith (2005) modified GPH Background to Tests Overview of Random Field Models Smith(2005) conducted monte-carlo studies on a mean plus noise model: y t = µ t + ɛ t and µ t = (1 ρ)µ t 1 + pη t 0 < p < 1 Smith shows that estimating d using GPH is biased upwards

13 Smith s modification Background Background to Tests Overview of Random Field Models Smith suggests estimating d from the following regression ˆf is the periodgram logf j = α + dx j + βz kj + û j where X j is the standard GPH term X j = log(2 2cos(ω j )) ω j = φj/t and Z kj is the additional term k is a nuisance parameter. ( ) (kj) 2 Z kj = log T 2 + ω2 j

14 Smith s test Background Background to Tests Overview of Random Field Models Let the modified GPH estimator be ˆd k then: if ˆd k < ˆd it is likely that the series contains a mena shift; if ˆd k ˆd it is unlikley that the series contains a mean shift.

15 The t d (a, c 1 ; b, c 2 ) tests Background to Tests Overview of Random Field Models Perron and Qu(2008) explored the behaviour of ˆd as m varies. Their basic statistic is: 24c1 [T t d (a, c 1 ; b, c 2 ) = a ] π 2 (ˆd a,c1 ˆd b,c2 ) Three tests: 1 t d ( 1 2, 1; 4 5, 1) 2 sup t d = sup c1 [1,2]t d ( 1 3, c 1; 1 2, 1) 3 mean t d = mean c1 [1,2]t d ( 1 3, c 1; 1 2, 1)

16 Shimotsu s Simple Wald Test Background to Tests Overview of Random Field Models Using the local Whittle or two step feasible Whittle, Shimotsu suggests the following test statistic: ( ) ( cm/n ) W c = 4m Aˆd n (AΩA ) + Aˆd n m/n where: ˆd d 0 ˆd ˆd (1) d n =., A = ˆd (n) d 0 c m = m i=1 v 2 i, v i = log(i) 1 m m log(j) j=1

17 Shimotsu s d th difference tests Background to Tests Overview of Random Field Models Shimotsu also suggests using both the Phillips and Perron Z t test and the KPSS test on the differenced series. Differencing carried out on a mean adjusted series. The adjustment is: µ(d) = w(d) X + (1 w(d))x 1 where w(d) is a smooth weight function such that w(d) = 1 for d 1 2 and w(d) = 0 for d 3 4. Shimotsu suggests using w(d) = 1 2 [1 + cos(4πd)] when d ( 1 2, 3 4 ).

18 Wu s Test Background Background to Tests Overview of Random Field Models If λ j are the frequencies and G(d) = 1 m m j=1 λ2d j I j then the test statistic is: m W = sup r [ɛ;1] vj 2 j=1 1 2 [mr] j=1 v j I j G(ˆd)λ 2ˆd j 1

19 Suggested Methodology Background to Tests Overview of Random Field Models Tradition classical regression Fractional testing Random Field non-linear tests (Random Field Modelling) Smith s Tests, Perron & Qu Tests, Qu s test; (Shimotsu s test)

20 Non-linear Modelling Background Background to Tests Overview of Random Field Models Increasingly Smooth Transition Regression being used rather abitary in choice of transition variable An alternative is random fields regression. Random field approach has relatively better small sample performance than a wide range of parametric and non-parametric alternatives including LSTR and ESTR.

21 Random Fields Background Background to Tests Overview of Random Field Models The basic model is of the form: y t = µ(x t ) + ɛ t where the functional form µ(x t ) is unknown and assumed to be the outcome of a random field. Hamilton s conditional mean function µ(x t ) as µ(x t ) = α 0 + α x t + λm( x t ) where x t = g x t, g is a kx1 vector of parameters and denotes the Hadamard product of matrices.

22 Gaussian random field Background to Tests Overview of Random Field Models Defined fully by its first two moments 0 and H k. so it follows that y t = α 0 + α 1 x t + u t where u t = λm( x t ) + ɛ t or in matrix form y = Xβ + u (1) where β = (α 0, α 1 ) so it follows that: u N(0, λ 2 H k + σ 2 I T ) (2)

23 Background to Tests Overview of Random Field Models Can be viewed as a generalised least squares problem The profile maximum likelihood function can be obtained and estimated. Only problem is that the form of the covariance matrix is unknown. Hamilton derives H k as a simple moving average representation of the random field based on g using a L 2 norm measure.

24 Random Field Non-Linearity Tests Background to Tests Overview of Random Field Models Simple method of testing is to check if λ is zero or not. MLE λ 2 for fixed g is consistent and asymptotically normal. Computationally complex. Simpler method is to construct Lagrange Multiplier test under the assumption of normality and linearity Provided the covariance function of the random field can be derived, for a fixed g this only requires a single linear regression to be estimated.

25 Hamilton s Test Background Background to Tests Overview of Random Field Models Hamilton(2001) used covariance function based on the L 2 norm derived the appropriate score vectors of first derivatives, for up to k=5, and the associated information matrix and proposed a form of the LM test for practical application. λ E H, the test statistic is distributed as χ2 1 under the null hypothesis. Problem of nuisance parameters under H 1

26 Dahl s alternatives Background Background to Tests Overview of Random Field Models Dahl(2003) λ E OP, covariance matrix based on the L 1 norm. λ A OP covariance function depicted by a Taylor expansion. g test makes no assumption about either the covariance function or λ. λ A OP and the g test have, in many circumstances, better power than other tests of nonlinearity.

27 Profile Likelihood Background Background to Tests Overview of Random Field Models estimating λ and g gives useful insights. need to estimate reparameterised likelihood η(y, X : g, ζ) = T 2 ln(2π) T 2 lnˆσ2 T (g, ζ) 1 2 ln W (X : g, ζ) T 2 ˆβ T (g, ζ) = [X W (X : g, ζ) 1 X] 1 [X W (X : g, ζ) 1 y] ˆσ T 2 (g, ζ) = 1 T [y X ˆβ T (g, ζ)] W (X : g, ζ) 1 [y X ˆβ T (g, ζ)] where ζ = λ/σ and W (X : g, ζ) = ζ 2 H k + I T. maximised with respect to (g, ζ)

28 Theoretical Background Results The Forward Rate Anomaly It is argued that the Forward Rate F t,k in period t is an unbiased predictor of the Spot Rate S t+k in period t + k. that is s t+k = E [ F t,k ] So in the regressions: s t+k = α 1 + β 1 F t,k + ɛ t and s t+k = α 2 + β 2 F t,k + ɛ t α 1 = α 2 = 0 β 1 = β 2 = 1 In practice ˆα i 0 and ˆβ i < 0 i = 1, 2

29 Results Background Theoretical Background Results See handout

30 Theoretical Background Results Simple regression results α and β - normally not within 2 SD of 0 and 1 R 2 decreases as time period increases D.W. close to zero Co-integration ADF test only significant for levels with low time period

31 Theoretical Background Results RF tests for non-linearity results majority of tests strongly support non-linearity of g op supports linearity for 1 one week - euro level 2 one week and one month us dollar level and return

32 Theoretical Background Results RF model results Strong support for non-linearity in exogenous variable except for nine month euro level all euro level results are within 2 S.D. of hypothesised values euro returns the βs are with 2 SD of one except for one year all dollar levels with 2 S.D. of hypothesised values results for dollar returns confused.

33 Fractional and Smith Results Theoretical Background Results See handout

34 Theoretical Background Results Figure 1: Expected value of likelihood function Euro-Sterling levels

35 Theoretical Background Results Figure 2: Expected value of likelihood function Dollar-Sterling levels

36 Theoretical Background Results Figure 3: Expected value of likelihood function Euro-Sterling returns

37 Theoretical Background Results Figure 4: Expected value of likelihood function Dollar-Sterling returns