Modelling Ireland s exchange rates: from EMS to EMU

Transcription

1 From the SelectedWorks of Derek Bond November, 2007 Modelling Ireland s exchange rates: from EMS to EMU Derek Bond, University of Ulster Available at:

2 Background : The Irish Experience Revisited Derek Bond 1 Mike Harrison 2 Ed O Brien 3 1 University of Ulster at Coleraine 2 Trinity College Dublin & University College Dublin 3 European Central Bank and Trinity College Dublin SUFE - 2 nd November Copies of Paper available from ECB Website

3 Background WORKING PAPER SERIES NO 823 / OCTOBER 2007 MODELLING IRELAND S EXCHANGE RATES FROM EMS TO EMU by Derek Bond, Michael J. Harrison and Edward J. O Brien

4 Outline Background 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. Recent work in the area include: Lee(2004) and Hong(2005).

5 Background The Aims of Presentation To explore the use two recent developments in econometric theory Dolado(2002) Fractional Augmented Dickey Fuller test (FADF) Hamilton(2001) method of random field estimation to investigate nonlinearity. To the behaviour of time series characteristics of financial series.

6 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

7 Background 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...

8 Background 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.

9 Background Testing for 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.

10 Non-linear Modelling Background Overview of Random Field Models Random Field Inference 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.

11 Random Fields Background Overview of Random Field Models Random Field Inference 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.

12 Background Gaussian random field Overview of Random Field Models Random Field Inference 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)

13 Background Random Field Inference Overview of Random Field Models Random Field Inference 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.

14 Background Random Field Non-Linearity Tests Overview of Random Field Models Random Field Inference 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.

15 Hamilton Test Background Overview of Random Field Models Random Field Inference 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

16 Dahl s alternatives Background Overview of Random Field Models Random Field Inference 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.

17 Profile Likelihood Background Overview of Random Field Models Random Field Inference 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)

18 Background Theoretical Background Ireland s experiences The theory of purchasing power parity(ppp) has become a major area of research in applied econometrics. Hypothesis is that national price levels should be equal when expressed in a common currency If s t is the logarithm of the nominal exchange rate, p t and p t are the logarithms of the domestic and foreign price levels respectively and q t the logarithm of the real exchange rate in period t = [1, 2,...T ], then: q t = s t p t + p t t

19 Background The empirical analysis Theoretical Background Ireland s experiences q t must be stationary for long-run PPP to hold Empirical work has been concerned with 1 whether q t has a mean reversion tendency overtime or 2 whether s t and p t, p t move together overtime. Often the second has involved the equation: s t = α 0 + β 1 p t + β 2 p t + ɛ t

20 Exploratory Results Background Theoretical Background Ireland s experiences Standard co-integration results do not support ppp Fractional analysis inconclusive non-linear test generally support non-linear causal model Confusing results for real exchange rate

21 Modelling Results Background Theoretical Background Ireland s experiences Indication that prices are non-linear Possibility that interest rate are also important Analysis suggests regime switching

22 s Background Non-linearity in relationships is important Discovering and Modelling is difficult Much more research needed Parametric inference Semi Parametric: e.g. relationship between d and frequency dommain