f and sFor example in FX markets CIP ⇒and UIP⇒(see lecture 4). Therefore if CIP and UIP hold then)1,1(~CIsftt−*ttthtrrsf−=−*tthtrrs−=Δ+)0(~Issfhttht+Δ=−The cointegratingvectoris (1 -1)Warwick Business School 14
Examples of cointegrationin empirical financeThe Expectations Hypo
thesis (EH) of the term structure (Brooks 7.12; Cuthbertson& Nitzsche20.2 and 22.1)The EH says that the expected one period holding yield on bonds of different maturities m should be equalized:If the yields are I(1) (and empirically they are) then the EH implies that the spreads are I(0).The implication is that the yields of different maturities are being driven by the same fundamentals (Rt+Τ): the yields share a common stochastic trend. ()()()()mttmttmtRRmRREε+Τ+=⇒Τ+=. maturities allfor ,()()()()).1,1(~CIRRmtntmtntεε−=−Second line follows assuming EH and rational expectations: is a martingaledifference error term.()mtεΤ≡Term premium (constant over timeand independent of m).R≡Known return on 1 period bond.The yields at different maturities are cointegrated. The cointegratingvectoris (1 -1).Warwick Business School 15
Examples of cointegrationin empirical finance
Purchasing Power Parity (PPP) (see Seminars 6-8)Adding a shock to the equilibrium at time tgives a stochastic equation For cointegrationthe equilibrium error must be stationary.Conversely if ε~I(1) then shocks to the equilibrium will have a permanent effect–No tendency for the system to revert back to equilibrium.–No long-run PPP (spurious relationship)..0lnlnln**=+−⇒=ttttttPPSPSPttttPPSε=+−*lnlnlnThe log real exchange rate=0in equilibrium (or a non-zero constant if the price indices are based In different years)Equilibrium error:The cointegratingvectoris:(1 -1 1)())0(~0IEttεε=lnS, lnPand lnP* are I(1) variables: see Seminar 6Warwick Business School 16
Dynamic equations: short run dynamics versus long run equilibrium
The cointegratingrelationship is a staticequation which relates only to the long run equilibrium.
We need to look at a dynamicmodel to gain information about the short-runand other dynamics in the system:
In equilibrium:
So the dynamic model has the following long-run form
tttttvYXXY++++=−−1110φδδμThe immediate impact of X on Y is *21*21... ,...XXXXYYYYtttttt≡===≡===−−−−()()*10**10***1*0*111XYXYYXXYφδδφμδδμφφδδμ−++−=⇒++=−⇒+++=
The long run impact of X on Y is:
There exists a stable long run relationship if φ<1. If φ=1 then y|xhas a unit root. In that case there is no cointegratingrelationshipbetween Y and X.
()()φδδ−+1100δAutoregressive distributed lag (ADL) modelWarwick Business School 17
Dynamic equations: impact, interim and long-run multiplierstttttvYXXY++++=−−1110φδδμ0δ=∂∂ttXY0111φδδφδ+=∂∂+=∂∂+ttttXYXY()0112φδδφφ+=∂∂=∂∂++ttttXYXYThe immediate impact of a unit change in X on Y:Impact multiplierThe impact after one period:Interim multiplier (after 1-period)()0111φδδφφ+=∂∂=∂∂−−++ntnttntXYXY()()()()()φδδδδφδδφδδφδδφφδδδ−+=++++++=+++++110102101001010KKThe impact after two periods:Interim multiplier (after 2-periods)The long-run (or equilibrium) multiplieris the cumulativeimpact of a unit change in X on Y The impact after n periods:Interim multiplier (after n-periods)Same result as before(see
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