Cointegration and Threshold Adjustment

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Cointegration and Threshold Adjustment

• Walter Enders- Iowa State University
• Pierre L. Siklos- Wilfrid Laurier University

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Asymmetries are Commonplace

• Asymmetric behavior over the business cycle is
  common (e.g., for GDP, unemployment)
• Neftci (1984)
• Falk (1986)
• Ramsey and Rothman (1996)
• Bradley and Jensen (1997)

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Asymmetries are Commonplace (contd)

• Gold Silver prices (Granger Escribano 1998)
• Money demand (Bohl 1999)
• Commodity prices (Abduali 1998 Hayenga Miller
  1998)
• retail vs farm prices (general, pork industry)
• Purchasing power parity (Enders-Dibooglu 1997)
• Fisher effect (Weidmann 1997)

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In what sense do asymmetries pose a problem?

• Most economic theories are based on equilibrium
  type relationships
• is the relationship linear? Non-linear?
• Are disequilibria linear? Non-linear?
• Quantifiying the relationship in regression
  analysis gives rise to the concept of
  cointegration
• The problem is that the existing tool kit is
  not adequate to deal with asymmetries

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Cointegration A Simple example

• y(t) x(t) e(t)
• where
• e(t) 0.95 e(t-1) w(t)
• x(t) x(t-1) v(t)
• y(t) and x(t) are attracted to each other
  cointegration
• Note If I replace 0.95 by 1 and write y(t)e(t)
  then y(t) and x(t) are independent from each
  other no cointegration

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Why were we interested in this problem?

• A particularly important theory in financial
  economics is the expectations hypothesis
• choosing to hold long-term vs. short-term bonds
  hinges on investors expectations about future
  interest rates
• hold liquidity and risk constant

Rk,t (1/k) ? Et R 1,tj1 ? Sk,1,t Rk,t -
R1,t ? long and short rates should be
attracted to each other
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A Plausible Scenario leading to Asymmetric
Behavior

• A rise in expected inflation, as proxied by
  long-term interest rates should signal an
  immediate rise in the Fed funds rate
• BUT, would lower expected inflation lead to an
  immediate fall in Fed funds?
• The Reverse is also possible...

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but there are other candidate scenarios

• The opportunistic view (Orphannides
  et.al.(various))
• fight inflation when its high and stabilize
  output when its low

inflation
time
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Taylors rule rules!
Rs r ? ?1 y ? 2 (? - ?)
Where ?1 if ? gt ? or ?-?gt? ?1 0 otherwise
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Asymmetries are commonplace (contd)

• And others .
• Accounting impact of changes in accounting
  standards the impact of surprise earnings
  announcements

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Parallel Developments

• Extension to the multivariate context a natural
  development
• Multi-cointegration of Granger and Lee (1989)
• Threshold cointegration of Balke and Fomby (1997)
• Non-linear error correction of Escribano and
  Granger (1998)
• and others..

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Econometric Contributions of the paper

• Generalize the Enders-Granger (1998) idea of
  asymmetric unit root behavior to asymmetric error
  correction
• Permit more than one form of asymmetric behavior
• simple asymmetric behavior and - disequilibria
• momentum model ? vs. ?-

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Basic Relationships
x1t ?0 ?2x2t ?t
?? t ??t-1 ?t
?xit ?i (x1t-1 - ?0- ?2x2 t-1) ?it
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TAR and Model
??t ?1 ?t-1 ?t if ?t-1 ? 0 ??t ?2 ?t-1
?t if ?t-1 lt 0
Indicator function (dummy variable)
It 1 if ?t-1 ?? 0 if ?t-1lt ?
Error correction form
?xit ?1.i It ?t-1 ?2.i (1-It) ?t-1
?it
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M-TAR Model
It 1 if ??t-1 ? ? It 0 if ??t-1 lt ?
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Monte Carlo Experiment
x1t x1 t-1 ?1t x2t x2 t-1 ?2t
Two cases ? 0 ? unknown
T 50, 100, 250, 500
Tables 1 through 6
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Consistent estimate of theThreshold
Chans (1993) methodology
1. Estimate ? 2. Sort ? in ascending order to
obtain ?1? lt?2? lt lt ?T? 3. Discard smallest
largest 15 of the ?i? 4. Re-estimate relevant
equations and consistent estimate is one that
yields lowest RSS
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An Application The U.S. Term Structure of
interest rates

• The expectations hypothesis
• short rates and long rates are cointegrated
• BUT Fed can only directly affect short rates
  expectations determine long rates also, Fed may
  want to send signals to market

Asymmetric relationship possible
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Data and Other details

• Fed Funds rate and 10 year yield on Govt
  securities
• Monthly data for the sample 7910-974
• Assume r I(1), non-stationary
• Compare Engle-Granger Johansen to TAR and M-TAR
  approaches

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How to Interpret the Results in Table 7

• There in NO long-run relationship between FFR
  and R10, based on the usual tests
• If equilibrium is defined as FFR-R100, there is
  still no long-run relationship between these
  series TAR case
• If the disequilibrium is defined in terms of how
  far we are from equilibrium M-TAR case then
  we do find a long-run relationship, and its even
  stronger when the threshold is estimated as .85

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Additional results of interest

• The error correction term is larger for POS
  disequilibria than NEG ones? when FFR is high it
  returns faster to equilibrium than when its low
• FFR is affected by R10 and vice-versa
• BUT, R10 is not caused by FFR the past history
  of R10 and FFR affect current FFR

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More Results

• Over the entire available data set we find that
  the Fed has acted in such a way as to
• raise the Fed funds rate when inflation rises (as
  proxied by a rise in the 10 year yield)
• is slow to reduce the Fed funds rate when
  inflation is expected to fall
• the 10 year yield is weakly exogenous, meaning
  that the spread does not influence changes in the
  long rate but
• a previous rise in Fed funds reduces the 10 year
  yield

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Conclusions

• Asymmetries are commonplace but there is, as yet,
  little available in the practitioners toolkit to
  estimate their significance in long-run
  relationships
• We develop a simple technique to estimate the
  significance of some commonly assumed forms of
  asymmetries between economic time series

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Conclusions (contd)

• CAVEATS
• exercise limited to 2 variables
• must have a priori a significant amount of
  asymmetry and choice of sample may matter
• equilibrium is assumed to be linear only
  disequilibria are non-linear
• BUT alternatives and extensions are VERY complex
  and research is at a very early stage.