2003 ames.models
TRANSCRIPT
1
Model 1: Lo-Zivot Threshold Cointegration Model
• Balke and Fomby (1997):
– univariate cointegrating residual behavior
– ad hoc model selection, no specification test is provided for TAR models
• Lo and Zivot (2001):
– multivariate setting of a threshold vector error correction model (TVECM) to investigate the dynamic
adjustment of individual series more efficiently, to uncover the overall dynamics of the whole
multivariate system, capture the long-run equilibrium relationship as well as the short-term
disequilibrium adjustment process to the long-run equilibrium
– use a specification test offered by Hansen (1997, 1999) to see which TAR model is appropriate to
capture the threshold cointegration relationships for the Treasury and corporate bond rates
• A TVECM will be estimated and used to evaluate the dynamic time paths of yield
spread adjustments to U.S. Treasury and corporate bond indices which allows:
•discontinuous adjustment relative to the thresholds
•nonlinear adjustments to the long-run equilibrium
•asymmetric adjusting speeds to the long-run equilibrium
2
A Bivariate Vector Error Correction Model (VECM)
• a bivariate vector autoregressive (VAR) model, where Xt is a 2 × 1 vector with )x,x(X t2t1
'
t = :
t
k
1iiti0t XAAX ε++= ∑
=− , (1)
where tε is a 2 × 1 white noise process, k is the order of autoregressive terms, A0 is a 2 × 1 parameter vector, and
Ai’s re 2 × 2 parameter matrices
• ,XXAX t
1k
1iiti1t0t ε+∆Γ+Π+=∆ ∑
−
=−− (2)
where
−−=Π ∑
=
k
1i2i IA , and ∑
+=
−=Γk
1ii A
ll , for i = 1, 2, … , k-1
• if elements of Xt are I(1) and cointegrated with a normalized cointegrating vector ),1(' 2ββ −= , then (2) has a vector
error-correction model (VECM) representation:
,XX'AX t
1k
1iiti1t0t εγβ +∆Γ++=∆ ∑
−
=−− (3)
where .),1('222
121
2
2
1
−−
=−
==Π
γβγγβγ
βγγ
γβ (4)
• γ is the speeds of adjustment, 1tX' −β denotes the error-correction terms or the cointegrating residuals
3
The BAND-TVECM (Band-Threshold Vector Error Correction Model)
• conventional VAR and VECM can only model linear relationships
• threshold autoregression (TAR) and threshold vector error correction model (TVECM) can overcome above drawback
• TAR and TVECM have the strength of modeling nonlinear and discontinuous phenomenon
• consider a simple three-regime bivariate TVECM for the threshold cointegrating relationship of the Treasury and
corporate bond rates, express the bivariate threshold vector autoregressive (TVAR) model for Xt as:
),cz(IXAA
)czc(IXAA
)cz(IXAAX
)2(
dtt3
)3(
tit
k
1i
)3(
i
)3(
0
)2(
dt
)1(
t2
)2(
tit
k
1i
)2(
i
)2(
0
)1(
dtt1
)1(
tit
k
1i
)1(
i
)1(
0t
>
+++
≤<
+++
≤
++=
−−=
−−=
−−=
∑
∑
∑
ε
ε
ε
(5)
where )j(
tε ’s are bivariate vector white noise processes, k is the autoregressive order, s'A )j(
0 are 2 × 1 parameter
vectors, and s'A )j(
i are 2 × 2 parameter matrices for regime j = 1, 2, 3, and for lag i = 1, 2, … , k; zt-d is called the
threshold variable; d is called the delay parameter, d is positive and usually less than or equal to the lag length k
• in general, ∞=<<<=∞− )3()2()1()0( cccc , the indicator function has the form:
4
=≤<
=≤< −−
−−
.otherwise,0
,3,2,1j,czcif,1)czc(I
)j(
dt
)1j(
)j(
dt
)1j(
jt (6)
• if elements of Xt are I(1) and they are cointegrated, then equation (5) can be expressed as a TVECM:
),cz(IXXA
)czc(IXXA
)cz(IXXAX
)2(
dtt3
)3(
tit
1k
1i
)3(
i1t
)3()3(
0
)2(
dt
)1(
t2
)2(
tit
1k
1i
)2(
i1t
)2()2(
0
)1(
dtt1
)1(
tit
1k
1i
)1(
i1t
)1()1(
0t
>
+∆Γ+Π++
≤<
+∆Γ+Π++
≤
+∆Γ+Π+=∆
−−
−
=−
−−
−
=−
−−
−
=−
∑
∑
∑
ε
ε
ε
(7)
where
−−=Π ∑
=
k
1i2
)j(
i
)j( IA , and ∑+=
−=Γk
1i
)j()j(
i Al
l for regime j = 1, 2, 3, and i = 1, 2, … , k-1
• if elements of Xt are cointegrated with a common (across regime) normalized cointegrating vector ),1(' 2ββ −= and if
the error terms )j(
tε share the same variance-covariance structure, then the TVECM may be written as:
,)cz(IXX'A
)czc(IXX'A
)cz(IXX'AX
t
)2(
dtt3it
1k
1i
)3(
i1t
)3()3(
0
)2(
dt
)1(
t2it
1k
1i
)2(
i1t
)2()2(
0
)1(
dtt1it
1k
1i
)1(
i1t
)1()1(
0t
εβγ
βγ
βγ
+>
∆Γ+++
≤<
∆Γ+++
≤
∆Γ++=∆
−−
−
=−
−−
−
=−
−−
−
=−
∑
∑
∑
(8)
5
where
−
−=−
=Π=
)j(
22
)j(
2
)j(
12
)j(
1
2)j(
2
)j(
1)j()j( ),1('γβγγβγ
βγ
γβγ , and j = 1, 2, 3 (9)
• note that although the three regimes share a common cointegrating vector ),1(' 2ββ −= , the speeds of adjustment
),(' )j(
2
)j(
1
)j( γγγ = are regime specific. For example, we may observe that )3(
1
)1(
1 γγ ≠ or )3(
2
)2(
2 γγ ≠
• the simplest form for the TVECM occurs when k = 1 in equation (8) so that all lag difference terms drop out of the
equation, the cointegrating residual tX'β follows a regime specific AR(1) process or threshold autoregressive (TAR)
process:
,X'X' )j(
t1t
)j()j(
t ηβρδβ ++= −
with )j(
22
)j(
1
)j()j( 1'1 γβγγβρ −+=+= , where )j(
0
)j( A'βδ = and )j(
t
)j(
t 'εβη =
Proof:
Set k = 1 in equation (8) to obtain:
[ ] [ ][ ] .)cz(IX'A
)czc(IX'A)cz(IX'AXXX
t
)2(
dtt31t
)3()3(
0
)2(
dt
)1(
t21t
)2()2(
0
)1(
dtt11t
)1()1(
01ttt
εβγ
βγβγ
+>++
≤<++≤+=−=∆
−−
−−−−−
Multiply both sides by 'β , then move 'β Xt-1 to the right-hand side, will obtain:
[ ] [ ][ ] .')cz(IX''A'
)czc(IX''A')cz(IX''A'X'X'
t
)2(
dtt31t
)3()3(
0
)2(
dt
)1(
t21t
)2()2(
0
)1(
dtt11t
)1()1(
01tt
εββγββ
βγβββγββββ
+>++
≤<++≤++=
−−
−−−−−
6
Split 1tX' −β and tε to each regime, then obtain:
[ ][ ][ ] ).cz(I'X''X'A'
)czc(I'X''X'A'
)cz(I'X''X'A'X'
)2(
dtt3
)3(
t1t
)3(
1t
)3(
0
)2(
dt
)1(
t2
)2(
t1t
)2(
1t
)2(
0
)1(
dtt1
)1(
t1t
)1(
1t
)1(
0t
>++++
≤<++++
≤+++=
−−−
−−−
−−−
εββγβββ
εββγβββ
εββγββββ
Collect terms to get:
[ ][ ][ ]
.X'
)cz(I'X')'1(A'
)czc(I'X')'1(A'
)cz(I'X')'1(A'X'
)j(
t1t
)j()j(
)2(
dtt3
)3(
t1t
)3()3(
0
)2(
dt
)1(
t2
)2(
t1t
)2()2(
0
)1(
dtt1
)1(
t1t
)1()1(
0t
ηβρδ
εββγββ
εββγββ
εββγβββ
++=
>++++
≤<++++
≤+++=
−
−−
−−
−−
Q.E.D.
• tX'β is stable within each regime if the stability condition 11 )j(
22
)j(
1
)j( <−+= γβγρ holds for each regime
• in equation (8), with k = 1, then we have:
[ ] [ ][ ] .)cz(IX'A
)czc(IX'A)cz(IX'AX
t
)2(
dtt31t
)3()3(
0
)2(
dt
)1(
t21t
)2()2(
0
)1(
dtt11t
)1()1(
0t
εβγ
βγβγ
+>++
≤<++≤+=∆
−−
−−−− (10)
• it is easier to capture the long-run equilibrium relationship if we rewrite (10) in:
[ ] [ ][ ] .)cz(IX'
)czc(IX')cz(IX'X
t
)2(
dtt3
)3(
1t
)3(
)2(
dt
)1(
t2
)2(
1t
)2()1(
dtt1
)1(
1t
)1(
t
εµβγµβγµβγ
+>−+
≤<−+≤−=∆
−−
−−−− (11)
explicitly we have:
7
[ ][ ][ ]
>+−−
≤<+−−
≤+−−
=∆
−−−
−−−
−−−
,czif,xx
,czcif,xx
,czif,xx
x)2(
dt
)3(
t1
)3(
1t221t1
)3(
1
)2(
dt
)1()2(
t1
)2(
1t221t1
)2(
1
)1(
dt
)1(
t1
)1(
1t221t1
)1(
1
t1
εµβγ
εµβγ
εµβγ and
[ ][ ][ ]
>+−−
≤<+−−
≤+−−
=∆
−−−
−−−
−−−
.czif,xx
,czcif,xx
,czif,xx
x)2(
dt
)3(
t2
)3(
1t221t1
)3(
2
)2(
dt
)1()2(
t2
)2(
1t221t1
)2(
2
)1(
dt
)1(
t2
)1(
1t221t1
)1(
2
t2
εµβγ
εµβγ
εµβγ
• the magnitudes and signs of the γ’s will provide fruitful information regarding the equilibrium relationships
• equation (11) offers the regime-specific means )j(µ , which is calculated as:
,1
AA
'
A')j(
)j(
)j(
22
)j(
1
)j(
2,02
)j(
1,0
)j(
)j(
0)j(
ρδ
γβγβ
γββ
µ−
=−
−−=−= (12)
where )A,A('A )j(
2,0
)j(
1,0
)j(
0 = , and ),1(' 2ββ −=
• it is also possible to eliminate the regime specific drift in Xt through the restriction:
)j()j()j(
0A µγ−= (13)
where )j(µ is calculated by (12)
• note that we may rewrite equation (11) as follows with 1t221t11t1t xxX'z −−−− −== ββ :
8
[ ][ ][ ]
>+−
≤<+−
≤+−
=∆
−−
−−
−−
.czif,z
,czcif,z
,czif,z
X)2(
dtt
)3(
1t
)3(
)2(
dt
)1(
t
)2(
1t
)2(
)1(
dtt
)1(
1t
)1(
t
εµγ
εµγ
εµγ (14)
• consider the case of d = 1, 0)2( =γ and )2(
0A = 0 in equation (14), this is the Band-TVECM structure which is the most
popular form in threshold cointegrating applications:
[ ]
[ ]
>+−
≤<
≤+−
=∆
−−
−
−−
.czif,z
,czcif,
,czif,z
X)2(
1tt
)3(
1t
)3(
)2(
1t
)1(
t
)1(
1tt
)1(
1t
)1(
t
εµγ
ε
εµγ (15)
• the stability conditions must hold for the outer regimes, i.e., 11 )j(
22
)j(
1
)j( <−+= γβγρ , for j = 1 and 3
• one may interpret above model as:
1. if the cointegrating residual (the error-correction term) 1t1t X'z −− = β lies within the inner band ]c,c[ )2()1( , then Xt
behaves like a random walk process without the drift, i.e., tX∆ has no tendency reverting to any long-term
equilibrium
2. if 1tz − is less than )1(c , then tz reverts to the regime specific mean )1(µ with adjustment coefficient )1(ρ while tX∆
adjusts with speed of adjustment vector )1(γ
9
3. if 1tz − is greater than )2(c , then tz reverts to the regime specific mean )3(µ with adjustment coefficient )3(ρ and
tX∆ adjusts with speed of adjustment vector )3(γ
4. expect ,0,0 )1(
i
)3(
i >≤ γγ for i = 1, 2, because of the force of the error correcting toward the long-term equilibrium
• if the regime specific means of the cointegrating residual tz are equal to the nearby threshold values (it is called the
“continuous” model): )2()3()1()1( c,c == µµ , then (15) may be written as:
[ ]
[ ]
>+−
≤<
≤+−
=∆
−−
−
−−
.czif,cz
,czcif,
,czif,cz
X)2(
1tt
)2(
1t
)3(
)2(
1t
)1(
t
)1(
1tt
)1(
1t
)1(
t
εγ
ε
εγ (16)
• the “symmetric” threshold model arises when the threshold values are symmetric against the origin ( ccc )1()2( =−= ):
[ ]
[ ]
>+−
≤<−
−≤++
=∆
−−
−
−−
.czif,cz
,czcif,
,czif,cz
X
1tt1t
)3(
1tt
1tt1t
)1(
t
εγ
εεγ
(17)
• if ,0)3()1( == µµ then we have the EQ-TVECM:
>+
≤<
≤+
=∆
−−
−
−−
.czif,z
,czcif,
,czif,z
X)2(
1tt1t
)3(
)2(
1t
)1(
t
)1(
1tt1t
)1(
t
εγ
ε
εγ (18)
10
Hansen’s Procedures for Testing Linearity
• once known that Xt is cointegrated with known cointegrating vector β, next to determine if the dynamics in the
cointegrating relationship is linear or exhibits threshold nonlinearity
• Hansen (1997,1999) developed a method for testing the null hypothesis of linearity (i.e., TAR(1)) versus the
alternative of a TAR(m) model, where m denotes the number of regimes based on nested hypothesis tests, m > 1
• a linear autoregressive model results under the restrictions that δδ =)j( and ρρ =)j( , ∀j
• consider the TAR(m) model for 1tt −− = X'z 1 β :
m...,,2,1j,zz )j(
t1t
)j()j(
t =++= − ηρδ (19)
• Hansen’s linearity test is a test using a sup-F (or sup-Wald) test constructed from the supremum over possible
threshold values of the F-statistic:
−=
m
m1
m,1 S
SSTF , (20)
where S1 and Sm denote the sum of squared residuals from the estimation of a TAR(1) model and a TAR(m) model
• Hansen provides a simple bootstrap procedure to compute p-values for this test
11
• Hansen’s method for testing linearity in univariate TAR models based on nested hypothesis tests can be easily
extended to test linearity in multivariate TVECMs
• to test the null hypothesis of a linear VECM against the alternative of a TVECM(m) for some m > 1, the test statistic is
the sup-LR statistic (which is asymptotically equivalent to the sup-Wald) constructed from:
|)))d,c(ˆln(||)ˆ(ln(|TLR mm,1 Σ−Σ= (21)
where Σ and )d,c(mΣ denote the estimated residual variance-covariance matrices from the linear VECM and the
m-regime TVECM
• in Hansen (1997), the distribution of the sup-LR statistic will be non-standard, a bootstrap procedure can be used to
compute p-values for this test
12
Hansen’s Procedures for Model Specification Test
• Hansen (1999) uses a sequential testing procedure based on nested hypotheses, we will adopt his nested hypotheses
tests based on unrestricted estimation of TAR models and TVECMs
• start with a typical three-regime continuous symmetric threshold and symmetric adjustment BAND-TAR model for zt
as well as a three-regime symmetric threshold and symmetric adjustment BAND-TVECM for Xt
• the symmetric BAND-TAR model is nested within an unrestricted TAR(3) model while the symmetric
BAND-TVECM is nested within an unrestricted TVECM(3)
• this nested structure allows for a systematic specification analysis
• consider first the determination of the number of regimes
• given that linearity is rejected in favor of threshold nonlinearity, in order to determine if a TAR(3) model for zt is
appropriate we test of the null of a TAR(2) model against the alternative of a TAR(3) model using the F-statistic:
−=
3
323,2
S
SSTF , (22)
13
where S2 and S3 denote the sum of squared residuals from the estimation of an unrestricted TAR(2) model and an
unrestricted TAR(3) model, respectively
• to determine if a TVECM(3) for Xt is appropriate we can test the null of a TVECM(2) against the alternative of a
TVECM(3) using the LR statistic:
|)),)d,c(ˆln(||))d,c(ˆ(ln(|TLR 323,2 Σ−Σ= (23)
where )d,c(2Σ and )d,c(3Σ denote the estimated residual variance-covariance matrices from the unrestricted
TVECM(2) and TVECM(3), respectively
• the asymptotic distributions of F2, 3 and LR2, 3 are nonstandard and bootstrap methods can be used to compute
approximate p-values
14
Model 2: Hansen-Seo Two-Regime Threshold Cointegartion Model
• Hansen and Seo (2001) propose a formal test procedure for threshold cointegration and they offer an algorithm to estimate model
parameters
• A two-regime vector error correction model with one cointegrating vector and with one built-in threshold effect in the error-
correction term
• Based on a fully specified joint model, they derive the maximum likelihood estimator of a threshold cointegration model
• Under the null hypothesis of linearity the threshold parameter is not identified, which causes a nuisance parameter problem, they
then:
1. base inference on a Sup-LM (Lagrange Multiplier) test statistic
2. derive the asymptotic null distribution for test statistic and discuss bootstrap approximations to the
sampling distribution: (a) the fixed regressor bootstrap
(b) the residual-based bootstrap
15
Two-Regime Threshold Cointegration Model
• xt is a p × 1 I(1) with one p × 1 cointegrating vector β, tt x')(w ββ = denotes the I(0) error-correction term
• A linear vector error correction model (VECM) of order (L+1):
t1tt u)(X'Ax +=∆ − β , (1)
where [ ]Lt2t1t1t
'
1t x...,,x,x),(w,1)(X −−−−− ∆∆∆= ββ , with dimensions: Xt-1(β) is k × 1, k = p × L + 2, A is k ×
p
• The error term ut is a p × 1 Martingale difference sequence with finite variance-covariance matrix )uu(E 'tt=Σ of
dimension p × p
• The approach is to estimate the parameters (β, A, Σ) by maximum likelihood estimation given the assumption
that the error terms ut’s are i.i.d. Gaussian distributed
• Let γ be the threshold parameter, a two-regime threshold cointegration model:
>+
≤+=∆
−−
−−
,)(wif,u)(XA
,)(wif,u)(XAx
1tt1t
'
2
1tt1t
'
1
t γββ
γββ,
16
or rewrite as
tt21t
'
2t11t
'
1t u),(d)(XA),(d)(XAx ++=∆ −− γββγββ , (2)
where ))(w(I),(d 1tt1 γβγβ ≤= − , ))(w(I),(d 1tt2 γβγβ >= − , and I(⋅) is the indicator function
• To ensure the nonlinearity, Hansen-Seo among others suggest imposing the boundary constraint:
01t0 1))(wPr( πγβπ −≤≤≤ − (3)
we will set 15.005.0 0 ≤≤ π
• The likelihood function is: ,),,A,A(u)',,A,A(u2
1log
2
n),,,A,A(L
n
1t21t
1
21t21n ∑=
−Σ−Σ−=Σ γβγβγβ where:
),(d)(XA),(d)(XAx),,A,A(u t21t
'
2t11t
'
1t21t γββγββγβ −− −−∆=
• The maximum likelihood estimators (MLEs) ),,,A,A( 21 γβΣ are the values that maximize the likelihood
function ),,,A,A(L 21n γβΣ
17
Estimation Procedure:
• First concentrate out ),A,A( 21 Σ by holding ),( γβ fixed and compute the constrained MLE for ),A,A( 21 Σ
• Through OLS estimation, since given Gaussian error terms the maximum likelihood estimators are the
same as ordinary least squared estimators:
∆
= ∑∑
=−
−
=−−
n
1tt1t1t
1n
1tt1
'
1t1t1 ),(dx)(X),(d)(X)(X),(A γββγβββγβ , (4)
∆
= ∑∑
=−
−
=−−
n
1tt2t1t
1n
1tt2
'
1t1t2 ),(dx)(X),(d)(X)(X),(A γββγβββγβ , (5)
),),,(A),,(A(u),(u 21tt γβγβγβγβ = , and ∑=
=Σn
1t
'
tt ),(u),(un
1),( γβγβγβ . (6)
• The concentrated likelihood function is:
2
np),(log
2
n),),,(),,(A),,(A(L),(L 21nn −Σ−=Σ= γβγβγβγβγβγβ . (7)
• Compute the vector of parameters: ),( γβ .
• The MLE ),(uu tt γβ= are the minimizers of ),(log γβΣ subject to the boundary constraint (3).
• The MLE for A1 and A2 are then: ),(AA 11 γβ= and ),(AA 22 γβ= .
18
Application to Term Structure of Interest Rates:
• Let x1t be the long rate and x2t be the short rate. Then a linear cointegrating VAR model is:
+
∆
∆
ΓΓΓΓ
+−
+
=
∆
∆
−
−
−−
t2
t1
1t2
1t1
2221
1211
1t21t1
2
1
2
1
t2
t1
u
u
x
x)xx(
x
xβ
αα
µµ
(8)
• Note: if set β = 1, then the error-correction term becomes the interest rate spread
• A two-regime model H1 will allow all coefficients to differ depending upon γβ ≤− −− 1t21t1 xx or
γβ >− −− 1t21t1 xx :
+
∆
∆
ΓΓ
ΓΓ+−
+
=
∆
∆
−
−
−−
t2
t1
1t2
1t1
)1(
22
)1(
21
)1(
12
)1(
11
1t21t1)1(
2
)1(
1
)1(
2
)1(
1
t2
t1
u
u
x
x)xx(
x
xβ
α
α
µ
µ, if γβ ≤− −− 1t21t1 xx , (9a)
+
∆
∆
ΓΓ
ΓΓ+−
+
=
∆
∆
−
−
−−
t2
t1
1t2
1t1
)2(
22
)2(
21
)2(
12
)2(
11
1t21t1)2(
2
)2(
1
)2(
2
)2(
1
t2
t1
u
u
x
x)xx(
x
xβ
α
α
µ
µ, if γβ >− −− 1t21t1 xx (9b)
19
Model 3: Enders-Siklos Threshold Cointegration Model
• standard unit-root and cointegration tests and their corresponding error correction representation may entail a
misspecification error if the adjustment process is asymmetric
• two types of asymmetric tests in the form of threshold autoregressive ( TAR ) and momentum threshold
autoregressive ( M-TAR ) adjustments representations were offered by Enders and Granger (1998) and Enders
and Siklos (2001)
Review of Engle-Granger Cointegration Test and Error Correction Representation
• conventional models often assume linearity and symmetric adjustment process for cointegrated variables
• Engle and Granger (1987) two-step cointegration test:
Step1. The first step is to apply ordinary least squares method (OLS) to estimate the regression model:
,x...xxx tntnt33t220t1 µββββ +++++= (1)
where xit are individual I(1) processes, βi’s are the parameters, with i = 0, 2, … , n, and µt is a stochastic
disturbance term that may be serially correlated
20
Step 2. The second step is a Dickey-Fuller (1979, 1981) type of unit root test applied to the OLS estimate of ρ in:
,ˆˆt1t1t εµρµ +=∆ −− (2)
where tµ is the residual from the OLS estimate of (1) and εt is a white noise process
• Engle-Granger cointegration test, the null hypothesis of no cointegration is H0: ρ = 0
• rejecting the null hypothesis of no cointegration (i.e., accepting the alternative hypothesis of HA: –2 < ρ < 0)
implies that the error process in (2) is stationary with mean zero
• this also implies the whole system of x1t, x2t, … , and xnt are cointegrated with a symmetric adjustment
mechanism towards the long run equilibrium (or the attractor) 0β
• the Granger Representation Theorem suggests that if ρ ≠ 0 (i.e., the system of x1t, x2t, … , and xnt are
cointegrated), then (1) and (2) will guarantee the existence of an error-correction representation in the form of:
t11ntn1t331t2201t11t1 )x...xxx(x εββββα +−−−−−=∆ −−−− (3)
• similar representations can be derived for x2t, x3t, … , and xnt
21
Enders-Siklos Cointegration Test
• cointegration tests by incorporating TAR and M-TAR adjustments into the unit-root tests of the residuals of the
cointegration regression such as equation (1)
• assuming the deviations from long run equilibrium behave as a TAR process:
,)I1(I t1t2t1t1tt εµρµρµ +−+=∆ −− (4)
where It is the Heaviside indicator function such that:
<≥
=−
−
.0if,0
0if,1I
1t
1t
t µµ
(5)
• in the M-TAR model, the Heaviside indicator function Mt is defined as:
<∆
≥∆=
−
−
.0if,0
0if,1M
1t
1t
t µµ
(6)
• the asymmetric adjustment coefficients of ρ1 and ρ2 allow a state-dependent autoregressive decay process, e.g.,
in the M-TAR model: if 01t ≥∆ −µ , the adjustment is 1t1 −µρ ; while if 1t −∆µ < 0, then the adjustment is 1t2 −µρ
22
Example: consider the TAR model, equations (4) and (5): the autoregressive decay to depend on µt-1
1. if |ρ2| > |ρ1|, say ρ1 = -0.2, ρ2 = -0.8, then positive deviations from the long-run cointegration equilibrium are
more persisted than negative deviations
2. there is a slow adjustment when the equilibrium error is above the attractor, while there is an accelerated
adjustment when the equilibrium error is below the attractor
3. this adjustment mechanism captures the feature of “deep” cyclical processes documented by Sichel (1993)
Example: consider the M-TAR model, equations (4) and (6): the autoregressive decay to depend on ∆µt-1
1. if |ρ2| > |ρ1|, say, ρ1 = -0.2, ρ2 = -0.8, there is little decay when ∆µt-1 is positive but substantial decay when ∆µt-1
is negative; then increases tend to persist but decreases tend to revert quickly toward the attractor
2. the M-TAR model could easily capture the “sharp” movements documented in DeLong and Summer (1986) and
Sichel (1993)
23
Extensions to modify the basic threshold cointegration model:
(1) allow a non-zero drift term as the linear attractor, which can be expressed as:
,)a()I1()a(I t01t2t01t1tt εµρµρµ +−−+−=∆ −− (7)
where It is the Heaviside indicator function such that:
<
≥=
−
−
.aif,0
aif,1I
01t
01t
t µµ
(8)
(2) allow a drift and linear trend as attractor with the expression:
,)]1t(aa[)I1()]1t(aa[I t101t2t101t1tt εµρµρµ +−−−−+−−−=∆ −− (9)
where It is the Heaviside indicator function such that:
−+<−+≥
=−
−
).1t(aaif,0
)1t(aaif,1I
101t
101t
t µµ
(10)
(3) involve higher-order terms of the error process to purge possible auto-correlation:
24
.)I1(I t
1p
1iiti1t2t1t1tt εµγµρµρµ +∆+−+=∆ ∑
−
=−−− (11)
• to ensure the stationarity of µt, all roots of the characteristic equation of (1 - γ1r - γ2r2 - … - γp-1r
p-1) = 0 must lie
outside the unit circle
• complex models can be built on combinations of the above modifications, e.g., an M-TAR model with a non-
zero attractor with p-th order process can be written as:
,)M1(M t
1p
1iiti1t2t1t1tt εµγµρµρµ +∆+−+=∆ ∑
−
=−−− (12)
Mt is the Heaviside indicator function and a0 is the linear attractor such that:
<∆
≥∆=
−
−
.aif,0
,aif,1M
01t
01t
t µµ
(13)
25
Chan’s consistent estimator of the threshold
• Tsay (1989) and Chan (1993) offer methodologies if the underlying variables are within the threshold
autoregressive framework
• Tong (1983) also demonstrates that if the adjustment process is asymmetric then the sample mean is a biased
estimator of the attractor
• Chan (1993) shows that searching over all values of a0 so as to minimize the sum of squared errors from the
fitted model yields a super-consistent estimator of the threshold
Estimation Procedures
Case 1: τ equals 0
Step1:
1. regress one of the variables on a constant and the other variable(s) and save the residuals sequence { tµ }
2. set the Heaviside indicator function according to (5) or (6) using τ = 0
26
3. estimate a regression equation in the form of (4) and record the larger of the t statistics for the null
hypothesis of ρi = 0 along with the F statistic for the null hypothesis H0: ρ1 = ρ2 = 0
4. compare the F-statistic with appropriate critical values simulated by Enders & Siklos (2001) in Tables 1 or 2
Step 2:
1. if the alternative hypothesis of stationarity is accepted, next to test for symmetric adjustment (i.e., ρ1 = ρ2)
2. when the value of threshold is known, Enders and Falk (1999) stated that bootstrap t intervals and classic t
intervals work well enough to be recommended in practice
Step3:
1. diagnostic checking of the residuals should be undertaken to ascertain whether the tε series could
reasonably be characterized by a white-noise process
2. for the TAR model, if the residuals are serially correlated, return to Step 2 and reestimate the model in the
form: tptp1t11t2t1t1ttˆˆˆ)I1(ˆIˆ εµγµγµρµρµ +∆+⋅⋅⋅+∆+−+=∆ −−−−
27
3. for the M-TAR case, replace It with Mt as specified in (6)
4. lag lengths can be determined by an analysis of the regression residuals and using model-selection criteria
such as AIC/BIC
28
Case 2: τ is unknown
Step 1:
regress one of the variables on a constant and the other variable(s) and save the residuals sequence { tµ }
Step 2:
1. for TAR case, the estimated residual series is sorted in ascending order and called τττ µµµ T21 <⋅⋅⋅<< , where
T denotes the number of usable observations
2. discard the largest and smallest 15% of the }{ i
τµ values and each of the remaining 70% of the values were
considered as possible thresholds
Step 3:
1. for each of these possible thresholds, estimate an equation in the form of (4) and (5)
29
2. the estimated threshold yielding the lowest residual sum of squares is the appropriate estimate of the
threshold
3. for the M-TAR case, the potential thresholds are τττ µµµ T21 ,,, ∆⋅⋅⋅∆∆ such that τττ µµµ T21 ∆<⋅⋅⋅<∆<∆
4. for each of these possible thresholds, estimate an equation in the form of (4) and (6)
5. the estimate of the threshold is the estimated threshold yielding the lowest residual sum of squares
Step 4:
reestimate the model by incorporating the estimated threshold
Step 5:
1. inference concerning the individual values of 1ρ and 2ρ , and the restriction 21 ρρ = , is problematic when
the true value of the threshold τ is unknown
30
2. since the property of asymptotic multivariate normality has not been established for this case, Chan and
Tong (1989) conjectured that utilizing a constant estimate should establish the asymptotic normality of the
coefficients
3. Enders and Falk (1999) found that the inversion of the bootstrap distribution for the likelihood ratio statistic
provides reasonably good coverage in small samples
Next, to perform the following tests:
1. Estimate error-correction model by incorporating the asymmetric adjustment
2. Estimate error-correction model based on Engle-Granger symmetric adjustment method
3. Conduct forecasting performance evaluation and simulation of Enders-Siklos method versus Engle-
Granger method