E(Yt) = m
Var(Yt) = g0 = s2
Cov(Yt,Ys) = g|t-s|, t ¹ s.
In other words, all the descriptive statistics about the time series: m, g0, g1, g2, ... are time invariant.
Yt ~ ii(m,s2) for each observation t = 1,2,...
That is, Yt is an individually independent data generating process with m mean and constant variance s2:
E(Yt) = m
Var(Yt) = g0 = s2
Cov(Yt,Ys) = 0, t ¹ s.
Yt = a + r1Yt-1 + r2Yt-2 + ... + rpYt-p + et
where r1, r2, ..., rp lie outside the unit circle of the p-th order polynomial function of B (ie. 1 - r1B - r2B2 - ... - rpBp = 0); and et ~ ii(0,s2), t = 1,2,...
Yt = m - q1et-1 - q2et-2 - ... - qqet-q + et
where q1, q2, ..., qp lie outside the unit circle of the q-th order polynomial function of B (ie. 1 - q1B - q2B2 - ... - qpBq = 0); and et ~ ii(0,s2), t = 1,2,...
Yt = d + r1Yt-1 + r2Yt-2 + ... + rpYt-p - q1et-1 - q2et-2 - ... - qqet-q + et
where r1, r2, ..., rp lie outside the unit circle of the p-th order polynomial function of B (ie. 1 - r1B - r2B2 - ... - rpBp = 0); q1, q2, ..., qq lie outside the unit circle of the q-th order polynomial function of B (ie. 1 - q1B - q2B2 - ... - qpBq = 0); and et ~ ii(0,s2), t = 1,2,...
Yt = a + bt + et where et ~ ii(0,s2), t = 1,2,.... Then
E(Yt) = a + bt
Var(Yt) = s2
As t ®¥, E(Yt) ®¥. This is the model with linear trend in the mean.
Yt = Yt-1 + et where et ~ ii(0,s2), t = 1,2,.... Equivalently,
Yt = Y0 + åi=1,2,...,t ei
Assuming Y0 exists and finite,
E(Yt) = Y0
Var(Yt) = ts2
As t ®¥, Var(Yt) ®¥.
This is the model with linear trend in the variance.
Yt = a + Yt-1 + et where et ~ ii(0,s2), t = 1,2,.... Equivalently,
Yt = Y0 + at + åi=1,2,...,t ei
Assuming Y0 exists and finite,
E(Yt) = Y0 + at
Var(Yt) = ts2
As t ®¥, E(Yt) ®¥ and
Var(Yt) ®¥.
This is the model with linear trend in the mean and variance.
Yt = a + bt + Yt-1 + et where et ~ ii(0,s2), t = 1,2,.... Equivalently,
Yt = Y0 + a t + b t2 + åi=1,2,...,t ei
where a = a + b/2 and b = b/2. Assuming Y0 exists and finite,
E(Yt) = Y0 + a t + b t2
Var(Yt) = ts2
As t ®¥, E(Yt) ®¥ and
Var(Yt) ®¥.
This is the model with exponential trend in the mean and
linear trend in the variance.
That is, Yt ~ I(d) if DdYt is stationary, where
DYt = Yt - Yt-1,
D2Yt = DYt - DYt-1, ...
For example, if Yt ~ I(1), then
| Yt | = DYt + Yt-1 |
| = DYt + DYt-1 + Yt-2 = ... | |
| = åj=0,...,t-1DYt-j with a known Y0 |
Similarly, if Yt ~ I(2), then
DYt-j =
åi=0,...,t-j-1D2Yt-j-i and
| Yt | = åj=0,...,t-1DYt-j |
| = åj=0,...,t-1åi=0,...,t-j-1D2Yt-j-i |
The white noise process is an integrated process of order 0, or I(0). A random walk process is an integrated process of order 1, or I(1).
Yt = a + bt +
et, or
Yt = a + bt +
gt2 + et
If et is stationary, then Yt is a trend stationary process.
Yt = Yt-1 + et, or
DYt =
Yt - Yt-1 = et
If et is stationary, then Yt is a difference stationary process.
Yt = a + Yt-1 + et, or
DYt =
Yt - Yt-1 = a + et
If et is stationary, then Yt is a difference stationary process.
Yt = a + bt +
Yt-1 + et, or
DYt =
Yt - Yt-1 = a + bt +
et
If et is stationary, then Yt is a difference stationary process (DYt is a trend stationary process).
High R2
Low DW (DW ® 0 or
r ® 1)
The purpose of the unit root test is to statistically test the data generating process for difference stationarity (trend nonstationarity) against trend stationarity. It is a formal test for Random Walk Hypothesis.
Dickey-Fuller (DF) and Augmented Dickey-Fuller (ADF) tests for unit roots (or random walk) depends on:
DYt = (r-1)Yt-1 + åj=1,2,...,J rjDYt-j+et
| Hypothesis | H0: r = 1 H1: r < 1 |
| Test Statistic | tr = (p-1)/se(p) p is the estimated r |
| Critical Value | ADFtr(I,N,e) |
DYt = a + (r-1)Yt-1 + åj=1,2,...,J rjDYt-j+et
| Hypothesis | H0: r = 1 H1: r < 1 | H0: a = 0,
given r = 1 H1: a ¹ 0 |
| Test Statistic | tr = (p-1)/se(p) p is the estimated r | ta = a/se(a) a is the estimated a |
| Critical Value | ADFtr(II,N,e) | ADFta(II,N,e) |
DYt = a + bt + (r-1)Yt-1 + åj=1,2,...,J rjDYt-j+et
| Hypothesis | H0: r = 1 H1: r < 1 | H0: a = 0,
given r = 1 H1: a ¹ 0 | H0: b = 0,
given r = 1 H1: b ¹ 0 |
| Test Statistic | tr = (p-1)/se(p) p is the estimated r | ta = a/se(a) a is the estimated a | tb = b/se(b) b is the estimated b |
| Critical Value | ADFtr(III,N,e) | ADFta(III,N,e) | ADFtb(III,N,e) |
DYt = a + (r-1)Yt-1 + åj=1,2,...,J rjDYt-j+et
| Hypothesis | H0: a = 0, r = 1 H1: not H0 |
| Restricted Model | DYt = åj=1,2,...,J rjDYt-j+et |
| Test Statistic | Fa,r = (RSSr-RSSur)/2 / RSSur/(N-J-2) |
| Critical Value | ADFFa,r(II,N,e) |
DYt = a + bt + (r-1)Yt-1 + åj=1,2,...,J rjDYt-j+et
| Hypothesis | H0: a = 0, b = 0, r = 1 H1: not H0 | H0: b = 0, r = 1 H1: not H0 |
| Restricted Model | DYt = åj=1,2,...,J rjDYt-j+et | DYt = a + åj=1,2,...,J rjDYt-j+et |
| Test Statistic | Fa,b,r = (RSSr-RSSur)/3 / RSSur/(N-J-3) | Fb,r = (RSSr-RSSur)/2 / RSSur/(N-J-3) |
| Critical Value | ADFFa,b,r(III,N,e) | ADFFb,r(III,N,e) |
DYt = a + bt + (r-1)Yt-1 + åj=1,2,...,J rjDYt-j+et
If r < 1 the stop (no unit root) else continue
If b ¹ 0 then
Step 2:
DYt = a + (r-1)Yt-1 + åj=1,2,...,J rjDYt-j+et
If r < 1 the stop (no unit root) else continue
If a ¹ 0 then
Step 3:
DYt = (r-1)Yt-1 + åj=1,2,...,J rjDYt-j+et
If r < 1 the stop (no unit root) else conclude (unit root)!
Model IIIa
H0: Yt = a + Yt-1 + qD(TB)t + et
H1: Yt = a1 + bt + (a2-a1)DUt + et
Model IIIb
H0: Yt = a1 + Yt-1 + (a1-a2)DUt + et
H1: Yt = a + b1t + (b2-b1)DTt + et
Model IIIc
H0: Yt = a1 + Yt-1 + qD(TB)t + (a1-a2)DUt + et
H1: Yt = a1 + b1t + (a2-a1)DUt + (b2-b1)DTt + et
Where et is stationary and possibly prescribed by an ARMA(p,q) process, and
| D(TB)t = | 1, if t = TB+1 |
| 0 otherwise | |
| DUt = | 1, if t>TB |
| 0 otherwise | |
| DTt = | t-TB, if t>TB |
| 0 otherwise |
Then the corresponding augmented testing equations are:
Model IIIa
DYt = a + bt + qD(TB)t + dDUt
+ (r-1)Yt-1 + åj=1,2,...,JrjDYt-j + et
Model IIIb
DYt = a + bt + dDUt + gDTt
+ (r-1)Yt-1 + åj=1,2,...,JrjDYt-j + et
Model IIIc
DYt = a + bt + qD(TB)t + dDUt + gDTt
+ (r-1)Yt-1 + åj=1,2,...,JrjDYt-j + et
For each version of testing equation, at the location of breakpoint l, t statistic of the lag parameter r or tr(l) is compared with the critical values of the asymptotic distribution of this statistic. We reject the null hypothesis of unit root if the computed tr(l) is less than the critical values for a given l.
Yt = a + Yt-1 + et
Therefore three versions of unit root test are:
Model IIIa
H0: Yt = a + Yt-1 + et
H1: Yt = a1 + bt + (a2-a1)DUt(l) + et
Model IIIb
H0: Yt = a + Yt-1 + et
H1: Yt = a + b1t + (b2-b1)DTt(l) + et
Model IIIc
H0: Yt = a + Yt-1 + et
H1: Yt = a1 + b1t + (a2-a1)DUt(l) + (b2-b1)DTt(l) + et
The corresponding augmented testing equations are:
Model IIIa
DYt = a + bt + dDUt(l)
+ (r-1)Yt-1 + åj=1,2,...,JrjDYt-j + et
Model IIIb
DYt = a + bt + gDTt(l)
+ (r-1)Yt-1 + åj=1,2,...,JrjDYt-j + et
Model IIIc
DYt = a + bt + dDUt(l) + gDTt(l)
+ (r-1)Yt-1 + åj=1,2,...,JrjDYt-j + et
We write the dummy variables DUt and DTt to depend on the breakpoint l, which is the outcome of fitting Yt to a certain trend stationary process with a one-time structural break at an unknown point of time. The purpose is to estimate the breakpoint that gives the most weight to the trend stationary alternative. In other words, l* is chosen to minimize the one-sided t statistic for testing the lag parameter r = 1. The estimate breakpoint l* and minimum t statistic are obtained as follows:
For Model IIIa, IIIb, IIIc, estimate the test equation for all possible values of l in (0,1). That is, from TB=2 to TB=T-1, run T-2 regressions and collect all the t statistics for testing r=1. We note that, the augmented lags J used in the test equation may be different for each l=TB/T.
Let tr* = minl in (0,1){tr(l)}, and l* is the estimated breakpoint corresponds to this minimum t statistic. Zivot and Andrews [1992] tabulates the critical values of the asymptotic distribution for tr*. The computed tr* is used to compared with these critical values. We reject the null hypothesis of unit root if the computed tr* is less than the critical value for a given level of significance.
Yt = a + Xtb + et
In general, if Yt, Xt ~ I(1), then et ~ I(1). If et can be shown to be I(0), then the set of variables [Yt, Xt] cointegrates, and the vector [1 -b]' (or any multiple of it) is called a cointegrating vector. Depending on the number of variables M, there are up to M-1 linearly independent cointegrating vectors. The number of linearly independent cointegrating vectors that exists in [Yt, Xt] is called cointegrating rank.
A simple way to test for cointegration is to apply unit root test on the residuals of the above regression equation. Let
N = Number of usable sample observations;
K = Number of variables in [Yt,Xt] for cointegration test
The unit root test for the regression residuals, or the cointegration test, is formulated as follows:
Det = (r-1)et-1 + ut
or with augmented lags:
Det =
(r-1)et-1 +
åj=1,2,...,J
rt-jDet-j + ut
| Hypothesis | H0: r = 1 H1: r < 1 | |
| Test Statistic | tr = (p-1)/se(p) where p is the estimate of r | |
| Critical Value | ADF(I,N,e) | |
If we can reject the null hypothesis of unit root on the residuals et, we can say that variables [Yt, Xt] in the regression equation are cointegrated. The cointegrating regression model may be generalized to include trend as follows:
Yt = a + gt + Xtb + et
Notice that the trend in the cointegrating regression equation may be the result of combined drifts in X and/or Y.
J. MacKinnon's table of critical values of cointegration tests for both cointegrating regression with and without trend (named Model 2 and Model 3, respectively) is provided in Table 5. It is based on simulation experiments by means of response surface regression in which critical values depend on the sample size. Therefore, this table is easier and more flexible to use than the original EG and AEG distributions.
Yt = a + Xtb + et
Det =
(r-1)et-1 + ut
where r < 1 and ut is stationary. Therefore the short-run dynamics of the model is
| DYt | = DXtb + Det |
| = DXtb + (r-1)et-1 + ut | |
| = DXtb + (r-1)(Yt-1-a-Xt-1b) + ut | |
This is exactly the Error Correction Model.
Zt = Zt-1P1 + Zt-2P2 + ... + Zt-pPp + P0 + Ut
where Pj, j=1,2,...M, are the MxM parameter matrices, P0 is a 1xM drift or constant vector, and the 1xM error vector Ut ~ normal(0,S) with a constant matrix S = Var(Ut) = E(Ut'Ut) denoting the covariance matrix across M variables.
The VAR(p) system can be transformed using the difference series of the variables, resemble the error correction model representation, as follows:
DZt = DZt-1G1 + DZt-2G2 + ... + DZt-(p-1)Gp-1 + Zt-1P + G0 + Ut
where P = åj=1,2,...,pPj - I, G1 = P1 - P - I , G2 = P2 + G1, ..., and G0 = P0 for notational convenience.
If Zt ~ I(1), then DZt ~ I(0). In order to have the variables in Zt cointegrated, we must have Ut ~ I(0). That is, we must show the term Zt-1P ~ I(0). By definition of cointegration, the parameter matrix P must contains 0 < r < M linearly independent cointegrating vectors such that ZtP ~ I(0). Therefore, the cointegration test amounts to check that Rank(P) = r > 0. If Rank(P) = r, we may impose the parameter restrictions P = -BA' where A and B are Mxr matrices. Given the existence of the constant vector G0, there can be up to M-r random walks or the drift trends. Such common trends in the variables may be removed in the case of Model II below. We consider the following three models:
For model estimation of the above VAR(p) system, where Ut ~ normal(0,S), we derive the log-likelihood function for Model III:
ll(G1,G2,..., Gp-1,G0,P,S) = - MN/2 ln(2p) - N/2 ln|det(S)| - ½ åt=1,2,...,NUtS-1Ut'
Since the maximum likelihood estimate of S is U'U/N, the concentrated log-likelihood function is written as:
ll*(G1,G2,..., Gp-1,G0,P) = - NM/2 (1+ln(2p)-ln(N)) - N/2 ln|det(U'U)|
The actual maximum likelihood estimation can be simplified by considering the following two auxiliary regressions:
Returning to the concentrated log-likelihood function, it is now written as
ll*(W(F1,F2,...,Fp-1,F0),
V(Y1,Y2,...,Yp-1,Y0),P)
= - NM/2 (1+ln(2p)-ln(N)) -
N/2 ln|det((W-VP)'(W-VP))|
Maximizing the above concentrated log-likelihood function is equivalent to minimize the sum-of-squares term det((W-VP)'(W-VP)). Conditional to W(F1,F2,...,Fp-1,F0) and V(Y1,Y2,...,Yp-1,Y0), the least squares estimate of P is (V'V)-1V'W. Thus,
det((W-VP)'(W-VP))
= det(W(I-V(V'V)-1V')W')
= det((W'W)(I-(W'W)-1(W'V)(V'V)-1(V'W))
= det(W'W) det(I-(W'W)-1(W'V)(V'V)-1(V'W))
= det(W'W) (Õi=1,2,...,M(1-li))
where l1, l2, ..., lM are the ascending ordered eigenvalues of the matrix (W'W)-1(W'V)(V'V)-1(V'W). Therefore the resulting double concentrated log-likelihood function (concentrating on both S and P) is
ll**(W(F1,F2,...,Fp-1,F0),
V(Y1,Y2,...,Yp-1,Y0))
= - NM/2 (1+ln(2p)-ln(N)) -
N/2 ln|det(W'W)| - N/2
åi=1,2,...,Mln(1-li)
Given the parameter constraints that there are 0 < r < M cointegrating vectors, that is P = -BA' where A and B are Mxr matrices, the restricted concentrated log-likelihood function is similarly derived as follows:
llr**(W(F1,F2,...,Fp-1,F0),
V(Y1,Y2,...,Yp-1,Y0))
= - NM/2 (1+ln(2p)-ln(N)) -
N/2 ln|det(W'W)| - N/2
åi=1,2,...,rln(1-li)
Therefore, with the degree of freedom M-r, the likelihood ratio test statistic for at least r cointegrating vectors is
-2(llr** - ll**) = -N åi=r+1,2,...,Mln(1-li)
Similarly the likelihood ratio test statistic for r cointegrating vectors against r+1 vectors is
-2(llr** - llr+1**) = -N ln(1-lr+1)
A more general form of the likelihood ratio test statistic for r1 cointegrating vectors against r2 vectors (0 £ r1 < r2 £ M) is
-2(llr1** - llr2**) = -N åi=r1+1,2,...,r2ln(1-li)
The following table summarizes the two popular cointegration test statistics: Eigenvalue Test Statistic lmax(r) and Trace Test Statistic ltrace(r). For the case of r = 0, they are the tests for no cointegration.
| Cointegrating Rank (r) | H0: r1 = r H1: r2 = r+1 | H0: r1 = r H1: r2 = M |
| 0 | -N ln(1-l1) | -N åi=1,2,...,Mln(1-li) |
| 1 | -N ln(1-l2) | -N åi=2,3,...,Mln(1-li) |
| ... | ... | ... |
| M-1 | -N ln(1-lM) | -N ln(1-lM) |
| Critical Value | lmax(r) | ltrace(r) |
1 - r1B - r2B2 - ... - rpBp = 0
must be great than 1 in absoulte value for the model to be stable.
For example, consider the AR(1) model. The characteristic equation is 1 - r1B = 0. The single root of this equation is B = 1/r1, which is greater than 1 in absolute value if |r1| < 1. Similarly, for an AR(2) model, the two roots of the characteristic equation 1 - r1B - r2B2 = 0 are B1,B2 = [r1±Ö(r12+4r2)]/2. Therefore, the stability conditions are:
A more general AR(p) model may be represented by VAR(1):
|
|
| = |
|
|
| + |
|
|
|
|
|
| + |
|
|
|
That is, Yt = a + r Yt-1 + et
By successive substitution, we obtain Yt = a + ra + r2a + ... (so that the equilibrium Y¥ = (I-r)-1a).
The roots of the asymmetric matrix r may be complex in the form a±bi, where i=Ö(-1). The stability requires that all the roots of r must be less than 1 in absolute value. That is, |a+bi| = Ö(a2+b2) < 1.
The unit circle refers to the two-dimensional set of values of a and b defined by a2+b2=1, which defines a circle centered at the origin with radius 1. Therefore, for a stable dynamic model, the roots of the characteristic equation
1 - r1B - r2B2 - ... - rpBp = 0
which are the reciprocals of the characteristic roots of the matrix r must lie outside the unit circle.
Probability to the Right of Critical Value
Model Statistic N 99% 97.5% 95% 90% 10% 5% 2.5% 1%
I ADFtr 25 -2.66 -2.26 -1.95 -1.60 0.92 1.33 1.70 2.16
50 -2.62 -2.25 -1.95 -1.61 0.91 1.31 1.66 2.08
100 -2.60 -2.24 -1.95 -1.61 0.90 1.29 1.64 2.03
250 -2.58 -2.23 -1.95 -1.61 0.89 1.29 1.63 2.01
500 -2.58 -2.23 -1.95 -1.61 0.89 1.28 1.62 2.00
>500 -2.58 -2.23 -1.95 -1.61 0.89 1.28 1.62 2.00
II ADFtr 25 -3.75 -3.33 -3.00 -2.62 -0.37 0.00 0.34 0.72
50 -3.58 -3.22 -2.93 -2.60 -0.40 -0.03 0.29 0.66
100 -3.51 -3.17 -2.89 -2.58 -0.42 -0.05 0.26 0.63
250 -3.46 -3.14 -2.88 -2.57 -0.42 -0.06 0.24 0.62
500 -3.44 -3.13 -2.87 -2.57 -0.43 -0.07 0.24 0.61
>500 -3.43 -3.12 -2.86 -2.57 -0.44 -0.07 0.23 0.60
III ADFtr 25 -4.38 -3.95 -3.60 -3.24 -1.14 -0.80 -0.50 -0.15
50 -4.15 -3.80 -3.50 -3.18 -1.19 -0.87 -0.58 -0.24
100 -4.04 -3.73 -3.45 -3.15 -1.22 -0.90 -0.62 -0.28
250 -3.99 -3.69 -3.43 -3.13 -1.23 -0.92 -0.64 -0.31
500 -3.98 -3.68 -3.42 -3.13 -1.24 -0.93 -0.65 -0.32
>500 -3.96 -3.66 -3.41 -3.12 -1.25 -0.94 -0.66 -0.33
Probability to the Right of Critical Value
Model Statistic N 1% 2.5% 5% 10% (Symmetric Distribution, given r = 1)
II ADFta 25 3.14 2.97 2.61 2.20
50 3.28 2.89 2.56 2.18
100 3.22 2.86 2.54 2.17
250 3.19 2.84 2.53 2.16
500 3.18 2.83 2.52 2.16
>500 3.18 2.83 2.52 2.16
III ADFta 25 4.05 3.59 3.20 2.77
50 3.87 3.47 3.14 2.78
100 3.78 3.42 3.11 2.73
250 3.74 3.39 3.09 2.73
500 3.72 3.38 3.08 2.72
>500 3.71 3.38 3.08 2.72
III ADFtb 25 3.74 3.25 2.85 2.39
50 3.60 3.18 2.81 2.38
100 3.53 3.14 2.79 2.38
250 3.49 3.12 2.79 2.38
500 3.48 3.11 2.78 2.38
>500 3.46 3.11 2.78 2.38
Probability to the Right of Critical Value
Model Statistic N 1% 2.5% 5% 10% 90% 95% 97.5% 99%
II ADFFa,r 25 7.88 6.30 5.18 4.12 0.65 0.49 0.38 0.29
50 7.06 5.80 4.86 3.94 0.66 0.50 0.30 0.29
100 6.70 5.57 4.71 3.86 0.67 0.50 0.30 0.29
250 6.52 5.45 4.63 3.81 0.67 0.51 0.39 0.30
500 6.47 5.41 4.61 3.79 0.67 0.51 0.39 0.30
>500 6.43 5.38 4.59 3.78 0.67 0.51 0.40 0.30
III ADFFa,b,r 25 8.21 6.75 5.68 4.67 1.10 0.89 0.75 0.61
50 7.02 5.94 5.13 4.31 1.12 0.91 0.77 0.62
100 6.50 5.59 4.88 4.16 1.12 0.92 0.77 0.63
250 6.22 5.40 4.75 4.07 1.13 0.92 0.77 0.63
500 6.15 5.35 4.71 4.05 1.13 0.92 0.77 0.63
>500 6.09 5.31 4.68 4.03 1.13 0.92 0.77 0.63
III ADFFb,r 25 10.61 8.65 7.24 5.91 1.33 1.08 0.90 0.74
50 9.31 7.81 6.73 5.61 1.37 1.11 0.93 0.76
100 8.73 7.44 6.49 5.47 1.38 1.12 0.94 0.76
250 8.43 7.25 6.34 5.39 1.39 1.13 0.94 0.76
500 8.34 7.20 6.30 5.36 1.39 1.13 0.94 0.76
>500 8.27 7.16 6.25 5.34 1.39 1.13 0.94 0.77
Model IIIa
DYt = a + bt + dDUt(l)
+ (r-1)Yt-1 + åj=1,2,...,JrjDYt-j + et
Model IIIb
DYt = a + bt + gDTt(l)
+ (r-1)Yt-1 + åj=1,2,...,JrjDYt-j + et
Model IIIc
DYt = a + bt + dDUt(l) + gDTt(l)
+ (r-1)Yt-1 + åj=1,2,...,JrjDYt-j + et
Where l = TB/T (T is the sample size and TB is the break point), and
| DUt(l) = | 1, if t>TB |
| 0 otherwise | |
| DTt(l) = | t-TB, if t>TB |
| 0 otherwise |
l* is the estimated breakpoint which minimizes the t statistic tr(l) for testing the unit root over the range of 0 < l < 1.
Source:
Perron (1989), Zivot and Andrews (1992).
Probability to the Right of Critical Value
Model Statistic l 99% 97.5% 95% 90% 50% 10% 5% 2.5% 1%
IIIa ADFtr(l) l* -5.34 -5.02 -4.80 -4.58 -3.75 -2.99 -2.77 -2.56 -2.32
0.1 -4.30 -3.93 -3.68 -3.40 -2.35 -1.38 -1.09 -0.78 -0.46
0.2 -4.39 -4.08 -3.77 -3.47 -2.45 -1.45 -1.14 -0.90 -0.54
0.3 -4.39 -4.03 -3.76 -3.46 -2.42 -1.43 -1.13 -0.83 -0.51
0.4 -4.34 -4.01 -3.72 -3.44 -2.40 -1.26 -0.88 -0.55 -0.21
0.5 -4.32 -4.01 -3.76 -3.46 -2.37 -1.17 -0.79 -0.49 -0.15
0.6 -4.45 -4.09 -3.76 -3.47 -2.38 -1.28 -0.92 -0.60 -0.26
0.7 -4.42 -4.07 -3.80 -3.51 -2.45 -1.42 -1.10 -0.82 -0.50
0.8 -4.33 -3.99 -3.75 -3.46 -2.43 -1.46 -1.13 -0.89 -0.57
0.9 -4.27 -3.97 -3.69 -3.38 -2.39 -1.37 -1.04 -0.74 -0.47
IIIb ADFtr(l) l* -4.93 -4.67 -4.42 -4.11 -3.23 -2.48 -2.31 -2.17 -1.97
0.1 -4.27 -3.94 -3.65 -3.36 -2.34 -1.35 -1.04 -0.78 -0.40
0.2 -4.41 -4.08 -3.80 -3.49 -2.50 -1.48 -1.18 -0.87 -0.52
0.3 -4.51 -4.17 -3.87 -3.58 -2.54 -1.59 -1.27 -0.97 -0.69
0.4 -4.55 -4.20 -3.94 -3.66 -2.61 -1.69 -1.37 -1.11 -0.75
0.5 -4.55 -4.20 -3.96 -3.68 -2.70 -1.74 -1.40 -1.18 -0.82
0.6 -4.57 -4.20 -3.95 -3.66 -2.61 -1.71 -1.36 -1.11 -0.78
0.7 -4.51 -4.13 -3.85 -3.57 -2.55 -1.61 -1.28 -0.97 -0.67
0.8 -4.38 -4.07 -3.82 -3.50 -2.47 -1.49 -1.16 -0.87 -0.54
0.9 -4.26 -3.96 -3.68 -3.35 -2.33 -1.34 -1.04 -0.77 -0.43
IIIc ADFtr(l) l* -5.57 -5.30 -5.08 -4.82 -3.98 -3.25 -3.06 -2.91 -2.72
0.1 -4.38 -4.01 -3.75 -3.45 -2.38 -1.44 -1.11 -0.82 -0.45
0.2 -4.65 -4.32 -3.99 -3.66 -2.67 -1.60 -1.27 -0.98 -0.67
0.3 -4.78 -4.46 -4.17 -3.87 -2.75 -1.78 -1.46 -1.15 -0.81
0.4 -4.81 -4.48 -4.22 -3.95 -2.88 -1.91 -1.62 -1.35 -1.04
0.5 -4.90 -4.53 -4.24 -3.96 -2.91 -1.96 -1.69 -1.43 -1.07
0.6 -4.88 -4.49 -4.24 -3.95 -2.87 -1.93 -1.63 -1.37 -1.08
0.7 -4.75 -4.44 -4.18 -3.86 -2.77 -1.81 -1.47 -1.17 -0.79
0.8 -4.70 -4.31 -4.04 -3.69 -2.67 -1.63 -1.29 -1.04 -0.64
0.9 -4.41 -4.10 -3.80 -3.46 -2.41 -1.44 -1.12 -0.80 -0.50
Model 2: E(Yt) = E(Xt) = 0 (both X and Y have no drift)
Model 2a: E(Xt) ¹ 0 (at least one variable in X has drift)
Model 3: E(Yt) ¹ 0 but E(Xt) = 0 (only Y has drift)
Note:
For the case of two variables in Model 2a, X is trended but Y is not. It is asymptotically
equivalent to ADF Unit Root Test for Model III (see Table 1, ADFtr for N=500).
If only Y has drift (Model 3), the cointegration equation can be expressed as
Yt = a + g t + Xt b + et.
Therefore, the same critical values of Model 2a apply to Model 3 for one extra variable t
(but not count for K).
Source:
Phillips and Ouliaris (1990)
Model K 1% 2.5% 5% 10% 2 2 -3.96 -3.64 -3.37 -3.07 3 -4.31 -4.02 -3.77 -3.45 4 -4.73 -4.37 -4.11 -3.83 5 -5.07 -4.71 -4.45 -4.16 6 -5.28 -4.98 -4.71 -4.43 2a 2 -3.98 -3.68 -3.42 -3.13 3 -4.36 -4.07 -3.80 -3.52 4 -4.65 -4.39 -4.16 -3.84 5 -5.04 -4.77 -4.49 -4.20 6 -5.36 -5.02 -4.74 -4.46 7 -5.58 -5.31 -5.03 -4.73 3 2 -4.36 -4.07 -3.80 -3.52 3 -4.65 -4.39 -4.16 -3.84 4 -5.04 -4.77 -4.49 -4.20 5 -5.36 -5.02 -4.74 -4.46 6 -5.58 -5.31 -5.03 -4.73
Critical values for unit root and cointegration tests can be computed from the equation:
CV(K, Model, N, sig) = b + b1 (1/N) + b2 (1/N)2
Notation:
Regression Model: 1=no constant; 2=no trend; 3=with trend;
K: Number of variables in cointegration tests (K=1 for unit root test);
N: Number of observations or sample size;
sig: Level of significance, 0.01, 0.05, 0.1.
Source:
J. G. MacKinnon, "Critical Values for Cointegration Tests," Cointegrated Time
Series, 267-276.
K Model sig b b1 b2
1 1 0.01 -2.5658 -1.960 -10.04
1 1 0.05 -1.9393 -0.398 0.00
1 1 0.10 -1.6156 -0.181 0.00
1 2 0.01 -3.4335 -5.999 -29.25
1 2 0.05 -2.8621 -2.738 -8.36
1 2 0.10 -2.5671 -1.438 -4.48
1 3 0.01 -3.9638 -8.353 -47.44
1 3 0.05 -3.4126 -4.039 -17.83
1 3 0.10 -3.1279 -2.418 -7.58
2 2 0.01 -3.9001 -10.534 -30.03
2 2 0.05 -3.3377 -5.967 -8.98
2 2 0.10 -3.0462 -4.069 -5.73
2 3 0.01 -4.3266 -15.531 -34.03
2 3 0.05 -3.7809 -9.421 -15.06
2 3 0.10 -3.4959 -7.203 -4.01
3 2 0.01 -4.2981 -13.790 -46.37
3 2 0.05 -3.7429 -8.352 -13.41
3 2 0.10 -3.4518 -6.241 -2.79
3 3 0.01 -4.6676 -18.492 -49.35
3 3 0.05 -4.1193 -12.024 -13.13
3 3 0.10 -3.8344 -9.188 -4.85
4 2 0.01 -4.6493 -17.188 -59.20
4 2 0.05 -4.1000 -10.745 -21.57
4 2 0.10 -3.8110 -8.317 -5.19
4 3 0.01 -4.9695 -22.504 -50.22
4 3 0.05 -4.4294 -14.501 -19.54
4 3 0.10 -4.1474 -11.165 -9.88
5 2 0.01 -4.9587 -22.140 -37.29
5 2 0.05 -4.4185 -13.461 -21.16
5 2 0.10 -4.1327 -10.638 -5.48
5 3 0.01 -5.2497 -26.606 -49.56
5 3 0.05 -4.7154 -17.432 -16.50
5 3 0.10 -4.4345 -13.654 -5.77
6 2 0.01 -5.2400 -26.278 -41.65
6 2 0.05 -4.7048 -17.120 -11.17
6 2 0.10 -4.4242 -13.347 0.00
6 3 0.01 -5.5127 -30.735 -52.50
6 3 0.05 -4.9767 -20.883 -9.05
6 3 0.10 -4.6999 -16.445 0.00
Probability to the Right of Critical Value
Model M-r 99% 97.5% 95% 90% 80% 50%
lmax 1 1 6.51 4.93 3.84 2.86 1.82 0.58
1 2 15.69 13.27 11.44 9.52 7.58 4.83
1 3 22.99 20.02 17.89 15.59 13.31 9.71
1 4 28.82 26.14 23.80 21.58 18.97 14.94
1 5 35.17 32.51 30.04 27.62 24.83 20.16
2 1 11.576 9.658 8.083 6.691 4.905 2.415
2 2 18.782 16.403 14.595 12.783 10.666 7.474
2 3 16.154 23.362 21.279 18.959 16.521 12.707
2 4 32.616 29.599 27.341 24.917 22.341 17.875
2 5 38.858 35.700 33.262 30.818 27.953 23.132
3 1 6.936 5.332 3.962 2.816 1.699 0.447
3 2 17.936 15.810 14.036 12.099 10.125 6.852
3 3 25.521 23.002 20.778 18.697 16.324 12.381
3 4 31.943 29.335 27.169 24.712 22.113 17.719
3 5 38.341 35.546 33.178 30.774 27.899 23.211
ltrace 1 1 6.51 4.93 3.84 2.86 1.82 0.58
1 2 16.31 14.43 12.53 10.47 8.45 5.42
1 3 29.75 26.64 24.31 21.63 18.83 14.30
1 4 45.58 42.30 39.89 36.58 33.16 27.10
1 5 66.52 62.91 59.46 55.44 51.13 43.79
2 1 11.586 9.658 8.083 6.691 4.905 2.415
2 2 21.962 19.611 17.844 15.583 13.038 9.355
2 3 37.291 34.062 31.256 28.436 25.445 20.188
2 4 55.551 51.801 48.419 45.248 41.623 34.873
2 5 77.911 73.031 69.977 65.956 61.566 53.373
3 1 6.936 5.332 3.962 2.816 1.699 0.447
3 2 19.310 17.299 15.197 13.338 11.164 7.638
3 3 35.397 32.313 29.509 26.791 23.868 18.759
3 4 53.792 50.424 47.181 43.964 40.250 33.672
3 5 76.955 72.140 68.905 65.063 60.215 52.588