Nonstationary Time Series

Table of Contents

Nonstationarity in Time Series

Trend in Time Series

Unit Roots Tests

Unit Roots Tests with Structural Break

Cointegration Tests

Statistical Tables

Readings and References:

Nonstationarity in Time Series

Economic data series follow random data generating process, stationary or nonstationary, although most of macroeconomic time series are nonstationary. Nonstationarity in time series can be identified with the presence of trend, seasonality, and structural change, etc..

Stationary Data Generating Process

For each data observation Y1, Y2, ...

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.

Nonstationary Data Generating Process

Integrated Process

A stationary process can be derived from a nonstationary process by differencing the series one or more times. Therefore the original level series is the integration of the differenced series. An integrated process of order d is denoted by I(d) for d=0,1,2,...

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).

Trend in Time Series

Trend Stationary Process

A stationary time series process can be derived by removing the linear or exponential trend from a nonstationary series. It is named trend stationarity.

Yt = a + bt + et, or
Yt = a + bt + gt2 + et

If et is stationary, then Yt is a trend stationary process.

Difference Stationary Process

A stationary time series process can be derived by differencing a nonstationary series. It is named difference stationarity. By removing the trend from a difference stationary series does not necessarily achieve trend stationarity (removing trend in the variance). However, a trend stationary process is also difference stationary.

Spurious Regression

Most of macroeconomic time series are nonstationary, and may have trend. That is, they are trend nonstationary. By removing the trend, only the trend stationary series are meaningful. By differencing a nonstationary time series doe not establish the trend stationarity, therefore a trend regression on such nonstationary time series has no meaning or spurious. A regression involves trend nonstationary time series may be spurious with the following characteristics:

High R2
Low DW (DW ® 0 or r ® 1)

Unit Roots Tests

Test for a difference stationary process is important since it is the potential source of spurious regression. That is, a trend nonstationay process should be estimated with difference data series, while a trend stationary process can be estimated with level data series.

The purpose of an unit roots 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:

  1. The Model: I, II, III
  2. The Sample Size: N
  3. The Level of Significance: e
The model error is assumed to be serial uncorrelated and homogeneously distributed. Extentions of DF tests include Said-Dickey on ARMA error structure, and Phillips-Perron on weakly dependent and heterogeneously distributed error structure. Both extentions of unit roots test have the same asymptotic distribution as the Dickey-Fuller distribution.

Augmented Dickey-Fuller t-Test

Simple Hypothesis Testing of Unit Root

Augmented Dickey-Fuller F-Test

Joint Hypothesis Testing of Unit Root

Unit Roots Test Procedure

Unit Roots Tests with Structural Break

The classical unit roots tests described above tend to not rejecting the unit root (or has low power) of a time series with changing mean or breaking trend. Let TB be the the break time of the sample period T, and define l = TB/T.

Exogenous Structural Break

If the breakpoint l is fixed (or given a prior), based on Model III (random walk with drift and trend), Perron [1989] considered three versions of hypothesis testing for unit roots and structural change:

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.

Endogenous Structural Break

If the breakpoint l is unknown and must be estimated, the null hypothesis is:

Yt = a + Yt-1 + et

Therefore three versions of unit roots 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 statitic. 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.

Homework

  1. Collect the following macroeconomic annual time series of China from China Statistical Yearbook: GNP, GDP, GDP1 (GDP of primary industry), GDP2 (GDP of secondary industry), GDP3 (GDP of teriary industry), and per capita GDP. From both the nominal and real terms of the above 6 definitions of national products, derive the corresponding price deflators.
  2. Define and test the unit roots for each of the 18 economic variables (nominal, real, and deflator of GDPs) assuming no structural break in the series.
  3. Define and test the unit roots for each of the 18 economic variables (nominal, real, and deflator of GDPs) assuming one-time structural break in the series.

Cointegration Tests

Consider a set of M variables Zt (a 1xM vector). If Zt ~ I(1), the column-wise linear combination of Zt is again usually I(1). Are there any suituations that one or more of such linear combinations will result a stationary process or I(0)? In other words, does the set of variables Zt cointegrate? A regression relationship involving Zt will only be meaningful or not spurious if the variables in Zt are cointegrated.

Cointegration Test: The Engle-Granger Approach

Without loss of generality, let Yt = Zt1 and Xt = [Zt2, ..., ZtM]. Consider the following regression equation:

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] cointergrates, 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 roots 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 roots 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 cointegreating 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.

Error Correction Model

When Yt and Xt are cointegrated, we have

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.

Cointegration Test: The Johansen Approach

Given a set of M variables Zt=[Zt1, Zt2, ..., ZtM], and considering their simultanenity, Johansen's FIML (Full Information Maximum Likelihood) approach of cointegration test is derived from

Similar to the random walk (unit roots) hypothsis testing for a single variable with argumented lags, we write a VAR(p) linear system for the M variables Zt:

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 vetors 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 simplied by considering the following two auxilary regressions:

  1. DZt = DZt-1F1 + DZt-2F2 + ... + DZt-(p-1)Fp-1 + F0 + Wt

  2. Zt-1 = DZt-1Y1 + DZt-2Y2 + ... + DZt-(p-1)Yp-1 + Y0 + Vt

Then Gj = Fj-YjP, for j=0,1,2,...,p-1, and Ut = Wt - VtP. If F0 = Y0 = 0, then G0 = 0 implying no drift in the VAR(p) representation. However, G0 = 0 will need only the restriction that F0 = Y0P.

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 similarily 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)

Statistical Tables

Table 1: Critical Values for the Dickey-Fuller Unit Root t-Test Statistics

                        Probabilty 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

                        Probabilty 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

Table 2: Critical Values for the Dickey-Fuller Unit Root F-Test Statistics

                        Probabilty 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

Table 3: Critical Values for the Dickey-Fuller Unit Root t-Test Statistics with One-Time Structural Break

Model:

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). 

                        Probabilty 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

Table 4: Critical Values for the Engle-Granger Cointegration t-Test Statistics Applied to Regression Residuals

Model:
Yt = a + Xt b + et
Det = (r-1)et-1 + åj=1,2,...,J rt-jDet-j + ut
K = Numbers of variables in the cointegration tests, i.e. [Yt, Xt].
t = 1,2,...,N (500).

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

Table 5: Critical Values for Unit Root and Cointegration Tests Based on Response Surface Estimates

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

Table 6: Critical Values for the Johansen's Cointegration Likelihood Ratio Test Statistics

Notation:
VAR Model: 1=no constant; 2=drift; 3=trend drift
N: Sample Size, 400
M: Number of Variables
r: Number of Cointegrating Vectors or Rank
Degree of Freedom = M-r 

                    Probabilty 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

Copyright © Kuan-Pin Lin
First edition: December 17, 1999
Last updated: June 6, 2002