Time Series Analysis I

Nonstationary Time Series

Table of Contents

Time Series as Data Generating Process

Trend in Time Series

Unit Roots Tests

Unit Roots Tests with Structural Break

Cointegration Tests

Appendix 1: Stability of a Dynamic Model

Appendix 2: Statistical Tables

Readings

Time Series as Data Generating Process

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..

Covariance Stationary Data Generating Process

For each data observation Y1, Y2, ...

E(Yt) = μ
Var(Yt) = γ0 = σ2
Cov(Yt,Ys) = γ|t-s|, t ≠ s.

In other words, all the descriptive statistics about the time series: μ, γ0, γ1, γ2, ... 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 ΔdYt is stationary, where

ΔYt = Yt - Yt-1,
Δ2Yt = ΔYt - ΔYt-1, ...

For example, if Yt ~ I(1), then

Yt= ΔYt + Yt-1
= ΔYt + ΔYt-1 + Yt-2 = ...
= ∑j=0,...,t-1ΔYt-j with a known Y0

Similarly, if Yt ~ I(2), then

ΔYt-j = ∑i=0,...,t-j-1Δ2Yt-j-i and
Yt = ∑j=0,...,t-1ΔYt-j
= ∑j=0,...,t-1∑i=0,...,t-j-1Δ2Yt-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 = α + βt + εt, or
Yt = α + βt + γt2 + εt

If εt 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 ρ → 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

Alternative Representation of Unit Roots Tests

Alternative Tests for Unit Roots

  1. Phillips-Perron (1987) Test

    Based on model selection criteria, ADF tests use lagged differenced terms to filter serial correlation in the test equation. The alternative Phillips-Perron unit root tests use Newey-West robust standard errors to account for serial correlation. Two statistics are computed: (1) T(p-1), (2) (p-1)/se*(p), where p is the OLS estimate of ρ and se*(p) is the estimated robust standard error of p, from the following three random walk model specifications:

    1. Yt = ρYt-1 + εt
    2. Yt = α + ρYt-1 + εt
    3. Yt = α + βt + ρYt-1 + εt
    Phillips-Perron test statistics can be viewed as Dickey-Fuller test statistics that have been made robust to serial correlation by estimating the Newey-West heteroscedasticity autocorrelation consistent variance-covariance matrix. Phillips-Perron test statistics have the same distribution as the Dickey-Fuller test statistics. Therefore, the ADF critical values can be used to carry out the test.

  2. ERS (Elliott, Rothenberg, and Stock, 1996) DF-GLS Test

    With the presence of deterministic trend in the test equation, it has been argued that ADF unit root tests had weak power (that is, it becomes more difficult to reject the null [incorrect] hypothesis of unit roots). In other words, the drift or the trend is not part of data generating process. It is necessary to distinguish the effects of unit roots from the deterministic trend. Elliott, Rothenberg, and Stock suggested to remove the trend or drift first using GLS, then perform unit roots test on the filtered data series. There are evidences that ERS's DF-GLS test has significant greater power than the ADF test.

    The idea of DF-GLS test is to estimate the trend of the data series {Yt} by GLS: a + bt. Then the filtered series is defined by: Yt* = Yt - (a + bt). Finally, we perform an ADF test on the filtered data series {Yt*} using tabulated critical values (see Elliott, Rothenberg, and Stock, 1996).

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 λ = TB/T.

Exogenous Structural Break

If the breakpoint λ 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 = α + Yt-1 + θD(TB)t + εt
H1: Yt = α1 + βt + (α2-α1)DUt + εt

Model IIIb
H0: Yt = α1 + Yt-1 + (α1-α2)DUt + εt
H1: Yt = α + β1t + (β2-β1)DTt + εt

Model IIIc
H0: Yt = α1 + Yt-1 + θD(TB)t + (α1-α2)DUt + εt
H1: Yt = α1 + β1t + (α2-α1)DUt + (β2-β1)DTt + εt

Where εt 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
ΔYt = α + βt + θD(TB)t + δDUt + (ρ-1)Yt-1 + ∑j=1,2,...,JρjΔYt-j + εt

Model IIIb
ΔYt = α + βt + δDUt + γDTt + (ρ-1)Yt-1 + ∑j=1,2,...,JρjΔYt-j + εt

Model IIIc
ΔYt = α + βt + θD(TB)t + δDUt + γDTt + (ρ-1)Yt-1 + ∑j=1,2,...,JρjΔYt-j + εt

For each version of testing equation, at the location of breakpoint λ, t statistic of the lag parameter ρ or tρ(λ) is compared with the critical values of the asymptotic distribution of this statistic. We reject the null hypothesis of unit root if the computed tρ(λ) is less than the critical values for a given λ.

Endogenous Structural Break

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

Yt = α + Yt-1 + εt

Therefore three versions of unit roots test are:

Model IIIa
H0: Yt = α + Yt-1 + εt
H1: Yt = α1 + βt + (α2-α1)DUt(λ) + εt

Model IIIb
H0: Yt = α + Yt-1 + εt
H1: Yt = α + β1t + (β2-β1)DTt(λ) + εt

Model IIIc
H0: Yt = α + Yt-1 + εt
H1: Yt = α1 + β1t + (α2-α1)DUt(λ) + (β2-β1)DTt(λ) + εt

The corresponding augmented testing equations are:

Model IIIa
ΔYt = α + βt + δDUt(λ) + (ρ-1)Yt-1 + ∑j=1,2,...,JρjΔYt-j + εt

Model IIIb
ΔYt = α + βt + γDTt(λ) + (ρ-1)Yt-1 + ∑j=1,2,...,JρjΔYt-j + εt

Model IIIc
ΔYt = α + βt + δDUt(λ) + γDTt(λ) + (ρ-1)Yt-1 + ∑j=1,2,...,JρjΔYt-j + εt

We write the dummy variables DUt and DTt to depend on the breakpoint λ, 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, λ* is chosen to minimize the one-sided t statistic for testing the lag parameter ρ = 1. The estimate breakpoint λ* and minimum t statistic are obtained as follows:

For Model IIIa, IIIb, IIIc, estimate the test equation for all possible values of λ in (0,1). That is, from TB=2 to TB=T-1, run T-2 regressions and collect all the t statistics for testing ρ=1. We note that, the augmented lags J used in the test equation may be different for each λ=TB/T.

Let tρ* = minλ in (0,1){tρ(λ)}, and λ* is the estimated breakpoint corresponds to this minimum t statitic. Zivot and Andrews [1992] tabulates the critical values of the asymptotic distribution for tρ*. The computed tρ* is used to compared with these critical values. We reject the null hypothesis of unit root if the computed tρ* is less than the critical value for a given level of significance.

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 = α + Xtβ + εt

In general, if Yt, Xt ~ I(1), then εt ~ I(1). If εt can be shown to be I(0), then the set of variables [Yt, Xt] cointergrates, and the vector [1 -β]' (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:

Δεt = (ρ-1)εt-1 + ut

or with augmented lags:

Δεt = (ρ-1)εt-1 + ∑j=1,2,...,J ρt-jΔεt-j + ut

Hypothesis H0: ρ = 1
H1: ρ < 1
Test
Statistic		tρ = (p-1)/se(p)
where p is the estimate of ρ
Critical
Value		ADF(I,N,e)

If we can reject the null hypothesis of unit root on the residuals εt, 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 = α + γt + Xtβ + εt

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 = α + Xtβ + εt
Δεt = (ρ-1)εt-1 + ut

where ρ < 1 and ut is stationary. Therefore the short-run dynamics of the model is

ΔYt = ΔXtβ + Δεt
= ΔXtβ + (ρ-1)εt-1 + ut
= ΔXtβ + (ρ-1)(Yt-1-α-Xt-1β) + 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-1Π1 + Zt-2Π2 + ... + Zt-pΠp + Π0 + Ut

where Πj, j=1,2,...M, are the MxM parameter matrices, Π0 is a 1xM drift or constant vector, and the 1xM error vector Ut ~ normal(0,Σ) with a constant matrix Σ = 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:

ΔZt = ΔZt-1γ1 + ΔZt-2γ2 + ... + ΔZt-(p-1)γp-1 + Zt-1Π + γ0 + Ut

where Π = ∑j=1,2,...,pΠj - I, γ1 = Π1 - Π - I , γ2 = Π2 + γ1, ..., and γ0 = Π0 for notational convenience.

If Zt ~ I(1), then ΔZt ~ 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-1Π ~ I(0). By definition of cointegration, the parameter matrix Π must contains 0 < r < M linearly independent cointegrating vetors such that ZtΠ ~ I(0). Therefore, the cointegration test amounts to check that Rank(Π) = r > 0.

If Rank(Π) = r, we may impose the parameter restrictions Π = BA' where A and B are Mxr matrices. Since A is a Mxr rank matrix, we can rewrite the constant γ0 = μA'+γ, where μ is 1xr and γ is 1xM. γ is orthogonal to μA'. That is, μA'γ = 0. Therefore,

ΔZt = ΔZt-1γ1 + ΔZt-2γ2 + ... + ΔZt-(p-1)γp-1 + γ + (Zt-1B+μ)A' + Ut

Given the existence of the constant vector γ0 = μA'+γ, 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,Σ), we derive the log-likelihood function for Model III:

ll(γ1,γ2,..., γp-1,γ0,Π,Σ) = - MN/2 ln(2π) - N/2 ln|det(Σ)| - ½ ∑t=1,2,...,NUtΣ-1Ut'

Since the maximum likelihood estimate of Σ is U'U/N, the concentrated log-likelihood function is written as:

ll*(γ1,γ2,..., γp-1,γ0,Π) = - NM/2 (1+ln(2π)-ln(N)) - N/2 ln|det(U'U)|

The actual maximum likelihood estimation can be simplied by considering the following two auxilary regressions:

  1. ΔZt = ΔZt-1Φ1 + ΔZt-2Φ2 + ... + ΔZt-(p-1)Φp-1 + Φ0 + Wt

  2. Zt-1 = ΔZt-1Ψ1 + ΔZt-2Ψ2 + ... + ΔZt-(p-1)Ψp-1 + Ψ0 + Vt

Then γj = Φj-ΨjΠ, for j=0,1,2,...,p-1, and Ut = Wt - VtΠ. If Φ0 = Ψ0 = 0, then γ0 = 0 implying no drift in the VAR(p) representation. However, γ0 = 0 will need only the restriction that Φ0 = Ψ0Π.

Returning to the concentrated log-likelihood function, it is now written as

ll*(W(Φ1,Φ2,...,Φp-1,Φ0), V(Ψ1,Ψ2,...,Ψp-1,Ψ0),Π)
= - NM/2 (1+ln(2π)-ln(N)) - N/2 ln|det((W-VΠ)'(W-VΠ))|

Maximizing the above concentrated log-likelihood function is equivalent to minimize the sum-of-squares term det((W-VΠ)'(W-VΠ)). Conditional to W(Φ1,Φ2,...,Φp-1,Φ0) and V(Ψ1,Ψ2,...,Ψp-1,Ψ0), the least squares estimate of Π is (V'V)-1V'W. Thus,

det((W-VΠ)'(W-VΠ))
= 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) (∏i=1,2,...,M(1-λi))

where λ1, λ2, ..., λM 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 Σ and Π) is

ll**(W(Φ1,Φ2,...,Φp-1,Φ0), V(Ψ1,Ψ2,...,Ψp-1,Ψ0))
= - NM/2 (1+ln(2π)-ln(N)) - N/2 ln|det(W'W)| - N/2 ∑i=1,2,...,Mln(1-λi)

Given the parameter constraints that there are 0 < r < M cointegrating vectors, that is Π = -BA' where A and B are Mxr matrices, the restricted concentrated log-likelihood function is similarily derived as follows:

llr**(W(Φ1,Φ2,...,Φp-1,Φ0), V(Ψ1,Ψ2,...,Ψp-1,Ψ0))
= - NM/2 (1+ln(2π)-ln(N)) - N/2 ln|det(W'W)| - N/2 ∑i=1,2,...,rln(1-λi)

Therefore, with the degree of freedom M-r, the likelihood ratio test statistic for at least r cointegrating vectors is

-2(llr** - ll**) = -N ∑i=r+1,2,...,Mln(1-λi)

Similarly the likelihood ratio test statistic for r cointegrating vectors against r+1 vectors is

-2(llr** - llr+1**) = -N ln(1-λr+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 ∑i=r1+1,2,...,r2ln(1-λi)

The following table summarizes the two popular cointegration test statistics: Eigenvalue Test Statistic λmax(r) and Trace Test Statistic λtrace(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-λ1) -N ∑i=1,2,...,Mln(1-λi)
1-N ln(1-λ2) -N ∑i=2,3,...,Mln(1-λi)
.........
M-1-N ln(1-λM) -N ln(1-λM)
Critical
Value		λmax(r) λtrace(r)

Appendix 1: Stability of a Dynamic Model

The stability of a dynamic model hinges on the characteristic equation for the autoregressive part of the model. The roots of the characteristic equation:

1 - ρ1B - ρ2B2 - ... - ρpBp = 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 - ρ1B = 0. The single root of this equation is B = 1/ρ1, which is greater than 1 in absolute value if |ρ1| < 1. Similarly, for an AR(2) model, the two roots of the characteristic equation 1 - ρ1B - ρ2B2 = 0 are B1,B2 = [ρ1±√(ρ12+4ρ2)]/2. Therefore, the stability conditions are:

A more general AR(p) model may be represented by VAR(1):

⌈
|
|
⌊
Yt
Yt-1
:
Yt-p+1
⌉
 |
 |
⌋
 =
⌈
|
|
⌊
α
0
:
0
⌉
 |
 |
⌋
 +
⌈
|
|
⌊
ρ1 ρ2.. ρp
10..0
::::
0..10
⌉
 |
 |
⌋
⌈
|
|
⌊
Yt-1
Yt-2
:
Yt-p
⌉
 |
 |
⌋
 +
⌈
|
|
⌊
εt
0
:
0
⌉
 |
 |
⌋

That is, Yt = α + ρ Yt-1 + εt

By successive substitution, we obtain Yt = α + ρα + ρ2α + ... (so that the equilibrium Y∞ = (I-ρ)-1α).

The roots of the asymmetric matrix ρ may be complex in the form a±bi, where i=√(-1). The stability requires that all the roots of ρ must be less than 1 in absolute value. That is, |a+bi| = √(a2+b2) < 1.

The unit circle refers to the two-dimentional 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 - ρ1B - ρ2B2 - ... - ρpBp = 0

which are the the reciprocals of the characteristic roots of the matrix ρ must lie outside the unit circle.

Appendix 2: 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   ADFtρ   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   ADFtρ   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   ADFtρ   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 ρ = 1)
  II   ADFtα   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   ADFtα   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   ADFtβ   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   ADFFα,ρ  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   ADFFα,β,ρ 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   ADFFβ,ρ  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
ΔYt = α + βt + δDUt(λ) + (ρ-1)Yt-1 + ∑j=1,2,...,JρjΔYt-j + εt

Model IIIb
ΔYt = α + βt + γDTt(λ) + (ρ-1)Yt-1 + ∑j=1,2,...,JρjΔYt-j + εt

Model IIIc
ΔYt = α + βt + δDUt(λ) + γDTt(λ) + (ρ-1)Yt-1 + ∑j=1,2,...,JρjΔYt-j + εt

Where λ = TB/T (T is the sample size and TB is the break point), and

DUt(λ) = 1, if t>TB
0 otherwise
DTt(λ) = t-TB, if t>TB
0 otherwise

λ* is the estimated breakpoint which minimizes the t statistic tρ(λ) for testing the unit root over the range of 0 < λ < 1.

Source:
Perron (1989), Zivot and Andrews (1992). 

                        Probabilty to the Right of Critical Value
Model Statistic λ    99%  97.5%    95%    90%    50%    10%     5%   2.5%     1%
 IIIa  ADFtρ(λ)  λ*  -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  ADFtρ(λ)  λ*  -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  ADFtρ(λ)  λ*  -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 = α + Xt β + εt
Δεt = (ρ-1)εt-1 + ∑j=1,2,...,J ρt-jΔεt-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) ≠ 0 (at least one variable in X has drift)
Model 3: E(Yt) ≠ 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, ADFtρ for N=500). If only Y has drift (Model 3), the cointegration equation can be expressed as Yt = α + γ t + Xt β + εt. 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%
λmax    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
λtrace  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
Last updated: 01/18/2010