Augmented Dickey-Fuller Unit Root and Engle-Granger Cointegration Testing

Common unit root tests have the null hypothesis that there is an autoregressive unit root , and the alternative is , where is the autoregressive coefficient of the time series

This is referred to as the zero mean model. The standard Dickey-Fuller (DF) test assumes that errors are white noise. There are two other types of regression models that include a constant or a time trend as follows:

These two models are referred to as the constant mean model and the trend model, respectively. The constant mean model includes a constant mean of the time series. However, the interpretation of depends on the stationarity in the following sense: the mean in the stationary case when is the trend in the integrated case when . Therefore, the null hypothesis should be the joint hypothesis that and . However for the unit root tests, the test statistics are concerned with the null hypothesis of . The joint null hypothesis is not commonly used. This issue is address in Bhargava (1986) with a different nesting model.

There are two types of test statistics. The conventional ratio is

and the second test statistic, called -test, is

For the zero mean model, the asymptotic distributions of the Dickey-Fuller test statistics are

For the constant mean model, the asymptotic distributions are

For the trend model, the asymptotic distributions are

where

One problem of the Dickey-Fuller and similar tests that employ three types of regressions is the difficulty in the specification of the deterministic trends. Campbell and Perron (1991) claimed that "the proper handling of deterministic trends is a vital prerequisite for dealing with unit roots". However the "proper handling" is not obvious since the distribution theory of the relevant statistics about the deterministic trends is not available. Hayashi (2000) suggests to using the constant mean model when you think there is no trend, and using the trend model when you think otherwise. However no formal procedure is provided.

The null hypothesis of the Dickey-Fuller test is a random walk, possibly with drift. The differenced process is not serially correlated under the null of I(1). There is a great need for the generalization of this specification. The augmented Dickey-Fuller (ADF) test, originally proposed in Dickey and Fuller (1979), adjusts for the serial correlation in the time series by adding lagged first differences to the autoregressive model,

where the deterministic terms and can be absent for the models without drift or linear trend. As previously, there are two types of test statistics. One is the OLS value

and the other is given by

The asymptotic distributions of the test statistics are the same as those of the standard Dickey-Fuller test statistics.

Nonstationary multivariate time series can be tested for cointegration, which means that a linear combination of these time series is stationary. Formally, denote the series by . The null hypothesis of cointegration is that there exists a vector such that is stationary. Residual-based cointegration tests were studied in Engle and Granger (1987) and Phillips and Ouliaris (1990). The latter are described in the next subsection. The first step regression is

where , , and = (,,. This regression can also include an intercept or an intercept with a linear trend. The residuals are used to test for the existence of an autoregressive unit root. Engle and Granger (1987) proposed augmented Dickey-Fuller type regression without an intercept on the residuals to test the unit root. When the first step OLS does not include an intercept, the asymptotic distribution of the ADF test statistic is given by

where is a vector standard Brownian motion and

is a partition such that is a scalar and is dimensional. The asymptotic distributions of the test statistics in the other two cases have the same form as the preceding formula. If the first step regression includes an intercept, then is replaced by the demeaned Brownian motion . If the first step regression includes a time trend, then is replaced by the detrended Brownian motion. The critical values of the asymptotic distributions are tabulated in Phillips and Ouliaris (1990) and MacKinnon (1991).

The residual based cointegration tests have a major shortcoming. Different choices of the dependent variable in the first step OLS might produce contradictory results. This can be explained theoretically. If the dependent variable is in the cointegration relationship, then the test is consistent against the alternative that there is cointegration. On the other hand, if the dependent variable is not in the cointegration system, the OLS residual do not converge to a stationary process. Changing the dependent variable is more likely to produce conflicting results in finite samples.

Phillips-Perron Unit Root and Cointegration Testing

Besides the ADF test, there is another popular unit root test that is valid under general serial correlation and heteroscedasticity, developed by Phillips (1997) and Phillips and Perron (1988). The tests are constructed using the AR(1) type regressions, unlike ADF tests, with corrected estimation of the long run variance of . In the case without intercept, consider the driftless random walk process

where the disturbances might be serially correlated with possible heteroscedasticity. Phillips and Perron (1988) proposed the unit root test of the OLS regression model,

Denote the OLS residual by . The asymptotic variance of can be estimated by using the truncation lag l.

where , for , and . This is a consistent estimator suggested by Newey and West (1987).

The variance of can be estimated by . Let be the variance estimate of the OLS estimator . Then the Phillips-Perron test (zero mean case) is written

The statistic is just the ordinary Dickey-Fuller statistic with a correction term that accounts for the serial correlation. The correction term goes to zero asymptotically if there is no serial correlation.

Note that as , which shows that the limiting distribution is skewed to the left.

Let be the statistic for . The Phillips-Perron (defined here as ) test is written

To incorporate a constant intercept, the regression model is used (single mean case) and null hypothesis the series is a driftless random walk with nonzero unconditional mean. To incorporate a time trend, we used the regression model and under the null the series is a random walk with drift.

The limiting distributions of the test statistics for the zero mean case are

where B() is a standard Brownian motion.

The limiting distributions of the test statistics for the intercept case are

Finally, The limiting distributions of the test statistics for the trend case are can be derived as

where for and for ,

The finite sample performance of the PP test is not satisfactory ( see Hayashi (2000) ).

When several variables are cointegrated, there exists a cointegrating vector such that is stationary and is a nonzero vector. The residual based cointegration test assumes the following regression model:

where , , and = (,,. You can estimate the consistent cointegrating vector by using OLS if all variables are difference stationary — that is, I(1). The estimated cointegrating vector is . The Phillips-Ouliaris test is computed using the OLS residuals from the preceding regression model, and it uses the PP unit root tests and developed in Phillips (1997), although in Phillips and Ouliaris (1990) the asymptotic distributions of some other leading unit root tests are also derived. The null hypothesis is no cointegration.

You need to refer to the tables by Phillips and Ouliaris (1990) to obtain the -value of the cointegration test. Before you apply the cointegration test, you may want to perform the unit root test for each variable (see the option STATIONARITY=( ADF)).

As in the Engle-Granger cointegration tests, the Phillips-Ouliaris test can give conflicting results for different choices of the regressand. There are other cointegration tests that are invariant to the order of the variables, including Johansen (1988), Johansen (1991), Stock and Watson (1988).

ERS and Ng-Perron Unit Root Tests

As mentioned earlier, ADF and PP both suffer severe size distortion and low power. There is a class of newer tests that improves both size and power, sometimes called efficient unit root tests, among which Elliott, Rothenberg, and Stock (1996) and Ng and Perron (2001) are prominent.

Elliott, Rothenberg, and Stock (1996) consider the data generating process

where is either or and is an unobserved stationary zero-mean process with positive spectral density at zero frequency. The null hypothesis is , and the alternative is . The key idea of Elliott, Rothenberg, and Stock (1996) is to study the asymptotic power and asymptotic power envelope of some new tests. Asymptotic power is defined with a sequence of local alternatives. For a fixed alternative hypothesis, the power of a test usually goes to one when sample size goes to infinity; however, this does not say anything about the finite sample performance. On the other hand, when the data generating process under the alternative moves closer to the null as the sample size increases, the power does not necessarily converge to one. The local to unity alternatives in ERS are

and the power against the local alternatives has a limit as goes to infinity, which is called asymptotic power. This value is strictly between 0 and 1. Asymptotic power indicates the adequacy of a test to distinguish small deviations from the null hypothesis.

Define

Let be the sum of squared residuals from a least squares regression of on . Then the point optimal test against the local alternative has the form

where is an estimator for . Note that the test rejects the null when is small. The asymptotic power function for the point optimal test constructed with under local alternatives with is denoted by . Then the power envelope is because the test formed with is the most powerful against the alternative . In other words, the asymptotic function , is always below the power envelope except that at one point they are tangent. Elliott, Rothenberg, and Stock (1996) show that choosing some specific values for can cause the asymptotic power function of the point optimal test to be very close to the power envelope. The optimal is when , and when . This choice of corresponds to the tangent point where . This is also true for the DF-GLS test.

Elliott, Rothenberg, and Stock (1996) also propose the DF-GLS test, given by the statistic for testing in the regression

where is obtained in a first step detrending

and is least squares regression coefficient of on . Regarding the lag length selection, Elliott, Rothenberg, and Stock (1996) favor the Schwartz Bayesian information criterion. The optimal selection of the lag length and the estimation of is further discussed in Ng and Perron (2001). The lag length is selected from the interval for some fixed by using the modified Akaike’s information criterion,

where and . For fixed lag length , an estimate of is given by

DF-GLS is indeed a superior unit root test, according to Stock (1994), Schwert (1989), and Elliott, Rothenberg, and Stock (1996). In terms of the size of the test, DF-GLS is almost as good as the ADF test and better than the PP and test. In addition, the power of the DF-GLS is larger than the ADF test and -test.

Ng and Perron (2001) also apply GLS detrending to obtain the following M-tests:

The first one is a modified version of Phillips-Perron test

where the detrended data is used. The second is a modified Bhargava (1986) test statistic. The third can be perceived as a modified Phillips-Perron statistic because of the relationship .

The modified point optimal tests using the GLS detrended data are

The DF-GLS test and the test have the same limiting distribution

The point optimal test and the modified point optimal test have the same limiting distribution

where is a standard Brownian motion and is an Ornstein-Uhlenbeck process defined by with , , and .

Overall, the M-tests has the smallest size distortion, with the ADF t test having the next smallest. The ADF -test, , and have the worst size distortion. In addition, the power of the DF-GLS and M-tests are larger than that of the ADF t test and -test. The ADF has more severe size distortion than the ADF , but larger power for a fixed lag length.

Kwiatkowski, Phillips, Schmidt, and Shin (KPSS) Unit Root Test

There are less existent tests for the null hypothesis of trend stationary I(0). The main reason is the difficulty in the theoretical development. The KPSS test was introduced in Kwiatkowski et al. (1992) to test the null hypothesis that an observable series is stationary around a deterministic trend. Please note, that for consistency reasons, the notation used here is different from the notation used in the original paper. The setup of the problem is as follows: it is assumed that the series is expressed as the sum of the deterministic trend, random walk , and stationary error ; that is,

where iid , and an intercept (in the original paper, the authors use instead of , here we assume .) The null hypothesis of trend stationary is specified by , while the null of level stationary is the same as above with the model restriction . Under the alternative that , there is a random walk component in the observed series .

Under stronger assumptions of normality and iid of and , a one-sided LM test of the null that there is no random walk () can be constructed as follows:

Notice that under the null hypothesis, can be estimated by ordinary least squares regression of on an intercept and the time trend. Following the original work of Kwiatkowski, Phillips, Schmidt, and Shin, under the null (), statistic converges asymptotically to three different distributions depending on whether the model is trend-stationary, level-stationary (), or zero-mean stationary (, ). The trend-stationary model is denoted by subscript and the level-stationary model is denoted by subscript . The case when there is no trend and zero intercept is denoted as 0. The last case, although rarely used in practice, is considered in Hobijn, Franses, and Ooms (2004).

with

where is a Brownian motion (Wiener process), and is convergence in distribution. Note that is a standard Brownian bridge, is a Brownian bridge of a second-level.

Using the notation of Kwiatkowski et al. (1992) the statistic is named as . This test depends on the computational method used to compute the long-run variance — that is, the window width and the kernel type . You can specify the kernel used in the test, using the KERNEL option:

• Newey-West/Bartlett (KERNEL=NW BART), default

• Quadratic spectral (KERNEL=QS)

You can specify the number of lags, , in three different ways:

• Schwert (SCHW = c) (default for NW, c=4)

• Manual (LAG = )

• Automatic selection (AUTO) (default for QS) Hobijn, Franses, and Ooms (2004)

The last option (AUTO) needs more explanation, summarized in the following table. For each of the kernel function, a formula for optimal window width is provided.

where if and 0, otherwise; .

Simulation evidence shows that the KPSS has size distortion in finite samples. For example, see Caner and Kilian (2001). The power is reduced when the sample size is large, which can be derived theoretically (see Breitung (1995)). Another problem of the KPSS test is that the power depends on the choice of the truncation lag used in the Newey-West estimator of the long run variance .

BDS Test

Broock, Dechert, and Scheinkman (1987) propose a test (BDS test) of independence based on the correlation dimension. Broock et al. (1996) show that the first-order asymptotic distribution of the test statistic is independent of the estimation error provided that the parameters of the model under test can be estimated -consistently. Hence, the BDS test can be used as a model selection tool and as a specification test.

Given the sample size , the embedding dimension , and the value of the radius , the BDS statistic is

where

The statistic has a standard normal distribution if the sample size is large enough. For small sample size, the distribution can be approximately obtained through simulation. Kanzler (1999) has a comprehensive discussion on the implementation and empirical performance of BDS test.

Runs Test and Turning Point Test

The runs test and turning point test are two widely used tests for independence (Cromwell, Labys, and Terraza; 1994).

The runs test needs several steps. First, convert the original time series into the sequence of signs, , that is, map into where is the sample mean of and is "" if is nonnegative and "" if is negative. Second, count the number of runs, , in the sequence. A run of a sequence is a maximal non-empty segment of the sequence that consists of adjacent equal elements. For example, the following sequence contains runs:

Third, count the number of pluses and minuses in the sequence and denote them as and , respectively. In the preceding example sequence, and . Note that the sample size . Finally, compute the statistic of runs test,

where

The statistic of the turning point test is defined as follows:

where the indicator function of the turning point is 1 if or (that is, both the previous and next values are greater or less than the current value); otherwise, 0.

The statistics of both the runs test and the turning point test have the standard normal distribution under the null hypothesis of independence.

Rank Version of von Neumann Ratio Test

Since the runs test completely ignores the magnitudes of the observations, Bartels (1982) proposes a rank version of the von Neumann Ratio test for independence:

where is the rank of th observation in the sequence of observations. For large sample, the statistic follows the standard normal distribution under the null hypothesis of independence. For small samples of size between 11 and 100, the critical values through simulation would be more precise; for samples of size no more than 10, the exact CDF is applied.

Generalized Durbin-Watson Tests

Consider the following linear regression model:

where is an data matrix, is a coefficient vector, and is a disturbance vector. The error term is assumed to be generated by the th-order autoregressive process where , is a sequence of independent normal error terms with mean 0 and variance . Usually, the Durbin-Watson statistic is used to test the null hypothesis against . Vinod (1973) generalized the Durbin-Watson statistic:

where are OLS residuals. Using the matrix notation,

where and is a matrix:

and there are zeros between and 1 in each row of matrix .

The QR factorization of the design matrix yields a orthogonal matrix :

where R is an upper triangular matrix. There exists an submatrix of such that and . Consequently, the generalized Durbin-Watson statistic is stated as a ratio of two quadratic forms:

where are upper n eigenvalues of and is a standard normal variate, and . These eigenvalues are obtained by a singular value decomposition of (Golub and Van Loan; 1989; Savin and White; 1978).

The marginal probability (or p-value) for given is

where

When the null hypothesis holds, the quadratic form has the characteristic function

The distribution function is uniquely determined by this characteristic function:

For example, to test given against , the marginal probability (p-value) can be used:

where

and is the calculated value of the fourth-order Durbin-Watson statistic.

In the Durbin-Watson test, the marginal probability indicates positive autocorrelation () if it is less than the level of significance (), while you can conclude that a negative autocorrelation () exists if the marginal probability based on the computed Durbin-Watson statistic is greater than . Wallis (1972) presented tables for bounds tests of fourth-order autocorrelation, and Vinod (1973) has given tables for a 5% significance level for orders two to four. Using the AUTOREG procedure, you can calculate the exact p-values for the general order of Durbin-Watson test statistics. Tests for the absence of autocorrelation of order p can be performed sequentially; at the th step, test given against . However, the size of the sequential test is not known.

The Durbin-Watson statistic is computed from the OLS residuals, while that of the autoregressive error model uses residuals that are the difference between the predicted values and the actual values. When you use the Durbin-Watson test from the residuals of the autoregressive error model, you must be aware that this test is only an approximation. See Autoregressive Error Model earlier in this chapter. If there are missing values, the Durbin-Watson statistic is computed using all the nonmissing values and ignoring the gaps caused by missing residuals. This does not affect the significance level of the resulting test, although the power of the test against certain alternatives may be adversely affected. Savin and White (1978) have examined the use of the Durbin-Watson statistic with missing values.

The Durbin-Watson probability calculations have been enhanced to compute the -value of the generalized Durbin-Watson statistic for large sample sizes. Previously, the Durbin-Watson probabilities were only calculated for small sample sizes.

Consider the following linear regression model:

where is an data matrix, is a coefficient vector, is a disturbance vector, and is a sequence of independent normal error terms with mean 0 and variance .

The generalized Durbin-Watson statistic is written as

where is a vector of OLS residuals and is a matrix. The generalized Durbin-Watson statistic DW can be rewritten as

where .

The marginal probability for the Durbin-Watson statistic is

where .

The -value or the marginal probability for the generalized Durbin-Watson statistic is computed by numerical inversion of the characteristic function of the quadratic form . The trapezoidal rule approximation to the marginal probability is

where is the imaginary part of the characteristic function, and are integration and truncation errors, respectively. Refer to Davies (1973) for numerical inversion of the characteristic function.

Ansley, Kohn, and Shively (1992) proposed a numerically efficient algorithm that requires O() operations for evaluation of the characteristic function . The characteristic function is denoted as

where and . By applying the Cholesky decomposition to the complex matrix , you can obtain the lower triangular matrix that satisfies . Therefore, the characteristic function can be evaluated in O() operations by using the following formula:

where . Refer to Ansley, Kohn, and Shively (1992) for more information on evaluation of the characteristic function.

Tests for Serial Correlation with Lagged Dependent Variables

When regressors contain lagged dependent variables, the Durbin-Watson statistic () for the first-order autocorrelation is biased toward 2 and has reduced power. Wallis (1972) shows that the bias in the Durbin-Watson statistic () for the fourth-order autocorrelation is smaller than the bias in in the presence of a first-order lagged dependent variable. Durbin (1970) proposes two alternative statistics (Durbin h and t ) that are asymptotically equivalent. The h statistic is written as

where and is the least squares variance estimate for the coefficient of the lagged dependent variable. Durbin’s t test consists of regressing the OLS residuals on explanatory variables and and testing the significance of the estimate for coefficient of .

Inder (1984) shows that the Durbin-Watson test for the absence of first-order autocorrelation is generally more powerful than the h test in finite samples. Refer to Inder (1986) and King and Wu (1991) for the Durbin-Watson test in the presence of lagged dependent variables.

Godfrey LM test

The GODFREY= option in the MODEL statement produces the Godfrey Lagrange multiplier test for serially correlated residuals for each equation (Godfrey 1978a and 1978b). is the maximum autoregressive order, and specifies that Godfrey’s tests be computed for lags 1 through . The default number of lags is four.

Portmanteau Q Test

For nonlinear time series models, the portmanteau test statistic based on squared residuals is used to test for independence of the series (McLeod and Li; 1983):

where

This Q statistic is used to test the nonlinear effects (for example, GARCH effects) present in the residuals. The GARCH process can be considered as an ARMA process. See the section Predicting the Conditional Variance later in this chapter. Therefore, the Q statistic calculated from the squared residuals can be used to identify the order of the GARCH process.

Engle’s Lagrange Multiplier Test for ARCH Disturbances

Engle (1982) proposed a Lagrange multiplier test for ARCH disturbances. The test statistic is asymptotically equivalent to the test used by Breusch and Pagan (1979). Engle’s Lagrange multiplier test for the qth order ARCH process is written

where

and

The presample values ( ,, ) have been set to 0. Note that the LM tests might have different finite-sample properties depending on the presample values, though they are asymptotically equivalent regardless of the presample values.

Lee and King’s Test for ARCH Disturbances

Engle’s Lagrange multiplier test for ARCH disturbances is a two-sided test; that is, it ignores the inequality constraints for the coefficients in ARCH models. Lee and King (1993) propose a one-sided test and prove that the test is locally most mean powerful. Let , denote the residuals to be tested. Lee and King’s test checks

where are in the following ARCH(q) model:

The statistic is written as

Wong and Li’s Test for ARCH Disturbances

Wong and Li (1995) propose a rank portmanteau statistic to minimize the effect of the existence of outliers in the test for ARCH disturbances. They first rank the squared residuals; that is, . Then they calculate the rank portmanteau statistic

where , , and are defined as follows:

The Q, Engle’s LM, Lee and King’s, and Wong and Li’s statistics are computed from the OLS residuals, or residuals if the NLAG= option is specified, assuming that disturbances are white noise. The Q, Engle’s LM, and Wong and Li’s statistics have an approximate distribution under the white-noise null hypothesis, while the Lee and King’s statistic has a standard normal distribution under the white-noise null hypothesis.