Other Tests for Nonstationarity

There are other tests for nonstationarity in gretl that you may find useful. The first is the DF-GLS test. It performs the modified Dickey-Fuller t-test (known as the DF-GLS test) proposed by Elliott et al. (1996). Essentially, the test is an augmented Dickey-Fuller test, similar to the test performed by gretl’s adf command, except that the time-series is transformed via a generalized least squares (GLS) regression before estimating the model. Elliott et al. (1996) and others have shown that this test has significantly greater power than the previous versions of the augmented Dickey-Fuller test. Consequently, it is not unusual for this test to reject the null of nonstationarity when the usual augmented Dickey-Fuller test does not.

The —gls option performs the DF-GLS test for a series of models that include 1 to k lags of the first differenced, detrended variable. The lag k can be set by the user or by the method described in Schwert (1989). As discussed above and in POE4, the augmented Dickey-Fuller test involves fitting a regression of the form

Ayt = a + eyt-i + St + Zi Ayt-i + … + Z k Ayt-k + ut (12.4)

and then testing the null hypothesis H0 : в = 0. The DF-GLS test is performed analogously but on GLS-demeaned or GLS-detrended data. The null hypothesis of the test is that the series is a random walk, possibly with drift. There are two possible alternative hypotheses: yt is stationary about a linear time trend or stationary with a possibly nonzero mean but with no linear time trend. Thus, you can use the –c or –ct options.

For the levels of the Fed funds rate:

Augmented Dickey-Fuller (GLS) test for f including 6 lags of (1-L)f (max was 12)

sample size 97

unit-root null hypothesis: a = 1 with constant and trend

model: (1-L)y = b0 + b1*t + (a-1)*y(-1) + … + e 1st-order autocorrelation coeff. for e: 0.010 lagged differences: F(6, 90) = 16.433 [0.0000] estimated value of (a – 1): -0.115012 test statistic: tau = -4.14427

10% 5% 2.5% 1%

Critical values: -2.74 -3.03 -3.29 -3.58

The test statistic is -4.14, which is in the 1% rejection region for the test. The series is nonstation­ary. Notice that the lag selected was 6 and that all available observations were used to estimate the model at this point. This is somewhat different from Stata’s implementation, which sets the sample to the maximum available for the largest model. Also, notice that we used a trend. This is optional.

For the levels of the Fed funds rate:

Augmented Dickey-Fuller (GLS) test for b including 5 lags of (1-L)b (max was 12) sample size 98

unit-root null hypothesis: a = 1 with constant and trend

model: (1-L)y = b0 + b1*t + (a-1)*y(-1) + … + e 1st-order autocorrelation coeff. for e: -0.005 lagged differences: F(5, 92) = 4.799 [0.0006] estimated value of (a – 1): -0.128209 test statistic: tau = -3.17998

10% 5% 2.5% 1%

Critical values: -2.74 -3.03 -3.29 -3.58

The test statistic is -3.18, which is in the 5% rejection region for the test. The series is nonsta­tionary. Notice that the lag selected was 5 and that one more observation is available to obtain the test statistic.

EL S2

T 2 <7 2

Gretl also can perform the KPSS test proposed by Kwiatkowski et al. (1992). The kpss command computes the KPSS test for each of the specified variables (or their first difference, if the —difference option is selected). The null hypothesis is that the variable in question is stationary, either around a level or, if the —trend option is given, around a deterministic linear trend.

where St = J= і es and <r2 is an estimate of the long-run variance of et = (yt — y). The long run

variance is estimated using a bandwidth parameter, m, that the user chooses.

52 = S (l — (mTi)) 7 (12’6)

i=-m

and where 7i is an empirical autocovariance of et from order — m to m.

The command calls for the a bandwidth parameter, m (see section 9.6.2 for a brief discussion). For this estimator to be consistent, m must be large enough to accommodate the short-run persis­tence of et, but not too large compared to the sample size T. If you supply a 0, gretl will compute an automatic bandwidth of 4(T/100)1/4.

kpss 0 f b

The KPSS statistics using automatic bandwidth selection results in:

KPSS test for f T = 104

Lag truncation parameter = 4 Test statistic = 1.36747

Critical values: 0.349 0.466 0.734

KPSS test for b T = 104

Lag truncation parameter = 4 Test statistic = 1.72833

Critical values: 0.349 0.466 0.734

Both are significantly different from zero and the stationary null hypothesis is rejected at any reasonable level of significance. Also note that the bandwidth was chosen to be 4.

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