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Variance Ratio Test

The question of whether asset prices are predictable has long been the subject of considerable interest. One popular approach to answering this question, the Lo and MacKinlay (1988, 1989) overlapping variance ratio test, examines the predictability of time series data by comparing variances of differences of the data (returns) calculated over different intervals. If we assume the data follow a random walk, the variance of a q -period difference should be q times the variance of the one-period difference. Evaluating the empirical evidence for or against this restriction is the basis of the variance ratio test.

EViews allows you to perform the Lo and MacKinlay variance ratio test for homoskedastic and heteroskedastic random walks, using the asymptotic normal distribution (Lo and MacKinlay, 1988) or wild bootstrap (Kim, 2006) to evaluate statistical significance. In addition, you may compute the rank, rank-score, or sign-based forms of the test (Wright, 2000), with bootstrap evaluation of significance. In addition, EViews offers Wald and multiple comparison variance ratio tests (Richardson and Smith, 1991; Chow and Denning, 1993), so you may perform joint tests of the variance ratio restriction for several intervals.

Performing Variance Ratio Tests in EViews

First, open the series which contains the data which you wish to test and click on View/ Variance Ratio Test… Note that EViews allows you to perform the test using the differences, log differences, or original data in your series as the random walk innovations.

The Output combo determines whether you wish to see your test output in Table or Graph form. (As we discuss below, the choices differ slightly in a panel workfile.)

The Data specification section describes the properties of the data in the series. By default, EViews assumes you wish to test whether the data in the series follow a Random walk, so that variances are computed for differences of the data. Alternately, you may assume that the data fol-

low an Exponential random walk so that the innovations are obtained by taking log differences, or that the series contains the Random walk innovations themselves.

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There are two ways to specify the periods to test. First, you may provide a user-specified list of values or name of a vector containing the values. The default settings, depicted above, are to compute the test for periods “2 4 8 16.” Alternately, you may click on the Equalspaced grid radio, and enter a minimum, maximum, and step.

If you are performing your test on a series in a panel workfile, the Output options differ slightly. If you wish to produce output in tabular form, you can choose to compute individual variance

ratio tests for each cross-section and form a Fisher Combined test (Table - Fisher Combined), or you can choose to stack the cross-sections into a single series and perform the test on the stacked panel (Table - Stacked Panel). Note that the stacked panel method assumes that all means and variances are the same across all cross-sections; the only adjustment for the panel structure is in data handling that insures that lags never cross the seams between cross-sections. There are two graphical counterparts to the table choices: Graph - Individual, which produces a graph for each cross-section, and Graph - Stacked Panel, which produces a graph of the results for the stacked analysis.

An Example

In our example, we employ the time series data on nominal exchange rates used by Wright (2000) to illustrate his modified variance ratio tests (“Wright.WF1”). The data in the first page (WRIGHT) of the workfile provide the relative-to-U.S. exchange rates for the Canadian dollar, French franc, German mark, Japanese yen, and the British pound for the 1,139 weeks from August 1974 through May 1996. Of interest is whether the exchange rate returns, as measured by the log differences of the rates, are i.i.d. or martingale difference, or alternately, whether the exchange rates themselves follow an exponential random walk.

We begin by performing tests on the Japanese yen. Open the JP series, then select

View/Variance Ratio... to display the dialog. We will make a few changes to the default settings to match Wright’s calculations. First, select Exponential random walk in the Data specification section to tell EViews that you wish to work with the log returns. Next, uncheck the Use unbiased variances and Use heteroskedastic robust S.E. check-

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Alternately, we may display a graph of the test statistics using the same settings. Simply click again on View/Variance Ratio Test..., change the Output combo from Table to

Graph, then fill out the dialog as before and click on OK:

EViews displays a graph of the variance ratio statistics and plus or minus two asymptotic standard error bands, along with a horizontal reference line at 1 representing the null hypothesis. Here, we see a graphical representation of the fact that with the exception of the test against period 2, the null reference line lies outside the bands.

Next, we repeat the previous analysis but allow for heteroskedasticity in the data and use bootstrapping to evaluate the statistical significance. Fill out the dialog as before, but enable the Use heteroskedastic

robust S.E. checkbox and use the Probabilities combo to select Wild bootstrap (with the two-point distribution, 5000 replications, the Knuth generator, and a seed for the random number generator of 1000 specified in the Options section). The top portion of the results is depicted here:

Null Hypothesis: Log JP is a martingale Date: 04/21/09 Time: 15:15

Sample: 8/07/1974 5/29/1996

Included observations: 1138 (after adjustments) Heteroskedasticity robust standard error estimates Use biased variance estimates

User-specified lags: 2 5 10 30

Test probabilities computed using wild bootstrap: dist=twopoint, reps=5000, rng=kn, seed=1000

Note that the Wald test is no longer displayed since the test methodology is not consistent with the use of heteroskedastic robust standard errors in the individual tests. The p-values

408—Chapter 30. Univariate Time Series Analysis

Null Hypothesis: Log EXCHANGE is a martingale Date: 04/21/09 Time: 15:18

Sample: 8/07/1974 5/29/1996 Cross-sections included: 5

Total panel observations: 5690 (after adjustments) Heteroskedasticity robust standard error estimates Use biased variance estimates

User-specified lags: 2 5 10 30

Test probabilities computed using wild bootstrap: dist=Two-point, reps=5000, rng=kn, seed=1000

shows the test settings and provides the joint Fisher combined test statistic which, in this case, strongly rejects the joint null hypothesis that all of the cross-sections are martingales.

The bottom portion of the output:

depicts the max z statistics for the individual cross-sections, along with corresponding wild bootstrap probabilities. Note that four of the five individual test statistics do not reject the joint hypothesis at conventional levels. It would therefore appear that the Japanese yen result is the driving force behind the Fisher combined test rejection.

Technical Details

where m is an arbitrary drift parameter. The key properties of a random walk that we would like to test are E(et) = 0 for all t and E(etet – j) = 0 for any positive j .

The Basic Test Statistic

Lo and MacKinlay (1988) formulate two test statistics for the random walk properties that are applicable under different sets of null hypothesis assumptions about et :

410—Chapter 30. Univariate Time Series Analysis

Joint Variance Ratio Tests

Since the variance ratio restriction holds for every difference q > 1 , it is common to evaluate the statistic at several selected values of q .

To control the size of the joint test, Chow and Denning (1993) propose a (conservative) test statistic that examines the maximum absolute value of a set of multiple variance ratio statistics. The p-value for the Chow-Denning statistic using m variance ratio statistics is bounded from above by the probability for the Studentized Maximum Modulus (SMM) distribution with parameter m and T degrees-of-freedom. Following Chow and Denning, we approximate this bound using the asymptotic (T = •) SMM distribution.

An second approach is available for variance ratio tests of the i.i.d. null. Under this set of assumptions, we may form the joint covariance matrix of the variance ratio test statistics as in Richardson and Smith (1991), and compute the standard Wald statistic for the joint hypothesis that all m variance ratio statistics equal 1. Under the null, the Wald statistic is asymptotic Chi-square with m degrees-of-freedom.

For a detailed discussion of these tests, see Fong, Koh, and Ouliaris (1997).

Wild Bootstrap

Kim (2006) offers a wild bootstrap approach to improving the small sample properties of variance ratio tests. The approach involves computing the individual (Lo and MacKinlay) and joint (Chow and Denning, Wald) variance ratio test statistics on samples of T observations formed by weighting the original data by mean 0 and variance 1 random variables, and using the results to form bootstrap distributions of the test statistics. The bootstrap p- values are computed directly from the fraction of replications falling outside the bounds defined by the estimated statistic.

EViews offers three distributions for constructing wild bootstrap weights: the two-point, the Rademacher, and the normal. Kim’s simulations indicate that the test results are generally insensitive to the choice of wild bootstrap distribution.

Rank and Rank Score Tests

Wright (2000) proposes modifying the usual variance ratio tests using standardized ranks of the increments, DYt . Letting r(DYt) be the rank of the DYt among all T values, we define the standardized rank (r1t) and van der Waerden rank scores (r2t ):

In cases where there are tied ranks, the denominator in r1t may be modified slightly to account for the tie handling.