Variance ratio test for random walk
h = vratiotest(
y
)
h = vratiotest(y
,'ParameterName'
,ParameterValue
,...)
[h,pValue] = vratiotest(...)
[h,pValue,stat] = vratiotest(...)
[h,pValue,stat,cValue] = vratiotest(...)
[h,pValue,stat,cValue,ratio] = vratiotest(...)
h = vratiotest(
assesses
the null hypothesis of a random walk in a univariate time series y
)
.y
h = vratiotest(
accepts
optional inputs as one or more commaseparated parametervalue pairs. y
,'ParameterName'
,ParameterValue
,...)'ParameterName'
is
the name of the parameter inside single quotation marks. ParameterValue
is
the value corresponding to 'ParameterName'
.
Specify parametervalue pairs in any order; names are caseinsensitive.
Perform multiple tests by passing a vector value for any parameter.
Multiple tests yield vector results.
[h,pValue] = vratiotest(...)
returns pvalues
of the test statistics.
[h,pValue,stat] = vratiotest(...)
returns
the test statistics.
[h,pValue,stat,cValue] = vratiotest(...)
returns
critical values for the tests.
[h,pValue,stat,cValue,ratio] = vratiotest(...)
returns
a vector of ratios.

Vector of timeseries
data. The last element is the most recent observation. The test ignores The input series 

Scalar or vector of nominal significance levels for the tests.
Set values between The test is twotailed, so Default: 

Scalar or vector of Boolean values indicating whether to assume independent identically distributed (IID) innovations. To strengthen the null model and assume that the e(t)
are independent and identically distributed (IID), set The IID assumption is often unreasonable for longterm macroeconomic or financial price series. Rejection of the randomwalk null due to heteroscedasticity is not interesting for these cases. Default: 

Scalar or vector of integers greater than one and less than
half the number of observations in When the period The test finds the largest integer Default: 

Vector of Boolean decisions for the tests, with length equal
to the number of tests. Values of 

Vector of pvalues of the test statistics, with length equal to the number of tests. Values are standard normal probabilities. 

Vector of test statistics, with length equal to the number of tests. Statistics are asymptotically standard normal. 

Vector of critical values for the tests, with length equal to the number of tests. Values are for standard normal probabilities. 

Vector of variance ratios, with length equal to the number of tests. Each ratio is the ratio of:
For a random walk, these ratios are asymptotically equal to one. For a meanreverting series, the ratios are less than one. For a meanaverting series, the ratios are greater than one. 
The vratiotest
test statistics are based
on a ratio of variance estimates of returns r(t) = y(t)–y(t–1) and
period q return horizons r(t) + ... + r(t–q+1).
Overlapping horizons increase the efficiency of the estimator and
add power to the test. Under either null, uncorrelated innovations e(t)
imply that the period q variance is asymptotically
equal to q times the period 1 variance. The variance
of the ratio, however, depends on the degree of heteroscedasticity,
and, therefore, on the null.
Rejection of the null due to dependence of the innovations does not imply that the e(t) are correlated. Dependence allows that nonlinear functions of the e(t) are correlated, even when the e(t) are not. For example, it can hold that Cov[e(t),e(t–k)] = 0 for all k ≠ 0, while Cov[e(t)^{2},e(t–k)^{2}] ≠ 0 for some k ≠ 0.
Cecchetti and Lam [2] show that sequential testing using multiple values of q results in smallsample size distortions beyond those that result from the asymptotic approximation of critical values.
[1] Campbell, J. Y., A. W. Lo, and A. C. MacKinlay. Chapter 12. “The Econometrics of Financial Markets.” Nonlinearities in Financial Data. Princeton, NJ: Princeton University Press, 1997.
[2] Cecchetti, S. G., and P. S. Lam. “VarianceRatio Tests: SmallSample Properties with an Application to International Output Data.” Journal of Business and Economic Statistics. Vol. 12, 1994, pp. 177–186.
[3] Cochrane, J. “How Big is the Random Walk in GNP?” Journal of Political Economy. Vol. 96, 1988, pp. 893–920.
[4] Faust, J. “When Are Variance Ratio Tests for Serial Dependence Optimal?” Econometrica. Vol. 60, 1992, pp. 1215–1226.
[5] Lo, A. W., and A. C. MacKinlay. “Stock Market Prices Do Not Follow Random Walks: Evidence from a Simple Specification Test.” Review of Financial Studies. Vol. 1, 1988, pp. 41–66.
[6] Lo, A. W., and A. C. MacKinlay. “The Size and Power of the Variance Ratio Test.” Journal of Econometrics. Vol. 40, 1989, pp. 203–238.
[7] Lo, A. W., and A. C. MacKinlay. A NonRandom Walk Down Wall St. Princeton, NJ: Princeton University Press, 2001.