kpsstest - KPSS stationarity test

Syntax

h = kpsstest(y) h = kpsstest(y,'ParameterName',ParameterValue,...) [h,pValue] = kpsstest(...) [h,pValue,stat] = kpsstest(...) [h,pValue,stat,cValue] = kpsstest(...) [h,pValue,stat,cValue,reg] = kpsstest(...)

Description

h = kpsstest(y) assesses the null hypothesis that a univariate time series y is trend stationary against the alternative that it is a nonstationary unit-root process.

h = kpsstest(y,'ParameterName',ParameterValue,...) accepts optional inputs as one or more comma-separated parameter-value pairs. 'ParameterName' is the name of the parameter inside single quotation marks. ParameterValue is the value corresponding to 'ParameterName'. Specify parameter-value pairs in any order; names are case-insensitive. Perform multiple tests by passing a vector value for any parameter. Multiple tests yield vector results.

[h,pValue] = kpsstest(...) returns p-values of the test statistics.

[h,pValue,stat] = kpsstest(...) returns the test statistics.

[h,pValue,stat,cValue] = kpsstest(...) returns critical values for the tests.

[h,pValue,stat,cValue,reg] = kpsstest(...) returns a structure of regression statistics.

Inputs

`y`	Vector of time-series data. The last element is the most recent observation. The test ignores `NaN` values, which indicate missing entries.

Parameter/Value Pairs

'alpha'

Scalar or vector of nominal significance levels for the tests. Set values between 0.01 and 0.1.

Default: 0.05

'Lags'

Scalar or vector of nonnegative integers indicating the number of autocovariance lags to include in the Newey-West estimator of the long-run variance.

For best results, give a suitable value for 'lags'. For information on selecting 'lags', see Determine Appropriate Lags.

Default: 0

'trend'

Scalar or vector of Boolean values indicating whether to include the deterministic trend term dt in the model.

Choose the value of 'trend' with a specific testing strategy in mind. If a series is growing, including a trend term (setting 'trend' to true) provides a reasonable comparison of a trend-stationary null and a unit-root process with drift. If a series does not exhibit long-term growth characteristics, do not include a trend term.

Default: false

Outputs

h

Vector of Boolean decisions for the tests, with length equal to the number of tests. Values of h equal to 1 indicate rejection of the trend-stationary null in favor of the unit-root alternative. Values of h equal to 0 indicate a failure to reject the null.

pValue

Vector of p-values of the test statistics, with length equal to the number of tests. Values are right-tail probabilities.

stat

Vector of test statistics, with length equal to the number of tests.

cValue

Vector of critical values for the tests, with length equal to the number of tests. Values are for right-tail probabilities.

reg

Structure of regression statistics for the OLS estimation of coefficients in the alternative model. The number of records equals the number of tests. Each record has the following fields:

num Length of input series with NaNs removed

size Effective sample size, adjusted for lags

names Regression coefficient names

coeff Estimated coefficient values

se Estimated coefficient standard errors

Cov Estimated coefficient covariance matrix

tStats t statistics of coefficients and p-values

FStat F statistic and p-value

yMu Mean of the lag-adjusted input series

ySigma Standard deviation of the lag-adjusted input series

yHat Fitted values of the lag-adjusted input series

res Regression residuals

autoCov Estimated residual autocovariances

NWEst Newey-West estimator

DWStat Durbin-Watson statistic

SSR Regression sum of squares

SSE Error sum of squares

SST Total sum of squares

MSE Mean square error

RMSE Standard error of the regression

RSq R^2 statistic

aRSq Adjusted R^2 statistic

LL Loglikelihood of data under Gaussian innovations

AIC Akaike information criterion

BIC Bayesian (Schwarz) information criterion

HQC Hannan-Quinn information criterion

Definitions

kpsstest computes the vector of test statistics, with length equal to the number of tests, using an OLS regression of y on an intercept. If 'trend' is true, the regression includes a trend. The test statistic stat is:

where s(t) = r₁ + ... + r_t, r is the vector of residuals from the regression, is the Newey-West estimator of the long-run variance, and T is the sample size.

Examples

Reproduce the first row of the second half of Table 5 in Kwiatkowski et al. [2]:

load NelsonPlosser
y = log(NPDataset.GNPR);
[~,~,stat] = kpsstest(y,'lags',0:8,'trend',true)

The resulting vector:

stat =
    0.6299    0.3366    0.2421    0.1976    0.1729    0.1578    0.1479    0.1412    0.1369

Algorithm

kpsstest performs a regression to find the LSQ fit between the data and the null model.

Test statistics follow nonstandard distributions under the null, even asymptotically. Kwiatkowski et al. [2] used Monte Carlo simulations, for models with and without a trend, to tabulate asymptotic critical values for a standard set of significance levels between 0.01 and 0.1. kpsstest interpolates critical values and p-values from these tables.

References

[1] Hamilton, J. D. Time Series Analysis. Princeton, NJ: Princeton University Press, 1994.

[2] Kwiatkowski, D., P. C. B. Phillips, P. Schmidt and Y. Shin. "Testing the Null Hypothesis of Stationarity against the Alternative of a Unit Root." Journal of Econometrics. Vol. 54, 1992, pp. 159–178.

[3] Newey, W. K., and K. D. West. "A Simple Positive Semidefinite, Heteroskedasticity and Autocorrelation Consistent Covariance Matrix." Econometrica. Vol. 55, 1987, pp. 703-708.