Finite Critical Values Research Articles

Rational Deficient Functions of Derivatives of Mappings in the Classes S and B

Let f be transcendental and meromorphic in the plane of finite lower order with a bounded set of finite critical and asymptotic values. It is shown that a rational deficient function of any derivative of f is zero at infinity. On the other hand, if f has arbitrary order and a finite set of critical and asymptotic values, then any rational deficient function of f ′ must have a multiple zero at infinity. Furthermore, if such f has finite lower order then f ′ admits no rational deficient functions other than 0.

Critical values of the augmented fractional Dickey–Fuller test

This paper presents response surface estimates of finite sample critical values of the Augmented Fractional Dickey–Fuller test of Dolado et al. (Testing I(1) against I(d) alternatives in the presence of deterministic components. Universidad Carlos III, Madrid, mimeo, 2006). The null hypothesis that a series contains a unit root is tested against the alternative that it is fractionally integrated. It is shown that finite sample critical values depend critically on the lag order used to construct the test statistic as well as the hypothesized value of the fractional differencing parameter under the alternative. Non-linear non-parametric response surfaces provide a novel approach to constructing estimates of finite sample critical values.

Joint Hypothesis Tests for a Unit Root When There is a Break in the Innovation Variance

Abstract. We develop extensions of the Dickey–Fuller F‐statistics for the joint null hypothesis of a unit root that allows for a break in the innovation variance. Our statistics are based on the modified generalized least squares (GLS) strategy outlined in Kim, Leybourne and Newbold [Journal of Econometrics (2002) Vol. 109, pp. 365–387] that requires estimation of the break‐date and corresponding pre‐break and post‐break variances. We derive the asymptotic distribution of the new F‐statistics, tabulate their finite sample and asymptotic critical values, and present finite sample simulation evidence regarding their size and power.

Testing for Unit Root Against Stationarity Using the Likelihood Ratio Test

In a first-order autoregressive model with drift, we derive the likelihood ratio test for a unit root against the stationary alternative. We also derive the test in a state space model with trend. Finite sample and asymptotic critical values are obtained by Monte Carlo simulations. A simulation study investigates the power performance of the likelihood ratio test and we also examine how a bias correction of the test affects the results.

A comment on specification searches in spatial econometrics: The relevance of Hendry's methodology: A reply

Hendry [Hendry, D.F., this issue. Specification searches in spatial econometrics: the relevance of Hendry's methodology. Regional Science and Urban Economics] argues that a comparison between the general-to-specific (Hendry) and specific-to-general (classical) approaches requires standardization of the null rejection frequencies. In this note, we show that the use of standardized finite sample critical values does not have practical implications for applied spatial econometric research, because the Hendry approach is not uniformly most powerful for the spatial error model and the classical approach still outperforms the Hendry approach for the spatial lag model. We also stress that the simulation setup in Florax et al. [Florax, R.J.G.M., Folmer, H., Rey, S.J., 2003. Specification searches in spatial econometrics: the relevance of Hendry's methodology. Regional Science and Urban Economics 33, 557–579] provides an adequate representation of the empirical practice in applied spatial econometric research.

On the use of Sub‐sample Unit Root Tests to Detect Changes in Persistence

Abstract. We investigate the behaviour of rolling and recursive augmented Dickey–Fuller (ADF) tests against processes which display changes in persistence. We show that the power of the tests depend crucially on the window width and warm up parameter for the rolling and recursive procedures respectively, on whether forward or reverse recursive sequences of tests are computed, and on the persistence change process generating the data. To ameliorate these dependencies we extend the available critical values for these tests, and propose a number of new sub‐sample unit root tests for which finite sample and asymptotic critical values are also provided. An empirical illustration on OECD real output data is also provided.

Bootstrap testing for detrended fluctuation analysis

Detrended fluctuation analysis (DFA) is a scaling method that allows the detection of long memory in a time series. Until now no asymptotic distribution has been found for this statistic. The bootstrap technique allows the simulation of the probability distribution of any statistic. In this paper the results of the Monte Carlo study using bootstrap method show that the DFA test has reasonably good power for short time series. Another advantage of the bootstrap technique is that allows the calculation of finite sample critical values. As an example we calculate bootstrap p -values for financial returns time series using DFA.

Seasonal cointegration for monthly data

In this paper we propose an extension of the maximum likelihood seasonal cointegration procedure developed by Johansen and Schaumburg [J. Economet. 88 (1999) 310] for monthly time series. We compute the finite sample critical values of the associated test statistics in monthly seasonal time series. We also apply this seasonal cointegration procedure to the retails and stocks in US monthly manufacturing industrial data.

Reconsidering LM unit root testing

Non-rejection of a unit root hypothesis by usual Dickey & Fuller (1979) (DF, hereafter) or Phillips & Perron (1988) (hereafter PP) tests should not be taken as strong evidence in favour of unit root presence. There are less popular, but more powerful, unit root tests that should be employed instead of DF-PP tests. A prime example of an alternative test is the LM unit root test developed by Schmidt & Phillips (1992) (hereafter SP) and Schmidt & Lee (1991) (hereafter SL). LM unit root tests are easy to calculate and invariant (similar); they employ optimal detrending and are more powerful than usual DF-PP tests. Asymptotic theory and finite sample critical values (with inaccuracies that we correct in this paper) are available for SP-SL tests. However, the usefulness of LM tests is not fully understood, due to ambiguity over test type recommendation, as well as potentially inefficient derivation of the test that might confuse applied researchers. In this paper, we reconsider LM unit root testing in a model with linear trend. We derive asymptotic distribution theory (in a new fashion), as well as accurate appropriate critical values. We undertake Monte Carlo investigation of finite sample properties of SP-SL LM tests, along with applications to the Nelson & Plosser (1982) time series and real quarterly UK GDP.

Seasonal Unit Root Tests Based on Forward and Reverse Estimation

In this paper, we suggest a new set of regression‐based statistics for testing the seasonal unit root null hypothesis. These tests are based on combining conventional Hylleberg et al. (1990) ‐type seasonal unit root test statistics calculated from both forward and reverse estimation of the auxiliary regression equation. We derive the asymptotic distributions of the new test statistics under the seasonal unit root null hypothesis. We provide finite sample critical values appropriate for the case of quarterly data together with asymptotic critical values, the latter appropriate for any seasonal aspect. Monte Carlo simulation of the finite‐sample size and power properties of the new tests reveals that, overall, they perform rather better than extant tests of the seasonal unit root hypothesis.

Tests for multiple change points under ordered alternatives

We consider the problem of testing the null hypothesis of no change against the alternative of multiple change points in a series of independent observations when the changes are in the same direction. We extend the tests of Terpstra (1952), Jonckheere (1954) and Puri (1965) to the change point setup. We obtain the asymptotic null distribution of the proposed tests. We also give approximations for their limiting critical values and tables of their finite sample Monte Carlo critical values. The results of Monte Carlo power studies conducted to compare the proposed tests with some competitors are reported.

LONG MEMORY IN FINANCIAL TIME SERIES DATA WITH NON-GAUSSIAN DISTURBANCES

In this article we propose the use of a version of the tests of Robinson [32] for testing unit and fractional roots in financial time series data. The tests have a standard null limit distribution and they are the most efficient ones in the context of Gaussian disturbances. We compute finite sample critical values based on non-Gaussian disturbances and the power properties of the tests are compared when using both, the asymptotic and the finite-sample (Gaussian and non-Gaussian) critical values. The tests are applied to the monthly structure of several stock market indexes and the results show that the if the underlying I(0) disturbances are white noise, the confidence intervals include the unit root; however, if they are autocorrelated, the unit root is rejected in favour of smaller degrees of integration. Using t-distributed critical values, the confidence intervals for the non-rejection values are generally narrower than with the asymptotic or than with the Gaussian finite-sample ones, suggesting that they may better describe the time series behaviour of the data examined.

Distributions of error correction tests for cointegration

Summary This paper provides densities and finite sample critical values for the singleequation error correction statistic for testing cointegration. Graphs and response surfaces summarize extensive Monte Carlo simulations and highlight simple dependencies of the statistic’s quantiles on the number of variables in the error correction model, the choice of deterministic components, and the sample size. The response surfaces provide a convenient way for calculating finite sample critical values at standard levels; and a computer program, freely available over the Internet, can be used to calculate both critical values and p-values. Two empirical applications illustrate these tools.

On time series with randomized unit root and randomized seasonal unit root

A time series model with possibly a randomized unit root and a randomized seasonal unit root is considered. Two statistical tests are developed for the null hypothesis of fixed unit roots against the alternative that the roots are random and fluctuate about the value of one. The testing problem is addressed via the score test approach. The asymptotic representations of the test statistics in terms of Brownian processes are obtained. Simulations are used to tabulate finite sample critical values and to investigate empirical sizes and powers. A Markov chain Monte Carlo approach is proposed for the estimation of model parameters. Both randomized unit root and randomized seasonal unit root are demonstrated to be present in a US money supply data.

Response surfaces of MOSUM critical values

The response surfaces of MOSUM critical values are computed, from which the finite sample and asymptotic critical values for different moving window bandwidths can easily be calculated. Results also show that the finite-sample critical values originally suggested by Bauer and Hackl and the asymptotic critical values are much too large in finite samples.

Orbits of Braid Groups on Cacti

One of the consequences of the classification of finite simple groups is the fact that non-rigid polynomials (those with more than two finite critical values), considered as branched coverings of the sphere, have exactly three exceptional monodromy groups (one in degree 7, one in degree 13 and one in degree 15). By exceptional here we mean primitive and not equal to S_n or A_n, where n is the degree. Motivated by the problem of the topological classification of polynomials, a problem that goes back to 19th century researchers, we discuss several techniques for investigating orbits of braid groups on ``cacti'' (ordered sets of monodromy permutations). Applying these techniques, we provide a complete topological classification for the three exceptional cases mentioned above.

Unit root tests with double trend breaks and the 1990s recession in Japan

This paper applies to Japanese macroeconomic series unit root tests that allow for the possibility of up to two endogenous break points. The presence of a single structural break around the first oil price shock in 1973 turns out to be very sensitive to the amplitude of the data sample and, in particular, it disappears when one extends the sample to the observations of the 1990s. This may indicate the presence of a second structural break, the existence of which is tested with a unit root test with a two-break alternative hypothesis for which we compute finite sample critical values. Interestingly enough, the hypothesis of the absence of a second break occurring in the 1990s can be rejected. Such results seem to indicate that the deep recession of the 1990s in Japan may not be the reflection of a negative output-gap, but that of a fall in the growth trend of output as a consequence of a huge productivity shock.

Distributions of Error Correction Tests for Cointegration

This paper provides cumulative distribution functions, densities, and finite sample critical values for the single-equation error correction statistic for testing cointegration. Graphs and response surfaces summarize extensive Monte Carlo simulations and highlight simple dependencies of the statistic's quantiles on the number of variables in the error correction model, the choice of deterministic components, and the estimation sample size. The response surfaces provide a convenient way for calculating finite sample critical values at standard levels; and a computer program, freely available over the Internet, can be used to calculate both critical values and p-values. Three empirical examples illustrate these tools.

Distributions of Error Correction Tests for Cointegration

This paper provides cumulative distribution functions, densities, and finite sample critical values for the single-equation error correction statistic for testing cointegration. Graphs and response surfaces summarize extensive Monte Carlo simulations and highlight simple dependencies of the statistic's quantiles on the number of variables in the error correction model, the choice of deterministic components, and the estimation sample size. The response surfaces provide a convenient way for calculating finite sample critical values at standard levels; and a computer program, freely available over the Internet, can be used to calculate both critical values and p-values. Three empirical examples illustrate these tools.

Robust tests for ARCH and GARCH components

Two robust one-sided tests are proposed for detecting ARCH and GARCH components. The first one is distribution free being a rank test based on the van der Waerden scores. The related test statistic has mean and variance known in closed form; moreover, it is shown to be asymptotically normal and its finite sample critical values have been estimated via the Monte Carlo method. The second test is the one sided modification of the Gregory test which is of course more powerful than the corresponding original two sided version. An extensive Monte Carlo analysis has been performed and - as long as it is generalizable - the following conclusions can be drawn. The new rank test performs well both for ARCH observations and ARCH residuals. From robustness point of view, under the null hypothesis, some bias appears for ARCH residuals from heavily asymmetric distributions. In this case the modified Gregory test is recommended. In the other situations i.e. ARCH observations from symmetric or asymmetric distributions and ARCH residuals from symmetric distributions the rank test is shown to be robust under the null hypothesis and more powerful with respect to the modified Gregory test. The rank test can be recommended as a useful alternative to the standard Lee and King test, which is the reference test for normal observations. As a matter of fact, it seems more robust under the null hypothesis and more powerful for non normal distributions. For normal distributions neither of these two dominates the other.