Fractional Integration Tests Research Articles

AFRIMA Model Forecast Performance: An Empirical Study using Naira-Yuan Exchange Rate

In financial time series and econometrics, some macroeconomic variables exhibit long memory features that may not be best described using short memory models like ARIMA. This paper, however, is structured to compare different fractional integration in AFRIMA forecast performance for the Naira-Yuan exchange rate. The empirical monthly data set used covered the period from January 1981 to December 2022. Fractional integration test are based on the ADF unit root test and the auxiliary autoregressive order three (AAR(3)) order of integration test. Model estimation is support by the Marquart algorithm for calculating least squares estimates and performance comparison is based on the Amaefula forecast criterion (AFC). The result specified that AFRIMA (1, d, 1) where I(d = 0.07891) is more appropriate and has the best forecast performance compared to others. The result also reveals that AFRIMA model yield better and more precise forecasts when fractional integration is closer to zero that is, I(d→0) than when I(d→½). Therefore, AFRIMA models can be useful in studying exchange rate dynamics for risk-averse and risk incline in times of investment and profitability in the long-run.

A new fractional integration approach based on neural network nonlinearity with an application to testing unemployment hysteresis

This paper proposes a nonlinear fractional unit root approach which is known as the autoregressive neural network–fractional integration (ARNN–FI) test. This new fractional integration test is based on a new multilayer perceptron of a neural network process, proposed in Yaya et al. (Oxf Bull Econ Stat 83(4):960–981, 2021). The asymptotic theory and the properties of the proposed test are given. By setting up a Monte Carlo simulation experiment, the simulation results reveal that as the number of observations increases, size and power distortions would disappear in the test. The empirical application based on this new test reveals that the unemployment rates of three European countries are neither stationary nor mean-reverting in line with the hysteresis hypothesis.

A sectoral perspective on the persistence of economic sentiment: mere transitory effect or a long memory process?

Opposing the mainstream view that economic sentiment conveys information of purely short-term relevance, we assess the long memory characteristics of confidence indicators derived from Business and Consumer Surveys (BCS). We utilize a battery of fractional integration tests on five sectoral BCS indicators in 28 European economies. Conforming to social learning theory, we find that the examined confidence indicators in all five sectors exhibit noteworthy persistence. This finding stays intact in various robustness checks: across countries and for different fractional integration test specifications. Nevertheless, post hoc comparisons reveal that confidence in the construction and consumer sectors exhibits considerably more intensive persistence than in the industry, retail trade, and services. Since it is evident that shocks in economic sentiment have long-term or permanent effects, this calls for a revival of Keynesian ideas and the stabilization role of well-thought and timely countercyclical economic policy measures.

The persistence of economic sentiment: a trip down memory lane.

Although various indicators of economic sentiment are often assessed in macroeconomic studies, the generating process of economic sentiment itself is still a puzzle. This paper offers pioneer evidence on the persistence of economic sentiment. Applying a battery of fractional integration tests on the European Economic Sentiment Indicator (ESI) of all individual EU member states, we reveal that ESI is dominantly a long-memory process. This finding is robust across several estimators, and it fairly contradicts the conventional wisdom of ESI as a purely transitory macroeconomic shock. Further on, this is true for both core EU economies and new member states, although the later ones exhibit slightly longer memory. Finally, we reveal that the end of the Great Moderation era has increased ESI’s persistence, but the effect is only marginal. As it seems, a series of macroeconomic turbulences recorded after the global financial crisis has not initiated a significant shift in agents’ collective memory and ESI will likely keep its pivotal role in governing business cycles in the future.

Multivariate fractional integration tests allowing for conditional heteroskedasticity with an application to return volatility and trading volume

SummaryWe introduce a new joint test for the order of fractional integration of a multivariate fractionally integrated vector autoregressive (FIVAR) time series based on applying the Lagrange multiplier principle to a feasible generalised least squares estimate of the FIVAR model obtained under the null hypothesis. A key feature of the test we propose is that it is constructed using a heteroskedasticity‐robust estimate of the variance matrix. As a result, the test has a standardχ2limiting null distribution under considerably weaker conditions on the innovations than are permitted in the extant literature. Specifically, we allow the innovations driving the FIVAR model to follow a vector martingale difference sequence allowing for both serial and cross‐sectional dependence in the conditional second‐order moments. We also do not constrain the order of fractional integration of each element of the series to lie in a particular region, thereby allowing for both stationary and non‐stationary dynamics, nor do we assume any particular distribution for the innovations. A Monte Carlo study demonstrates that our proposed tests avoid the large oversizing problems seen with extant tests when conditional heteroskedasticity is present in the data. We report an empirical case study for a sample of major US stocks investigating the order of fractional integration in trading volume and different measures of volatility in returns, including realised variance. Our results suggest that both return volatility and trading volume are fractionally integrated, but with the former generally found to be more persistent (having a higher fractional exponent) than the latter, when more reliable proxies for volatility such as the range or realised variance are used.

Semiparametric Tests for the Order of Integration in the Possible Presence of Level Breaks

Lobato and Robinson developed semiparametric tests for the null hypothesis that a series is weakly autocorrelated, or I(0), about a constant level, against fractionally integrated alternatives. These tests have the advantage that the user is not required to specify a parametric model for any weak autocorrelation present in the series. We extend this approach in two distinct ways. First, we show that it can be generalized to allow for testing of the null hypothesis that a series is for any δ lying in the usual stationary and invertible region of the parameter space. The second extension is the more substantive and addresses the well-known issue in the literature that long memory and level breaks can be mistaken for one another, with unmodeled level breaks rendering fractional integration tests highly unreliable. To deal with this inference problem, we extend the Lobato and Robinson approach to allow for the possibility of changes in level at unknown points in the series. We show that the resulting statistics have standard limiting null distributions, and that the tests based on these statistics attain the same asymptotic local power functions as infeasible tests based on the unobserved errors, and hence there is no loss in asymptotic local power from allowing for level breaks, even where none is present. We report results from a Monte Carlo study into the finite-sample behavior of our proposed tests, as well as several empirical examples.

Bubble Detection in Financial Markets - A Survey of Theoretical Bubble Models and Empirical Bubble Detection Tests

Asset price bubbles have fascinated economists for decades. In consequence, the literature on bubbles and their detection is abundant, with many researchers taking very opposite positions on the topic, however. This survey gives a structured overview of the two branches of research that have received the most attention in bubble research. First, we describe the theoretical models that have been developed to model bubble phenomena. These can be divided into rational bubble models and behavioral bubble models. Second, we provide a structured overview of empirical methods for the detection of rational bubbles. We focus in particular on recently developed bubble detection methods, namely recursive unit root tests, fractional integration tests, and regime-switching tests. These tests are predominantly advanced stationarity and cointegration-based tests and as such are not based on the fundamental factors of the assets but rather focus on the time series of the asset prices. As a result, they avoid testing a joint hypothesis of the presence of rational bubbles and the validity of the model used to determine the asset’s value. Furthermore, they are capable of detecting multiple, periodically collapsing bubbles. While a consensus both on the appropriate theoretical bubble model as well as on the most applicable empirical bubble detection test has not been reached, especially empirical research has made significant progress. Different bubble detection tests now increasingly find overlapping evidence of rational bubbles when used to analyze the same time series. Nonetheless, many results are still inconclusive and bubbles remain an interesting avenue for further research.

The random-walk hypothesis revisited: new evidence on multiple structural breaks in emerging markets

We examine whether stock prices in 18 emerging markets follow random-walk or mean-reversion processes in the presence of sudden and gradual multiple structural breaks. Our tests endogenously determined the structural shifts and are more powerful than either the traditional random-walk (unit root) tests or the single structural break tests. In all emerging markets, we find strong evidence for multiple structural breaks. When we use single break tests, the random-walk hypothesis is rejected. However, when we use tests of double level shifts in the mean and make due allowance for multiple structural breaks, the results are consistent with the random-walk hypothesis in the vast majority of the sampled markets. The evidence proves robust to using price indexes whether denominated in U.S. dollars, in local currencies or in real terms, and also to using fractional integration tests. Our results contradict some previous studies for emerging markets which restrict structural breaks to only one-time shift.

English

We extend the rational speculative bubbles literature to the frontier emerging stock markets. For this purpose, this paper employs fractional integration tests and duration dependence tests based on the ARFIMA models and nonparametric smoothed hazard functions. Unlike traditional bubble tests, fractional integration tests and duration dependence tests do not show strong evidence of rational speculative bubbles in the frontier emerging stock markets.

Persistence in the banking industry: Fractional integration and breaks in memory

Certain “spurious long memory” processes mimic the behavior of fractional integration in that the variance of their sample mean behaves like that of a fractionally integrated process of some order D . We show, however, experimentally that a fractional integration test may discriminate between spurious long memory of order D and integration of order D . Further, we suggest a test for the null hypothesis that the order of integration does not change from one subperiod to another. It simply builds on the difference of the estimates from the respective subsamples that are split exogenously. Upon appropriate normalization a limiting standard normal distribution arises. With these methods we tackle the question whether international and sectoral bank equity index returns are fractionally integrated and whether the memory parameters have changed. The daily data are split into three regimes: one pre-crises subsample, a second including the collapse of the Lehman Brothers bank, and a third covering the Euro area sovereign debt crisis. In particular, we provide evidence that both turmoils had differing international effects. • Integration test may discriminate between spurious long memory and true fractional integration. • Test for the null hypothesis that the order of integration does not change. • International and sectoral bank equity index return. • Collapse of the Lehman Brothers bank and the Euro area sovereign debt crisis had differing effects.

An analysis of the time series properties of the UKex-postreal interest rate: fractional integration, breaks or nonlinear

In this article we examine the persistent nature of the 3-month UK real interest rate for the period 1957:Q2 to 2008:Q2. We employ unit root and cointegration tests, confidence intervals for the sum of the Autoregressive (AR) coefficient, fractional integration tests, structural break tests and threshold modelling. Evidence from both unit root tests and AR modelling support the view that the real rate is nonstationary. Similarly, and in contrast to previous literature, the fractional integration test supports covariance nonstationarity, although there is evidence of mean reversion. Evidence from structural break tests support stationary behaviour, but only if we allow for three or more breaks, which may not be defendable on economic grounds. Finally, stationary behaviour is supported by a nonlinear exponential smooth transition model, which suggests that the real rate behaves in a random walk fashion when the rate is close to equilibrium but exhibits strong mean reversion when the disequilibrium becomes l...

Rational speculative bubbles in MENA stock markets

PurposeThe purpose of this paper is to examine the existence of rational speculative bubbles in the Middle East and North African (MENA) stock markets.Design/methodology/approachTo complement shortcomings of the traditional bubble tests, such as unit root tests and cointegration tests, mainly relying on expectations of future steams of dividends, the authors employ fractional integration tests and duration dependence tests.FindingsDespite recent extreme fluctuations of MENA stock markets, fractional integration tests built on autoregressive fractionally integrated moving average models do not support the possibility of bubbles in the MENA stock markets. Similarly, duration dependence tests based on nonparametric Nelson‐Aalen hazard functions not only reject the existence of bubbles but also support equality of hazard functions between domestic and the US‐based investors without regard to the rapid financial liberalization and integration in the MENA stock markets.Originality/valueThe reliable results of bubble tests of the MENA stock markets provide domestic and international investors as well as policy makers with invaluable benchmark to better understand the irregular and highly fluctuating stock market behaviors of the MENA stock markets compared to other developed and emerging stock markets. For domestic and international investors, the formal analysis of MENA stock markets behavior including rational speculative bubbles will help them in their portfolio decisions and hedging purposes. Similarly, the empirical results of bubble tests in the paper will be also helpful to policymakers in MENA countries to take actions to improve the functioning of these dynamic markets.

Efficiency of the IBEX spot–futures basis: The impact of the mini‐futures

AbstractThis paper considers whether the introduction of the mini‐futures contract for the Spanish Ibex index affects overall market efficiency. Using linear, non‐linear, and fractional integration modeling techniques for the basis term, results of this study suggest the following salient points. First, the equilibrium speed of adjustment is reduced after the introduction of the mini‐futures contract. This effect is particularly pronounced in the mini‐futures second year when its contracts are more heavily traded. Second, fractional integration tests support longer memory in the basis term after the contract introduction, again particularly in the second year. Third, the relationship between the full‐size and mini‐futures contracts appears highly efficient, with a quick speed of adjustment and short memory. Finally, an examination of the volatility dynamics suggests that in the second year of the mini‐futures contract shocks to spot return volatility exhibit longer memory. The results reported here suggest that the increased use of the mini‐futures contract after its introduction has had a detrimental impact on price discovery. © 2008 Wiley Periodicals, Inc. Jrl Fut Mark 28: 398–415, 2008

Bias correction for the regression-based LM fractional integration test

This paper examines the finite-sample behavior of the Lagrange Multiplier (LM) test for fractional integration proposed by Breitung and Hassler (J. Econom. 110:167–185, 2002). We find by extensive Monte Carlo simulations that size distortions can be quite large in small samples. These are caused by a finite-sample bias towards the alternative. Analytic expressions for this bias are derived, based on which the test can easily be corrected.

Rational Speculative Bubbles: An Empirical Investigation of the Middle East and North African Stock Markets

Despite recent extreme fluctuations of the Middle East and North African (MENA) stock markets, we do not find strong evidence of rational speculative bubbles in the perspective of both domestic and U.S.-based investors. Fractional integration tests built on ARFIMA models do not support the possibility of bubbles in the MENA stock markets. Similarly, duration dependence tests based on nonparametric Nelson-Aalen hazard functions not only reject the existence of bubbles but also support equality of hazard functions between domestic and U.S.-based investors without regard to the rapid financial liberalization and integration in the MENA stock markets.

Comment on “The uncertain unit root in real GNP: A re-examination”

With the objective to settle the often conflicting and inconclusive extant debate on whether the observed secular growth can be characterized by a deterministic or stochastic trend, Darné comes up with an alternative method – the outlier detection methodology and studies the persistence properties of US real GNP for the period 1869–1993. Given that the presence of outliers in a time series – unless detected and adjusted – often render wrong impression on the persistence properties and their implications for long-run growth, Darné’s contribution in this regard is significant. Outliers of varied types assume varying persistence profiles, therefore detection of them in the GNP growth is a pre-requisite for characterizing the series in a stochastic or deterministic setting. To further intuition on the exact nature of persistence, I applied fractional integration test of the four real GNP series and found that outlier adjusted GNP series are mean-reverting and are more persistent than original. The rate of convergence of stochastic shocks to long-run mean have important implications for growth dynamics.

Univariate time series behaviour of the real exchange rate: evidence from colonial India

This paper empirically examines the long-run behaviour of the real exchange rate in colonial India between the British pound and the Indian rupee using a battery of unit root tests. The unit root tests based on the KPSS test, the GPH fractional integration test, and the non-linear KSS test indicate that the real exchange rate series is stationary and mean-reverting, which tends to support the validity of the purchasing power parity (PPP) hypothesis in the long run.

Long memory and structural breaks in hyperinflation countries

In this paper we examine the stochastic behavior of prices in hyperinflation countries by using a fractional integration test (Robinson 1994) that lends itself to incorporating structural breaks into the model. We focus on Argentina, Brazil, and Israel and find that when allowing for structural breaks, in the form of slope dummies (or squared slope dummies), the order of integration of the series decreases considerably. Especially in the case of Brazil, the degree of persistence of inflation seems to be less substantial than estimated in other studies, which might be interpreted as evidence against “heterodox” inflation stabilization. (JEL C22, E31)

Parity reversion persistence in real exchange rates: middle income country case

This article investigates purchasing power parity (PPP) in the specific context of middle income countries. To circumvent the low power of traditional stationarity tests (Augmented Dickey-Fuller and Phillips-Perron tests), it performs variance ratio and fractional integration tests in addition to Perron's test that accounts for potential structural changes in real exchange rate processes. Beyond estimating half-life shocks to PPP, this article attempts to explain these estimates using a set of country specific variables as suggested by economic theory. The evidence suggests that reversion to parity tends to be faster in high inflation countries and that productivity improvement leads to a higher level of persistence. Openness to trade tends to reduce the extent of deviations from parity but this result does not appear to be statistically robust. Evidence shows also that deviations are less persistent under a fixed exchange rate regime and under unrestricted capital mobility.

The Stochastic Structure of the Time–Varying Beta: Evidence from UK Companies

The stochastic structure of time–varying betas from 15 companies in the UK is investigated. Time–varying betas are estimated by means of the bivariate MA–GARCH model. The stochastic structure is investigated by means of two fractional integration tests, the Geweke and Porter–Hudak and the Robinson tests, and a structural–break–oriented unit root test. Results show that time–varying betas are mean–reverting but only few have a long memory and thus are mean–reverting at a slow rate. This result is further backed by the structural break unit root test. These results contradict earlier studies, which fail to find a stationary beta. Stationary betas may imply that stock returns may be forecast in the long run.