Long Memory Components Research Articles

Multivariate Fractional Components Analysis

Abstract We propose a setup for fractionally cointegrated time series which is formulated in terms of latent integrated and short-memory components. It accommodates nonstationary processes with different fractional orders and cointegration of different strengths and is applicable in high-dimensional settings. In an application to realized covariance matrices, we find that orthogonal short- and long-memory components provide a reasonable fit and competitive out-of-sample performance compared with several competing methods.

The Effect of Economic Policy Uncertainty Under Fractional Integration

One of the most popular measures of economic policy uncertainty (EPU) is an index based on newspapers coverage of particular keywords. The constructed index is often then included in VAR models to examine the extent to which EPU affects economic activity. Researchers, however, have not investigated how the possibility of fractional integration in this index may affect the results. Under fractional integration, the effects of EPU on output in a standard VAR setting may be biased. In this paper, I first show that all EPU series posted in policyuncertainty.com are indeed fractionally integrated processes. In addition, I show that the residuals from Baker, Bloom, and Davis (2016)’s VAR model that includes output, EPU, and other covariates are also fractionally integrated. Next, I estimate a fractionally cointegrated VAR (FCVAR) model to evaluate the dynamic effects of EPU. Likelihood ratio tests indicate that fractional cointegration cannot be rejected at low lag orders. Lastly, I investigate the effect of EPU on ouput after applying various filters to remove the long-memory component. While I still find that EPU imparts a negative effect on output, relative to Baker et al. (2016)’s results, the effects are 50 to 90% smaller in magnitude and take place several months sooner. Long memory, therefore, appears to explain the bulk of the dynamic effect.

Remittances in Mexico and their unobserved components

ABSTRACTThe present study aims to determine the common trends and the permanent and transitory components of remittances received by Mexican households. This is done by estimating a small Dynamic Factor Model (DFM), using the approach first proposed by Gonzalo and Granger [1995. “Estimation of Common Long-memory Components in Cointegrated Systems.” Journal of Business and Economic Statistics 13 (1): 27–35], determining the number of common trends subject to the cointegration results. The study also shows the similarities between this small DFM with respect to large DFM, which are widely used in the econometric literature. The results indicate the presence of one cointegration relationship. Consequently, there are four common trends. The cointegration relationship is negatively dominated by Mexico's economic activity and positively by the US industrial production. The effects of the exchange rate and the US unemployment rate are positive, but less relevant. This economic scenario leads to remittances exceeding its permanent component.

How Unemployment Affects Bond Prices: A Mixed Frequency Google Nowcasting Approach

We analyze the relationship between unemployment rate changes and government bond yields during and after the most recent financial crisis across ten industrialized countries. The study is conducted on a weekly basis and we therefore nowcast unemployment data, which are only available once a month, on a weekly frequency using Google search query data. In order to account for the time series’ long-memory components during the first-stage nowcasting and the second-stage modeling, we draw on Corsi’s (J Financ Economet 7:174–196, 2009) heterogeneous autoregressive time series model. In particular, we adapt this idea to a setting of mixed-frequency nowcasting. Our results indicate that Google searches greatly increase the nowcasting accuracy of unemployment rate changes. The impact of an idiosyncratic rise in unemployment on bond yields turns out to be positive for European countries while it is negative for the United States and Australia. The speed of the response also varies. Not unexpectedly, bond yields do not have an impact on unemployment. Our findings have interesting implications for the way shocks are absorbed in economic systems that differ, in particular, with respect to the central bank’s core tasks.

Fractional Black–Scholes option pricing, volatility calibration and implied Hurst exponents in South African context

Background: Contingent claims on underlying assets are typically priced under a framework that assumes, inter alia, that the log returns of the underlying asset are normally distributed. However, many researchers have shown that this assumption is violated in practice. Such violations include the statistical properties of heavy tails, volatility clustering, leptokurtosis and long memory. This paper considers the pricing of contingent claims when the underlying is assumed to display long memory, an issue that has heretofore not received much attention.Aim: We address several theoretical and practical issues in option pricing and implied volatility calibration in a fractional Black–Scholes market. We introduce a novel eight-parameter fractional Black–Scholes-inspired (FBSI) model for the implied volatility surface, and consider in depth the issue of calibration. One of the main benefits of such a model is that it allows one to decompose implied volatility into an independent long-memory component – captured by an implied Hurst exponent – and a conditional implied volatility component. Such a decomposition has useful applications in the areas of derivatives trading, risk management, delta hedging and dynamic asset allocation.Setting: The proposed FBSI volatility model is calibrated to South African equity index options data as well as South African Rand/American Dollar currency options data. However, given the focus on the theoretical development of the model, the results in this paper are applicable across all financial markets.Methods: The FBSI model essentially combines a deterministic function form of the 1-year implied volatility skew with a separate deterministic function for the implied Hurst exponent, thus allowing one to model both observed implied volatility surfaces as well as decompose them into independent volatility and long-memory components respectively. Calibration of the model makes use of a quasi-explicit weighted least-squares optimisation routine.Results: It is shown that a fractional Black–Scholes model always admits a non-constant implied volatility term structure when the Hurst exponent is not 0.5, and that 1-year implied volatility is independent of the Hurst exponent and equivalent to fractional volatility. Furthermore, we show that the FBSI model fits the equity index implied volatility data very well but that a more flexible Hurst exponent parameterisation is required to fit accurately the currency implied volatility data.Conclusion: The FBSI model is an arbitrage-free deterministic volatility model that can accurately model equity index implied volatility. It also provides one with an estimate of the implied Hurst exponent, which could be very useful in derivatives trading and delta hedging.

Term Structure Persistence

Stationary I(0) models employed in yield curve analysis typically imply an unrealistically low degree of volatility in long-run, short-rate expectations due to fast mean reversion. In this article, we propose a novel multivariate affine term structure model with a two-fold source of persistence in the yield curve: long memory and short memory. Our model, based on an I(d) specification, nests the I(0) and I(1) models as special cases and the I(0) model is decisively rejected by the data. Our model estimates imply both mean reversion in yields and quite volatile long-distance, short-rate expectations, due to the higher persistence imparted by the long-memory component. Our implied term premium estimates differ from those of the I(0) model during some relevant periods by more than 3 percentage points and exhibit a realistic counter-cyclical pattern. (JEL Classification: C3, E4, G1.)

A Cross-Country Analysis of Unemployment and Bonds with Long-Memory Relations

We analyze the relationship between unemployment rate changes and government bond yields during and after the most recent financial crisis across nine industrialized countries. The study is conducted on a weekly basis and we therefore nowcast unemployment data, which are only available once a month, on a weekly frequency using Google search query data. In order to account for the time series' long-memory components during the first-stage nowcasting and the second-stage modeling, we draw on Corsi's (2009) heterogeneous autoregressive time series model. In particular, we adapt this idea to a setting of mixed-frequency nowcasting. Our results indicate that Google searches greatly increase the nowcasting accuracy of unemployment rate changes. The impact of an idiosyncratic rise in unemployment on bond yields turns out to be positive for European countries while it is negative for the United States and Australia. The speed of the response also varies. Not unexpectedly, bond yields do not have an impact on unemployment. Our findings have interesting implications for the way shocks are absorbed in economic systems that differ, in particular, with respect to the central bank's core tasks.

Information Transmission in the World Money Markets

Using 1,966 daily observations since the introduction of the euro, we apply cointegration and error correction tests to examine information transmission in the major world money markets as represented by the domestic CD markets and the Eurocurrency market for the US dollar, euro, Japanese yen, and British pound sterling. Our inter-market tests show a high degree of integration and interdependency among inter-market interest rates. Our intra-market results show that $ LIBOR and LIBOR rates drive LIBOR and £ LIBOR. Application of Johansen's (1988) multivariate test procedure and Gonzalo and Granger's (1995) long-memory components technique confirms and reinforces our intra-market findings that the system of four LIBOR rates is fully integrated (i.e., three cointegrating vectors), with the single common trend driven by $ LIBOR and LIBOR. These results are consistent with the strength of the dollar and yen relative to the pound sterling and the euro during the developing world financial crisis in late 2008.

Empirical aspects of the Whittle-based maximum likelihood method in jointly estimating seasonal and non-seasonal fractional integration parameters

This paper addresses the efficiency of the maximum likelihood ( ML) method in jointly estimating the fractional integration parameters d s and d , respectively associated with seasonal and non-seasonal long-memory components in discrete stochastic processes. The influence of the size of non-seasonal parameter over seasonal parameter estimation, and vice versa, was analyzed in the space d × d s ∈ ( 0 , 1 ) × ( 0 , 1 ) by using mean squared error statistics M S E ( d ˆ s ) and M S E ( d ˆ ) . This study was based on Monte Carlo simulation experiments using the ML estimator with Whittle’s approximation in the frequency domain. Numerical results revealed that efficiency in jointly estimating each integration parameter is affected in different ways by their sizes: as d s and d increase simultaneously to 1, M S E ( d ˆ s ) and M S E ( d ˆ ) become larger; however, effects on M S E ( d ˆ s ) are much stronger than the effects on M S E ( d ˆ ) . Moreover, as each parameter tends individually to 1, M S E ( d ˆ ) becomes larger, but M S E ( d ˆ s ) is barely influenced.

Four factors that explain both the rise and fall of US crime, 1970–2003

Previous research has failed to explain the rise and fall of US crime since 1970. This study uses cointegration, error correction and common long-memory components analyses to demonstrate that four basic crime factors explaining both the increases in US violent and property crime between 1970 and 1991 and the dramatic declines in crime after 1991. The four factors include arrest rates, income per capita, the proportion of criminal-justice resources devoted to drug crime and alcohol consumption. Error correction models and common long-memory factors show an especially close link between crime rates and the percentage of prison resources devoted to drug offenders. Similar factors result in cointegrated models for murder, rape, robbery, assault and larceny. Additional modelling shows that effective abortion rates computed along the lines of Donohue and Levitt (2001) do not help in explaining the rise and fall of US crime.

The Making of "Estimation of Common Long-Memory Components in Cointegrated Systems"

The eighties were very good years for music as well as econometrics. Intime-series econometrics, the first half of that decade was dominated by research on unit roots whilecointegration was the queen of the second half. Estimation and testing of a cointegrated system werethe key questions to answer. When I started my dissertation at the end of the eighties, under the supervision of CliveGranger and Robert Engle, you could sense that everyone was of the opinion that the testing problemof the cointegration rank had been solved by Johansen (1988). Johansen applied reduced rankregression techniques to the following error correction model (ECM) (Granger’s old notation is used tokeep the spirit of the eighties) and obtained the corresponding asymptotics. Nevertheless, estimation was still under close scrutiny,and new estimators appeared on a regular basis in the literature. It was also the case howeverthat different methods could lead to very different estimates of the cointegrating vector,and this was the reason Clive and Rob wanted me to start working on the problem. In Gonzalo (1994), I obtained the asymptotic distribution of the principal components (method proposed in Stock and Watson, 1988) and canonical correlations (between the vectorXt andXt−1proposed by Bossaerts, 1988) based estimators of cointegrating vectors and compared theirperformance to the most popular alternatives at the time: ordinary least square (Engle and Granger, 1987), nonlinear least square (Stock, 1987), and maximum likelihood (Johansen, 1988). Thelatter was the clear winner once the dispersion of the finite sample distributions was measured by theinterquartile range instead of the simple variance.

International stock market linkages: Evidence from Latin America

This paper examines long-run relationships between four Latin America stock markets and a mature stock market that of the US. We estimate both the autoregressive and moving average representations of a VAR model as suggested by Johansen [Johansen, S. (1988). Statistical analysis of cointegrating vectors, Journal of Economic Dynamics and Control, 12, 231–254, Johansen, S. (1991). Estimation and hypothesis testing of cointegration vectors in Gaussian vector autoregressive models, Econometrica, 59, 1551–1580.] using weekly observations for the period January 1988–July 2006. The main finding of the analysis suggests that there is one long-run relationship among the five equity markets. Application of the Gonzalo and Granger [Gonzalo, J., & Granger, C. W. J. (1995). Estimation of common long-memory components in cointegrated systems, Journal of Business and Economic Statistics, 13, 27–35.] methodology indicates that the examined stock markets are partially integrated, while there is also evidence that the four stock markets of Latin America (Argentina, Brazil, Chile and Mexico) together with the US stock market have four significant common permanent components, which drive this system of equity markets in the long run. Finally, we show that although cointegration exists there are small long-run benefits from international portfolio diversification since the stock prices adjust very slowly to these common trends.

Information contents misjudged: Digressive convergence to equilibrium in cointegrated prices

The spot price on the Taiwan stock index is richer in information than the futures price judged by the price discovery measures of Gonzalo and Granger [Gonzalo, J., & Granger, C.W.J. (1995). Estimation of common long-memory components in cointegrated systems. Journal of Business and Economic Statistics, 13, 27–35.] and Hasbrouck [Hasbrouck, J. (1995). One security, many markets: Determining the contributions to price discovery. Journal of Finance, 50, 1175–1199.]. What is special about the markets is that both the spot and futures error-correction coefficients are positive, implying a digressive convergence to their long-run equilibrium in the error-correction (EC) process. Innovation accounting suggests that the cause of this digressive equilibrium adjustment is that investors systematically overreact to news in the less informative futures market but under-react to the more informative spot market. Our contribution is in identifying the digressive convergence implied by same-sign EC coefficients, comparing it to the normal convergence widely found in opposite-sign EC models, and providing short-run mispricing interpretations for both types of convergence to equilibrium.

The Effect of Long Memory in Volatility on Stock Market Fluctuations

Recent empirical evidence demonstrates the presence of an important long-memory component in realized asset return volatility. We specify and estimate multivariate models for the joint dynamics of stock returns and volatility that allow for long memory in volatility without imposing this property on returns. Asset pricing theory imposes testable cross-equation restrictions on the system that are not rejected in our preferred specifications, which include a strong financial leverage effect. We show that the impact of volatility shocks on stock prices is small and short lived, in spite of a positive risk-return tradeoff and long memory in volatility.

Determining the contributions to price discovery for Chinese cross-listed stocks

We study the price-discovery process for a number of Chinese cross-listed stocks. For the stocks cross-listed on the New York Stock Exchange (NYSE) and the Stock Exchange of Hong Kong (SEHK), we find that the stock prices of these two exchanges are cointegrated and mutually adjusting, and that the SEHK makes more contributions than the NYSE to the price-discovery process. The SEHK contributions are 81.6% and 89.4%, computed from Gonzalo and Granger [Gonzalo, J., Granger, C., 1995. Estimation of common long-memory components in cointegrated systems. Journal of Business and Economics Statistics 13, 27–35] permanent–transitory (PT) and Hasbrouck [Hasbrouck, J., 1995. One security, many markets: Determining the contributions to price discovery. Journal of Finance 50, 1175–1119] information share (IS) models respectively.

Non-Linearities and Fractional Integration in the US Unemployment Rate

This paper proposes a model of the US unemployment rate which accounts for both its asymmetry and its long memory. Our approach, based on the tests of Robinson (1994), introduces fractional integration and nonlinearities simultaneously into the same framework (unlike earlier studies employing a sequential procedure), using a Lagrange Multiplier procedure with a standard limit distribution. The empirical results suggest that the US unemployment rate can be specified in terms of a fractionally integrated process, which interacts with some non-linear functions of the labour demand variables (real oil prices and real interest rates). We also find evidence of a long-memory component. Our results are consistent with a hysteresis model with path dependency rather than a NAIRU model with an underlying unemployment equilibrium rate, hence giving support to more activist stabilisation policies. However, any suitable model should also include business cycle asymmetries, with implications for both forecasting and policymaking.

Term structure linkages surrounding the Plaza and Louvre accords: Evidence from Euro-rates and long-memory components

Eurocurrency deposit rates for Germany, Japan, the UK and the US are used to investigate term structure linkages over the 1979–85 and 1986–01 periods. A single common trend drives the term structure in each country, with 1-month rates generally having little or no influence in setting the long-term trend. Cointegration and common factor analyses based on the intracurrency common factors suggest weaker integration of economic fundamentals following the Plaza Accord in 1986. However, using a 10-year moving window to examine the 1986–01 period, similar tests show that integration increased between the 1986–95 and 1988–97 periods, but declined again by 1992–01.

Testing for Neglected Nonlinearity in Long-Memory Models

This article constructs tests for the presence of nonlinearity of unknown form in addition to a fractionally integrated, long-memory component in a time series process. The tests are based on artificial neural network approximations and do not restrict the parametric form of the nonlinearity. Some theoretical results for the new tests are obtained, and detailed simulation evidence on the power of the tests is presented. The new methodology is then applied to a wide variety of economic and financial time series.

Is it Possible to Study Chaotic and ARCH Behaviour Jointly? Applicationof a Noisy Mackey–Glass Equation with Heteroskedastic Errors to theParis Stock Exchange Returns Series

Most recent empirical works that apply sophisticated statistical procedures such as a correlation-dimension method have shown that stock returns are highly complex. The estimated correlation dimension is high and there is little evidence of low-dimensional deterministic chaos. Taking the complex behaviour in stock markets into account, we think it is more robust than the traditional stochastic approach to model the observed data by a nonlinear chaotic model disturbed by dynamic noise. In fact, we construct a model having negligible or even zero autocorrelations in the conditional mean, but a rich structure in the conditional variance. The model is a noisy Mackey–Glass equation with errors that follow a GARCH(p,q) process. This model permits us to capture volatility-clustering phenomena. Its characteristic is that volatility clustering is interpreted as an endogenous phenomenon. The main objective of this article is the identification of the underlying process of the Paris Stock Exchange returns series CAC40. To this end, we apply several different tests to detect longmemory components and chaotic structures. Forecasting results for the CAC40 returns series, will conclude this paper.

SEMIFAR models—a semiparametric approach to modelling trends, long-range dependence and nonstationarity

Time series in many areas of application often display local or global trends. Statistical “explanations” of such trends are, for example, polynomial regression, smooth bounded trends that are estimated nonparametrically, and difference-stationary processes such as, for instance, integrated ARIMA processes. In addition, there is a fast growing literature on stationary processes with long memory which generate spurious local trends. Visual distinction between deterministic, stochastic and spurious trends can be very difficult. For some time series, several “trend generating” mechanisms may occur simultaneously. Here, a class of semiparametric fractional autoregressive models ( SEMIFAR) is proposed that includes deterministic trends, difference stationarity and stationarity with short- and long-range dependence. The components of the model can be estimated by combining maximum likelihood estimation with kernel smoothing in an iterative plug-in algorithm. The method helps the data analyst to decide whether the observed process contains a stationary short- or long-memory component, a difference stationary component, and/or a deterministic trend component. Data examples from climatology, economics and dendrochronology illustrate the method. Finite sample behaviour is studied in a small simulation study.