Short Memory Processes Research Articles

Asymptotic Theory for Near Integrated Process Driven by Tempered Linear Process

This paper develops an asymptotic theory for near-integrated random processes and some associated regressions when the errors are tempered linear processes. Tempered processes are stationary time series that have a semi-long memory property in the sense that the autocovariogram of the process resembles that of a long memory model for moderate lags but eventually diminishes exponentially fast according to the presence of a decay factor governed by a tempering parameter. When the tempering parameter is sample size dependent, the resulting class of processes admits a wide range of behavior that includes both long memory, semi-long memory, and short memory processes. The paper develops asymptotic theory for such processes and associated regression statistics thereby extending earlier findings that fall within certain subclasses of processes involving near-integrated time series. The limit results relate to tempered fractional processes that include tempered fractional Brownian motion and tempered fractional diffusions. The theory is extended to provide the limiting distribution for autoregressions with such tempered near-integrated time series, thereby enabling analysis of the limit properties of statistics of particular interest in econometrics, such as unit root tests, under more general conditions than existing theory. Some extensions of the theory to the multivariate case are reported.

An Overview of Modified Semiparametric Memory Estimation Methods

Several modified estimation methods of the memory parameter have been introduced in the past years. They aim to decrease the upward bias of the memory parameter in cases of low frequency contaminations or an additive noise component, especially in situations with a short-memory process being contaminated. In this paper, we provide an overview and compare the performance of nine semiparametric estimation methods. Among them are two standard methods, four modified approaches to account for low frequency contaminations and three procedures developed for perturbed fractional processes. We conduct an extensive Monte Carlo study for a variety of parameter constellations and several DGPs. Furthermore, an empirical application of the log-absolute return series of the S&P 500 shows that the estimation results combined with a long-memory test indicate a spurious long-memory process.

Model order selection in periodic long memory models

An automatic model order selection procedure for k-factor Gegenbauer processes is proposed. The procedure is based on sequential tests of the maximum of the periodogram and semiparametric estimators of the model parameters. As a byproduct, a generalized version of Walker’s large sample g-test is introduced that allows to test for persistent periodicity in stationary short memory processes. Simulation studies show that the model order selection procedure performs well in identifying the correct order under various circumstances. An application to Californian electricity load data illustrates its value in empirical analyses and allows new insights into the periodicity of this process that has been the subject of several studies.

Harmonically Weighted Processes: The Case of U.S. Inflation

We suggest a model for long memory in time series that amounts to harmonically weighting short memory processes, ∑ j x t − j / ( j 1). A nonstandard rate of convergence is required to establish a Gaussian functional central limit theorem. Further, we study the asymptotic least squares theory when harmonically weighted processes are regressed on each other. The regression estimators converge to Gaussian limits upon the conventional normalization with square root of the sample size, and standard testing procedures apply. Harmonically weighted processes do not allow - or require - to choose a memory parameter. Nevertheless, they may well be able to capture dynamics that have been modelled by fractional integration in the past, and the conceptual simplicity of the new model may turn out to be a worthwhile advantage in practice. The harmonic inverse transformation that removes this kind of long memory is also developed. We successfully apply the procedure to monthly U.S. inflation, and provide simulation evidence that fractional integration of order d is well captured by harmonic weighting over a relevant range of d in finite samples.

On the persistence of prices in Mexico: a fractional integration approach

ABSTRACTA relevant yet often overlooked characteristic of the inflation rate is its mean-reverting property. If a series has this feature, shocks eventually dissipate, whereas if it does not, they have a permanent effect on the series. The usual I(1) versus I(0) dichotomy in time-series econometrics goes only so far towards disentangling this issue. By employing a methodology that estimates the persistence of inflation by allowing (i) fractional integration and (ii) persistence and level shifts in the series, we aim to define whether it is stationary and/or mean reverting and, if so, during which periods. The results of our analysis for the period 1987–2015 are threefold: firstly, inflation in the eighties and nineties should be seen as a highly persistent yet mean-reverting process (not a random walk); secondly, inflation remained mean reverting, though became a short-memory (less persistent) process around the date of the implementation of the inflation-targeting framework of 2001; thirdly, during the later phase, the level of inflation also decreased and is now within the inflation target range set by Banco de México, namely 3 per cent with an interval of ±1 percentage point.

Long-Memory and Spurious Breaks in Ecological Experiments

The impact of long-memory on the Before-After-Control-Impact (BACI) design and a commonly used nonparametric alternative, Randomized Intervention Analysis (RIA), is examined. It is shown the corrections used based on short-memory processes are not adequate. Long-memory series are also known to exhibit spurious structural breaks that can be mistakenly attributed to an intervention. Two examples from the literature are used as illustrations.

Modelling exchange rate volatility with random level shifts

ABSTRACTRecent literature has shown that the volatility of exchange rate returns displays long memory features. It has also been shown that if a short memory process is contaminated by level shifts, the estimate of the long memory parameter tends to be upward biased. In this article, we directly estimate a random level shift model to the logarithm of the absolute returns of five exchange rates series, in order to assess whether random level shifts (RLSs) can explain this long memory property. Our results show that there are few level shifts for the five series, but once they are taken into account the long memory property of the series disappears. We also provide out-of-sample forecasting comparisons, which show that, in most cases, the RLS model outperforms popular models in forecasting volatility. We further support our results using a variety of robustness checks.

Cross-response in correlated financial markets: individual stocks

Previous studies of the stock price response to trades focused on the dynamics of single stocks, i.e. they addressed the self-response. We empirically investigate the price response of one stock to the trades of other stocks in a correlated market, i.e. the cross-responses. How large is the impact of one stock on others and vice versa? -- This impact of trades on the price change across stocks appears to be transient instead of permanent as we discuss from the viewpoint of market efficiency. Furthermore, we compare the self-responses on different scales and the self- and cross-responses on the same scale. We also find that the cross-correlation of the trade signs turns out to be a short-memory process.

Spectral Characteristics of Instrumental and Climate Model Surface Temperatures

Abstract The spatiotemporal temperature variability for several gridded instrumental and general circulation climate model data is characterized, contrasting power spectra of local and global temperatures, land and sea temperatures, and temperatures of different regions. There is generally a high degree of agreement between the spectral characteristics of instrumental and climate model data. All but the equatorial spectra exhibit a power-law shape and are hence more consistent with the spectra expected from long-memory processes than from short-memory processes. The power-law exponent β of the spectra is a measure of memory, or persistence, of the temperatures and is observed to be about twice as large for global temperature than for local temperatures. However, there are large variations, in particular between land and sea surface temperatures. This is shown by estimates of the spectra for different regions and global maps of β. It is also demonstrated that global spectra are related to local spectra via teleconnections between local temperatures.

Cross-Response in Correlated Financial Markets: Individual Stocks

Previous studies of the stock price response to trades focused on the dynamics of single stocks, i.e. they addressed the self-response. We empirically investigate the price response of one stock to the trades of other stocks in a correlated market, i.e. the cross-responses. How large is the impact of one stock on others and vice versa? This impact of trades on the price change across stocks appears to be transient instead of permanent as we discuss from the viewpoint of market efficiency. Furthermore, we compare the self-responses on different scales and the self- and cross-responses on the same scale. We also find that the cross-correlation of the trade signs turns out to be a short-memory process.

On the estimation of short memory components in long memory time series models

AbstractA substantial amount of recent time series research has emphasized semi-parameteric estimators of a long memory parameter and we provide a selective review of the literature on this issue. We consider such estimators applied to the issue of estimating the parameters relating to a short memory process which is embedded within the long memory process. We consider the fractional differencing filter and the subsequent properties of a two step estimator of the short memory parameters. We conclude that while the semi-parametric estimators can have excellent properties in terms of estimating the long memory parameter, they do not have good properties when applied to the two step estimator of short memory

Exponential decay rate of partial autocorrelation coefficients of ARMA and short-memory processes

We present a short proof of the fact that the exponential decay rate of partial autocorrelation coefficients of a short-memory process, in particular an ARMA process, is equal to the exponential decay rate of the coefficients of its infinite autoregressive representation.

Does the Forward Discount Represent a Long Memory Process or Short Memory Process with Multiple Changes in the Mean?

Summary This study aims to determine whether forward discount represents a long memory or short memory process with multiple changes in the mean. Based on the samples of six currencies from November 3, 1986, to March 6, 1998, using Baek and Pipiras’s (2012, 2014) statistical procedures, our findings suggest that forward discount is a short memory process with multiple changes in the mean rather than long memory. These changes in mean are the result of an intervention by monetary authorities in the forex market. Thus, earlier findings of long memory in forward discount, as reported in the extant literature, are questionable.

An application of a random level shifts model to the volatility of Peruvian stock and exchange rate returns

The literature has shown that the volatility of stock and forex rate market returns shows the characteristic of long memory. Another fact that is shown in the literature is that this feature may be spurious and volatility actually consists of a short memory process contaminated with random level shifts (RLS). In this paper, we follow recent econometric approaches estimating an RLS model to the logarithm of the absolute value of stock and forex returns. The model consists of the sum of a short-term memory component and a component of level shifts. The second component is specified as the cumulative sum of a process that is zero with probability ‘1-alpha’ and is a random variable with probability ‘alpha’. The results show that there are level shifts that are rare, but once they are taken into account, the characteristic or property of long memory disappears. Also, the presence of General Autoregressive Conditional Heteroscedasticity (GARCH) effects is eliminated when included or deducted level shifts. An exercise of out-of-sample forecasting shows that the RLS model has better performance than traditional models for modelling long memory such as the models ARFIMA (p,d,q).

Optimal eigen expansions and uniform bounds

Let \(\{X_k\}_{k \in \mathbb {Z}} \in {\mathbb {L}}^2({\mathcal {T}})\) be a stationary process with associated lag operators \({\varvec{\mathcal {C}}}_h\). Uniform asymptotic expansions of the corresponding empirical eigenvalues and eigenfunctions are established under almost optimal conditions on the lag operators in terms of the eigenvalues (spectral gap). In addition, the underlying dependence assumptions are optimal in a certain sense, including both short and long memory processes. This allows us to study the relative maximum deviation of the empirical eigenvalues under very general conditions. Among other things, convergence to an extreme value distribution is shown. We also discuss how the asymptotic expansions transfer to the long-run covariance operator \({\varvec{\mathcal {G}}}\) in a general framework.

Evolution of low flows in Czechia revisited

Abstract. Although a nationwide study focusing on the evolution of low flows in Czechia was conducted in the past, a need for the revision of the results has arisen. By means of the trend analysis, which specifically considers the presence of significant serial correlation at the first lag, the former study highlighted areas where 7-day low flows increase or decrease. However, taking into account only the lag-one autoregressive process might still have led to the detection of so-called pseudo-trends because, besides short-term persistence, also long-term persistence may adversely influence the variance of the test statistic when the independence among data is required. Therefore, one should carefully investigate the presence of persistence in time series. Before the trend analysis itself, the authors' previous studies aimed at the discrimination between short memory processes and long memory processes employing jointly the Phillips–Perron test and the Kwiatkowski–Phillips–Schmidt–Shin test. This analysis was accompanied by the Hurst exponent estimation. Here, the subsequent identification of trends is carried out using three modifications of the Mann–Kendall test that allow different kinds of persistence. These include the Bayley–Hammersley–Matalas–Langbein–Lettenmaier equivalent sample size approach, the trend-free pre-whitening approach and a block bootstrap with automatic selection of the block length, which was applied for the first time in hydrology. The general results are similar to those presented in the former study on trends. Nevertheless, the divergent minimum discharges evolution in the western part of Czechia is now much clear. Moreover, no significant increasing trend in series incorporating Julian days was found.

Forecasting return volatility: Level shifts with varying jump probability and mean reversion

We extend the random level shift (RLS) model of Lu and Perron (2010) to the volatility of asset prices, which consists of a short memory process and a random level shift component. Motivated by empirical features, (a) we specify a time-varying probability of shifts as a function of large negative lagged returns; and (b) we incorporate a mean reverting mechanism so that the sign and magnitude of the jump component change according to the deviations of past jumps from their long run mean. This allows the possibility of forecasting the sign and magnitude of the jumps. We estimate the model using twelve different series, and compare its forecasting performance with those of a variety of competing models at various horizons. A striking feature is that the modified RLS model has the smallest mean square forecast errors in 64 of the 72 cases, while it is a close second for the other 8 cases. The improvement in forecast accuracy is often substantial, especially for medium- to long-horizon forecasts. This is strong evidence that our modified RLS model offers important gains in forecasting performance.

Marked empirical processes for non-stationary time series

Consider a first-order autoregressive process $X_{i}=\beta X_{i-1}+\varepsilon_{i}$, where $\varepsilon_{i}=G(\eta_{i},\eta_{i-1},\ldots)$ and $\eta_{i}$, $i\in\mathbb{Z}$ are i.i.d. random variables. Motivated by two important issues for the inference of this model, namely, the quantile inference for $H_{0}\colon\ \beta=1$, and the goodness-of-fit for the unit root model, the notion of the marked empirical process $\alpha_{n}(x)=\frac{1}{n}\sum_{i=1}^{n}g(X_{i}/a_{n})I(\varepsilon_{i}\leq x)$, $x\in\mathbb{R}$ is investigated in this paper. Herein, $g(\cdot)$ is a continuous function on $\mathbb{R}$ and $\{a_{n}\}$ is a sequence of self-normalizing constants. As the innovation $\{\varepsilon_{i}\}$ is usually not observable, the residual marked empirical process $\hat{\alpha}_{n}(x)=\frac{1}{n}\sum_{i=1}^{n}g(X_{i}/a_{n})I(\hat{\varepsilon}_{i}\leq x)$, $x\in\mathbb{R}$, is considered instead, where $\hat{\varepsilon}_{i}=X_{i}-\hat{\beta}X_{i-1}$ and $\hat{\beta}$ is a consistent estimate of $\beta$. In particular, via the martingale decomposition of stationary process and the stochastic integral result of Jakubowski (Ann. Probab. 24 (1996) 2141–2153), the limit distributions of $\alpha_{n}(x)$ and $\hat{\alpha}_{n}(x)$ are established when $\{\varepsilon_{i}\}$ is a short-memory process. Furthermore, by virtue of the results of Wu (Bernoulli 95 (2003) 809–831) and Ho and Hsing (Ann. Statist. 24 (1996) 992–1024) of empirical process and the integral result of Mikosch and Norvaiša (Bernoulli 6 (2000) 401–434) and Young (Acta Math. 67 (1936) 251–282), the limit distributions of $\alpha_{n}(x)$ and $\hat{\alpha}_{n}(x)$ are also derived when $\{\varepsilon_{i}\}$ is a long-memory process.

On asymptotic distributions of weighted sums of periodograms

We establish asymptotic normality of weighted sums of periodograms of a stationary linear process where weights depend on the sample size. Such sums appear in numerous statistical applications and can be regarded as a discretized versions of quadratic forms involving integrals of weighted periodograms. Conditions for asymptotic normality of these weighted sums are simple, minimal, and resemble Lindeberg-Feller condition for weighted sums of independent and identically distributed random variables. Our results are applicable to a large class of short, long or negative memory processes. The proof is based on sharp bounds derived for Bartlett type approximation of these sums by the corresponding sums of weighted periodograms of independent and identically distributed random variables.

The Sensitivity of Detrended Long-Memory Processes

Long-memory tests are often complicated by the presence of deterministic trends. Hence, an additional step of detrending the data is necessary. The typical way to detrend a suspected long-memory series is to use OLS or BSP residuals. Applying the method of sensitivity analysis we address the of question of how robust these residuals are in presence of potential long memory components. Unlike short-memory ARMA process long-memory I(d) processes causes sensitivity to OLS/BSP residuals. Therefore, we develop a finite sample measure of the sensitivity of a detrended series based on the residuals. Based on our sensitivity measure we propose a “rule of thumb” for practitioners to choose between the two methods of detrending, has been provided in this article.