Econometric Theory

In a series of recent publications of the author, three interpolation procedures, denoted IMPE, IMMPE, and ITEA, were proposed for vector-valued functions F (z), where F : C → C N , and their algebraic properties were studied. The convergence studies of two of the methods, namely, IMPE and IMMPE, were also carried out as these methods are being applied to meromorphic functions with simple poles, and de Montessus and König type theorems for them were proved. In the present work, we concentrate on ITEA. We study its convergence properties as it is applied to meromorphic functions with simple poles, and prove de Montessus and König type theorems analogous to those obtained for IMPE and IMMPE.

In this paper we show that product type designs are optimal in partially heteroscedastic multi-factor linear models. This result is applied to obtain lo- cally D-optimal designs in multi-factor generalized linear models by means of a canonical transformation. As a consequence we can construct optimal designs for logistic response in paired comparison experiments which are close to the concept of utility balance.

This paper determines bounds on the asymptotic orders of the coverage probability errors of parametric bootstrap confidence intervals (CIs) and tests for the covariance parameters of a time series generated by a regression model with Gaussian, stationary, and strongly dependent errors. The CIs and tests are based on the plug-in Whittle maximum likelihood (PWML) estimators. It is shown that, under some sets of conditions on the regression coefficients, the spectral density function, and the parameter values, the bounds on the coverage probability errors of symmetric two-sided and one-sided parametric bootstrap confidence intervals on the plug-in Whittle log-likelihood function are shown to be O(n^{-3/2}\ln{n}) and O(n^{-1}\ln{n}), respectively. Apart from the \ln{n} term, the magnitudes of the coverage probability errors of the one-sided bootstrap confidence intervals for our model is shown to be essentially the same as that of the independent and identically distributed (iid) data. ...

In this paper, we propose methods of the determination of the rank of matrix. We consider a rank test for an unobserved matrix for which an estimate exists having normal asymptotic distribution of order N 1/2 where N is the sample size. The test statistic is based on the smallest estimated singular values. Using Matrix Perturbation Theory, the smallest singular values of random matrix converge asymptotically to zero in the order O(N 1 ) and the corresponding left and right singular vectors converge asymptotically in the order O(N 1/2 ). Moreover, the asymptotic distribution of the test statistic is seen to be chi-squared. The test has advantages over standard tests in being easier to compute. Two approaches to be considered are sequential testing strategy and information theoretic criterion. We establish a strongly consistent of the determination of the rank of matrix using both approaches. Some economic applications are discussed and simulation evidence is given We are grateful for the comments of D. Belsley, M-C. Pichery.

The paper presents the asymptotic theory of the efficient method of moments when the model of interest is not correctly specified+ The paper assumes a sequence of independent and identically distributed observations and a global misspecifica-tion+ It is found that the limiting distribution of the estimator is still asymptotically normal, but it suffers a strong impact in the covariance matrix+ A consistent estimator of this covariance matrix is provided+ The large sample distribution on the estimated moment function is also obtained+ These results are used to discuss the situation when the moment conditions hold but the model is misspecified+ It also is shown that the overidentifying restrictions test has asymptotic power one whenever the limit moment function is different from zero+ It is also proved that the bootstrap distributions converge almost surely to the previously mentioned distributions and hence they could be used as an alternative to draw inferences under misspecification+ Interestingly, it is also shown that bootstrap can be reliably applied even if the number of bootstrap replications is very small+ The authors express their gratitude to Professor A+R+ Gallant for his support and for helpful discussions about this topic+ They also thank Professor Donald Andrews for his revision and for the suggestion to read the work by Hall and Inoue ~2003!+ They thank Hall and Inoue for sending them their technical report+ Finally the authors thank two anonymous referees for their helpful comments, which led to a significant improvement of the paper+ The views expressed in this paper are our sole responsibility+ Víctor Aguirre-Torres acknowledges partial support from the Asociación Mexicana de la Cultura A+C+ Manuel A+ Domínguez acknowledges funding from CONACYT, research project J38076-D, and partial support from Asociación Mexicana de la Cultura

This paper proposes a notion of near cointegration and generalizes several existing results from the cointegration literature to the case of near cointegration. In particular, the properties of conventional cointegration methods under near cointegration are characterized, thereby investigating the robustness of cointegration methods. In addition, we obtain local asymptotic power functions of five cointegration tests that take cointegration as the null hypothesis.

We show that the order of integration of a vector autoregressive process is equal to the difference between the multiplicity of the unit root in the characteristic equation and the multiplicity of the unit root in the adjoint matrix polynomial. The equivalence with the standard I(1) and I( ) conditions is proved and polynomial cointegration discussed in the general setup.

In the following paper we analyze the strategic competition between fast and slow traders. The model of Kyle (1985) is adapted to analyze the effect of speed in such a model. A High Frequency Trader (HFT) is defined as a trader that has the ability to react to information faster than other informed traders and as a consequence can trade more than other traders. This trader benefits from low latency compared to slower trader. In such a setting, we prove the existence and the unicity of an equilibrium with fast and slow traders. We find that the speed advantage of High Frequency Traders (HFTs) has a beneficial effect on market liquidity as well as price efficiency. The positive effect on liquidity is present only if there are 2 or more HFTs. However, despite those effects slower traders are at a disadvantage as they are not able to trade on their private information as many times as their HFTs counterpart . When they are able to trade on it, most of their private information has been incorporated into prices due to the latency of HFTs. This implies that slower traders are worse off when HFTs are present. The speed differential benefits HFTs as they earn higher expected profits than their slower counterparts and also benefits liquidity traders. We find the existence of an optimal level of speed for HFT.

Non technical summary 1 Introduction 2 The model 3 Identification of the structural shocks 3.1 Response heterogeneity, n large and fundamentalness 3.2 Economic conditions for shocks identification 4 Estimation 4.1 Population formulas 4.2 Estimators 4.3 Consistency 4.4 Standard errors and confidence bands 5 Empirical application 5.1 The data 5.2 The choice of r and the number of common shocks 5.3 Fundamentalness 5.4 Impulse-response functions and variance decomposition 6 Conclusions Appendix 1: Proof of Proposition 3 References European Central Bank Working Paper Series

Economic data are frequently generated by stochastic processes that can be modeled as realizations of random functions (functional data). This paper adapts the specification test for functional data developed by Bugni, Hall, Horowitz, and Neumann (2009, Econometrics Journal12, S1–S18) to the presence of missing observations. By using a worst case scenario approach, our method is able to extract the information available in the observed portion of the data while being agnostic about the nature of the missing observations. The presence of missing data implies that our test will not only result in the rejection or lack of rejection of the null hypothesis, but it may also be inconclusive.Under the null hypothesis, our specification test will reject the null hypothesis with a probability that, in the limit, does not exceed the significance level of the test. Moreover, the power of the test converges to one whenever the distribution of the observations conveys that the null hypothesis is ...

Statistical inference for unknown distributions of statistics or estimators may be based on asymptotic distributions. Unfortunately, in the case of dependent data the structure of such statistical procedures is often ineffective. There are different reasons that, e.g. too small a number of observations, the unknown form of the asymptotic distribution or too slow convergence to the asymptotic distribution. In the last three decades we can observe an intensive development the of so-called resampling methods. Using these methods, it is possible to directly approximate the unknown distributions of statistics and estimators. The idea of resampling is simple, i.e. we calculate the estimator replications and the replications determine the empirical distribution called resampling distribution. A problem that needs to be solved during the study of the resampling procedures is their consistency, i.e. whether the resampling distribution is close enough to the true one? There are many resampling methods. Their consistency for independent observations has been extensively studied. The case of the stationary data with strong mixing dependence structure has also been well investigated. Resampling for time series with a specific non-stationarity, i.e. the periodic and almost periodic also been the subject of research. Recent research on resampling methods focus mainly on the time series with the weak dependency structure, defined by Paul Doukhan. The thesis presents a time series model with specific features i.e.: long memory, heavy tails (stable or GED) and periodic structure. Such a model can be naturally used in many areas like energy, vibromechanics, telecommunications, climatology and economics. The objective of this thesis is to present several consistency theorems for the resampling method for the estimator of the mean function in the abovementioned time series. Only one of the resampling techniques can be used for the long-range dependent data. This method is subsampling. It involves selecting from the observation all possible subsequences of a some length and calculate the estimator on these subsequences. In the thesis,we introduce and prove theorems that are necessary to establish consistency of resampling. Moreover, a brief overview of the previous results in inference for non-stationary time series is presented.

Statistical inference for unknown distributions of statistics or estimators may be based on asymptotic distributions. Unfortunately, in the case of dependent data the structure of such statistical procedures is often ineffective. There are different reasons that, e.g. too small a number of observations, the unknown form of the asymptotic distribution or too slow convergence to the asymptotic distribution. In the last three decades we can observe an intensive development the of so-called resampling methods. Using these methods, it is possible to directly approximate the unknown distributions of statistics and estimators. The idea of resampling is simple, i.e. we calculate the estimator replications and the replications determine the empirical distribution called resampling distribution. A problem that needs to be solved during the study of the resampling procedures is their consistency, i.e. whether the resampling distribution is close enough to the true one? There are many resampling methods. Their consistency for independent observations has been extensively studied. The case of the stationary data with strong mixing dependence structure has also been well investigated. Resampling for time series with a specific non-stationarity, i.e. the periodic and almost periodic also been the subject of research. Recent research on resampling methods focus mainly on the time series with the weak dependency structure, defined by Paul Doukhan. The thesis presents a time series model with specific features i.e.: long memory, heavy tails (stable or GED) and periodic structure. Such a model can be naturally used in many areas like energy, vibromechanics, telecommunications, climatology and economics. The objective of this thesis is to present several consistency theorems for the resampling method for the estimator of the mean function in the abovementioned time series. Only one of the resampling techniques can be used for the long-range dependent data. This method is subsampling. It involves selecting from the observation all possible subsequences of a some length and calculate the estimator on these subsequences. In the thesis,we introduce and prove theorems that are necessary to establish consistency of resampling. Moreover, a brief overview of the previous results in inference for non-stationary time series is presented.

Forecasts from a univariate autoregressive model estimated by OLS are unbiased, irrespective of whether the model fitted has the correct order; this property only requires symmetry of the distribution of the innovations. In this paper, this result is generalized to vector autoregressions and a wide class of multivariate stochastic processes (which include Gaussian stationary multivariate stochastic processes) is described for which unbiasedness of predictions holds: specifically, if a vector autoregression of arbitrary finite order is fitted to a sample from any process in this class, the fitted model will produce unbiased forecasts, in the sense that the prediction errors have distributions symmetric about zero. Different numbers of lags may be used for each variable in each autoregression and variables may even be missing, without unbiasedness being affected. This property is exact in finite samples. Similarly, the residuals from the same autoregressions have distributions symmetric about zero. 1 .

We study two Durbin-Watson type tests for serial correlation of errors in regression models when observations are missing. We derive them by applying standard methods used in time series and linear models to deal with missing observations. The first test may be viewed as a regular Durbin-Watson test in the context of an extended model. We discuss appropriate adjustments that allow one to use all available bounds tables. We show that the test is locally most powerful invariant against the same alternative error distribution as the Durbin-Watson test. The second test is based on a modified Durbin-Watson statistic suggested by Ring (1981a) and is locally most powerful invariant against a first-order autoregressive process.

This paper provides a simple method to estimate both univariate and multivariate MA processes. Similar to Durbin's method, it rests on the recursive relation between the parameters of the MA process and those of its AR representation. This recursive relation is shown to be valid both for invertible / stable and non invertible / unstable processes under the assumption that the process has no constant and started from zero. This makes the method suitable for unit root processes, too.

This paper provides a simple method to estimate both univariate and multivariate MA processes. Similar to Durbin's method, it rests on the recursive relation between the parameters of the MA process and those of its AR representation. This recursive relation is shown to be valid both for invertible / stable and non invertible / unstable processes under the assumption that the process has no constant and started from zero. This makes the method suitable for unit root processes, too.

This paper provides a simple method to estimate both univariate and multivariate MA processes. Similar to Durbin's method, it rests on the recursive relation between the parameters of the MA process and those of its AR representation. This recursive relation is shown to be valid both for invertible / stable and non invertible / unstable processes under the assumption that the process has no constant and started from zero. This makes the method suitable for unit root processes, too.

This paper provides a simple method to estimate both univariate and multivariate MA processes. Similar to Durbin's method, it rests on the recursive relation between the parameters of the MA process and those of its AR representation. This recursive relation is shown to be valid both for invertible / stable and non invertible / unstable processes under the assumption that the process has no constant and started from zero. This makes the method suitable for unit root processes, too.

In this paper we provide an asymptotic theoretical power comparison in the Bahadur sense, between the portmanteau and Breusch-Godfrey Lagrange Multiplier (LM) tests for the goodness-of-fit checking of vector autoregressive (VAR) models. The merits and the drawbacks of the studied tests are illustrated using Monte Carlo experiments.

An interval-valued observation in a time period contains more information than a point-valued observation in the same time period. Examples of interval data include the maximum and minimum temperatures in a day, the maximum and minimum GDP growth rates in a year, the maximum and minimum asset prices in a trading day, the bid and ask prices in a trading period, the long term and short term interests, and the 90%-tile and 10%-tile incomes of a cohort in a year, etc. Interval forecasts may be of direct interest in practice, as it contains information on the range of variation and the level or trend of economic processes. Moreover, the informational advantage of interval data can be exploited for more e¢ cient econometric estimation and inference. We propose a new class of autoregressive conditional interval (ACI) models for interval-valued time series data. A minimum distance estimation method is proposed to estimate the parameters of an ACI model, and the consistency, asymptotic normality and asymptotic e¢ ciency of the proposed estimator are established. It is shown that a two-stage minimum distance estimator is asymptotically most e¢ cient among a class of minimum distance estimators, and it achieves the Cramer-Rao lower bound when the left and right bounds of the interval innovation process follow a bivariate normal distribution. Simulation studies show that the two-stage minimum distance estimator outperform conditional least squares estimators based on the ranges and/or midpoints of the interval sample, as well as the conditional quasi-maximum likelihood estimator based on the bivariate left and right bound information of the interval sample. In an empirical study on asset pricing, we document that when return interval data is used, some bond market factors, particularly the default risk factor, are signi…cant in explaining excess stock returns, even after the stock market factors are controlled in regressions. This di¤ers from the previous …ndings (e.g., ) in the literature.

proposed an efficient method to estimate the conditional variance of heteroskedastic regression models. Chen, Cheng, and Peng (2009) applied variance reduction techniques to the estimator of Fan and Yao (1998) and proposed a new estimator for conditional variance to account for the skewness of financial data. In this paper, we apply empirical likelihood methods to construct confidence intervals for the conditional variance based on the estimator of Fan and Yao (1998) and the reduced variance modification of . Simulation studies and data analysis demonstrate the advantage of the empirical likelihood method over the normal approximation method. We are grateful to two referees and Prof. Kitamura, one of the co-editors, for their constructive comments.

The article discusses statistical inference in parametric models for panel data. The models feature dynamics of a general nature, individual effects, and possible explanatory variables. The focus is on large-cross-section inference on Gaussian pseudo maximum likelihood estimates with temporal dimension kept fixed, partially complementing and extending recent work of the authors. We focus on a particular kind of initial condition but go on to discuss implications of alternative initial conditions. Some possible further developments are briefly reviewed.

We propose tests of the null of spurious relationship against the alternative of fractional cointegration among the components of a vector of fractionally integrated time series+ Our test statistics have an asymptotic chi-square distribution under the null and rely on generalized least squares-type of corrections that control for the short-run correlation of the weak dependent components of the fractionally integrated processes+ We emphasize corrections based on nonparametric modelization of the innovations' autocorrelation, relaxing important conditions that are standard in the literature and, in particular, being able to consider simultaneously ~asymptotically! stationary or nonstationary processes+ Relatively weak conditions on the corresponding short-run and memory parameter estimates are assumed+ The new tests are consistent with a divergence rate that, in most of the cases, as we show in a simple situation, depends on the cointegration degree+ Finite-sample properties of the tests are analyzed by means of a Monte Carlo experiment+

This paper proposes bootstrap assisted specification tests for the autoregressive fractionally integrated moving average model based on the Bartlett Tp-process with estimated parameters whose limiting distribution under the null depends on the estimated model and the estimation method employed. The computation of the asymptotic critical values is not easy if at all possible under these circumstances. To circumvent this problem Delgado, Hidalgo, and Velasco (2005, Annals of Statistics 33, 2568–2609) proposed an asymptotically pivotal transformation of the Tp-process with estimated parameters. The aim of this paper is twofold. First, to examine alternative methods based on bootstrap algorithms for estimating the distribution of the test under the null, showing its validity. And second, to study the finite-sample performance of the different alternative procedures via Monte Carlo simulation.

The construction of asymptotically distribution free time series model specification tests using as statistics the estimated residual autocorrelations is considered from a general view point. We focus our attention on Box-Pierce type tests based on the sum of squares of a few estimated residual autocorrelations. This type of tests belongs to the class defined by quadratic forms of weighted residual autocorrelations, where weights are suitably transformed resulting in asymptotically distribution free tests. The weights can be optimally chosen to maximize the power function when testing in the direction of local alternatives. The optimal test in this class against MA, AR or Bloomfield alternatives is a Box-Pierce type test based on the sum of squares of a few transformed residual autocorrelations. Such transformations are, in fact, the recursive residuals in the projection of the residual autocorrelations on a certain score function.

This paper analyzes the classical linear regression model with measurement errors in all the variables. First, we provide necessary and sufficient conditions for identification of the coefficients. We show that the coefficients are not identified if and only if an independent normally distributed linear combination of regressors can be transferred from the regressors to the errors. Second, we introduce a new estimator for the coefficients using a continuum of moments that are based on second derivatives of the log characteristic function of the observables. In Monte Carlo simulations, the estimator performs well and is robust to the amount of measurement error and number of mismeasured regressors. In an application to firm investment decisions, the estimates are similar to those produced by a generalized method of moments estimator based on third to fifth moments.

This paper identifies and estimates the coefficients in a multivariate errors-invariables linear model when the unobserved arbitrarily dependent regressors are not jointly normal and independent of errors. To identify the coefficients, we use variation in the second-order partial derivatives of the log characteristic function of the unobserved regressors; a property of only not jointly normal distributions. A root-n consistent and asymptotically normal extremum estimator performs well in simulations relative to third and fourth order moment estimators.

The study empirically assesses the impact of monetary policy on economic growth in Nigeria.
The study employed time series data ranging from 1982 to 2023. Economic growth is
expressed by Real Gross Domestic Product (RGDP) with interest rate, exchange rate, money
supply and foreign direct investment serving as monetary policy variables or instruments.
The study adopted the ex post facto research design due to its unique quality of being able to
examine the cause –and- effect relationship between dependent variables and regressors. The
method of data analysis was Auto regressive distributed lag (ARDL) model. The empirical
result shown, while foreign direct investment and exchange rate have positive and significant
effect on real GDP, money supply (M2) has negative significant impact on real GDP and
interest rate was found to be positive and insignificant to real GDP. The study concludes that
monetary policy has positive impact on economic growth in Nigerian during the reviewed
period. Owing to the findings, the study recommends that monetary authority should
implement policy that will enhance foreign direct investment and optimize exchange rate
management in the country.

The research is aimed at exploring Foreign Direct Investment and its impact on economic growth of Nigeria. The study covers 31-year period between 1985-2016. Simple ordinary least-square regression model is used to measure the effects and relationships between the independent variable and the dependent variable using E-views 9.0. Foreign Direct Investment (FDI) serve as the independent variable while economic growth as the dependent variable. GDP, exchange rate, inflation rate, unemployment rate, total savings and interest rate were used as proxies for economic growth. Data on FDI, GDP, exchange rate, unemployment rate, savings and interest rate were retrieved from the CBN Annual Statistical Bulletin, World bank Report and National Bureau of Statistics. The stationarity property of a time series data can be examined by conducting unit root test in order to ascertain the stationarity or otherwise of the series variables (Akinola,2016). Augmented Dickey-Fuller (ADF) test due to Dickey and Fuller (1979, 1981), and the Phillip-Perron (PP) due to Phillips (1987) and Phillips and Perron (1988) were used to ensure the stationarity of the time series data i.e dependent and independent variable. The finding showed that there is a strong and positive relationship between FDI and economic growth in Nigeria. The government of Nigeria must put all hands-on desk, formulating policies and necessary reforms to ensure that foreign direct investments are attracted to benefit the populace at large. It also recommended that Institutionalized corruption both in private and public sectors must be fought, if the nation must attract FDI, we must change our ways of doing things.

This paper proposes a robust moment selection method aiming to pick the best model even if this is a moment condition model with mixed identification strength, that is, moment conditions including moment functions that are local to zero uniformly over the parameter set. We show that the relevant moment selection procedure of Hall et al. (2007, Journal of Econometrics 138, 488–512) is inconsistent in this setting as it does not explicitly account for the rate of convergence of parameter estimation of the candidate models which may vary. We introduce a new moment selection procedure based on a criterion that automatically accounts for both the convergence rate of the candidate model’s parameter estimate and the entropy of the estimator’s asymptotic distribution. The benchmark estimator that we consider is the two-step efficient generalized method of moments estimator, which is known to be efficient in this framework as well. A family of penalization functions is introduced that guaran...

We investigate the validity of the standard specification tests for assessing the exogeneity of subvectors in the linear IV regression. Our results show that ignoring the endogeneity of the regressors whose exogeneity is not being tested leads to invalid tests (level is not controlled). When the fitted values from the first stage regression of these regressors are used as instruments under the partial null hypothesis of interest, as suggested Hausman and Taylor (1980, 1981), some versions of these tests are invalid when identification is weak and the number of instruments is moderate. However, all tests are overly conservative and have no power when the number of instruments increases, even for moderate identification strength.

Dans cette these, nous etudions des modeles de moyennes conditionnelles de series temporelles a valeurs entieres. Tout d’abord, nous proposons l’estimateur de quasi maximum de vraisemblance de Poisson (EQMVP) pour les parametres de la moyenne conditionnelle. Nous montrons que, sous des conditions generales de regularite, cet estimateur est consistant et asymptotiquement normal pour une grande classe de modeles. Etant donne que les parametres de la moyenne conditionnelle de certains modeles sont positivement contraints, comme par exemple dans les modeles INAR (INteger-valued AutoRegressive) et les modeles INGARCH (INteger-valued Generalized AutoRegressive Conditional Heteroscedastic), nous etudions la distribution asymptotique de l’EQMVP lorsque le parametre est sur le bord de l’espace des parametres. En tenant compte de cette derniere situation, nous deduisons deux versions modifiees du test de Wald pour la significativite des parametres et pour la moyenne conditionnelle constante. ...

This paper investigates change point detection within financial time series volatility using the NonParametric AutoRegressive Conditionally Heteroscedastic (NPARCH) process. It introduces a classical CUSUM type change point test following the test procedure by [19]. In the absence of change (Null Hypothesis), the test statistic converges to a well-established distribution, simplifying the determination of the critical test value. Conversely, under the Alternative Hypothesis, the probability that the test statistic tends to infinity is one, indicating asymptotic power one. Simulation results are presented to confirm the efficacy of the proposed methodology.

It is well-known that if the forcing variable of a present value (PV) model is an integrated process, then the model will give rise to a particular cointegrating restriction. In this paper we demostrate that if the PV relation is exact, such that no additive error term appears in the specification, then te variables will be multicointegrated such that the cumlation of cointegration errors at one level of cointegration will cointegrate with the forcing variable. Multicointegration thus delivers a statistical property of the data that is necessary, though not sufficient, for this class of models to be valido Estimation and inference of the model are discussed and it is shown that, provided me PV relation is exact, the discount factor of the model can be estimated with arate of convergence that is faster than the usual super-consistent rate characterising estimators in the cointegration literature. Finally, the paper is completed with two empirical analyses of PV models using term structure data and farmland data, respectively.

This paper analyses the behaviour of a Wald-type test, i.e., the (Efficient) Fractional Dickey-Fuller (EFDF) test of I(1) against I(d), d<1, relative to LM tests. Further, it extends the implementation of the EFDF test to the presence of deterministic trending components in the DGP. Tests of these hypotheses are important in many macroeconomic applications where it is crucial to distinguish between permanent and transitory shocks because shocks die out in I(d) processes with d<1. We show how simple the implementation of the EFDF in these situations is and argue that, under fixed alternatives, it is preferred to the LM test in Bahadur's sense. Finally, an empirical application is provided where the EFDF approach allowing for deterministic components is used to test for long-memory in the GDP p.c. of several OECD countries, an issue that has important consequences to discriminate between alternative growth theories.

In this paper we propose a new nonparametric test for conditional heteroskedasticity based on a measure of nonparametric goodness-of-fit (R2) that is obtained from the local polynomial regression of the residuals from a parametric regression on some covariates. We show that after being appropriately standardized, the nonparametric R2 is asymptotically normally distributed under the null hypothesis and a sequence of Pitman local alternatives. We also prove the consistency of the test and propose a bootstrap method to obtain the bootstrap p-values. We conduct a small set of simulations and compare our test with some popular parametric and nonparametric tests in the literature.

In this article, some models for random replication of character strings are considered that involve random mutations, deletions and insertions of characters. We derive some sufficient conditions on the replication process and the ancestor chain that ensure stationarity and mixing properties of the replicated chain. We also give examples of replication processes which lead to descendant chains not having any mixing properties even if the ancestor chain is i.i.d. in nature. Stationarity and mixing properties are two properties of dependent processes that are of fundamental importance and well studied in the literature. These properties are quite useful in generalizing many asymptotic results for i.i.d. processes to dependent processes and, in many situations, they are useful in justifying statistical estimation and inference based on dependent data. The presence of random deletions and insertions makes our stochastic replication model considerably different from simpler models that involve only mutations, and it leads to some interesting theoretical problems.

We improve, first, a strong invariance principle from for nonconventional sums of the form [Nt] n=1 F X(n), X(2n), ..., X(ℓn) (normalized by 1/ √ N ) where X(n), n ≥ 0's is a sufficiently fast mixing vector process with some moment conditions and stationarity properties and F satisfies some regularity conditions. Applying this result we obtain next a version of the law of iterated logarithm for such sums, as well as an almost sure central limit theorem. Among motivations for such results are their applications to multiple recurrence for stochastic processes and dynamical systems.

This papers offers a theoretical explanation for the stylized fact that forecast combinations with estimated optimal weights often perform poorly in applications. The properties of the forecast combination are typically derived under the assumption that the weights are fixed, while in practice they need to be estimated. If the fact that the weights are random rather than fixed is taken into account during the optimality derivation, then the forecast combination will be biased (even when the original forecasts are unbiased) and its variance is larger than in the fixed-weights case. In particular, there is no guarantee that the 'optimal' forecast combination will be better than the equal-weights case or even improve on the original forecasts. We provide the underlying theory, some special cases and an application in the context of model selection.

We consider the problem of combining forecasts from two different levels ~called "macro" and "micro"!, where we have access to the forecasts and their precisions but not to the full data set+ We develop a theoretical framework and provide Monte Carlo evidence in the cases of both perfect and imperfect aggregation+ Our proposed procedure is simple and robust+ We also extend the procedure to time series and propose a forecast model for the European zero rates, combining quarterly and monthly observations+ We show that forecast accuracy is improved at both levels+ Data are often available at different levels or at different frequencies+ In cross sections, relevant data may be available at a macro level ~industries! and at a micro level ~firms!, where the different levels give rise to different models, estimates, and forecasts+ The models are often estimated by separate offices and statisticians, although they are clearly related to each other+ For example, Marcellino, Stock, and Watson ~2003! forecast aggregate Euro-wide inflation ~macro! and country-specific inflation indexes ~micro!, and Hendry and Hubrich ~2005! study aggregate forecasts of Euro-area inflation ~macro! and five subcomponents: unprocessed food, processed food, industrial goods, energy, and services prices ~micro!+ In the empirical part of this paper we shall consider an application in the time domain: quarterly ~"macro"! and monthly ~"micro"! forecasts for the Euro yield curve+ If all data were available to us at both levels, combining data would lead to efficiency gains+ We shall assume, however, that we have no access to the underlying data but that we do have information on the one-period-ahead forecasts, both at the micro level and at the macro level+ We now wish to combine these forecasts to obtain better forecasts at both levels, taking into account our knowledge of the relationship between the two levels+ In a linear world, the models at the two levels would be analogous, 1 but in a nonlinear world the models could ~and probably should! be quite different+ 2 We are grateful to the editors of this special issue and to two referees for their constructive comments, which greatly helped improve the paper+ Earlier versions of this paper were presented at the NAKE workshop in Amsterdam, at Tilburg University, and at ESEM 2006 in Vienna+ We thank the participants for their useful comments+ In addition, we thank Steffan Berridge, John Einmahl,

We consider the Breitung (2002, Journal of Econometrics 108, 343–363) statistic ξn, which provides a nonparametric test of the I(1) hypothesis. If ξ denotes the limit in distribution of ξn as n → ∞, we prove (Theorem 1) that 0 ≤ ξ ≤ 1/π2, a result that holds under any assumption on the underlying random variables. The result is a special case of a more general result (Theorem 3), which we prove using the so-called cotangent method associated with Cauchy&#39;s residue theorem.

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