Econometric Theory

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.

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.

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.

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.

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.

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...

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.

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.

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's residue theorem.

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A formal statistical test of stationary-ergodicity is developed for known Markovian processes on ℝd This makes it applicable to testing models and algorithms, as well as estimated time series processes ignoring the estimation error. The analysis is conducted by examining the asymptotic properties of the Markov operator on density space generated by the transition in the state space. The test is developed under the null of stationary-ergodicity, and it is shown to be consistent against the alternative of nonstationary-ergodicity. The test can be easily performed using any of a number of standard statistical and mathematical computer packages.

It is known that Efron's resampling bootstrap of the mean of random variables with common distribution in the domain of attraction of the stable laws with infinite variance is not consistent, in the sense that the limiting distribution of the bootstrap mean is not the same as the limiting distribution of the mean from the real sample. Moreover, the limiting distribution of the bootstrap mean is random and unknown. The conventional remedy for this problem, at least asymptotically, is either the m out of n bootstrap or subsampling. However, we show that both these procedures can be quite unreliable in other than very large samples. A parametric bootstrap is derived by considering the distribution of the bootstrap P value instead of that of the bootstrap statistic. The quality of inference based on the parametric bootstrap is examined in a simulation study, and is found to be satisfactory with heavy-tailed distributions unless the tail index is close to 1 and the distribution is heavily skewed.

We study the problem of identifying a forecaster's loss function from observations on forecasts, realizations, and the forecaster's information set. Essentially different loss functions can lead to the same forecasts in all situations, though within the class of all continuous loss functions, this is strongly nongeneric. With the small set of exceptional cases ruled out, generic nonparametric preference recovery is theoretically possible, but identification depends critically on the amount of variation in the conditional distributions of the process being forecast. There exist processes with sufficient variability to guarantee identification, and much of this variation is also necessary for a process to have universal identifying power. We also briefly address the case in which the econometrician does not fully observe the conditional distributions used by the forecaster, and in this context we provide a practically useful set identification result for loss functions used in forecasting binary variables. We thank David Hendry, Ivana Komunjer, Michael McCracken, Ulrich Müller, and two anonymous referees for useful comments.

The paper investigates the relationship between inflation and unemployment in the South African setting, encompassing a comprehensive analysis of historical, theoretical, and empirical evidence. Existing literature surveys demonstrate that several studies have been conducted in multiple countries to ascertain the correlation between inflation and unemployment. The review incorporated the postulations of many economic schools of thought as they are applied to South Africa. The Phillips Curve was examined in accordance with the research purpose. The process involves gathering secondary data from reputable sources such as Statistics South Africa, the South African Reserve Bank, and other reliable secondary sources to analyse this relationship. The findings are presented using linear regression, tables, and figures. The study utilised annual inflation and unemployment data in South Africa spanning from 2000 to 2022 to elucidate the correlation between inflation and unemployment. The study discovered that the link was nonlinear and did not exhibit any meaningful association or correlation.

Many popular estimators for duration models require independent competing risks or independent censoring. In contrast, copula based estimators are also consistent in presence of dependent competing risks. In this paper we suggest a computationally convenient extension of the Copula Graphic Estimator (Zheng and Klein, 1995) to a model with more than two dependent competing risks. We analyse the applicability of this estimator by means of simulations and real world unemployment duration data from Germany. We obtain evidence that our estimator yields nice results if the dependence structure is known and that it is a powerful tool for the assessment of the relevance of (in-)dependence assumptions in applied duration research.

The order of integration is valid to characterize linear processes; but it is not appropriate for non-linear worlds. We propose the concept of summability (a re-scaled partial sum of the process being Op(1)) to handle non-linearities. The paper shows that this new concept, S(): (i) generalizes I(); (ii) measures the degree of persistence as well as of the evolution of the variance; (iii) controls the balancedness of non-linear regressions; (iv) co-summability represents a generalization of co-integration for non-linear processes. To make this concept empirically applicable asymptotic properties of estimation and inference methods for the degree of summability, , are provided. The …nite sample performance of these methods is analyzed via a Monte Carlo experiment. The paper …nishes with the estimation of the degree of summability for the Nelson-Plosser extended database.

Some statistical properties of a vector autoregressive process with Markov-switching coefficients are considered. Sufficient conditions for this nonlinear process to be covariance stationary are given. The second moments of the process are derived under the conditions. The autocovariance matrix decays at exponential rate, permitting the application of the law of large numbers. Under the stationarity conditions, although sharing the “mean-reverting” property with conventional linear stationary processes, the process offers richer short-run dynamics such as conditional heteroskedasticity, asymmetric responses, and occasional nonstationary behavior.

The online meta-learning framework has arisen as a powerful tool for the continual lifelong learning setting. The goal for an agent is to quickly learn new tasks by drawing on prior experience, while it faces with tasks one after another. This formulation involves two levels: outer level which learns meta-learners and inner level which learns task-specific models, with only a small amount of data from the current task. While existing methods provide static regret analysis for the online meta-learning framework, we establish performance in terms of dynamic regret which handles changing environments from a global prospective. We also build off of a generalized version of the adaptive gradient methods that covers both ADAM and ADAGRAD to learn meta-learners in the outer level. We carry out our analyses in a stochastic setting, and in expectation prove a logarithmic local dynamic regret which depends explicitly on the total number of iterations T and parameters of the learner. Apart from, we also indicate high probability bounds on the convergence rates of proposed algorithm with appropriate selection of parameters, which have not been argued before.

Reduced-rank regression is a dimensionality reduction method with many applications. The asymptotic theory for reduced rank estimators of parameter matrices in multivariate linear models has been studied extensively. In contrast, few theoretical results are available for reduced-rank multivariate generalised linear models. We develop M-estimation theory for concave criterion functions that are maximised over parameters spaces that are neither convex nor closed. These results are used to derive the consistency and asymptotic distribution of maximum likelihood estimators in reduced-rank multivariate generalised linear models, when the response and predictor vectors have a joint distribution. We illustrate our results in a real data classification problem with binary covariates.

This paper studies the two-step sieve M estimation of general semi/nonparametric models, where the second step involves sieve estimation of unknown functions that may use the nonparametric estimates from the …rst step as inputs, and the parameters of interest are functionals of unknown functions estimated in both steps. We establish the asymptotic normality of the plug-in two-step sieve M estimate of a functional that could be root-n estimable. The asymptotic variance may not have a closed form expression, but can be approximated by a sieve variance that characterizes the e¤ect of the …rst-step estimation on the second-step estimates. We provide a simple consistent estimate of the sieve variance and hence a Wald type inference based on the Gaussian approximation. The …nite sample performance of the two-step estimator and the proposed inference procedure are investigated in a simulation study.

In this study, we consider preliminary test and shrinkage estimation strategies for quantile regression models. In classical Least Squares Estimation (LSE) method, the relationship between the explanatory and explained variables in the coordinate plane is estimated with a mean regression line. In order to use LSE, there are three main assumptions on the error terms showing white noise process of the regression model, also known as Gauss-Markov Assumptions, must be met: (1) The error terms have zero mean, (2) The variance of the error terms is constant and (3) The covariance between the errors is zero i.e., there is no autocorrelation. However, data in many areas, including econometrics, survival analysis and ecology, etc. does not provide these assumptions. First introduced by Koenker, quantile regression has been used to complement this deficiency of classical regression analysis and to improve the least square estimation. The aim of this study is to improve the performance of quan...

In a classical regression model, it is usually assumed that the explanatory variables are independent of each other and error terms are normally distributed. But when these assumptions are not met, situations like the error terms are not independent or they are not identically distributed or both of these, LSE will not be robust. Hence, quantile regression has been used to complement this deficiency of classical regression analysis and to improve the least square estimation (LSE). In this study, we consider preliminary test and shrinkage estimation strategies for quantile regression models with independently and non-identically distributed (i.ni.d.) errors. A Monte Carlo simulation study is conducted to assess the relative performance of the estimators. Also, we numerically compare their performance with Ridge, Lasso, Elastic Net penalty estimation strategies. A real data example is presented to illustrate the usefulness of the suggested methods. Finally, we obtain the asymptotic re...

This study has empirically investigated the impact of macroeconomic variables on stock
returns in Nigeria. The method of analysis used in this study is Autoregressive Distributed
Lag (ARDL) model. The study spanned from 1987 up to 2022. The objective of the study is
to assess the impact of macroeconomic variables on stock returns in Nigeria. Using
econometrics techniques, a good number of pre-diagnosis tests and post-diagnosis tests were
conducted. The study revealed that macroeconomic variables have significant impact on
stock returns in Nigeria. Therefore, it is recommended that the government should prioritise
the price stability with the view of availing the investors to plan ahead and optimally harness
the market opportunities. The government should also direct and manage interest rate level
with the sole aim of encouraging the investors to keep patronising the securities market and
not dispel them. Exchange rate determination must be managed to strengthen the value of the
naira in relation to foreign currencies. Also, the quantity of money supply to the circulation
should be monitored and must be seen adequate so as not to create unnecessary scarcity of
naira to the detriment of the stock market.

By pointing out the spurious regression problem, Granger and Newbold ~1974! have shown the importance of stochastic trends in time series data in the context of linear regression models+ At the time, removing trends by differencing was already common practice in univariate time series modeling as part of the Box-Jenkins approach ~Box and Jenkins, 1976!+ These new developments, however, emphasized the importance of autoregressive ~AR! unit roots and motivated the development of statistical procedures for their detection+ Dickey and Fuller ~1979! and Fuller ~1976! were pioneers in developing tests for unit roots that became widely used+ The foundation of asymptotic theory for regressions involving stochastic trends was led by Phillips ~1986, 1987! with the introduction of the functional limit theory, weak convergence methods, convergence to stochastic integrals, nonparametric unit root testing, and continuous record asymptotics+ Phillips and Durlauf ~1986! extended some of these results to the multivariate setting by presenting the multivariate invariance principles and the asymptotic theory of multivariate nonstationary and cointegrating regressions+ These contributions provided the asymptotic tools that have served as the basis for most of the limit results derived in the context of unit root nonstationarity, and they have stimulated extensive subsequent research+ Over the last 30 years many other important contributions were made, of which the following are of particular relevance+ Augmented Dickey-Fuller ~ADF! tests We are grateful to Peter Phillips for proposing a special issue of Econometric Theory for papers from our conference Unit Root and Cointegration Testing+ We thank all participants of the conference who contributed their papers to this special issue+ We are also very grateful to those colleagues who agreed to serve as referees for the papers+ They not only provided generous help through comments but also respected our rather tight deadlines+ We are grateful to all those who helped us to complete the special issue unusually fast+ To protect their identities, we do not list them here+ They will be included in the general list of referees for Econometric Theory+ Our conference would not have been possible without the generous financial support of a number of sponsors+ We are grateful to the Bank of Portugal, to the Portuguese Science Foundation ~FCT!, to the Luso-American Foundation

In this note we show that, when the true data generating process is a stationary one around a constant term with a break, the stationarity test of Kim ~2000, Journal of Econometrics 95, 97-116! against the alternative hypothesis of change of persistence rejects the null of stationarity asymptotically with probability one+

duals or institutions. Indeed, as long as public and private agents must make decisions that require judgement as to the future of economic events, forecasts will not only continue to be made, but acted upon. Considered as a branch of econometrics, the broad purpose of macroeconometric models is to improve our decision making in the context of the uncertain environment in which we must make our way, All forecasts generated by these models are wrong in the simple sense that they are not as accurate as they could be. There is always either some error or some relevant aspect of reality they do not address, or both. It should be understood that econometric models, even structural econometric models, are simplified images of the "true" economy that highlight certain aggregate relationships for the purpose of generating useful insights into the possible evolution of the economy. In constructing such macroeconometric models, we are always choosing among alternative approximations. The exact nature of these approximations will depend principally on the overall structure of the economy, the intended use of the model, the availability of data and the stability of particular relationships. Further, it is important to remember that the size of the available data sample is small relative to the variety of events that can be expected to have an important impact on the economy. It will remain true, therefore, that these models will normally be in a stage of transition as they are adapted to changing circumstances and interests. Their value hinges on whether the "core of stability" that characterizes macroeconometric models is sufficient to continue to provide useful information throaugh the ever-changing circumstances that characterize our economy. That is, can econometric models continue to embody over time an adequate understanding of relevant economic behavior.

This paper extends the simple threshold regression framework of Hansen (2000) and Caner and Hansen (2004) to allow for endogeneity of the threshold variable. We develop a concentrated least squares estimator of the threshold parameter based on an inverse Mills ratio bias correction. We show that our estimator is consistent and investigate its performance using a Monte Carlo simulation that indicates the applicability of the method in …nite samples.

We consider a parameter estimation problem for a diffusion with killing, starting at a point in an open and bounded set. The infinitesimal killing rate function depends on a control variable and parameters. Values of the control variable are known while parameters have unknown values which have to be estimated from data. The minimum of three times: the maximum observation time, the first exit time from the open set, and the killing time, is observed. Instead of the maximum likelihood estimation method we propose and use the minimum χ 2-estimation method that is based on the conditional mean of the data observed before the maximum observation time is reached, and on the frequency of data that are equal to the maximum observation time. We prove that the estimator exists and is consistent and asymptotically normal. The method is illustrated by an example.

The variational auto-encoder (VAE) is a deep latent variable model that has two neural networks in an autoencoder-like architecture; one of them parameterizes the model's likelihood. Fitting its parameters via maximum likelihood (ML) is challenging since the computation of the marginal likelihood involves an intractable integral over the latent space; thus the VAE is trained instead by maximizing a variational lower bound. Here, we develop a ML training scheme for VAEs by introducing unbiased estimators of the log-likelihood gradient. We obtain the estimators by augmenting the latent space with a set of importance samples, similarly to the importance weighted auto-encoder (IWAE), and then constructing a Markov chain Monte Carlo coupling procedure on this augmented space. We provide the conditions under which the estimators can be computed in finite time and with finite variance. We show experimentally that VAEs fitted with unbiased estimators exhibit better predictive performance.

When seasonal time series are periodically integrated, we show that the any cointegration is either full periodic cointegration or full nonperiodic cointegration, with no possibility of cointegration applying for only some seasons. In contrast, seasonally integrated series can be seasonally, periodically or nonperiodically cointegrated, with the possibility of cointegration applying for a subset of seasons. Cointegration tests are analysed for periodically integrated series. A residual-based test is examined and its asymptotic distribution is derived under the null hypothesis of no cointegration. A Monte Carlo analysis confirms the size properties of the test and shows it to have good power. The role of deterministic terms in the cointegrating test regression are also investigated. Further, we show that the asymptotic distribution of the errorcorrection test for periodic cointegration derived by Boswijk and Franses (1995) does not apply for periodically integrated processes, and we derive this distribution for the case of uncorrelated periodically integrated processes.

In this paper we extend the large-sample results provided for the augmented Dickey–Fuller test by Said and Dickey (1984, Biometrika 71, 599–607) and Chang and Park (2002, Econometric Reviews 21, 431–447) to the case of the augmented seasonal unit root tests of Hylleberg, Engle, Granger, and Yoo (1990, Journal of Econometrics 44, 215–238), inter alia. Our analysis is performed under the same conditions on the innovations as in Chang and Park (2002), thereby allowing for general linear processes driven by (possibly conditionally heteroskedastic) martingale difference innovations. We show that the limiting null distributions of the t-statistics for unit roots at the zero and Nyquist frequencies and joint F-type statistics are pivotal, whereas those of the t-statistics at the harmonic seasonal frequencies depend on nuisance parameters that derive from the lag parameters characterizing the linear process. Moreover, the rates on the lag truncation required for these results to hold are sh...

This paper examines the implications of applying the Hylleberg, Engle, Granger, and Yoo (1990,Journal of Econometrics44, 215–238) (HEGY) seasonal root tests to a process that is periodically integrated. As an important special case, the random walk process is also considered, where the zero-frequency unit roott-statistic is shown to converge to the Dickey–Fuller distribution and all seasonal unit root statistics diverge. For periodically integrated processes and a sufficiently high order of augmentation, the HEGYt-statistics for unit roots at the zero and semiannual frequencies both converge to the same Dickey–Fuller distribution. Further, the HEGY joint test statistic for a unit root at the annual frequency and all joint test statistics across frequencies converge to the square of this distribution. Results are also derived for a fixed order of augmentation. Finite-sample Monte Carlo results indicate that, in practice, the zero-frequency HEGY statistic (with augmentation) captures ...