Econometric Theory

I provide a systematic treatment of the asymptotic properties of weighted M-estimators under variable probability stratified sampling. The characterization of the sampling scheme and representation of the objective function allow for a straightforward analysis. Simple, consistent asymptotic variance matrix estimators are proposed for a large class of problems. When stratification is based on exogenous variables, I show that the unweighted M-estimator is more efficient than the weighted estimator under a generalized conditional information matrix equality. When population frequencies are known, a more efficient weighting is possible. I also show how the results carry over to multinomial sampling.

The paper provides a survey of the work that has been done in financial econometrics in the past decade. It proceeds by first establishing a set of stylized facts that are characteristics of financial series and then by detailing the range of techniques that have been developed to model series which possess these characteristics. Both univariate and multivariate models are considered.

In this paper, new fully nonparametric estimators of the diffusion coefficient of continuous time models are introduced. The estimators are based on Fourier analysis of the state variable trajectory observed and on the estimation of quadratic variation between observations by means of realized volatility. The estimators proposed are shown to be consistent and asymptotically normally distributed. Moreover, the Fourier estimator can be iterated to get a fully nonparametric estimate of the diffusion coefficient in a bivariate model in which one state variable is the volatility of the other. The estimators are shown to be unbiased in small samples using Monte Carlo simulations, and are used to estimate univariate and bivariate models for interest rates. * I wish to acknowledge especially Marzia De Donno for pointing out an error in an older version, as well as Emma Thorley for revising the manuscript. Two anonymous referees and the editor substantially contributed to improving the paper. All errors and omissions are my own responsibility.

Time series merupakan salah satu jenis data yang sering kita jumpai. Karakteristik data time series melibatkan lebih dari satu titik waktu dengan unit analisis dapat berupa kota, kabupaten, perusahaan, negara dan sebagainya. Sedangkan periode analisa dapat berupa harian, mingguan, bulanan, triwulanan, atau tahunan. Contoh data time series adalah data pendapatan nasional, inflasi, suku bunga, dan sebagainya.
Handout ini membahas empat topik yaitu data non stasioner, stasioneritas, akar unit, dan kointegrasi.

Unit root tests for time series with level shifts of general form are considered when the timing of the shift is unknown. It is proposed to estimate the nuisance parameters of the data generation process including the shift date in a first step and apply standard unit root tests to the residuals. The estimation of the nuisance parameters is done in such a way that the unit root tests on the residuals have the same limiting distributions as for the case of a known break date. Simulations are performed to investigate the small sample properties of the tests, and empirical examples are discussed to illustrate the procedure.

Analisis regresi adalah analisis statistika yang bertujuan untuk memodelkan hubungan antara variabel independent dengan variabel dependent. Istilah regresi pertamakali dikenalkan oleh Francis Galton (1886) melalui artikelnya yang berjudul Regression Towards Mediocrity In Hereditary Stature, di dalam artikel ini Galton mengkaji hubungan antara tinggi badan anak dengan tinggi badan orang tua. Dari hasil kajian ini diperoleh informasi adanya hubungan antara tinggi badan anak dengan tinggi orang-tuanya. Model yang menggambarkan hubungan antara variabel independent (X) dengan variabel dependent (Y) adalah : Y= f(X, ) +   

This paper extends the Integrated Conditional Moment (ICM) test for the functional form of nonlinear regression models to tests for para- metric conditional distributions. This test is formed on the basis of the integrated squared difference between the empirical characteristic function of the actual data and the characteristic function implied by the model. This test is consistent, and has nontrivial

Bu çalışmada çok kategorili, nitel değişkenlerin sırasız bir şekilde kullanıldığı ve bağımlı değişkenin ikiden fazla değer aldığı Multinomial Logit Modeli üzerinde durulmuştur. Kavramsal açıdan çok değişkenli modellerin arasındaki farklar ortaya konulduktan sonra, Multinomial Logit Modelinin çeşitleri ve varsayımları ile modelin tahmini ve yorumlanması teorik çerçevede ve uygulamalı örneklerle açıklanmıştır.

This paper proposes a nonparametric simultaneous test for parametric specification of the conditional mean and variance functions in a time series regression model. The test is based on an empirical likelihood (EL) statistic that measures the goodness of fit between the parametric estimates and the nonparametric kernel estimates of the mean and variance functions. A unique feature of the test is its ability to distribute natural weights automatically between the mean and the variance components of the goodness-of-fit measure. To reduce the dependence of the test on a single pair of smoothing bandwidths, we construct an adaptive test by maximizing a standardized version of the empirical likelihood test statistic over a set of smoothing bandwidths. The test procedure is based on a bootstrap calibration to the distribution of the empirical likelihood test statistic. We demonstrate that the empirical likelihood test is able to distinguish local alternatives that are different from the null hypothesis at an optimal rate.

This paper shows that if the errors in a multiple regression model are heavy-tailed, the ordinary least squares (OLS) estimators for the regression coefficients are tail-dependent. The tail dependence arises, because the OLS estimators are stochastic linear combinations of heavy-tailed random variables. Moreover, tail dependence also exists between the fitted sum of squares (FSS) and the residual sum of squares (RSS), because they are stochastic quadratic combinations of heavy-tailed random variables.

The least absolute shrinkage and selection operator (LASSO) is a widely used statistical methodology for simultaneous estimation and variable selection. It is a shrinkage estimation method that allows one to select parsimonious models. In other words, this method estimates the redundant parameters as zero in the large samples and reduces variance of estimates. In recent years, many authors analyzed this technique from a theoretical and applied point of view. We introduce and study the adaptive LASSO problem for discretely observed multivariate diffusion processes. We prove oracle properties and also derive the asymptotic distribution of the LASSO estimator. This is a nontrivial extension of previous results by Wang and Leng (2007, Journal of the American Statistical Association, 102(479), 1039-1048) on LASSO estimation because of different rates of convergence of the estimators in the drift and diffusion coefficients. We perform simulations and real data analysis to provide some evidence on the applicability of this method.

The paper provides a survey of the work that has been done in financial econometrics in the past decade. It proceeds by first establishing a set of stylized facts that are characteristics of financial series and then by detailing the range of techniques that have been developed to model series which possess these characteristics. Both univariate and multivariate models are considered. 0927-5398(95)00020-8 , and, when var(Yrt) -~ var(~,) = 0, the true MA(1) coefficient et equals 0. Inverting the MA to produce an infinite AR, (1 -aL + o~2L 2 -ot3L3...)xt = "1: t creates a J Many earnings series exhibit a weak first order dependence because of this measurement feature. J~ Returns will include the dividend yield so that Yt needs to be re-defined as beginning of period price plus dividend. In fact, the stability of the dividend yield means that most variation in returns is from the capital gains component.

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

The authors thank the two referees and the co-editor for their valuable suggestions. Thanks to Professor Qiwei Yao for helpful discussion on programming of bootstrap method. Oliver Linton was supported by the ESRC (UK), and Jiazhu Pan was partially supported by the starter ...

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We consider cointegration tests when a Vector AutoRegressive (VAR) process of order k is used to approximate a more general linear process with a possibly infinite VAR representation. Traditional methods to select the lag order, such as Akaike's (AIC) or the Bayesian information criteria, often lead to too parsimonious a model with the implication that the cointegration tests suffer from substantial size distortions in finite samples. We extend the analysis of Ng and Perron (2001) to derive a Modified Akaike's Information Criterion (MAIC) in this multivariate setting. The idea is to use the information specified by the null hypothesis as it relates to restrictions on the parameters of the model to keep an extra term in the penalty function of the AIC. This MAIC takes a very simple form for which this extra term is simply the likelihood ratio test for testing the null hypothesis of r against more than r cointegrating vectors. We provide theoretical analyses of its validity and of the fact that cointegration tests constructed from a VAR whose lag order is selected using the MAIC have the same limit distribution as when the order is finite and known. We also provide theoretical and simulation analyses to show how the MAIC leads to VAR approximations that yield tests with drastically improved size properties with little loss of power.

The paper provides a survey of the work that has been done in financial econometrics in the past decade. It proceeds by first establishing a set of stylized facts that are characteristics of financial series and then by detailing the range of techniques that have been developed to model series which possess these characteristics. Both univariate and multivariate models are considered.

This paper considers a nonparametric time series regression model with a nonstationary regressor. We construct a nonparametric test for testing whether the regression is of a known parametric form indexed by a vector of unknown parameters. We establish the asymptotic distribution of the proposed test statistic. Both the setting and the results differ from earlier work on nonparametric time series regression with stationarity. In addition, we develop a bootstrap simulation scheme for the selection of suitable bandwidth parameters involved in the kernel test as well as the choice of simulated critical values. An example of implementation is given to show that the proposed test works in practice.

An integration test against fractional alternatives is suggested for univariate time series. The new test is a completely regression based, lag augmented version of the LM test by Robinson (1991, Journal of Econometrics 47, 67-84). Our main contributions, however, are the following. First, we let the short memory component follow a general linear process. Second, the innovations driving this process are martingale differences with eventual conditional heteroskedasticity that is accounted for by means of White's standard errors. Third, we assume the number of lags to grow with the sample size, thus approximating the general linear process. Under these assumptions limiting normality of the test statistic is retained. The usefulness of the asymptotic results for finite samples is established in Monte Carlo experiments. In particular, we study several strategies of model selection.

In this paper nearly unstable AR(p) processes (in other words, models with characteristic roots near the unit circle) are studied. Our main aim is to describe the asymptotic behaviour of the least squares estimators of the coecients. A convergence result is presented for the general complex-valued case. The limit distribution is given by the help of some continuous time AR processes. We apply the results for real-valued nearly unstable AR(p) models. In this case the limit distribution can be identied with the maximum likelihood estimator of the coecients of the corresponding continuous time AR processes.

This examination has four parts. Weights applied to the four parts will be 15, 15, 30 and 40. This is an open book exam. You may use any source of information that you have with you. You may not phone or text message or email or Bluetooth (is that a verb?) to "a friend," however. Part I. Fixed and Random Effects Define the two basic approaches to modeling unobserved effects in panel data. What are the different assumptions that are made in the two settings? What is the benefit of the fixed effects assumption? What is the cost? Same for the random effects specification. Now, extend your definitions to a model in which all parameters, not just the constant term, are heterogeneous. For the random parameters case, describe the estimators that one would use under the two assumptions. Two approaches are fixed effects and random effects. In the "effects model," y it = x it ′β + c i + ε it , x it is exogenous with respect to ε it. FE: c i may be correlated with x it. Benefits: General approach, Robust-estimator of β is consistent even if RE is the right model. Cost: Many parameters, inefficient if RE is correct. Precludes time invariant variables. RE: c i is uncorrelated with x it Benefits: Tight parameterization-only one new parameter Efficient estimation-use GLS Allows time invariant parameters Cost Unreasonable orthogonality assumption Inconsistent if RE is the right model. Random parameters case. Replace the model statement with y it = x it ′β i + ε it , β i = β + w i. C a s e 1 : w i may be correlated with x it. This is the counterpart to FE. In this case, it is necessary to fit the equations one at a time. Requires that there be enough observations to do so, so T > K. The efficient estimator is equation by equation OLS. Same benefits (robustness) and costs (inefficiency) as FE C a s e 2 ; w i is uncorrelated with x it. This RP model can be fit An efficient estimator will be the matrix weighted FGLS estimator. (Swamy et al.) This would be a two step estimator, just like FGLS for the RE model. This model can also be fit by simulation-we mentioned this briefly in class, and

Most model selection mechanisms work in an 'overall' modus, providing models without specific concern for how the selected model is going to be used afterwards. The focussed information criterion (FIC), on the other hand, is geared towards optimum model selection when inference is required for a given estimand. In this paper the FIC method is extended to weighted versions. This allows one to rank and select candidate models for the purpose of handling a range of similar tasks well, as opposed to being forced to focus on each task separately. Applications include selecting regression models that perform well for specified regions of covariate values. We derive these wFIC criteria, give asymptotic results, and apply the methods to real data. Formulae for easy implementation are provided for the class of generalised linear models.

We consider the problem of hypothesis testing in a modified version of the stochastic integration and cointegration framework of Harris, . This nonlinear setup allows for volatility in excess of that catered for by the standard integration/cointegration paradigm through the introduction of nonstationary heteroscedasticity. We propose a test for stochastic cointegration against the alternative of no cointegration and a secondary test for stationary cointegration against the heteroscedastic alternative. Asymptotic distributions of these tests under their respective null hypotheses are derived and consistency under their respective alternatives is established. Monte Carlo evidence suggests the tests will perform well in practice. An empirical application to the term structure of interest rates is also given.

In this paper we introduce a linear programming estimator (LPE) for the slope parameter in a constrained linear regression model with a single regressor. The LPE is interesting because it can be superconsistent in the presence of an endogenous regressor and, hence, preferable to the ordinary least squares estimator (LSE). Two different cases are considered as we investigate the statistical properties of the LPE. In the first case, the regressor is assumed to be fixed in repeated samples. In the second, the regressor is stochastic and potentially endogenous. For both cases the strong consistency and exact finite-sample distribution of the LPE is established. Conditions under which the LPE is consistent in the presence of serially correlated, heteroskedastic errors are also given. Finally, we describe how the LPE can be extended to the case with multiple regressors and conjecture that the extended estimator is consistent under conditions analogous to the ones given herein. Finite-sample properties of the LPE and extended LPE in comparison to the LSE and instrumental variable estimator (IVE) are investigated in a simulation study. One advantage of the LPE is that it does not require an instrument.

The main uniform convergence results of are generalized in two directions: Data is allowed to (i) be heterogenously dependent and (ii) depend on a (possibly unbounded) parameter. These results are useful in semiparametric estimation problems involving time-inhomogenous models and/or sampling of continuous-time processes. The usefulness of these results are demonstrated by two applications: Kernel regression estimation of a time-varying AR(1) model , and the kernel density estimation of a Markov chain that has not been intialized at its stationary distribution.

We study some methods of combining procedures for forecasting a continuous random variable. Statistical risk bounds under the square error loss are obtained under mild distributional assumptions on the future given the current outside information and the past observations. The risk bounds show that the combined forecast automatically achieves the best performance among the candidate procedures up to a constant factor and an additive penalty term. In term of the rate of convergence, the combined forecast performs as well as if one knew which candidate forecasting procedure is the best in advance.

In this paper we propose an estimation procedure for a censored regression model where the latent regression function has a partially linear form. Based on a conditional quantile restriction, we estimate the model by a two stage procedure. The first stage nonparametrically estimates the conditional quantile function at insample and appropriate out-of-sample points, and the second stage involves a simple weighted least squares procedure. The proposed procedure is shown to have desirable asymptotic properties under regularity conditions that are standard in the literature. A small scale simulation study indicates that the estimator performs well in moderately sized samples. We are grateful to J. Powell, B. Honore, L.-F. Lee, two anonymous referees, and the co-editor Joel Horowitz for helpful comments. Songnian Chen gratefully acknowledges the financial support of grant HKUST6175/98H from the Research Grants Council of Hong Kong.

This paper derives the limiting distributions of alternative jackknife IV (JIV ) estimators and gives formulae for accompanying consistent standard errors in the presence of heteroskedasticity and many instruments. The asymptotic framework includes the many instrument sequence of Bekker (1994) and the many weak instrument sequence of . We show that JIV estimators are asymptotically normal; and that standard errors are consistent provided that √ Kn rn → 0, as n → ∞, where K n and r n denote, respectively, the number of instruments and the rate of growth of the concentration parameter. This is in contrast to the asymptotic behavior of such classical IV estimators as LIM L, B2SLS, and 2SLS, all of which are inconsistent in the presence of heteroskedasticity, unless K n rn → 0. We also show that the rate of convergence and the form of the asymptotic covariance matrix of the JIV estimators will in general depend on strength of the instruments as measured by the relative orders of magnitude of r n and K n .

We consider a model $Y\_t=\sigma\_t\eta\_t$ in which $(\sigma\_t)$ is not independent of the noise process $(\eta\_t)$, but $\sigma\_t$ is independent of $\eta\_t$ for each $t$. We assume that $(\sigma\_t)$ is stationary and we propose an adaptive estimator of the density of $\ln(\sigma^2\_t)$ based on the observations $Y\_t$. Under various dependence structures, the rates of this nonparametric estimator coincide with the

The hat matrix maps the vector of response values in a regression to its predicted counterpart. The trace of this hat matrix is the workhorse for calculating the effective number of parameters in both parametric and nonparametric regression settings. Drawing on the regression literature, the standard kernel density estimate is transformed to mimic a regression estimate thus allowing extraction of a usable hat matrix for calculating the effective number of parameters of the kernel density estimate. Asymptotic expressions for the trace of this hat matrix are derived under standard regularity conditions for mixed, continuous, and discrete densities. Simulations validate the theoretical contributions. Several empirical examples demonstrate the usefulness of the method suggesting that calculating the effective number of parameters of a kernel density estimator may be useful in interpreting differences across estimators.

The matrix that transforms the response variable in a regression to its predicted value is commonly referred to as the hat matrix. The trace of the hat matrix is a standard metric for calculating degrees of freedom. The two prominent theoretical frameworks for studying hat matrices to calculate degrees of freedom in local polynomial regressions-ANOVA and non-ANOVA-abstract from both mixed data and the potential presence of irrelevant covariates, both of which dominate empirical applications. In the multivariate local polynomial setup with a mix of continuous and discrete covariates, which include some irrelevant covariates, we formulate asymptotic expressions for the trace of both the non-ANOVA and ANOVA-based hat matrices from the estimator of the unknown conditional mean. The asymptotic expression of the trace of the non-ANOVA hat matrix associated with the conditional mean estimator is equal up to a linear combination of kernel-dependent constants to that of the ANOVA-based hat matrix. Additionally, we document that the trace of the ANOVA-based hat matrix converges to 0 in any setting where the bandwidths diverge. This attrition outcome can occur in the presence of irrelevant continuous covariates or it can arise when the underlying data generating process is in fact of polynomial order.

We propose an efficient semiparametric estimator for the coefficients of a multivariate linear regression model -with a conditional quantile restriction for each equation -in which the conditional distributions of errors given regressors are unknown. The procedure can be used to estimate multiple conditional quantiles of the same regression relationship. The proposed estimator is asymptotically as efficient as if the true optimal instruments were known. Simulation results suggest that the estimation procedure works well in practice and dominates an equation-by-equation efficiency correction if the errors are dependent conditional on the regressors.

I provide a systematic treatment of the asymptotic properties of weighted M-estimators under variable probability stratified sampling. The characterization of the sampling scheme and representation of the objective function allow for a straightforward analysis. Simple, consistent asymptotic variance matrix estimators are proposed for a large class of problems. When stratification is based on exogenous variables, I show that the unweighted M-estimator is more efficient than the weighted estimator under a generalized conditional information matrix equality. When population frequencies are known, a more efficient weighting is possible. I also show how the results carry over to multinomial sampling.

In this paper, we propose a new diagnostic test for residual cross-section uncorrelatedness in a nonparametric panel data model. The proposed nonparametric cross-section uncorrelatedness (CU) test is a nonparametric counterpart of an existing parametric cross-section dependence (CD) test proposed in Pesaran for the parametric case. We establish asymptotic distributions of the proposed test statistic for several different cases. Without assuming cross-section independence, we establish asymptotic distributions for the proposed test for the case where both the cross-section dimension and the time dimension go to infinity simultaneously.

This paper shows how large-dimensional dynamic factor models are suitable for structural analysis. We argue that all identification schemes employed in structural vector autoregression (SVAR) analysis can be easily adapted in dynamic factor models. Moreover, the “problem of fundamentalness,” which is intractable in SVARs, can be solved, provided that the impulse-response functions are sufficiently heterogeneous. We provide consistent estimators for the impulse-response functions and for (n, T) rates of convergence. An exercise with U.S. macroeconomic data shows that our solution of the fundamentalness problem may have important empirical consequences.

Here, G is a known link, a, J3 are unknown parameters, and ml, ... , I1ld are unknown (smooth) functions of possibly higher dimensional covariates T}, ... , T d. Estimates of m}, ... , 11ld, a and J3 are presented and asymptotic distribution theory for both the nonparametric and the parametric part is given. The main focus is the application of bootstrap methods. It is shown that bootstrap can be used for bias correction, hypothesis testing (e.g. component-wise analysis) and the construction of uniform confidence bands. Various bootstrap tests for model specification and parametrization are given, in particular for testing additivity and link function specification. The practical performance of our methods is illustrated in simulations and in an application to East-West German migration.

Colin Blyth's paradox <<of three pies>>: setwise vs. pairwise event preferences

The pairwise independence of events does not entail their setwise independence (Bernstein's example, 1910-1917). The probability
distributions of all pairs of events do not determine the probability distribution of the whole set of events (the paradox of a triangular room
\cite[2001]{Vorobyev2007ru,Vorobyev2016famems4ru}). The pairwise preferences of events do not determine the setwise preferences of these events
(Blyth's paradox, 1972). The eventological theory of setwise preferences, proposed in \cite[2007]{Vorobyev2007ru}, gives an eventual justification
and extension of the classical theory of preferences and explains Blyth's paradox <<of three pies>> (that is already well-known to Yule) by human ability to use triplewise preferences.

The fact that pairwise independence of events does not imply setwise independence of events was mentioned for the first time in the correspondence in the years 1910 to 1917 between A.A.Chuprov\footnote{\emph{Alexander Alexandrovich Chuprov (1974--1926)} was a Russian statistician who worked on mathematical statistics, sample survey theory and demography.} and A.A.Markov\footnote{\emph{Andrey Andreyevich Markov (1856--1922)} was a Russian mathematician. He is best known for his work on stochastic processes. A primary subject of his research later became known as Markov chains and Markov processes.}. The bright example of the fact is attributted usually to Bernstein\footnote{Sergei Natanovich Bernstein (1880--1968) was a Russian and Soviet mathematician of Jewish origin known for contributions to partial differential equations, differential geometry, probability theory, and approximation theory.} with reference to \cite[1946, page 48]{Bernstein1946}.

In \cite[2016]{Vorobyev2016famems4} an improved generalization of Feynman's paradox of negative probabilities \cite{Feynman1982,Feynman1987} for observing three events which is directly related to the theory of quantum computing is presented.
This generalization, first proposed in \cite[2001]{Vorobyev2001a} and called the <<paradox of the triangular room of negative probabilities of events>>, clearly demonstrates the fact that three probability distributions of pairs of events from a given triplet are insufficient to determine the probabilistic distribution of the whole triplet of events. In other words, three pairwise (partial) probability distributions of events do not determine the triplewise (joint) probability distribution of the whole triplet of events.

In this paper, I intend to briefly show the main advantages of the new \emph{theory of setwise event preferences} developed in \cite{Vorobyev2007ru}, and at the same time to demonstrate once again the failure of the pairwise to describe the whole. This time, using the example of preferences, when each of us is forced to make a comparison every time, hitting the next situation of choice, which requires a decision. The theory of setwise event preferences shows that the familiar pairwise comparisons do not provide a complete description of the whole set of preferences that can be trapped at every step. A vivid example of this failure of the pairwise to describe the whole is the explanation of the eventological theory of setwise event preferences of the famous Blyth's paradox <<of three pies>> \cite[1972]{Blyth1972}.

How does a person make a decision? Clarify:
<<How does the person make an event decision?>>, because from the point of view of the eventology theory \cite{Vorobyev2007ru} any decision accepted by a person is always an eventual decision.
Let $(\Omega,\mathcal F,\mathbf P)$ be a probability space in which the eventological theory of setwise event preferences answers this question.

If you are interested in my work, and you have the opportunity to provide financial support for the publication of my book "Theory of experience and of chance as a theory of being," I ask you to use my PayPal account: paypal.me/EventologyTheory

Bootstrap tests are tests for which the significance level is calculated by some sort of bootstrap procedure, which may be parametric or nonparametric. We provide a theoretical framework in which to study the size distortions of bootstrap P values. We show that, in many circumstances, the size distortion of a bootstrap test will be one whole order of magnitude smaller than that of the corresponding asymptotic test. We also show that, at least in the parametric case, the magnitude of the distortion will depend on the shape of what we call the P value function. Monte Carlo results are presented for the case of nonnested hypothesis tests. These results confirm and illustrate the utility of our theoretical results, and they also suggest that bootstrap tests may often work extremely well in practice.