Asymptotic Normality of the Least-Squares Estimates for Higher Order Autoregressive Integrated Processes with Some Applications

1993 ◽  
Vol 9 (2) ◽  
pp. 263-282 ◽  
Author(s):  
In Choi
Keyword(s):  
Asymptotic Normality ◽  
Least Squares ◽  
Confidence Intervals ◽  
Unit Roots ◽  
Small Sample ◽  
Asymptotic Distributions ◽  
Standard Normal Distribution ◽  
Finite Sample ◽  
Standard Normal ◽  
Least Squares Estimates

Using the asymptotic normality of the least-squares estimates for the autoregressive (AR) process with real, positive unit roots and at least one stable root, we consider the asymptotic distributions of the Wald and t ratio tests on AR coefficients. In addition, we propose a method of constructing confidence intervals for the sum of AR coefficients possibly in the presence of a unit root. Using simulation methods, we compare the finite-sample cumulative distributions of the t ratios for individual autoregressive coefficients with those of standard normal distributions, and investigate the finite-sample performance of our confidence intervals and t ratios. Our simulation results show that the t ratios for nonstationary processes converge to a standard normal distribution more slowly than those for stationary processes. Further, the confidence intervals are shown to work reasonably well in moderately large samples, but they display unsatisfactory performance at small sample sizes.

CAUCHY ESTIMATORS FOR AUTOREGRESSIVE PROCESSES WITH APPLICATIONS TO UNIT ROOT TESTS AND CONFIDENCE INTERVALS

1999 ◽  
Vol 15 (2) ◽  
pp. 165-176 ◽  
Author(s):  
Beong Soo So ◽  
Dong Wan Shin
Keyword(s):  
Least Squares ◽  
Confidence Intervals ◽  
Unit Root ◽  
Unit Roots ◽  
Ordinary Least Squares ◽  
Unit Root Tests ◽  
Autoregressive Processes ◽  
Seasonal Time Series ◽  
Least Squares Estimators ◽  
Standard Normal

For autoregressive processes, we propose new estimators whose pivotal statistics have the standard normal limiting distribution for all ranges of the autoregressive parameters. The proposed estimators are approximately median unbiased. For seasonal time series, the new estimators give us unit root tests that have limiting normal distribution regardless of period of the seasonality. Using the estimators, confidence intervals of the autoregressive parameters are constructed. A Monte-Carlo simulation for first-order autoregressions shows that the proposed tests for unit roots are locally more powerful than the tests based on the ordinary least squares estimators. It also shows that the proposed confidence intervals have shorter average lengths than those of Andrews (1993, Econometrica 61, 139–165) based on the ordinary least squares estimators when the autoregressive coefficient is close to one.

scholarly journals Integral Least-Squares Inferences for Semiparametric Models with Functional Data

2014 ◽  
Vol 2014 ◽  
pp. 1-8
Author(s):  
Limian Zhao ◽  
Peixin Zhao
Keyword(s):  
Data Analysis ◽  
Asymptotic Normality ◽  
Least Squares ◽  
Confidence Intervals ◽  
Functional Data ◽  
Estimation Method ◽  
Real Data ◽  
Semiparametric Models ◽  
Simulation Studies ◽  
Finite Sample

The inferences for semiparametric models with functional data are investigated. We propose an integral least-squares technique for estimating the parametric components, and the asymptotic normality of the resulting integral least-squares estimator is studied. For the nonparametric components, a local integral least-squares estimation method is proposed, and the asymptotic normality of the resulting estimator is also established. Based on these results, the confidence intervals for the parametric component and the nonparametric component are constructed. At last, some simulation studies and a real data analysis are undertaken to assess the finite sample performance of the proposed estimation method.

scholarly journals The Shortest Confidence Interval for the Mean of a Normal Distribution

2018 ◽  
Vol 7 (2) ◽  
pp. 33
Author(s):  
Traoré Boubakar ◽  
Diabaté Lassina ◽  
Touré Belco ◽  
Fané Abdou
Keyword(s):  
Confidence Interval ◽  
Normal Distribution ◽  
Confidence Intervals ◽  
Mathematical Statistics ◽  
Standard Normal Distribution ◽  
Pivotal Quantity ◽  
Construction Technique ◽  
Standard Normal ◽  
The Mean ◽  
Shortest Confidence Intervals

An interesting topic in mathematical statistics is that of the construction of the confidence intervals. Two kinds of intervals which are both based on the method of pivotal quantity are the shortest confidence interval and the equal tail confidence intervals. The aim of this paper is to clarify and comment on the finding of such intervals and to investigation the relation between the two kinds of intervals. In particular, we will give a construction technique of the shortest confidence intervals for the mean of the standard normal distribution. Examples illustrating the use of this technique are given.

scholarly journals A Note on the Asymptotic Normality Theory of the Least Squares Estimates in Multivariate HAR-RV Models

Mathematics ◽  
2020 ◽  
Vol 8 (11) ◽  
pp. 2083
Author(s):  
Won-Tak Hong ◽  
Jiwon Lee ◽  
Eunju Hwang
Keyword(s):  
Asymptotic Normality ◽  
Least Squares ◽  
Average Length ◽  
Real Data ◽  
Spot Price ◽  
Correlated Errors ◽  
Sample Mean ◽  
Q Model ◽  
Multiple Assets ◽  
Least Squares Estimates

In this work, multivariate heterogeneous autoregressive-realized volatility (HAR-RV) models are discussed with their least squares estimations. We consider multivariate HAR models of order p with q multiple assets to explore the relationships between two or more assets’ volatility. The strictly stationary solution of the HAR(p,q) model is investigated as well as the asymptotic normality theories of the least squares estimates are established in the cases of i.i.d. and correlated errors. In addition, an exponentially weighted multivariate HAR model with a common decay rate on the coefficients is discussed together with the common rate estimation. A Monte Carlo simulation is conducted to validate the estimations: sample mean and standard error of the estimates as well as empirical coverage and average length of confidence intervals are calculated. Lastly, real data of volatility of Gold spot price and S&P index are applied to the model and it is shown that the bivariate HAR model fitted by selected optimal lags and estimated coefficients is well matched with the volatility of the financial data.

Small-Sample Accuracy of Approximate Distributions of Functions of Observed Probabilities From t Tests

1991 ◽  
Vol 16 (4) ◽  
pp. 345-369
Author(s):  
Betsy Jane Becker
Keyword(s):  
Small Sample ◽  
Distribution Functions ◽  
Asymptotic Distributions ◽  
Cumulative Distribution ◽  
Statistical Hypothesis ◽  
Small Samples ◽  
P Value ◽  
Natural Logarithm ◽  
Standard Normal ◽  
The One

The observed probability p is the social scientist’s primary tool for evaluating the outcomes of statistical hypothesis tests. Functions of p s are used in tests of “combined significance,” meta-analytic summaries based on sample probability values. This study examines the nonnull asymptotic distributions of several functions of one-tailed sample probability values (from t tests). Normal approximations were based on the asymptotic distributions of z(p), the standard normal deviate associated with the one-sided p value; of ln(p), the natural logarithm of the probability value; and of several modifications of ln(p). Two additional approximations, based on variance-stabilizing transformations of ln(p) and z(p), were derived. Approximate cumulative distribution functions (cdfs) were compared to the computed exact cdf of the p associated with the one-sample t test. Approximations to the distribution of z(p) appeared quite accurate even for very small samples, while other approximations were inaccurate unless sample sizes or effect sizes were very large. Approximations based on variance-stabilizing transformations were not much more accurate than those based on ln(p) and z(p). Generalizations of the results are discussed, and implications for use of the approximations conclude the article.

scholarly journals On the small sample properties of variants of Mardia’s and Srivastava’s kurtosis-based tests for multivariate normality

2012 ◽  
Vol 49 (2) ◽  
pp. 159-175 ◽  
Author(s):  
Zofia Hanusz ◽  
Joanna Tarasińska ◽  
Zbigniew Osypiuk
Keyword(s):  
Small Sample ◽  
Multivariate Normality ◽  
Small Samples ◽  
Standard Normal Distribution ◽  
Test Statistics ◽  
Significance Level ◽  
Nominal Significance ◽  
Standard Normal ◽  
Small Sample Properties ◽  
Mean And Variance

Summary The kurtosis-based tests of Mardia and Srivastava for assessing multivariate normality (MVN) are considered. The asymptotic standard normal distribution of their test statistics, under normality, is often misused for too small samples. The purpose of this paper is to suggest mean-and-variance corrected versions of the Mardia and Srivastava test statistics. Simulation studies evaluating both the true sizes and the powers of original and corrected tests against selected alternatives are presented and compared to the size and the power of the Henze-Zirkler test. The proposed corrected statistics have empirical sizes closer to a nominal significance level than the original ones. It is also shown that the corrected versions of the tests can be more powerful than the original ones.

scholarly journals ASYMPTOTIC INFERENCE FOR NONSTATIONARY FRACTIONALLY INTEGRATED AUTOREGRESSIVE MOVING-AVERAGE MODELS

2001 ◽  
Vol 17 (4) ◽  
pp. 738-764 ◽  
Author(s):  
Shiqing Ling ◽  
W.K. Li
Keyword(s):  
Moving Average ◽  
Unit Roots ◽  
Asymptotic Distributions ◽  
Autoregressive Moving Average ◽  
Finite Sample ◽  
Asymptotic Inference ◽  
Moving Average Models ◽  
Special Cases ◽  
Finite Sample Properties ◽  
Memory Time

This paper considers nonstationary fractional autoregressive integrated moving-average (p,d,q) models with the fractionally differencing parameter d ∈ (− 1/2,1/2) and the autoregression function with roots on or outside the unit circle. Asymptotic inference is based on the conditional sum of squares (CSS) estimation. Under some suitable conditions, it is shown that CSS estimators exist and are consistent. The asymptotic distributions of CSS estimators are expressed as functions of stochastic integrals of usual Brownian motions. Unlike results available in the literature, the limiting distributions of various unit roots are independent of the parameter d over the entire range d ∈ (− 1/2,1/2). This allows the unit roots and d to be estimated and tested separately without loss of efficiency. Our results are quite different from the current asymptotic theories on nonstationary long memory time series. The finite sample properties are examined for two special cases through simulations.

SHARP BOUNDS ON THE DISTRIBUTION OF TREATMENT EFFECTS AND THEIR STATISTICAL INFERENCE

2009 ◽  
Vol 26 (3) ◽  
pp. 931-951 ◽  
Author(s):  
Yanqin Fan ◽  
Sang Soo Park
Keyword(s):  
Sample Size ◽  
Statistical Inference ◽  
Confidence Intervals ◽  
Simulation Study ◽  
Treatment Effect ◽  
Treatment Effects ◽  
Asymptotic Distributions ◽  
Finite Sample ◽  
Sharp Bounds ◽  
Nonparametric Estimators

In this paper, we propose nonparametric estimators of sharp bounds on the distribution of treatment effects of a binary treatment and establish their asymptotic distributions. We note the possible failure of the standard bootstrap with the same sample size and apply the fewer-than-nbootstrap to making inferences on these bounds. The finite sample performances of the confidence intervals for the bounds based on normal critical values, the standard bootstrap, and the fewer-than-nbootstrap are investigated via a simulation study. Finally we establish sharp bounds on the treatment effect distribution when covariates are available.

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