Iid Errors Research Articles

Temporal aggregation of random walk processes and implications for economic analysis

Abstract This paper examines the impact of time averaging and interval sampling data assuming that the data generating process for a given series follows a random walk with iid errors. We provide exact expressions for the corresponding variances, and covariances, for both levels and higher order differences of the aggregated series, as well as that for the variance ratio, demonstrating exactly how the degree of temporal aggregation impacts these properties. We empirically investigate this issue on exchange rates and find that the values of the variance ratios and autocorrelation coefficients at different frequencies are consistent with our theoretical results. We also conduct a simulation exercise that illustrates the potential effect that conditional heteroskedasticity and fat tails may have on the temporal aggregation of a random walk and of a highly persistent autoregressive process.

Robust second-order least-squares estimation for regression models with autoregressive errors

Rosadi and Peiris (Comput Stat 29:931–943, 2014) applied the second-order least squares estimator (SLS), which was proposed in Wang and Leblanc (Ann Inst of Stat Math 60:883–900, 2008), to regression models with autoregressive errors. In case of autocorrelated errors, it shows that the SLS performs well for estimating the parameters of the model and gives small bias. For less correlated data, the standard error (SE) of the SLS lies between the SE of the ordinary least squares estimator (OLS) and the generalized least squares estimator, however, for more correlated data, the SLS has higher SE than the OLS estimator. In case of a regression model with iid errors, Chen, Tsao and Zhou (Stat Pap 53:371–386, 2012) proposed a method to improve the robustness of the SLS against X-outliers. In this paper, we consider a new robust second-order least squares estimator (RSLS), which extends the study in Chen et al. (2012) to the case of regression with autoregressive errors, and the data may be contaminated with all types of outliers (X-, y-, and innovation outliers). Besides the regression coefficients, here we also propose a robust method to estimate the parameters of the autoregressive errors and the variance of the errors. We evaluate the performance of the RSLS by means of simulation studies. In the simulation study, we consider both a linear and a nonlinear regression model. The results show that the RSLS performs very well. We also provide guidelines to use the RSLS in practice and present a real example.

The relation between working memory capacity and auditory lateralization in children with auditory processing disorders

ObjectiveAuditory processing disorder (APD) describes a complex and heterogeneous disorder characterized by poor speech perception, especially in noisy environments. APD may be responsible for a range of sensory processing deficits associated with learning difficulties. There is no general consensus about the nature of APD and how the disorder should be assessed or managed. This study assessed the effect of cognition abilities (working memory capacity) on sound lateralization in children with auditory processing disorders, in order to determine how “auditory cognition” interacts with APD. MethodsThe participants in this cross-sectional comparative study were 20 typically developing and 17 children with a diagnosed auditory processing disorder (9–11 years old). Sound lateralization abilities investigated using inter-aural time (ITD) differences and inter-aural intensity (IID) differences with two stimuli (high pass and low pass noise) in nine perceived positions. Working memory capacity was evaluated using the non-word repetition, and forward and backward digits span tasks. Linear regression was employed to measure the degree of association between working memory capacity and localization tests between the two groups. ResultsChildren in the APD group had consistently lower scores than typically developing subjects in lateralization and working memory capacity measures. The results showed working memory capacity had significantly negative correlation with ITD errors especially with high pass noise stimulus but not with IID errors in APD children. ConclusionsThe study highlights the impact of working memory capacity on auditory lateralization. The finding of this research indicates that the extent to which working memory influences auditory processing depend on the type of auditory processing and the nature of stimulus/listening situation.

The effect of recursive detrending on panel unit root tests

This paper analyzes the properties of panel unit root tests based on recursively detrended data. The analysis is conducted while allowing for a (potentially) non-linear trend function, which represents a more general consideration than the current state of affairs with (at most) a linear trend. A new test statistic is proposed whose asymptotic behavior under the unit root null hypothesis, and the simplifying assumptions of a polynomial trend and iid errors are shown to be surprisingly simple. Indeed, the test statistic is not only asymptotically independent of the true trend polynomial, but also is in fact unique in that it is independent also of the degree of the fitted polynomial. However, this invariance property does not carry over to the local alternative, under which it is shown that local power is a decreasing function of the trend degree. But while power does decrease, the rate of shrinking of the local alternative is generally constant in the trend degree, which goes against the common belief that the rate of shrinking should be decreasing in the trend degree. The above results are based on simplifying assumptions. To compensate for this lack of generality, a second, robust, test statistic is proposed, whose validity does not require that the trend function is a polynomial or that the errors are iid.

Optimal convergence rates in non‐parametric regression with fractional time series errors

Consider the estimation of g(ν), the νth derivative of the mean function, in a fixed‐design non‐parametric regression model with stationary time series errors ξi. We assume that , ξi are obtained by applying an invertible linear filter to iid innovations, and the spectral density of ξi has the form as λ → 0 with constants cf > 0 and α ∈ (−1,1). Under regularity conditions, the optimal convergence rate of is shown to be with r = (1 − α)(k − ν)/(2k+1 − α). This rate is achieved by local polynomial fitting. Moreover, in spite of including long memory and antipersistence, the required conditions on the innovation distribution turn out to be the same as in non‐parametric regression with iid errors.

Goodness-of-fit test using residuals in infinite-order autoregressive models

In this paper we study the asymptotic behavior of the Bickel–Rosenblatt test in infinite-order autoregressive models. Under some mild conditions, the test statistic based on residuals is shown to have the same limiting distribution as that based on true iid errors. It is also proved that this result remains the same when the nuisance parameters in the model are to be estimated, which means that the Bickel–Rosenblatt test is easily applicable in composite hypothesis goodness-of-fit testing unlike the Kolmogorov–Smirnov type tests. The result of a simulation study is supplemented to verify the result of this paper.

Adaptive Estimators and Tests of Stationary and Nonstationary Short- and Long-Memory ARFIMA–GARCH Models

This article considers the fractionally autoregressive integrated moving average [ARFIMA(p,d,q)] models with GARCH errors. The process generated by this model is short memory, long memory, stationary, and nonstationary, respectively, when d ∈ (-1/2, 0), d ∈ (0, 1/2), d ∈ (-1/2, 1/2), and d ∈ (1/2, ∞). Using a unified approach, the local asymptotic normality of the model is established for d ∈ ∪∞j = 0(J - 1/2, J + 1/2). The adaptivity and efficiency of the estimating parameters are discussed. In a class of loss functions, the asymptotic minimax bound of the estimators for the model is given when the density f of rescaled residuals is unknown. An adaptive estimator is constructed for the parameters in the ARFIMA part when f is symmetric, and a general form of the efficient estimator is also constructed for all the parameters in the ARFIMA and GARCH parts. When the density f is unknown, Wald tests are constructed for testing the unit root + 1 against the class of fractional unit roots. It is shown that these tests asymptotically follow the chi-squared distribution and are locally most powerful. These results are new contributions to the literature, even for the ARFIMA model with iid errors, except for the adaptive estimator in this case with d ∈ (0, 1/2). The performance of the asymptotic results in finite samples is examined through Monte Carlo experiments. An application to the U.S. Consumer Price Index inflation series is given, and a clear conclusion from this is that the series is neither an I(0) nor an I(1), but rather than an I(d) process with d ≈ 0.288.

End-of-Sample Instability Tests

This paper considers tests for structural instability of short duration, such as at the end of the sample. The key feature of the testing problem is that the number, m, of observations in the period of potential change is relatively small—possibly as small as one. The well-known F test of Chow (1960) for this problem only applies in a linear regression model with normally distributed iid errors and strictly exogenous regressors, even when the total number of observations, n+m, is large. We generalize the F test to cover regression models with much more general error processes, regressors that are not strictly exogenous, and estimation by instrumental variables as well as least squares. In addition, we extend the F test to nonlinear models estimated by generalized method of moments and maximum likelihood. Asymptotic critical values that are valid as n→∞ with m fixed are provided using a subsampling-like method. The results apply quite generally to processes that are strictly stationary and ergodic under the null hypothesis of no structural instability.

Energy efficiency of TCP in a local wireless environment

The focus of this paper is to analyze the energy consumption performance of various versions of TCP, namely, Tahoe, Reno and NewReno, for bulk data transfer in an environment where channel errors are correlated. We investigate the performance of a single wireless TCP connection by modeling the correlated packet loss/error process (e.g., as induced by a multipath fading channel) as a first-order Markov chain. Based on a unified analytical approach, we compute the throughput and energy performance of various versions of TCP. The main findings of this study are the (1) error correlations significantly affect the energy performance of TCP (consistent with analogous conclusions for throughput), and in particular they result in considerably better performance for Tahoe and NewReno than iid errors, and (2) the congestion control mechanism implemented by TCP does a good job at saving energy as well, by backing off and idiling during error bursts. An interesting conclusion is that, unlike throughput, the energy efficiency metric may be very sensitive to the TCP version used and to the choice of the protocol parameters, so that large gains appear possible.

SMALL SAMPLE EVIDENCE OF REGRESSION QUANTILE ESTIMATES FOR STRUCTURAL MODELS: ESTIMATION AND TESTING

A small Monte Carlo study is conducted to investigate the small sample properties of regression quantiles estimates in structural econometric models with iid errors. Two versions of median regression (the 50th quantile) estimates using instrumented regressors are used, with the first stage estimated by least squares and by median regression. These estimators are compared to two stage least squares, ordinary least squares and simple median regression. It is found that the performance of the two stage regression quantile estimators is equal, if not better, than two stage least squares in terms of bias and that t tests have good size and power properties, under a variety of error distributions.

Classical and Bayesian Inference Robustness in Multivariate Regression Models

We consider a one-to-one correspondence between points z ∈ Rn - {0} and pairs (y, r), where r > 0 and y lies in some space y, through z = r y. As an immediate consequence, we can represent random variables Z that take values in Rn - {0} as Z = R Y, where R is a positive random variable and Y takes values in y. By fixing the distribution of either R or Y while imposing independence between them, we generate classes of distributions on Rn . Many families of multivariate distributions (e.g., spherical, elliptical, lq spherical, v spherical, and anisotropic) can be interpreted in this unifying framework. Some classical inference procedures can be shown to be completely robust in these classes of multivariate distributions. We use these findings in the practically relevant context of regression models. Finally, we present a robust Bayesian analysis and indicate the links between classical and Bayesian results. In particular, for the regression model with iid errors up to a scale, we provide a formal characterization for both classical and Bayesian robustness results concerning inference on the regression parameters.

Adaptive parametric test in a semiparametric regression model

Consider the semiparametric regression model Yi = x′iβ +g(ti)+ei for i=1,2, …,n. Here the design points (xi,ti) are known and nonrandom and the ei are iid random errors with Ee1 = 0 and Ee2 1 = α2<∞. Based on g(.) approximated by a B-spline function, we consider using atest statistic for testing H0 : β = 0. Meanwhile, an adaptive parametric test statistic is constructed and a large sample study for this adaptive parametric test statistic is presented.

Deconvolving a density from contaminated dependent observations

The paper studies the performance of deconvoluting kernel density estimators for estimating the marginal density of a linear process. The data stem from the linear process and are partially, respectively fully contaminated by iid errors with a known distribution. If 1−p denotes the proportion of contaminated observations (and it is, of course, unknown which observations are contaminated and which are not) then for 1−p ∈ (0, 1) and under mild conditions almost sure deconvolution rates of orderO(n−2/5(logn)9/10) can be achieved for convergence in\(\mathcal{L}_\infty\). This rate compares well with the existing rates foriid uncontaminated observations. Forp=0 and exponentially decreasing error characteristic function the corresponding rates are of merely logarithmic order. As a by-product the paper also gives a rate of convergence result for the empirical characteristic function in the linear process context and utilizes this to demonstrate that deconvoluting kernel density estimators attain the optimal rate in the dependence case with exponentially decreasing error characteristic function.

Strong Consistency of Regression Quantiles and Related Empirical Processes

The strong consistency of regression quantile statistics (Koenker and Bassett [4]) in linear models with iid errors is established. Mild regularity conditions on the regression design sequence and the error distribution are required. Strong consistency of the associated empirical quantile process (introduced in Bassett and Koenker [1]) is also established under analogous conditions. However, for the proposed estimate of the conditional distribution function of Y, no regularity conditions on the error distribution are required for uniform strong convergence, thus establishing a Glivenko-Cantelli-type theorem for this estimator.

An Empirical Quantile Function for Linear Models with iid Errors

Abstract The regression quantile statistics of Koenker and Bassett (1978) are employed to construct an estimate of the error quantile function in linear models with iid errors. Some finite sample properties and the asymptotic behavior of the proposed estimator are derived. Comparisons with procedures based on residuals are made. The stackloss data of Brownlee (1965) is reanalyzed to illustrate the technique