kinkyreg: Instrument-free inference for linear regression models with endogenous regressors

In models with endogenous regressors, a standard regression approach is to exploit just-identifying or overidentifying orthogonality conditions by using instrumental variables. In just-identified models, the identifying orthogonality assumptions cannot be tested without the imposition of other nontestable assumptions. While formal testing of overidentifying restrictions is possible, its interpretation still hinges on the validity of an initial set of untestable just-identifying orthogonality conditions. We present the kinkyreg command for kinky least-squares inference, which adopts an alternative approach to identification. By exploiting nonorthogonality conditions in the form of bounds on the admissible degree of endogeneity, feasible test procedures can be constructed that do not require instrumental variables. The kinky least-squares confidence bands can be more informative than confidence intervals obtained from instrumental-variables estimation, especially when the instruments are weak. Moreover, the approach facilitates a sensitivity analysis for standard instrumental-variables inference. In particular, it allows the user to assess the validity of previously untestable just-identifying exclusion restrictions. Further instrument-free tests include linear hypotheses, functional form, heteroskedasticity, and serial correlation tests.

On Five-Diagonal Splitting for Cubic Spline Wavelets with Six Vanishing Moments on a Segment

In this study, we use the vanishing property of the first six moments for constructing a splitting algorithm for cubic spline wavelets. First, we construct the corresponding wavelet space that satisfies the orthogonality conditions for all fifth-degree polynomials. Then, using the homogeneous Dirichlet boundary conditions, we adapt spaces to the closed interval. The originality of the study consists in obtaining implicit relations connecting the coefficients of the spline decomposition at the initial scale with the spline coefficients and wavelet coefficients at the nested scale by a tape system of linear algebraic equations with a non-degenerate matrix. After excluding the even rows of the system, in contrast to the case with two zero moments, the resulting transformation matrix has five (instead of three) diagonals. The results of numerical experiments on calculating the derivatives of a discrete function are presented.

Fast computation and practical use of amplitudes at non-Fourier frequencies

In this paper, it is shown that the performance of various frequency-domain estimators of the memory parameter can be boosted by the inclusion of non-Fourier frequencies in addition to the regular Fourier frequencies. A fast two-stage algorithm for the efficient computation of the amplitudes at these additional frequencies is presented. In the first stage, the naïve sine and cosine transforms are computed with a modified version of the Fast Fourier Transform. In the second stage, these transforms are amended by taking the violation of the standard orthogonality conditions into account. A considerable number of auxiliary quantities, which are required in the second stage, do not depend on the data and therefore only need to be computed once. The superior performance (in terms of root-mean-square error) of the estimators based also on non-Fourier frequencies is demonstrated by extensive simulations. Finally, the empirical results obtained by applying these estimators to financial high-frequency data show that significant long-range dependence is present only in the absolute intraday returns but not in the signed intraday returns.

Multiple Meixner Polynomials on a Non-Uniform Lattice

We consider two families of type II multiple orthogonal polynomials. Each family has orthogonality conditions with respect to a discrete vector measure. The r components of each vector measure are q-analogues of Meixner measures of the first and second kind, respectively. These polynomials have lowering and raising operators, which lead to the Rodrigues formula, difference equation of order r+1, and explicit expressions for the coefficients of recurrence relation of order r+1. Some limit relations are obtained.

A Breakdown-Free Block COCG Method for Complex Symmetric Linear Systems with Multiple Right-Hand Sides

The block conjugate orthogonal conjugate gradient method (BCOCG) is recognized as a common method to solve complex symmetric linear systems with multiple right-hand sides. However, breakdown always occurs if the right-hand sides are rank deficient. In this paper, based on the orthogonality conditions, we present a breakdown-free BCOCG algorithm with new parameter matrices to handle rank deficiency. To improve the spectral properties of coefficient matrix A, a precondition version of the breakdown-free BCOCG is proposed in detail. We also give the relative algorithms for the block conjugate A-orthogonal conjugate residual method. Numerical results illustrate that when breakdown occurs, the breakdown-free algorithms yield faster convergence than the non-breakdown-free algorithms.

A generalized Walsh system for arbitrary matrices

In this paper we study in detail a variation of the orthonormal bases (ONB) of L2[0, 1] introduced in [Dutkay D. E., Picioroaga G., Song M. S., Orthonormal bases generated by Cuntz algebras, J. Math. Anal. Appl., 2014, 409(2), 1128-1139] by means of representations of the Cuntz algebra ON on L2[0, 1]. For N = 2 one obtains the classic Walsh system which serves as a discrete analog of the Fourier system. We prove that the generalized Walsh system does not always display periodicity, or invertibility, with respect to function multiplication. After characterizing these two properties we also show that the transform implementing the generalized Walsh system is continuous with respect to filter variation. We consider such transforms in the case when the orthogonality conditions in Cuntz relations are removed. We show that these transforms which still recover information (due to remaining parts of the Cuntz relations) are suitable to use for signal compression, similar to the discrete wavelet transform.

Generic behavior of a measure-preserving transformation

Del Junco–Lemańczyk [Generic spectral properties of measure-preserving maps and applications. Proc. Amer. Math. Soc., 115 (3) (1992)] showed that a generic measure-preserving transformation satisfies certain orthogonality conditions. More precisely, there is a dense $G_{\unicode[STIX]{x1D6FF}}$ subset of measure preserving transformations such that, for every $T\in G$ and $k(1),k(2),\ldots ,k(l)\in \mathbb{Z}^{+}$, $k^{\prime }(1),k^{\prime }(2),\ldots ,k^{\prime }(l^{\prime })\in \mathbb{Z}^{+}$, the convolutions $$\begin{eqnarray}\unicode[STIX]{x1D70E}_{T^{k(1)}}\ast \cdots \ast \unicode[STIX]{x1D70E}_{T^{k(l)}}\quad \text{and}\quad \unicode[STIX]{x1D70E}_{T^{k^{\prime }(1)}}\ast \cdots \ast \unicode[STIX]{x1D70E}_{T^{k^{\prime }(l^{\prime })}},\end{eqnarray}$$ where $\unicode[STIX]{x1D70E}_{T^{k}}$ is the maximal spectral type of $T^{k}$, are mutually singular, provided that $(k(1),k(2),\ldots ,k(l))$ is not a rearrangement of $(k^{\prime }(1),k^{\prime }(2),\ldots ,k^{\prime }(l^{\prime }))$. We will introduce analogous orthogonality conditions for continuous unitary representations of the group of all measurable functions with values in the circle, $L^{0}(\unicode[STIX]{x1D707},\mathbb{T})$, which we denote by the DL-condition. We connect the DL-condition with a result of Solecki [Unitary representations of the groups of measurable and continuous functions with values in the circle. J. Funct. Anal., 267 (2014), pp. 3105–3124] which identifies continuous unitary representations of $L^{0}(\unicode[STIX]{x1D707},\mathbb{T})$ with a collection of measures $\{\unicode[STIX]{x1D706}_{\unicode[STIX]{x1D705}}\}$, where $\unicode[STIX]{x1D705}$ runs over all increasing finite sequence of non-zero integers. In particular, we show that the ‘probabilistic’ DL-condition translates to ‘deterministic’ orthogonality conditions on the measures $\unicode[STIX]{x1D706}_{\unicode[STIX]{x1D705}}$. As a corollary, we show that the same orthogonality conditions as in the result by Del Junco–Lemańczyk hold for a generic unitary operator on a separable infinite-dimensional Hilbert space.

Linear maps on *-algebras acting on orthogonal elements like derivations or anti-derivations

Let U be a unital *-algebra and ? : U ? U be a linear map behaving like a derivation or an anti-derivation at the following orthogonality conditions on elements of U: xy = 0, xy* = 0, xy = yx = 0 and xy* = y*x = 0. We characterize the map ? when U is a zero product determined algebra. Special characterizations are obtained when our results are applied to properly infinite W*-algebras and unital simple C*-algebras with a non-trivial idempotent.

A constrained optimization solution for Caughey damping coefficients in seismic analysis

Purpose Extended from the classic Rayleigh damping model in structural dynamics, the Caughey damping model allows the damping ratios to be specified in multiple modes while satisfying the orthogonality conditions. Despite these desirable properties, Caughey damping suffers from a few major drawbacks: depending on the frequency distribution of the significant modes, it can be difficult to choose the reference frequencies that ensure reasonable values for all damping ratios corresponding to the significant modes; it cannot ensure all damping ratios are positive. This paper aims to present a constrained quadratic programming approach to address these issues. Design/methodology/approach The new method minimizes the error of the structural displacement peak based on the response spectrum theory, while all modal damping ratios are constrained to be greater than zero. Findings Several comprehensive examples are presented to demonstrate the accuracy and effectiveness of the proposed method, and comparisons with existing approaches are provided whenever possible. Originality/value The proposed method is highly efficient and allows the damping ratios to be conveniently specified for all significant modes, producing optimal damping coefficients in practical applications.