Natural Cubic Spline Model for Estimating Volatility

Volatility measures the dispersion of returns for a market variable since a reasonable estimation of the volatility is an appropriate starting point for assessing investment risks and monetary policymaking. These risks are usually assessed by using the GARCH (1,1) model. However, the recursive term in this model makes finding the derivatives of the likelihood function mathematically intractable. In this study, the natural cubic spline model is used to estimate the volatility by fitting it to the absolute returns of the data. In estimating the parameters, the Maximum Likelihood method was applied while a simple algebra was used to find its derivatives. The damped Newton-Raphson method was then used to maximize the likelihood function with the R programming software. The proposed method was illustrated using the absolute returns of the crude oil prices data from West Texas Intermediate, and it showed similar results with the popular GARCH (1,1) model. The natural cubic spline can be an alternative for estimating the volatility of any financial time series data

Interpolating the Trace of the Inverse of Matrix A+tB\mathbf{A} + t \mathbf{B}

We develop heuristic interpolation methods for the function $t \mapsto \operatorname{trace}\left( (\mathbf{A} + t \mathbf{B})^{-1} \right)$, where the matrices $\mathbf{A}$ and $\mathbf{B}$ are symmetric and positive definite and $t$ is a real variable. This function is featured in many applications in statistics, machine learning, and computational physics. The presented interpolation functions are based on the modification of a sharp upper bound that we derive for this function, which is a new trace inequality for matrices. We demonstrate the accuracy and performance of the proposed method with numerical examples, namely, the marginal maximum likelihood estimation for linear Gaussian process regression and the estimation of the regularization parameter of ridge regression with the generalized cross-validation method

Efficient algorithms for robust generalized cross-validation spline smoothing

Generalized cross-validation (GCV) is a widely used parameter selection criterion for spline smoothing, but it can give poor results if the sample size n is not sufficiently large. An effective way to overcome this is to use the more stable criterion called robust GCV (RGCV). The main computational effort for the evaluation of the GCV score is the trace of the smoothing matrix, tr A, while the RGCV score requires both tr A and tr A(2). Since 1985, there has been an efficient O(n) algorithm to compute tr A. This paper develops two pairs of new O(n) algorithms to compute tr A and tr A(2), which allow the RGCV score to be calculated efficiently. The algorithms involve the differentiation of certain matrix functionals using banded Cholesky decomposition