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Tests based on the CUSUM statistics were originally introduced to investigate whether a regression relationship is stable over time Brown and Durbin Brown, Durbin, and Evans test constancy of the regression coefficients based on the cumulated sum of squared recursive residuals.
Studies in Nonlinear Dynamics & Econometrics
Although used for different purposes, it should be emphasized that the cointegration models studied by Hao and Inder and Xiao and Phillips have the same behavior under the null hypotheses, but have different behaviors under the alternatives.
This paper extends the analysis of Xiao and Phillips to the case of conditional quantiles, since the long run relationship among nonstationary time series may not be uniform. In practice, locations in the distribution other than the mean may matter for cointegration analysis. To examine the equilibrium relationships across different quantiles of the distribution of the response variable, the CUSUM test is employed to test the null hypothesis of quantile cointegration. Similar to the OLS regression, in the quantile regression with I 1 regressors, due to serial correlation between the regression disturbance and the innovation of the integrated regressors and long run endogeneity in the data, the quantile estimator for the cointegrating coefficients is second-order biased and depends on nuisance parameters.
macroeconomics - Why Is Cointegration Important In Practice? - Economics Stack Exchange
In this case, it is difficult to make inference. To solve this problem, this paper uses a Phillips-Hansen type fully modified quantile estimator. The resulting limit distribution is mixed normal so that it provides a standard inference procedure. The CUSUM test statistic is composed of partial sums of the residuals from the fully modified quantile regression. Under the null of quantile cointegration, the test statistic has the same limit distribution as that from Xiao and Phillips A great number of empirical papers find that the interest rate process is I 1 and test for cointegration among interest rates of different terms to maturity.
The seminal paper of Campbell and Shiller demonstrates that present value models of the term structure imply cointegration of short- and long-term interest rates. Engle and GrangerStock and WatsonBootheHansenHall, Anderson, and GrangerMandeno and Gilesand Downing and Oliner among others also discuss the expectations theory of the term structure of interest rates.
A more complete review of work on the expectations hypothesis of the term structure is provided in Iacone These papers consider various US interest rate series, including different yield series on the federal funds rate, Treasury bill rate, and commercial paper rate. The results are mixed. This paper applies the residual based quantile cointegration test to several sets of US interest rate data.
The quantile version of the CUSUM test rejects the expectations hypothesis of the term structure in certain quantiles of the interest rate distributions. The remainder of this paper is organized as follows: In Section 2, the model is set up.
The asymptotic theory for the fully modified quantile estimator and test statistic is developed. Finally, we analyze a set of EEG recordings and discuss the current applicability of cointegration analysis in the field of neuroscience. To a statistician this presents a fascinating challenge of modelling complex behavior in large scale systems and how to infer the data-generating mechanisms.
Many innovative ideas have been presented since Winfree began a mathematical treatment of the subject. When Kuramoto first presented his model of coupled oscillators, this made a huge impact in the field and spawned a new generation of research on synchronization. A long standing problem in neuroscience is to recover the network structure in a coupled system. This could for example be to infer the functional connectivity between units in a network of neurons from multiple extracellularly recorded spike trains, or how traces of EEG signals from different locations on the scalp affect each other, which we will treat in this paper.
To the authors knowledge, this challenge is still lacking a sound statistical framework to model and test for interaction in a system, as well as impose statistical hypotheses on the network structure.
Oscillating systems with cointegrated phase processes
For this task, cointegration analysis offers a refined statistical toolbox, where detailed information on the connections can be inferred, such as the direction and proportional strength of the coupling. The theory of cointegration was originally conceived by Grangerand has since then also been the subject of intense research, most notably within the field of econometrics.
In the monograph by Johansenthe full likelihood theory for linear cointegration models with Gaussian i. This well acknowledged framework is popularly termed the Johansen procedure. Even though cointegration analysis has developed from within the field of econometrics, it may potentially be used for different models outside economics, such as biological models in continuous time as we explore here.