Probability and Measure

https://link.springer.com/content/pdf/10.1057/jors.1979.126.pdf

Forecasting Economic Time Series

Many common approaches to detecting changepoints, for example based on statistical criteria such as penalised likelihood or minimum description length, can be formulated in terms of minimising a cost over segmentations. We focus on a class of dynamic programming algorithms that can solve the resulting minimisation problem exactly, and thus find the optimal segmentation under the given statistical criteria. The standard implementation of these dynamic programming methods have a computational cost that scales at least quadratically in the length of the time-series. Recently pruning ideas have been suggested that can speed up the dynamic programming algorithms, whilst still being guaranteed to be optimal, in that they find the true minimum of the cost function. Here we extend these pruning methods, and introduce two new algorithms for segmenting data: FPOP and SNIP. Empirical results show that FPOP is substantially faster than existing dynamic programming methods, and unlike the existing methods its computational efficiency is robust to the number of changepoints in the data. We evaluate the method for detecting copy number variations and observe that FPOP has a computational cost that is even competitive with that of binary segmentation, but can give much more accurate segmentations.

/pdf/on-optimal-multiple-changepoint-algorithms-for-large-data-2evhv3ua8b.pdf

On optimal multiple changepoint algorithms for large data

This chapter covers methodological issues related to estimation, testing and computation for models involving structural changes Our aim is to review developments as they relate to econometric applications based on linear models Substantial advances have been made to cover models at a level of generality that allow a host of interesting practical applications These include models with general stationary regressors and errors that can exhibit temporal dependence and heteroskedasticity, models with trending variables and possible unit roots and cointegrated models, among others Advances have been made pertaining to computational aspects of constructing estimates, their limit distributions, tests for structural changes, and methods to determine the number of changes present A variety of topics are covered The first part summarizes and updates developments described in an earlier review, Perron (2006), with the exposition following heavily that of Perron (2008) Additions are included for recent developments: testing for common breaks, models with endogenous regressors (emphasizing that simply using least-squares is preferable over instrumental variables methods), quantile regressions, methods based on Lasso, panel data models, testing for changes in forecast accuracy, factors models and methods of inference based on a continuous records asymptotic framework Our focus is on the so-called off-line methods whereby one wants to retrospectively test for breaks in a given sample of data and form confidence intervals about the break dates The aim is to provide the readers with an overview of methods that are of direct usefulness in practice as opposed to issues that are mostly of theoretical interest

Structural Breaks in Time Series

This book is fundamentally about the estimation of risk. At an intuitive level, risk is easy to understand: given an asset with a current price of say $100, what is the likelihood that at some future time—the risk horizon—its price will be less than $90? Or more than $120? And how does the probability of observing a price below $90 change as a function of the risk horizon? Such heuristic notions can be formalized in several different ways. Suppose that the (log) price of the asset follows Xt “ X0 ` σWt where Wt is a standard Brownian motion and σ ą 0 is a parameter quantifying the risk of X, which needs to be estimated. Suppose also that the price is observed at a sequence of fixed points in time, say at the end of each trading day. Then a simple measure of risk is the squared change in the observed price from one day to the next, possibly averaged over several adjacent intervals to obtain a smoothed estimate—this is, of course, the (daily) variance. As the number of intervals increases, and assuming the statistical properties of the underlying price process remain unchanged, this estimator eventually yields the true value of σ (or in fact its square). But asymptotic results can be considered in another sense: suppose that the total length of the observation interval is fixed (say one day) but the distance between observations within the interval becomes increasingly small—that is, one is in the realm of high-frequency data. Then another way to estimate risk on the interval is to sum the squared incremental price changes. This estimator is called the realized volatility, and it is this estimator, along with its variants and applications, that form the main focus of the book. Formally, if the observation interval has length T and the distance between observations is ∆ then the realized volatility is:

High-Frequency Financial Econometrics

Continuous record Laplace-based inference about the break date in structural change models

For a partial structural change in a linear regression model with a single break, we develop a continuous record asymptotic framework to build interference methods for the break date. We have T observations with a sampling frequency h over a fixed time horizon [0, N], and let T â â with h â 0 wile keeping the time span N fixed. We impose very mild regularity conditions on an underlying continuous-time model assumed to generate the data. We consider the least-squares estimate of the break date and establish consistency and convergence rate. We providea limit theory for shrinking magnitudes of shifts and locally increasing variances. The asymptotic distribution corresponds to the location of the extremum of a function of the quadratic variation of the regressors and of a Gaussian centered martingale process over a certain time interval. We can account for the asymmetric informational content provided by the pre- and post-break regimes and show how the location of the break and shift magnitude are key ingredients in shaping the distribution. We consider a feasible version based on plug-in estimates, which provides a very good approximation to the finite sample distribution. We use the concept of Highest Density Region to construct confidence sets. Overall, our method is reliable and delivers accurate coverage probabilities and relatively short average length of the confidence sets. Importantly, it does so irrespective of the size of the break.

Continuous record asymptotics for structural change models

We develop a novel continuous-time asymptotic framework for inference on whether the predictive ability of a given forecast model remains stable over time. We formally define forecast instability from the economic forecaster's perspective and highlight that the time duration of the instability bears no relationship with stable period. Our approach is applicable in forecasting environment involving low-frequency as well as high-frequency macroeconomic and financial variables. As the sampling interval between observations shrinks to zero the sequence of forecast losses is approximated by a continuous-time stochastic process (i.e., an Ito semimartingale) possessing certain pathwise properties. We build an hypotheses testing problem based on the local properties of the continuous-time limit counterpart of the sequence of losses. The null distribution follows an extreme value distribution. While controlling the statistical size well, our class of test statistics feature uniform power over the location of the forecast failure in the sample. The test statistics are designed to have power against general form of insatiability and are robust to common forms of non-stationarity such as heteroskedasticty and serial correlation. The gains in power are substantial relative to extant methods, especially when the instability is short-lasting and when occurs toward the tail of the sample.

Tests for Forecast Instability and Forecast Failure under a Continuous Record Asymptotic Framework

Under the classical long-span asymptotic framework we develop a class of Generalized Laplace (GL) inference methods for the change-point dates in a linear time series regression model with multiple structural changes analyzed in, e.g., Bai and Perron (1998). The GL estimator is defined by an integration rather than optimization-based method and relies on the least-squares criterion function. It is interpreted as a classical (non-Bayesian) estimator and the inference methods proposed retain a frequentist interpretation. Since inference about the change-point dates is a nonstandard statistical problem, the origional insight of Laplace to interpret a certain transformation of a least-squares criterion function as a statistical believe over the parameter space provides a better approximation about the uncertainty in the data about the change-points relative to existing methods. Simulations show that the GL estimator is in general more precise than the OLS estimator. On the theoretical side, depending on some input (smoothing) parameter, the class of GL estimators exhibits a dual limiting distribution; namely, the classical shrinkage asymptotic distribution of Bai an Perron (1998), or a Bayes-type asymptotic distribution.

Alessandro Casini

Papers

Structural Breaks in Time Series

Continuous record Laplace-based inference about the break date in structural change models

Continuous record asymptotics for structural change models

Tests for Forecast Instability and Forecast Failure under a Continuous Record Asymptotic Framework

Generalized Laplace inference in multiple change-points models