**Pierre Perron**

**WORKING PAPERS **

Testing
for Changes in Forecasting Performance (with Yohei Yamamoto), May 2018,
revised December 2018, submitted.

，
We
consider the issue of forecast failure (or breakdown) and propose methods to
assess retrospectively whether a given forecasting model provides forecasts
which show evidence of changes with respect to some loss function. We adapt the
classical structural change tests to the forecast failure context. First, we
recommend that all tests should be carried with a fixed scheme to have best
power. This ensures a maximum difference between the fitted in and out-of sample means of the losses and avoids contamination issues
under the rolling and recursive schemes. With a fixed scheme, Giacomini and
Rossi's (2009) (GR) test is simply a Wald test for a one-time change in the
mean of the total (the in-sample plus out-of-sample) losses at a known break
date, say m, the value that
separates the in and out-of-sample periods. To
alleviate this problem, we consider a variety of tests: maximizing the GR test
over all possible values of m within a pre-specified range; a Double sup-Wald
(DSW) test which for each m performs a sup-Wald test for a change in the mean
of the out-of-sample losses and takes the maximum of such tests over some
range; we also propose to work directly with the total loss series to define
the Total Loss Sup-Wald test (TSLW) and the Total Loss UDmax
test (TLUD). Using extensive simulations, we show that with forecasting models
potentially involving lagged dependent variables, the only tests having a
monotonic power function for all Data Generating Processes are the DSW and TLUD
tests, constructed with a fixed forecasting window scheme. Some explanations
are provided and two empirical applications illustrate the relevance of our
findings in practice.

Causality from Long-lived Radiative Forcings
to the Climate Trend (with Francisco Estrada),
forthcoming in the *Annals of the New York
Academy of Sciences*.

，
We
present a purely statistical observations-based model-free analysis that
provides evidence about causality, defined as in Granger (1969), from
long-lived radiative forcings (LLRF) to the climate
trend. This relies on having locally ordered breaks in the slopes of the trend
functions of LLRF and the climate trend, with the break for LLRF occurring
before that of the climate trend and with the slope changes being of the same
sign. The empirical evidence indicates that these conditions are satisfied
empirically using standard global surface temperature series and an aggregate
measure of LLRF (CO2, N2O, and CFCs). We also discuss why the presence of
broken trends can lead one to conclude in favor of Granger causality when using
standard methods even if the noise function in LLRF is negligible.

Structural Breaks in Time Series (with
Alessandro Casini), forthcoming in the *Oxford Research Encyclopedia of Economics
and Finance*.

，
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.

Continuous Record Asymptotics for
Structural Change Models (with Alessandro Casini), November 2017, submitted.

，
For
a partial structural change in a linear regression model with a single break,
we develop a continuous record asymptotic framework to build inference methods
for the break date. We have T observations with a sampling frequency h over a
fixed time horizon [0, N] , and let T ★ ± with h ◎ 0
while 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 provide a 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 Laplace-based Inference about the Break Date in Structural Change Models
(with Alessandro Casini), December 2017, submitted.

，
Building
upon the continuous record asymptotic framework recently introduced by Casini and Perron (2017a) for inference in structural
change models, we propose a Laplace-based (Quasi-Bayes) procedure for the
construction of the estimate and confidence set for the date of a structural
change. The procedure relies on a Laplace-type estimator defined by an integration-based
rather than an optimization-based method. A transformation of the least-squares
criterion function is evaluated in order to derive a proper distribution,
referred to as the Quasi-posterior. For a given choice of a loss function, the
Laplace-type estimator is defined as the minimizer of the expected risk with
the expectation taken under the Quasi-posterior. Besides providing an
alternative estimate that is more precise！lower mean absolute error (MAE) and
lower root-mean squared error (RMSE)！than the usual least-squares one, the
Quasi-posterior distribution can be used to construct asymptotically valid
inference using the concept of Highest Density Region. The resulting
Laplace-based inferential procedure proposed is shown to have lower MAE and
RMSE, and the confidence sets strike the best balance between empirical
coverage rates and average lengths of the confidence sets relative to
traditional long-span methods, whether the break size is small or large.

Generalized Laplace Inference in Multiple Change-Points Models (with Alessandro Casini), March 2018,
submitted.

，
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 original insight
of Laplace to interpret a certain transformation of a least-squares criterion
function as a statistical belief 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 and Perron (1998), or a Bayes-type asymptotic distribution.

Inference
Related to Common Breaks in a Multivariate System with Joined Segmented Trends
with Applications to Global and Hemispheric Temperatures (with Dukpa Kim, Tatsushi Oka and Francisco Estrada), January
2017; Revised April 2018. Forthcoming in the *Journal of Econometrics*.

，
What
transpires from recent research is that temperatures and forcings
seem to be characterized by a linear trend with two changes in the rate of
growth. The first occurs in the early 60s and indicates a very large increase
in the rate of growth of both temperatures and radiative forcings.
This was termed as the "onset of sustained global warming". The
second is related to the more recent so-called hiatus period, which suggests
that temperatures and total radiative forcings have
increased less rapidly since the mid-90s compared to the larger rate of
increase from 1960 to 1990. There are two issues that remain unresolved. The
first is whether the breaks in the slope of the trend functions of temperatures
and radiative forcings are common. This is important
because common breaks coupled with the basic science of climate change would
strongly suggest a causal effect from anthropogenic factors to temperatures.
The second issue relates to establishing formally via a proper testing
procedure that takes into account the noise in the series, whether there was
indeed a `hiatus period' for temperatures since the mid 90s.
This is important because such a test would counter the widely held view that
the hiatus is the product of natural internal variability. Our paper provides
tests related to both issues. The results show that the breaks in temperatures
and forcings are common and that the hiatus is
characterized by a significant decrease in the rate of growth of temperatures
and forcings. The statistical results are of
independent interest and applicable more generally.

Forecasting in the presence of in and out of sample breaks (with Jiawen Xu), Revised November, 2018.

，
We
present a frequentist-based approach to forecast time series in the presence of
in-sample and out-of-sample breaks in the parameters of the forecasting model.
We first model the parameters as following a random level shift process, with
the occurrence of a shift governed by a Bernoulli process. In order to have a
structure so that changes in the parameters be
forecastable, we introduce two modifications. The first models the probability
of shifts according to some covariates that can be forecasted. The second
incorporates a built-in mean reversion mechanism to the time path of the
parameters. Similar modifications can also be made to model changes in the
variance of the error process. Our full model can be cast into a non-linear
non-Gaussian state space framework. To estimate it, we use particle filtering
and a Monte Carlo expectation maximization algorithm. Simulation results show
that the algorithm delivers accurate in-sample estimates,
in particular the filtered estimates of the time path of the parameters follow
closely their true variations. We provide a number of empirical applications
and compare the forecasting performance of our approach with a variety of
alternative methods. These show that substantial gains in forecasting accuracy
are obtained.

Temporal
Aggregation, Bandwidth Selection and Long Memory for Volatility Models
(with Wendong Shi), June 2014.

，
The
effects of temporal aggregation and choice of sampling frequency are of great
interest in modeling the dynamics of asset price volatility. We show how the
squared low-frequency returns can be expressed in terms of the temporal
aggregation of a high-frequency series. Based on the theory of temporal
aggregation, we provide the link between the spectral density function of the
squared low-frequency returns and that of the squared high-frequency returns.
Furthermore, we analyze the properties of the spectral density function of
realized volatility series, constructed from squared returns with different
frequencies under temporal aggregation. Our theoretical results allow us to
explain some findings reported recently and uncover new features of volatility
in financial market indices. The theoretical findings are illustrated via the
analysis of both low-frequency daily S&P 500 returns from 1928 to 2011 and
high-frequency 1-minute S&P 500 returns from 1986 to 2007.

Testing Jointly for Structural Changes in the Error Variance and
Coefficients of a Linear Regression Model (with Jing Zhou), July 2008.

，
We
provide a comprehensive treatment of the problem of testing jointly for
structural change in both the regression coefficients and the variance of the errors
in a single equation regression involving stationary regressors.
Our framework is quite general in that we allow for general mixing-type regressors and the assumptions imposed on the errors are
quite mild. The errors' distribution can be non-normal and conditional heteroskedasticity is permissable.
Extensions to the case with serially correlated errors are also treated. We
provide the required tools for addressing the following testing problems, among
others: a) testing for given numbers of changes in regression coefficients and
variance of the errors; b) testing for some unknown number of changes less than
some pre-specified maximum; c) testing for changes in variance (regression
coefficients) allowing for a given number of changes in regression coefficients
(variance); and d) estimating the number of changes present. These testing
problems are important for practical applications as witnessed by recent
interests in macroeconomics and finance for which documenting structural change
in the variability of shocks to simple autoregressions
or vector autoregressive models has been a concern.

Testing
for Breaks in Coefficients and Error Variance: Simulations and Applications (with
Jing Zhou), July 2008.

，
In
a companion paper, Perron and Zhou (2008) provided a comprehensive treatment of
the problem of testing jointly for structural change in both the regression
coefficients and the variance of the errors in a single equation regression model
involving stationary regressors, allowing the break
dates for the two components to be different or overlap. The aim of this paper
is twofold. First, we present detailed simulation analyses to document various
issues related to their procedures: a) the inadequacy of the two step
procedures that are commonly applied; b) which particular version of the
necessary correction factor exhibits better finite sample properties; c)
whether applying a correction that is valid under more general conditions than necessary
is detrimental to the size and power of the tests; d) the finite sample size
and power of the various tests proposed; e) the performance of the sequential
method in determining the number and types of breaks present. Second, we apply
their testing procedures to various macroeconomic time series studied by Stock
and Watson (2002). Our results reinforce the prevalence of change in mean,
persistence and variance of the shocks to these series, and the fact that for
most of them an important reduction in variance occurred during the 1980s. In
many cases, however, the so-called "great moderation" should instead
be viewed as a "great reversion".

An
Analytical Evaluation of the Log-periodogram Estimate
in the Presence of Level Shifts (with Zhongjun Qu), November
2007.

，
Recently,
there has been an upsurge of interest on the possibility of confusing long
memory and structural changes in level. Many studies have shown that when a
stationary short memory process is contaminated by level shifts the estimate of
the fractional differencing parameter is biased away from zero and the autocovariance function exhibits a slow rate of decay, akin
to a long memory process. We analyze the properties of the log periodogram estimate of the memory parameter when the jump
component is specified by a simple mixture model. Our theoretical results
explain many findings reported and uncover new features. Simulations are
presented to highlight the properties of the distributions and to assess the
adequacy of our approximations. We also show the usefulness of our results to
distinguish between long memory and level shifts via an application to the
volatility of daily returns for wheat commodity futures.

Note: This is a revised version of parts of a working paper entitled "An
Analytical Evaluation of the Log-periodogram Estimate
in the Presence of Level Shifts and its Implications for Stock Returns
Volatility".

**Some of this work was
supported by the National Science Foundation under Grant No.** 0649350 and 0078492**.**