**Pierre Perron**

**WORKING PAPERS **

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.

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 November 2017.

，
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.

Testing for
Common Breaks in a Multiple Equations System (with Tatsushi
Oka), July 2011; Revised January 2018. Forthcoming in
the *Journal of Econometrics*.

，
The
issue addressed in this paper is that of testing for common breaks across or
within equations of a multivariate system. Our framework is very general and
allows integrated regressors and trends as well as
stationary regressors. The null hypothesis is that
breaks in different parameters occur at common locations and are separated by
some positive fraction of the sample size unless they occur across different
equations. Under the alternative hypothesis, the break dates across parameters
are not the same and also need not be separated by a positive fraction of the
sample size whether within or across equations. The test considered is the
quasi-likelihood ratio test assuming normal errors, though as usual the limit
distribution of the test remains valid with non-normal errors. Of independent
interest, we provide results about the rate of convergence of the estimates
when searching over all possible partitions subject only to the requirement
that each regime contains at least as many observations as some positive
fraction of the sample size, allowing break dates not separated by a positive
fraction of the sample size across equations. Simulations show that the test
has good finite sample properties. We also provide an application to issues
related to level shifts and persistence for various measures of inflation to
illustrate its usefulness.

__ A
Comparison of Alternative Methods to Construct Confidence Intervals for the
Estimate of a Break Date in Linear Regression Models__ (with Seong
Yeon Chang), Revised October 2015. Forthcoming in

，
This
paper considers constructing confidence intervals for the date of a structural
break in linear regression models. Using extensive simulations, we compare the
performance of various procedures in terms of exact coverage rates and lengths
of the confidence intervals. These include the procedures of Bai (1997) based
on the asymptotic distribution under a shrinking shift framework, Elliott and
M┨ller (2007) based on inverting a test locally invariant to the magnitude of
break, Eo and Morley (2014) based on inverting a
likelihood ratio test, and various bootstrap procedures. On the basis of
achieving an exact coverage rate that is closest to the nominal level, Elliott
and M┨ller's (2007) approach is by far the best one. However, this comes with a
very high cost in terms of the length of the confidence intervals. When the
errors are serially correlated and dealing with a change in intercept or a
change in the coefficient of a stationary regressor
with a high signal to noise ratio, the length of the confidence interval
increases and approaches the whole sample as the magnitude of the change
increases. The same problem occurs in models with a lagged dependent variable,
a common case in practice. This drawback is not present for the other methods,
which have similar properties. Theoretical results are provided to explain the
drawbacks of Elliott and M┨ller's (2007) method.

Combining Long Memory and Level Shifts in Modeling and Forecasting
the Volatility of Asset Returns (with Rasmus T. Varneskov),
Revised April 2017 (supplementary
material). Forthcoming in *Quantitative
Finance*.

，
We provide a
framework for modeling and forecasting the volatility of asset returns. In
particular, we propose a parametric state space model with an accompanying
estimation and forecasting framework that allows for ARFIMA dynamics, random
level shifts and measurement errors. The Kalman _lter is used to construct the likelihood function after
augmenting the probability of states by a mixture of normally distributed
processes. A new forecasting framework for random level shift models is
proposed, which utilizes the information in the Kalman
recursions to generate mean- and path-corrected forecasts. We apply our model
to eight daily volatility series constructed from: (1) Tick-by-tick trades on
the BAC, MRK and SPY stocks, (2) one-minute returns on
S&P 500 and 10-year Treasury Bond futures contracts, and (3) daily returns
on the USD-AUD, USD-CHF and USD-YEN exchange rates. The full sample parameter
estimates reveal that random level shifts are present in all series. A genuine
long memory component is present in the measures of volatility constructed
using high-frequency data. On the other hand, the residual dynamics of the
volatility series proxied by log-daily absolute
returns may be characterized as a combination of short memory dynamics and
measurement errors. We conduct extensive out-of-sample forecast evaluations and
compare the results with six popular models in the literature. Interestingly,
our ARFIMA model with random level shifts is the only model that consistently
belongs to the 10% Model Confidence Set of Hansen et al. (2011) across a
variety of forecast periods, forecast horizons, asset classes, and volatility
measures. The gains in forecast accuracy can be very pronounced, especially at
longer horizons.

Forecasting in the presence of in and out of sample breaks (with Jiawen Xu), Revised January 30, 2017.

，
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.

Robust testing of time trend and mean with unknown integration order errors (with Jiawen Xu); March 2013.

，
We
provide tests to perform inference on the coefficients of a linear trend
assuming the noise to be a fractionally integrated process with memory
parameter d（(-0.5,1.5) by applying a
quasi-GLS procedure using d-differences of the data. Doing so, the error term
is short memory, the asymptotic distribution of the OLS estimators applied to
quasi-differenced data and their t-statistics are unaffected by the value of d
and standard procedures have a limit normal distribution. No truncation or
pre-test is needed given the continuity with respect to d.
To have feasible tests, we use the Exact Local Whitlle
estimator of Shimotsu (2010), valid for processes
with a linear trend. The finite sample size and power of the tests are
investigated via simulations. We also provide a comparison with the tests of
Perron and Yabu (2009) valid for a noise component that is I(0)
or I(1). The results are encouraging in that our test is valid under more
general conditions, yet has similar power as those that apply to the
dichotomous cases with d either 0 or 1. We apply our tests to construct
confidence intervals for the growth rate of temperature series pre and post
1960, which show that the slope is significantly higher in the post-1960 period
consistent with global warming.

Breaks, trends and the
attribution of climate change: a time-series analysis (with Francisco Estrada), March
2012.

，
Climate
change detection and attribution have been the subject
of intense research and debate over at least four decades. However, direct
attribution of climate change to anthropogenic activities using observed
climate and forcing variables through statistical methods has remained elusive,
partly caused by the difficulties for correctly identifying the time-series
properties of these variables and by the limited availability of methods for
relating nonstationary variables. This paper provides strong evidence
concerning the direct attribution of observed climate change to anthropogenic
greenhouse gases emissions by first investigating the univariate time-series properties
of observed global and hemispheric temperatures and forcing variables and then
by proposing statistically adequate multivariate models. The results show that
there is a clear anthropogenic fingerprint on both global and hemispheric
temperatures. The signal of the well-mixed GHG forcing in all temperature
series is very clear and accounts for most of their secular movement since the
beginning of observations. Both temperature and forcing variables are
characterized by piecewise linear trends with abrupt changes in their slopes
estimated to occur at different dates. Nevertheless, their long-term movements
are so closely related that the observed temperature and forcing trends cancel
out. The warming experimented during the last century was mainly due to the
increase in GHG which was partially offset by the effect of tropospheric
aerosols. Other forcing sources, such as solar, are shown to only contribute to
(shorter-term) variations around the GHG forcing trend.

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**.**