**Pierre Perron**

**WORKING PAPERS (all
files in pdf format) (some of this work was supported by the National Science
Foundation under Grant No.** 0649350
and 0078492**)**

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 May 2017 (supplementary
material).

，
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".