Handbook of Empirical Economics and Finance _8 pot

198 Handbook of Empirical Economics and Finance7.4 Kernel Methods with Mixed Data Types So far we have presumed that the categorical variable is of the “unordered”“nominal” data type.. 7

198 Handbook of Empirical Economics and Finance

7.4 Kernel Methods with Mixed Data Types

So far we have presumed that the categorical variable is of the “unordered”(“nominal” data type) We shall now distinguish between categorical (dis-crete) data types and real-valued (continuous) data types Also, for categor-ical data types we could have unordered or ordered (“ordinal” data type)

variables For an ordered discrete variable ˜x d, we could use Wang and vanRyzin (1981) kernel given by

We shall now refer to the unordered kernel defined in Equation 7.2 as ¯l(·) so

as to keep each kernel type separate notationally speaking We shall denotethe traditional kernels for continuous data types such as the Epanechnikov

of Gaussian kernels by W(·).

A generalized product kernel for one continuous, one unordered, and oneordered variable would be defined as follows,

Using such product kernels, we can modify any existing kernel-based method

to handle the presence of categorical variables, thereby extending the reach

of kernel methods We define K ( X i , x) to be this product, where

is the vector of bandwidths for the continuous and categorical variables

7.4.1 Kernel Estimation of a Joint Density Defined over Categorical

and Continuous Data

Estimating a joint probability/density function defined over mixed data lows naturally using these generalized product kernels For example, for one

fol-unordered discrete variable ¯x d and one continuous variable x c, our kernelestimator of the PDF would be

Nonparametric Kernel Methods for Qualitative and Quantitative Data 199

1 2 3 4 5 6 7 0.0

0.1 0.2 0.3 0.4

observa-earnings for an individual “numdep” the number of dependents (0, 1, ).

We use likelihood cross-validation to obtain the bandwidths, and the resultingestimate is presented in Figure 7.1

Note that this is indeed a case of “sparse” data for some cells (see Table 7.4),and the traditional approach would require estimation of a nonparametricunivariate density function based upon only two observations for the last cell

(c= 6)

TABLE 7.4

Summary of the ber of Dependents in theWooldridge (2002) “wage1”

200 Handbook of Empirical Economics and Finance

7.4.2 Kernel Estimation of a Conditional PDF

Let f (·) and (·) denote the joint and marginal densities of (X, Y) and X, respectively, where we allow Y and X to consist of continuous, unordered, and ordered variables For what follows we shall refer to Y as a dependent variable (i.e., Y is explained), and to X as covariates (i.e., X is the explanatory variable) We use ˆf and ˆ to denote kernel estimators thereof, and we estimate

the conditional density g( y | x) = f (x, y)/(x) by

ˆg( y | x) = ˆf(x, y)

ˆ

The kernel estimators of the joint and marginal densities f (x, y) and (x) are

described in the previous sections; see Hall, Racine, and Li (2004) for details onthe theoretical underpinnings of a data-driven method of bandwidth selectionfor this method

7.4.2.1 The Presence of Irrelevant Covariates

Hall, Racine, and Li (2004) proposed the estimator defined in Equation 7.18,but choosing appropriate smoothing parameters in this setting can be tricky,not least because plug-in rules take a particularly complex form in the case ofmixed data One difficulty is that there exists no general formula for the op-timal smoothing parameters A much bigger issue is that it can be difficult to

determine which components of X are relevant to the problem of conditional inference For example, if the jth component of X is independent of Y then that component is irrelevant to estimating the density of Y given X, and ideally

should be dropped before conducting inference Hall, Racine, and Li (2004)show that a version of least-squares cross-validation overcomes these difficul-ties It automatically determines which components are relevant and whichare not, through assigning large smoothing parameters to the latter and con-sequently shrinking them toward the uniform distribution on the respectivemarginals This effectively removes irrelevant components from contention,

by suppressing their contribution to estimator variance; they already have

very small bias, a consequence of their independence of Y Cross-validation

also gives us important information about which components are relevant; therelevant components are precisely those that cross-validation has chosen tosmooth in a traditional way, by assigning them smoothing parameters of con-ventional size Cross-validation produces asymptotically optimal smoothingfor relevant components, while eliminating irrelevant components by over-smoothing

Hall, Racine, and Li (2004) demonstrate that, for irrelevant conditioning

variables in X, their bandwidths in fact ought to behave exactly the site, namely, h → ∞ as n → ∞ for optimal smoothing The same has been

oppo-demonstrated for regression as well; see Hall, Li, and Racine (2007) for furtherdetails Note that this result is closely related to the Bayesian results described

in detail in Section 7.3

Nonparametric Kernel Methods for Qualitative and Quantitative Data 201

7.4.3 Kernel Estimation of a Conditional CDF

Li and Racine (2008) propose a nonparametric conditional CDF kernel tor that admits a mix of discrete and categorical data along with an associatednonparametric conditional quantile estimator Bandwidth selection for ker-nel quantile regression remains an open topic of research, and they employ amodification of the conditional PDF-based bandwidth selector proposed byHall, Racine, and Li (2004)

estima-We use F ( y | x) to denote the conditional CDF of Y given X = x, while f (x)

is the marginal density of X We can estimate F ( y | x) by

Li and Racine (2008) demonstrate that

7.4.4 Kernel Estimation of a Conditional Quantile

Estimating regression functions is a popular activity for practitioners times, however, the regression function is not representative of the impact ofthe covariates on the dependent variable For example, when the dependentvariable is left (or right) censored, the relationship given by the regressionfunction is distorted In such cases, conditional quantiles above (or below)the censoring point are robust to the presence of censoring Furthermore, theconditional quantile function provides a more comprehensive picture of theconditional distribution of a dependent variable than the conditional meanfunction

Some-Once we can estimate conditional CDFs, estimating conditional quantilesfollows naturally That is, having estimated the conditional CDF we simplyinvert it at the desired quantile as described below A conditionalth quantile

202 Handbook of Empirical Economics and Finance

of a conditional distribution function F (· | x) is defined by ( ∈ (0, 1))

(since = F (q(x) | x)).

7.4.5 Binary Choice and Count Data Models

Another application of kernel estimates of PDFs with mixed data involves theestimation of conditional mode models By way of example, consider some

discrete outcome, say Y ∈ S = {0, 1, , c − 1}, which might denote by way

of example the number of successful patent applications by firms We define

the conditional mode of y | x by

m(x)= max

In order to estimate a conditional mode m(x), we need to model the

con-ditional density Let us call ˆm(x) the estimated conditional mode, which is

given by

ˆ

m(x)= max

where ˆg( y | x) is the kernel estimator of g(y | x) defined in Equation 7.18.

7.4.6 Kernel Estimation of Regression Functions

The local constant (Nadaraya 1965; Watson 1964) and local polynomial (Fan1992) estimators are perhaps the most well-known of all kernel methods.Racine and Li (2004) and Li and Racine (2004) propose local constant andlocal polynomial estimators of regression functions defined over categoricaland continuous data types To extend these popular estimators so that they canhandle both categorical and continuous regressors requires little more thanreplacing the traditional kernel function with the generalized kernel given inEquation 7.16 That is, the local constant estimator defined in Equation 7.7would then be

Nonparametric Kernel Methods for Qualitative and Quantitative Data 203

Racine and Li (2004) demonstrate that

$

n ˆh p

ˆg(x) − g(x) − ˆB( ˆh, ˆ)/

ˆ

(x) → N(0, 1) in distribution. (7.25)See Racine and Li (2004) for further details

7.5 Summary

We survey recent developments in the kernel estimation of objects definedover categorical and continuous data types We focus on theoretical underpin-nings, and focus first on kernel methods for categorical data only We pay closeattention to recent theoretical work that draws links between kernel methodsand Bayesian methods and also highlight the behavior of kernel methods inthe presence of irrelevant covariates Each of these developments leads to ker-nel estimators that diverge from more traditional kernel methods in a number

of ways, and sets the stage for mixed data kernel methods which we brieflydiscuss We hope that readers are encouraged to pursue these methods, anddraw the readers attention to an R package titled “np” (Hayfield and Racine2008) that implements a range of the approaches discussed above A number

of relevant examples can also be found in Hayfield and Racine (2008), and wedirect the interested reader to the applications contained therein

References

Aitchison, J., and C G G Aitken 1976 Multivariate binary discrimination by the

kernel method Biometrika 63(3): 413–420.

Efron, B., and C Morris 1973 Stein’s estimation rule and its competitors–an empirical

Bayes approach Journal of the American Statistical Association 68(341): 117–130 Fan, J 1992 Design-adaptive nonparametric regression Journal of the American Statis-

tical Association 87: 998–1004.

Hall, P., Q Li, and J S Racine 2007 Nonparametric estimation of regression

func-tions in the presence of irrelevant regressors The Review of Economics and Statistics

89: 784–789

Hall, P., J S Racine, and Q Li 2004 Cross-validation and the estimation of conditional

probability densities Journal of the American Statistical Association 99(468): 1015–

1026

Hayfield, T., and J S Racine 2008 Nonparametric econometrics: the np package

Journal of Statistical Software 27(5) http://www.jstatsoft.org/v27/i05/

Heyde, C 1997 Quasi-Likelihood and Its Application New York: Springer-Verlag Kiefer, N M., and J S Racine 2009 The smooth colonel meets the reverend Journal of

Nonparametric Statistics 21: 521–533.

Li, Q., and J S Racine 2003 Nonparametric estimation of distributions with

categor-ical and continuous data Journal of Multivariate Analysis 86: 266–292.

204 Handbook of Empirical Economics and Finance

Li, Q., and J S Racine 2004 Cross-validated local linear nonparametric regression

Statistica Sinica 14(2): 485–512.

Li, Q., and J S Racine 2007 Nonparametric Econometrics: Theory and Practice Princeton,

NJ: Princeton University Press

Li, Q., and J S Racine 2008 Nonparametric estimation of conditional CDF and

quan-tile functions with mixed categorical and continuous data Journal of Business and

Economic Statistics 26(4): 423–434.

Lindley, D V., and A F M Smith 1972 Bayes estimates for the linear model Journal

of the Royal Statistical Society 34: 1–41.

Nadaraya, E A 1965 On nonparametric estimates of density functions and regression

curves Theory of Applied Probability 10: 186–190.

Ouyang, D., Q Li, and J S Racine 2006 Cross-validation and the estimation of

probability distributions with categorical data Journal of Nonparametric Statistics

18(1): 69–100

Ouyang, D., Q Li, and J S Racine 2008 Nonparametric estimation of regression

functions with discrete regressors Econometric Theory 25(1): 1–42.

R Development Core Team 2008 R: A Language and Environment for Statistical

Comput-ing, R Foundation for Statistical ComputComput-ing, Vienna, Austria ISBN 3-900051-07-0.

http://www.R-project.orgRacine, J S and Q Li 2004 Nonparametric estimation of regression functions with

both categorical and continuous data Journal of Econometrics 119(1): 99–130 Simonoff, J S 1996 Smoothing Methods in Statistics New York: Springer Series in

Statistics

Wand, M., and B Ripley, 2008 KernSmooth: Functions for Kernel

KernSmoothWang, M C., and J van Ryzin, 1981 A class of smooth estimators for discrete distri-

butions Biometrika 68: 301–309.

Watson, G S 1964 Smooth regression analysis Sankhya 26:(15): 359–372.

Wooldridge, J M 2002 Econometric Analysis of Cross Section and Panel Data Cambridge,

MA: MIT Press

The Unconventional Dynamics of Economic and Financial Aggregates

Karim M Abadir and Gabriel Talmain

CONTENTS

8.1 Introduction 205

8.2 The Economic Origins of the Nonlinear Long-Memory 206

8.3 Modeling Co-Movements for Series with Nonlinear Long-Memory 209

8.3.1 Econometric Model 209

8.3.2 Empirical Implications 210

8.3.3 Special Case: co-CM 211

8.4 Further Developments 212

8.5 Acknowledgments 212

References 212

8.1 Introduction

Time series models have provided econometricians with a rich toolbox from which to choose Linear ARIMA models have been very influential and have enhanced our understanding of many empirical features of economics and finance As with any scientific endeavor, data have emerged that show the need for refinements and improvements over existing models

Nonlinear models have gained popularity in recent times, but which one

do we choose from? Once we move away from linear models, there is a huge variety on offer Surely, economic theory should provide the guiding light, insofar as economics and finance are the subject in question Abadir and Talmain (2002) provided one possible answer This chapter is mainly a sum-mary of the econometric aspects of the line of research started by that paper The main result of that literature is that macroeconomic and aggregate financial series follow a nonlinear long-memory process that requires new econometric tools It also shows that integrated series (which are a special case of the new process) are not the norm in our subject, and proposes a new approach to econometric modeling

205

206 Handbook of Empirical Economics and Finance

8.2 The Economic Origins of the Nonlinear Long-Memory

Abadir and Talmain (AT) started with a micro-founded macro model It was

a standard real business cycle (RBC) model, except that it allowed for geneity: the “representative firm” assumption was dropped They workedout the intertemporal general equilibrium solution for the economy, and theresult was an explicit dynamic equation for GDP and all the variables thatmove along with it

hetero-It was well known, long before AT, that heterogeneity and aggregation led

to long-memory; e.g., see Robinson (1978) and Granger (1980) for a start of theliterature on linear aggregation of ARIMA models, and Granger and Joyeux(1980) and Hosking (1981) for the introduction of long-memory models.1But in economics, there is an inherent nonlinearity which makes linear ag-gregation results incomplete Let us illustrate the nonlinearity in the sim-plest possible aggregation context; see AT for the more general CES-typeaggregation

Decompose GDP, denoted by Y, into the outputs Y(1), Y(2), of firms

(alternatively, sectors) in the economy as

Y : = Y(1) + Y(2) + · · · = e y(1)+ ey(2) + · · · , where we write the expression in terms of y(i) := log Y(i) (i = 1, 2, ) to consider percentage changes in Y(i) (and to make sure that models to be chosen for y(i) keep Y(i) > 0, but this can be achieved by other methods too).

With probability 1,

ey(1)+ ey(2)+ · · ·
= ey(1) +y(2)+··· ,

where the right-hand side is what linear aggregation entails The right-hand

side is the aggregation considered in the literature, typically with y(i) ∼

ARIMA( p i , d i , q i), but it is not what is needed in macroeconomics AT cially p 765) show that important features are missed by linearization whenaggregating dynamic series

(espe-One implication of the nonlinear aggregation is that the auto-correlationfunction (ACF)
of the logarithm of GDP and other variables moving with

it take the common form

:= cov( y t , y t−)

√

var( y t )var( y t−) = 1− a [1 − cos ()]

1 A time series is said to have long memory if its autocorrelations dampen very slowly, more

so than the exponential decay rate of stationary autoregressive models but faster than the permanent memory of unit roots Unlike the latter, long-memory series revert to their (possibly trending) means.

The Unconventional Dynamics of Economic and Financial Aggregates 207

0.75 0.8 0.85 0.9 0.95 1

ACF of the log of U.S real GDP per capita over 1929–2004.

where the subscript of y denotes the time period and a , b, c,  depend on theparameters of the underlying economy but differ across variables.2 Abadir,Caggiano, and Talmain (2006) tried this on all the available macroeconomicand aggregate financial data, about twice as many as (and including the ones)

in Nelson and Plosser (1982) The result was an overwhelming rejection ofAR-type models and the shape they imply for ACFs, as opposed to the oneimplied by Equation 8.1 For example, for the ACF of the log of U.S real

GDP per capita over 1929–2004, Figure 8.1 presents the fit of the best AR( p) model (it turns out that p= 2 with one root of almost 1) by the undecoratedsolid line, compared to the fit of Equation 8.1 by nonlinear LS Linear models,like ARIMA, are simply incapable of allowing for sharp turning points that

we can see in the decay of memory The empirical ACFs found that there istypically an initial period where persistence is high, almost like a unit-rootwith a virtually flat ACF, then a sudden loss of memory We can illustrate thisalso in the time domain in Figure 8.2, where we see that the log of real GDPper capita is evolving around a linear time trend, well within small variancebands that don’t expand over time (unlike unit-root processes whose varianceexpands linearly to infinity as time passes)

ACFs of this shape have important implications for macroeconomic cymakers, as Abadir, Caggiano, and Talmain (2006) show For example, if aneconomy is starting to slow down, such ACFs predict that it will produce along sequence of small signs of a slowdown followed by an abrupt decline.When only the small signs have appeared, no-one fitting a linear (e.g., AR)

poli-2The restrictions b, c,  > 0 apply, but the restriction on a cannot be expressed explicitly.

208 Handbook of Empirical Economics and Finance

6 6.5 7 7.5 8 8.5 9 9.5 10 10.5 11

Time series of the log of U.S real GDP per capita over 1929–2004.

model would be able to guess the substantial turning point that is about tooccur Another implication is that any stimulus that is applied to the economyshould be timed to start well before the abrupt decline of the economy hastaken place, and will take a long time to have an impact (and will eventuallywear off unlike in unit root models) Consequently, a gradualist macroeco-nomic policy will not yield the desired results because it will be a case oftoo little and too late In other words, a gradualist approach can be compati-ble with linear models but will be disastrous in the context of the ACFs thatarise from macroeconomic data and that are compatible with the nonlineardynamics generated by the general-equilibrium model of AT

The ACF shape has important implications for econometric methods also.The long-memory cycles it generates require the consideration of singulari-

ties at frequencies other than 0 in spectral analysis In fact, if a is close to 1

in the ACF (Equation 8.1), Fourier inversion produces a spectrum f () that

is approximately proportional to| − |c−1; that is, at frequency, there is a

singularity when c ∈ (0, 1) For I(d) series having d ∈ (0,1

2), the spectrum has

a singularity at the origin that is proportional to||−2d, giving the

correspon-dence c = 1 − 2d in the special case of  = 0 This correspondence holds also

in the tails of the ACFs of the two processes when = 0

I(d) models are a special case of the new process We therefore need to go beyond I(d) models and consider the estimation of spectral densities near sin-

gularities that are not necessarily located at the origin, as a counterpart (when

a ≈ 1) to the ACF-domain estimation mentioned earlier Giraitis, Hidalgo,and Robinson (2001) and Hidalgo (2005) give a frequency-domain method ofestimating and d, when d ∈ (0,1

2)

The Unconventional Dynamics of Economic and Financial Aggregates 209

For a ≈ 1, we introduce the following definition

Definition 8.1 A process is said to be of cyclical long-memory, respectively with parameters  ∈ [0, ] and d ∈ (0,1

2), if it has a spectrum f () that is proportional

to| − |−2d as  →  and is bounded elsewhere Such a process is denoted by

CM(, d), with the special case CM(0, d) = I(d)

It is no wonder that a statistical model with cycles arises from a real ness cycle model Note that integrated processes cannot generate cycles thathave long memory because their spectrum is bounded at
= 0 They canonly generate short transient cycles that are not sufficiently long for macroe-conomics

busi-When a is not close to 1 in the ACF (Equation 8.1), the result of the Fourier inversion is approximately a linear combination of one I(d) and one CM(, d)

when
= 0 Here, too, the approximation arises from the inversion focusingmore on the tail of the ACF and neglecting to some extent the initial concavepart of the ACF in Equation 8.1

But if the individual series are not of the integrated type, can we talk of

co-integrated series? It is an approximation that many not be adequate enough.

What about the modification of co-integration modeling for variables thathave this new type of dynamics? Abadir and Talmain (2008) propose a solu-tion We summarize it in the next section, and present an additional definition

an extension of co-integration to allow for co-movements of CM processes

8.3.1 Econometric Model

Suppose we have a sample of t = 1, , T observations To simplify the

exposition, consider the model

where z is T × 1 and  is k × 1 The matrix X can contain lagged

depen-dent variables, so that we cover autoregressive distributed-lag models (e.g.,

used in co-integration analysis) as one of the special cases The vector u tains the residual dynamics of the adjustment of z toward its fundamental

con-210 Handbook of Empirical Economics and Finance

value X By definition, u is centered around zero and is mean-reverting, otherwise z will not revert to its fundamental value We write u ∼ D(0, Σ), where Σ is the T × T autocovariance matrix of the u’s The autocorrelation

matrix of u is denoted by R, and Abadir and Talmain (2008) use Equation 8.1

to parameterize the typical i jth element
|i− j| of R There are two

implica-tions to u tbeing mean-reverting (which is a testable assumption) First, Σ is

proportional to R Second, the ML estimator of and the ACF parameters

in u is consistent The asymptotic distribution will depend on the properties

of the variables, but if the estimated residuals are found to satisfy c > 1

2(implying square-summability of
), then standard t, F, LR tests are justi-fied asymptotically.3This condition on c is sufficient but not necessary, and

we have found it to hold in practice when dealing with macro and financialseries

The quasi maximum likelihood (QML) procedure of Abadir and Talmain(2008) estimates jointly the parameters  and the ACF parameters in
 of

u They remove the sample mean of each variable in Equation 8.2 to avoid multicollinearity in practice, with the constant term in X redefined accord- ingly They also assume that X is weakly exogenous (see Engle, Hendry, and

Richard 1983) for the parameters of Equation 8.2

For any given R, define

with respect to the parameters of the ACF: the optimization of the joint

like-lihood (for Σ and) now depends on only four parameters that are given in

Equation 8.1 and that determine the whole autocorrelation matrix R Once

the optimal value
R of R is obtained, the QMLE of is
 ≡

R

8.3.2 Empirical Implications

One is often interested in detecting the presence of co-movements betweenseries This may be for the purpose of empirically validating theoretical work,producing predictions, or determining optimal policies In practice, one is of-ten frustrated by the results produced by co-integration analysis The theory

of purchasing power parity (PPP) is typically tested using co-integration

3 This is a case where the results of Tsay and Chung (2000) on the divergent behavior of

t-statistics do not apply, since the condition on c corresponds to the case of the series ing d < 1

hav-The Unconventional Dynamics of Economic and Financial Aggregates 211

Generally, the findings are that PPP does not hold in the short run anddeviations from PPP are cycling around the theoretical value at very lowfrequency, implying that the estimated reversion to PPP is, if at all, unrealis-tically slow

Even when the series have less memory, dynamic modeling of movements can spring surprises According to the uncovered interest par-ity (UIP) theory, no contemporaneous variable should be able to predict thefuture excess returns in investing in a foreign asset However, researchershave consistently found a strong negative relation between future excess re-turns and the forward premium on a currency With the usual interpretation

co-of the forward rate as a predictor co-of the future spot exchange rate, this wouldimply the irrational result that a currency is expected to depreciate in periodswhen assets denominated in this currency actually do produce systematicexcess returns!

These “anomalies” or “paradoxes” are what one would find if the truenature of the relation between the variables is of the type in Equation 8.2,

but the possibility of unconventional dynamics for u has been neglected.

Co-integration would try to force a noncyclical zero-frequency pattern onthis residual term which, in reality, is slowly cycling By allowing for thepossibility of long-memory cycles, the methodology described above brings

to light the true nature of the residuals and, thus, of the true relation betweenthe co-moving variables The “long-run” relation between economic variablesoften involves long cycles of adjustment

8.3.3 Special Case: co-CM

The model in Equation 8.2 avoids the question of the individual, d in each

of the series contained in z, X It just states that the dynamics of adjustment

to the fundamental value (through changes in u) is of the general AT type A

way in which this can arise is through the following special CM case of the

AT process, where we use a bivariate context to simplify the illustration and

to show how it generalizes the notion of co-integration

Definition 8.2 Two processes are said to be linearly co-CM if they are both CM( , d)

and there exists a linear combination that is CM( , s) with s < d.

This follows by the same spectral methods used in Granger (1981, Section 4).The definition can be extended to allow for nonlinear co-CM, for example,

if z t = g(x t)+ u t with g a nonlinear function For the effect on the ACF

(hence on , d) of parametric nonlinear transformations, see Abadir and

Talmain (2005)

In Equation 8.2, it was not assumed that s < d In fact, in the UIP application

in Abadir and Talmain (2008), we had the ACF equivalent of s = d because

it was a trivial co-CM case where the right-hand side variable had a zero

coefficient and z t = u t

212 Handbook of Empirical Economics and Finance

8.4 Further Developments

Work is currently being carried out on a number of developments of thesemodels and the tools required to estimate them and test hypotheses abouttheir parameters The topic is less than a decade old, at the time of writingthis chapter, but we hope to have demonstrated its potential importance

A simple time-domain parameterization of the CM(, d) process has beendeveloped in preliminary work by Abadir, Distaso, and Giraitis The frequency-domain estimation of this process is also being considered, generalizing theFELW estimator of Abadir, Distaso, and Giraitis (2007) to the case where isnot necessarily zero

Abadir, K M., W Distaso, and L Giraitis 2007 Nonstationarity-extended local Whittle

estimation Journal of Econometrics 141: 1353–1384.

Abadir, K M., and G Talmain 2002 Aggregation, persistence and volatility in a macro

model Review of Economic Studies 69: 749–779.

Abadir, K M., and G Talmain 2005 Autocovariance functions of series and of their

transforms Journal of Econometrics 124: 227–252.

Abadir, K M., and G Talmain 2008 Macro and financial markets: the memory of anelephant? Working Paper Series 17-08, Rimini Centre for Economic Analysis

Engle, R F., D F Hendry, and J -F Richard 1983 Exogeneity Econometrica 51: 277–304.

Giraitis, L., J Hidalgo, and P M Robinson 2001 Gaussian estimation of parametric

spectral density with unknown pole Annals of Statistics 29: 987–1023.

Granger, C W J 1980 Long memory relationships and the aggregation of dynamic

models Journal of Econometrics 14: 227–238.

Granger, C W J 1981 Some properties of time series data and their use in

econo-metric model specification Journal of Econoecono-metrics 16: 121–130.

Granger, C W J., and R Joyeux 1980 An introduction to long-range time series

models and fractional differencing Journal of Time Series Analysis 1: 15–29.

Hidalgo, J 2005 Semiparametric estimation for stationary processes whose spectra

have an unknown pole Annals of Statistics 33: 1843–1889.

Hosking, J R M 1981 Fractional differencing Biometrika 68: 165–176.

212 Handbook of Empirical Economics and Finance< /i>

8.4 Further Developments

Work is currently being carried out on a number of developments of thesemodels and the... class="page_container" data-page="9">

206 Handbook of Empirical Economics and Finance< /i>

8.2 The Economic Origins of the Nonlinear Long-Memory

Abadir and Talmain (AT) started with... class="page_container" data-page="11">

208 Handbook of Empirical Economics and Finance< /i>

6 6.5 7.5 8.5 9.5 10 10.5 11

Time series of the log of U.S real GDP per capita over