MODEL SELECTION AND PREDICTION FOR LONG MEMORY MODELS IN ECONOMIC
TIME SERIES
by
Chaleampong Kongcharoen

A DISSERTATION
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
DOCTOR OF PHILOSOPHY
ECONOMICS
2011

ABSTRACT
MODEL SELECTION AND PREDICTION FOR LONG MEMORY MODELS IN ECONOMIC
TIME SERIES
by
Chaleampong Kongcharoen
This dissertation considers methodological issues involving both prediction and model selection
in the application of long memory processes to economic time series data. We also have a section
which discusses some practical aspects of the implementation of the model selection methodology to real GNP, monthly CPI inflation series, interest rates and realized volatility from currency
markets.
Chapter 1 is the Introduction and sets the scene, established notation for the approaches to
be used later. Chapter 2 examines different approaches to estimate the optimal, minimum mean
square errors (MSE) multi step predictor to univariate fractionally autoregressive moving average
(ARFIMA) models. We consider two competing approaches of (i) using the MLE of the ARFIMA
parameters inserted into the optimal predictor, and (ii) using an alternative predictor using estimates
arising from the Local Whittle Two Step Estimator (LW T SE) are applied. In the latter approach, an
initial semi parametric estimate of the long memory parameter is used to feasibly fractionally filter
the series to obtain an estimate of the short memory series, which is then used to estimate the short
memory ARMA parameters. This latter approach seems very relevant given that the most popular
methodology in current econometric literature is to obtain an initial semi parametric estimate of
a long memory parameter. We found that the predictor based on MLE is generally superior in
MSE sense to the predictor based on the LW T SE estimation. In general the “optimal” bandwidth
LW T SE brings about worse predictions than the LW T SE based on the conventional one of using
an agnostic bandwidth of the square root of the sample size.
Chapter 3 is concerned with the issue of model selection and considers the choice of lag length
of the p and q parameters in an ARFIMA(p, d, q) model. Perhaps surprisingly, there appears to be a

major gap in the previous literature, which has only considered this problem for the very restrictive
case of the ARFIMA(p, d, 0) model. We show that when estimation is performed by either MLE or
QMLE the AIC is inconsistent and show that the widely used BIC and HQIC are consistent model
selection procedures. We accordingly extend the theory to be applied to the ARFIMA(p, d, q)
model. The implementation of the technique involves selection of the maximum orders of the short
memory process that the investigator is willing to consider. In our simulation study we restrict the
maximum values of p and q to be 8, which involves 81 different ARFIMA models to be estimated
by MLE and then assessed from a model selection perspective. In our detailed simulation we
consider data generating processes up to orders of an ARFIMA(4, d, 1) model. The BIC works
well for some low order models; but has a mixed performance on high order models with p = 4
and is quite dependent on the part of the true parameter space. We also show how the theory can
be extended for seasonal ARFIMA models.
Chapter 3 also considers the two step procedure, which in this context amounts to the application of the LW T SE where an initial estimate of the long memory is obtained by Local Whittle
and is then used to fractionally filter the series. The short memory ARMA(p, q) model is then
estimated by the minimization of the conditional sum of squares (CSS), which has been shown by
Robinson (2006) and Granger and Newbold (1996) among others to be a valid procedure. Our
simulations show that this procedure can also work quite well, but generally not as well as the
previously discussed direct estimation of ARFIMA by MLE.
However, the above procedure arguably suffers from the “generated regressor” problem and
following recent work on factor models in VARs, it seems possibly desirable to also include a
small sample adjustment to the BIC which reflects the sub optimal rate of convergence of the Local
Whittle estimate to the true value of the long memory parameter. We also report some simulations
to assess the effectiveness of this procedure. There are some cases when this appears to be a useful
procedure.
Chapter 4 provides a detailed set of applications of the model selection with MLE and LW T SE
to real economic and financial time series.

ACKNOWLEDGMENTS

I would like to gratefully thank Professor Richard T. Baillie for his guidance, understanding, support and patience. His mentorship was paramount in completing this dissertation. I would like
to thank Professor Christine Amsler, Professor Robert J. Myers, and Professor Peter Schmidt for
their guidance and advice. I also would like to thank Professor Hira L. Koul and Professor Emma
Iglesias for helping me at the beginning of this dissertation. I also want to thank Ruth Mendel for
helping me prepare for the dissertation defense.
The financial support from both Faculty of Economics, Thammasat University, and Thammasat University for pursuing graduate study is gratefully acknowledged.
Finally, and most importantly, I would like to deeply thank my wife Prae and my daughter
Prim for their support, encouragement, patience and love. I thank my parents, patents-in-law,
sister, and sister-in-law for their support and encouragement.

iv

TABLE OF CONTENTS

LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

vi

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ix

CHAPTER 1 INTRODUCTION
Appendix 1A: Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1
7

CHAPTER 2 PREDICTION FROM ARFIMA MODELS
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . .
2.2 Prediction from the ARFIMA Model . . . . . . . . .
2.3 Estimation of ARFIMA Models . . . . . . . . . . . .
2.4 Practical Issues and Simulation . . . . . . . . . . . .
2.5 Conclusion . . . . . . . . . . . . . . . . . . . . . .
Appendix 2A: The Prediction Weight Formula . . . .
Appendix 2B: Tables . . . . . . . . . . . . . . . . .
CHAPTER 3
3.1
3.2
3.3
3.4
3.5
3.6

INFORMATION CRITERIA METHODS
ARFIMA MODELS
Introduction . . . . . . . . . . . . . . . . . . . . . .
Information Criteria for Model Selection . . . . . . .
Simulation Study . . . . . . . . . . . . . . . . . . .
Modified Information Criteria for LW T SE . . . . . .
Inflation Example . . . . . . . . . . . . . . . . . . .
Conclusion . . . . . . . . . . . . . . . . . . . . . .
Appendix 3A: Tables . . . . . . . . . . . . . . . . .

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FOR SELECTION OF
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METHODS FOR SELECTION OF ARFIMA MODELS: APPLICATIONS
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Application to Real Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Application to Inflation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Application to Realized Volatility of Exchange Rate . . . . . . . . . . . . . . . .
Application to Interest Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Appendix 4A: Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

CHAPTER 4
4.1
4.2
4.3
4.4
4.5
4.6

CHAPTER 5

CONCLUSION

85

BIBLIOGRAPHY

87

v

LIST OF TABLES

2.1

Biases of Different Predictors, ARFIMA(1, d, 0) with d = 0.4, ϕ = 0.8, and T = 200 . 22

2.2

Biases of Different Predictors, ARFIMA(1, d, 0) with d = 0.4, ϕ = 0.9, and T = 200 . 22

2.3

Biases of Different Predictors, ARFIMA(1, d, 0) with d = 0.4, ϕ = 0.8, and T = 1, 000

23

2.4

Biases of Different Predictors, ARFIMA(1, d, 0) with d = 0.4, ϕ = 0.9, and T = 1, 000

23

2.5

MSEs of Different Predictors, ARFIMA(1, d, 0) with d = 0.4, ϕ = 0.8, and T = 200 . . 24

2.6

MSEs of Different Predictors, ARFIMA(1, d, 0) with d = 0.4, ϕ = 0.9, and T = 200 . . 24

2.7

MSEs of Different Predictors, ARFIMA(1, d, 0) with d = 0.4, ϕ = 0.8, and T = 1, 000

25

2.8

MSEs of Different Predictors, ARFIMA(1, d, 0) with d = 0.4, ϕ = 0.9, and T = 1, 000

25

2.9

MSEs of Different Predictors, ARFIMA(0, d, 1) with d = 0.4, θ = 0.8, and T = 200 . . 26

2.10 MSEs of Different Predictors, ARFIMA(2, d, 1) with d = 0.4, ϕ1 = −0.5, ϕ2 = −0.9,
θ = 0.6, and T = 200 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.11 Biases of Different Predictors, ARFIMA(2, d, 0) process with ϕ1 = 0.08, ϕ2 = 0.9025,
d = 0.2, and T = 400 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.12 Biases of Different Predictors, ARFIMA(2, d, 0) process with ϕ1 = 0.08, ϕ2 = 0.9025,
d = 0.4, and T = 400 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.13 Biases of Different Predictors, ARFIMA(2, d, 0) process with ϕ1 = 0.08, ϕ2 = 0.9025,
d = 0.49, and T = 400 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.14 MSEs of Different Predictors, ARFIMA(2, d, 0) process with ϕ1 = 0.08, ϕ2 = 0.9025,
d = 0.2 and T = 400 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.15 MSEs of Different Predictors, ARFIMA(2, d, 0) process with ϕ1 = 0.08, ϕ2 = 0.9025,
d = 0.4 and T = 400 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.16 MSEs of Different Predictors, ARFIMA(2, d, 0) process with ϕ1 = 0.08, ϕ2 = 0.9025,
d = 0.49 and T = 400 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.1

Percentage of Model Selected, DGP ARFIMA(1, d, 0), ϕ1 = 0.8, d = 0.2, T = 400 . . 43

vi

3.2

Percentage of Model Selected, DGP ARFIMA(1, d, 0), ϕ1 = 0.8, d = 0.4, T = 400 . . 44

3.3

Percentage of Model Selected, DGP ARFIMA(1, d, 0), ϕ1 = 0.8, d = 0.8, T = 400 . . 45

3.4

Percentage of Model Selected, DGP ARFIMA(1, d, 0), ϕ1 = 0.9, d = 0.4, T = 400 . . 46

3.5

Percentage of Model Selected, DGP ARFIMA(0, d, 1), θ1 = 0.9, d = 0.4, T = 400 . . 47

3.6

Percentage of Model Selected, DGP ARFIMA(1, d, 1), ϕ1 = 0.5, θ1 = −0.7, d = 0.4,
T = 400 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.7

Percentage of Model Selected, DGP ARFIMA(2, d, 0), ϕ1 = 0.25, ϕ2 = 0.375, d = 0.4,
T = 400 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

3.8

Percentage of Model Selected, DGP ARFIMA(2, d, 0), ϕ1 = 0.25, ϕ2 = 0.375, d = 0.4,
T = 1, 000 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

3.9

Percentage of Model Selected, DGP ARFIMA(2, d, 0), ϕ1 = 1, ϕ2 = −0.5, d = 0.4,
T = 400 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.10 Percentage of Model Selected, DGP ARFIMA(2, d, 1), ϕ1 = 0.25, ϕ2 = 0.375 θ1 =
1/3, d = 0.4, T = 400 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.11 Percentage of Model Selected, DGP ARFIMA(2, d, 1), ϕ1 = 1, ϕ2 = −0.5, θ1 = 1/3,
d = 0.4, T = 400 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.12 Percentage of Model Selected, DGP ARFIMA(2, d, 1), ϕ1 = 1.4, ϕ2 = −0.6 θ1 = −0.8,
d = 0.4, T = 400 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.13 Percentage of Model Selected, DGP ARFIMA(2, d, 1), ϕ1 = −0.5, ϕ2 = −0.9, θ1 =
−0.6, d = 0.4, T = 400 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.14 Percentage of Model Selected, DGP ARFIMA(3, d, 0), ϕ1 = −13/20, ϕ2 = 3/5, ϕ3 =
27/80, d = 0.4, T = 400 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.15 Percentage of Model Selected, DGP ARFIMA(3, d, 0), ϕ1 = −0.56163, ϕ2 = 0.57986,
ϕ3 = −0.92705, d = 0.4, T = 400 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.16 Percentage of Model Selected, DGP ARFIMA(4, d, 0), ϕ1 = 5/4, ϕ2 = −3/8, ϕ3 =
−1/4, ϕ4 = 3/16, d = 0.4, T = 400 . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
3.17 Percentage of Model Selected, DGP ARFIMA(4, d, 0), ϕ1 = −9/20, ϕ2 = 73/100,
ϕ3 = 87/400, ϕ4 = 27/400, d = 0.4, T = 400 . . . . . . . . . . . . . . . . . . . . . . 59
3.18 Percentage of Model Selected, DGP ARFIMA(4, d, 0), ϕ1 = 1.350, ϕ2 = −0.7, ϕ3 =
0.4, ϕ4 = −0.31, d = 0.4, T = 400 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

3.19 Percentage of Model Selected, DGP ARFIMA(4, d, 0), ϕ1 = 1.350, ϕ2 = −0.7, ϕ3 =
0.4, ϕ4 = −0.31, d = 0.3, T = 400 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.20 Percentage of Model Selected, DGP ARFIMA(4, d, 1), ϕ1 = 5/4, ϕ2 = −3/8, ϕ3 =
−1/4, ϕ4 = 3/16, θ1 = 1/3,d = 0.4, T = 400 . . . . . . . . . . . . . . . . . . . . . . 62
3.21 Modified BIC and HQ for LW T SE, Percentage of Model Selected, DGP
ARFIMA(2, d, 0) with ϕ1 = 1,ϕ2 = −0.5, d = 0.4 and T = 400 . . . . . . . . . . . . . 63
3.22 Modified BIC and HQ for LW T SE, Percentage of Model Selected, DGP
ARFIMA(2, d, 0) with ϕ1 = 1,ϕ2 = −0.5, d = 0.4 and T = 1, 000 . . . . . . . . . . . . 64
3.23 Modified BIC and HQ for LW T SE, Percentage of Model Selected, DGP
ARFIMA(1, d, 0) with ϕ1 = 0.8, d = 0.4 and T = 400 . . . . . . . . . . . . . . . . . . 65
3.24 Modified BIC and HQ for LW T SE, Percentage of Model Selected, DGP
ARFIMA(4, d, 0), ϕ1 = 5/4, ϕ2 = −3/8, ϕ3 = −1/4, ϕ4 = 3/16, d = 0.4, and T = 1, 000 66
3.25 Modified BIC and HQ for LW T SE, Percentage of Model Selected, DGP
ARFIMA(4, d, 0), ϕ1 = 1.350, ϕ2 = −0.7, ϕ3 = 0.4, ϕ4 = −0.31, d = 0.4, and T = 1, 000 67
3.26 Estimated ARFIMA Models for U.S. Inflation . . . . . . . . . . . . . . . . . . . . . . 68
3.27 Estimated ARFIMA Models for U.S. Inflation . . . . . . . . . . . . . . . . . . . . . . 69
4.1

Model selection for U.S. Real Output . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

4.2

Estimated ARFIMA Models for U.S. Real Output . . . . . . . . . . . . . . . . . . . . 78

4.3

Model Selection for G-7 Monthly Inflation . . . . . . . . . . . . . . . . . . . . . . . . 79

4.4

Estimated ARFIMA Models for G-7 Monthly Inflation . . . . . . . . . . . . . . . . . 80

4.5

Model Selection for Realized Volatility of Exchange Rates . . . . . . . . . . . . . . . 81

4.6

Estimated ARFIMA Models for Realized Volatility of Exchange Rates . . . . . . . . . 81

4.7

Model Selection for Interest Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

4.8

Model Selection for Interest Rate Spreads . . . . . . . . . . . . . . . . . . . . . . . . 82

4.9

Estimated ARFIMA Models for U.S. Interest Rates . . . . . . . . . . . . . . . . . . . 83

4.10 Estimated ARFIMA models for U.S. Interest Rate Spreads . . . . . . . . . . . . . . . 84

LIST OF FIGURES

1.1

1.2

Infinite AR Representations, Infinite MA Representations, and Theoretical MSE of Predictors, ARFIMA(1, d, 0) process with d = 0.4 and ϕ = 0.8 . . . . . . . . . . . . . . .

7

Infinite AR Representations, Infinite MA Representations, and Theoretical MSE of Predictors, ARFIMA(1, d, 0) process with d = 0.4 and ϕ = 0.9 . . . . . . . . . . . . . . .

8

ix

Chapter 1

INTRODUCTION

There is a burgeoning literature on the theoretical and statistical aspects of long memory models
in economic and financial time series; where the series possess hyperbolically decaying autocorrelations. The work of Granger (1980), Granger and Joyeux (1980) and Hosking (1981) has been
particularly influential in this regard. These models have proved extremely relevant for many time
series in finance and macroeconomics; see Baillie (1996) and Henry and Zaffaroni (2003) for a
survey of some of the literature.
A time series is said to have long memory, (or long-range dependence or strong dependence),
if its autocorrelations decay to zero at very slow hyperbolic rates. For a more thorough and detailed
definitions of long memory see Baillie (1996) and Beran (1994). The univariate long memory,
fractionally autoregressive moving average (ARFIMA) models have been introduced by Granger
and Joyeux (1980) and Hosking (1981). A fractionally integrated or long memory process generates hyperbolic rates of decay in its autocorrelation function and impulse response weights. A
univariate time series process, yt , which is said to be fractionally integrated of order d, or I(d), if
(1 − L)d yt = ut ,

t = 1, ...T

(1.1)

where L is the lag operator and ut is a short memory, I(0) process. The d parameter represents the
degree of “long memory”, or persistence in the series. If −0.5 < d < 0.5, the process is stationary
and invertible. If 0.5 < d < 1, the process has an infinite variance but it is mean reverting. If the
short memory component is represented as a stationary and invertible ARMA(p, q) process, then
equation (1.1) becomes the well-known ARFIMA(p, d, q) model,

ϕ (L)(1 − L)d yt = θ (L)εt

(1.2)

where L is the lag operator, and E(εt ) = 0, E(εt2 ) = σ 2 , E(εt εs ) = 0, s ̸= t. Also, the lag polynoq

p

mial operators are defined as ϕ (L) = 1 − ∑ j=1 ϕ j L j and θ (L) = 1 + ∑ j=1 θ j L j . For a stationary
1

long memory ARFIMA(p, d, q) process with −0.5 < d < 0.5, the ARFIMA model can be represented by the AR(∞) and MA(∞) representations of

π (L)yt = εt ,
where

π (L) = 1 −

(1.3)

∞

∑ π j L j = ϕ (L)(1 − L)d θ (L)−1

j=1

and,
yt = ψ (L)εt ,
where

ψ (L) = 1 +

(1.4)

∞

∑ ψ j L j = θ (L)ϕ (L)−1(1 − L)−d .

j=1

It is also assumed that all the roots of ϕ (L) and θ (L) lie outside the unit circle. For large lags
k, these coefficients decay at the very slow hyperbolic rates of ψk ∼ c1 kd−1 and similarly the
infinite autoregressive representation coefficients decay at the rate of c2 k−d−1 and autocorrelation
coefficients at the rate of c3 k2d−1 , where c1 , c2 and c3 are constants. The hyperbolic decay that is
generated by such a process is known as the ‘Hurst effect’, after Hurst (1956), who first discovered
the phenomenon in hydrological time series data. The full impact of these different rates of decay
can be visually seen in Figures 1.1 and 1.2, where the infinite AR weights decline far more quickly
that the Wold decomposition, or infinite MA weights. This has relevance for the construction of
predictions from these models, as compared with Impulse Response Weights (IRW s).
Much applied work, such as Barkoulas and Baum (2006) for monetary indices, and Doornik
and Ooms (2004) for US and UK inflation, have stressed the usefulness of ARFIMA models for
forecasting macroeconomic and financial time series. Because of the possibility of misspecification of ARFIMA models, there is a burgeoning literature on estimating the single long memory
parameter using semi parametric estimators (SPE), such as the GPH’s local periodogram estimator, proposed by Geweke and Porter-Hudak (1983) and the Local Whittle (LW ) estimation,
proposed by Kunsch (1987) and Robinson (1995).

2

The standard methodology for constructing dynamic multi step ahead predictions from any
dynamic model is to first derive the theoretical minimum MSE predictor with known parameter
values. The feasible minimum MSE predictor is then obtained by substituting estimated parameters for the theoretical unknown parameters. Of course, this approach assumes a known parametric
model, whereas it may be desirable to use some neural net or non parametric approach; see Diebold
and Mariano (1995). In our chapter 2 we continue this approach with ARFIMA models where the
obvious method of using MLEs of the parameters are then substituted into the minimum MSE
predictor with known theoretical values.
However, in recent years the dominant paradigm in estimation of long memory models is
to use the semi parametric estimators of the long memory parameter e.g. the Local Whittle of
Robinson (1995). It is not quite clear how to use this in finding the optimum predictor since the
SPE, or the Local Whittle estimator, only gives us information on one parameter, and a typical
model may also involve many short run parameters, such as ARMA and nonlinear terms such as
Exponential Smooth Transition Autoregressive (ESTAR) or Logistic Smooth Transition Autoregressive (LSTAR). Therefore we use the recent suggestion of Baillie and Kapetanios (2007) and
Baillie and Kapetanios (2008) to use a Local Whittle Two Step Estimator, or LW T SE. Their
suggestions have been in terms of first testing for nonlinearity within a long memory process and
second to estimate a long memory non linear autoregression with smooth transition regimes; or
FI − NLAR − ESTAR model. Both approaches compare the one step MLE approach to testing and
to the LW T SE. Chapter 2 of this dissertation extends this type of investigation to the prediction
situation and we compare the MLE and the LW T SE used for substituting into the theoretical minimum MSE predictor. We consider various representations for the minimum mean square errors
(MSE) predictor with known parameters. We then conduct a detailed simulation study when the
true parameters are replaced by estimates that are either based on MLE or LW T SE. The predictors
are compared in terms of their bias and MSE for different prediction horizons. In general the predictors based on MLE are found to be sometimes only slightly superior to the predictors based on
the LW T SE method. Since LW T SE estimates depend on the bandwidth selection, the “optimal”

3

bandwidth LW T SE is also considered. In general the “optimal” bandwidth LW T SE brings about
worse predictions than the LW T SE based on the conventional one of using an agnostic bandwidth
of the square root of the sample size. Although the MLE based predictor is generally better, the
LW T SE nevertheless is better than expected given its rather poor performance on estimating IRW s
and short memory parameters as documented by Baillie and Kapetanios (2009). We also present
some results on prediction where the MLE is based on a mis-specified models. A summary of the
results then concludes chapter 2.
Chapter 3 is concerned with a completely different problem; namely the total model specification of an ARFIMA model. There are now many techniques available for determining if a
univariate time series has long memory; and in many cases the initial use of SPEs like GPH or
Local Whittle may conform the presence of long memory. While the basic long memory ARFIMA
model is widely used in time series analysis, the problem of selection of the ARMA orders of the
model appears to have been unresolved. In Chapter 3, we show the Bayesian (BIC) and HannanQuinn (HQ) information criteria are consistent model selection procedures for the class of long
memory ARFIMA models when the models are estimated by Quasi-MLE (QMLE). Very surprisingly, it seems that this very basic problem has never been resolved in time series analysis before.
Our end result is that the BIC is consistent and is an easily implemented procedure for model selection of an ARFIMA(p, d, q) model. The implementation of the technique involves selection of
the maximum orders of the short memory process that the investigator is willing to consider. In our
simulation study we restrict the maximum values of p and q to be 8, which involves 81 different
ARFIMA models to be estimated by MLE and then assessed from a model selection perspective. In
our detailed simulation we consider data generating processes up to orders of an ARFIMA(4, d, 1)
model. The BIC works well for some low order models; but has a mixed performance on high
order models with p = 4 and is quite dependent on the part of the true evidence indicates that the
BIC based on QMLE works very well for low order models but can be rather mixed for high order
models.
The initial focus of this chapter is thus on comparing the IC where all the models are estimated

4

by MLE. We also consider a modified approach where the AIC, BIC and HQIC are all based on
the LW T SE. Although this seems on face value a rather strange method, it does provide a role for
the popular LW T SE. In many of the designs the use of the LW T SE does surprisingly well. One
interesting aspect of using the LW T SE for the purpose of ICs is that we are in the domain of a
generated regressor problem and we investigate the possibility of introducing an adjustment that
takes into account the fact that the IC should be appropriately modified. Some simulation evidence
suggests that this may provide an improvement over the traditional IC with LW T SE.
Moving ahead, chapter 4 is concerned with some illustrations of the previous methodology.
In particular, we show how the model selection methods can be used to select ARFIMA models
for the very seasonal monthly US CPI inflation series. The applications of model selection criteria
are compared with MLE and LW T SE are considered in Chapter 4. We examine U.S. real output,
G-7 inflation, U.S. interest rates and spreads, and daily volatility of exchange rate. Our empirical
results confirm long memory features found in literature. For model selection issue, the BIC with
MLE estimators, generally, selects the parsimonious model. The model selection and degree of
persistence from MLE mostly differ from ones from LW estimator.

5

APPENDIX

6

Appendix 1A: Figures
Figure 1.1: Infinite AR Representations, Infinite MA Representations, and Theoretical MSE of
Predictors, ARFIMA(1, d, 0) process with d = 0.4 and ϕ = 0.8

Key: Author’s calculation. Dash line represents infinite AR representation coefficients. Solid line
represents infinite MA representation coefficients. Dot line represents theoretical MSE (right axis).

7

Figure 1.2: Infinite AR Representations, Infinite MA Representations, and Theoretical MSE of
Predictors, ARFIMA(1, d, 0) process with d = 0.4 and ϕ = 0.9

Key: See Figure 1.1.

8

Chapter 2

PREDICTION FROM ARFIMA MODELS

2.1 Introduction
A currently important research issue concerns the estimation of the long memory parameter in long
memory, ARFIMA models. The method of semi parametric estimation (SPE) of the long memory
parameter has been popular in the theoretical literature with the Local Whittle (LW ) estimator
emerging as the most widely recommended; in particular see Robinson (1995), Dalla, Giraitis,
and Hidalgo (2005) and others. Also, Phillips (2007), Phillips and Shimotsu (2004) and many
others have provided results on further extensions and refinements. The excessive attention to just
one parameter in the model building, or research program of an applied econometrician has been
questioned by Baillie and Kapetanios (2008), Baillie and Kapetanios (2009) and others. They
suggest that an empirical investigator is typically interested in a wider range of issues that will
generally include (i) constructing a model involving both short and long memory components, (ii)
estimating Impulse Response Weights (IRW s) and (iii) making predictions over short and long
horizons. In particular Baillie and Kapetanios (2009) show that the SPE Local Whittle (LW )
estimator of the long memory parameter in a univariate time series is shown to perform poorly
in the presence of persistent short memory component. These problems with the LW estimator
extend to the estimation of short run parameters and Impulse Response Weights (IRW s) and the
difficulties remain even when optimal bandwidths, or local polynomial Whittle methods are used.
Furthermore, Baillie and Kapetanios (2009) have suggested an alternative time domain SPE of
estimating a high order autoregressive approximation, which turns out to have desirable properties
for estimating IRW s with the method also being relevant for certain non-stationary processes.
This chapter considers the equally important practical issue of the impact of estimation strategy on the properties of predictions from a long memory process. Hence this chapter brings to9

gether two previously unconnected areas of the literature; namely the choice of using MLE or LW
estimation procedures and the impact on prediction.
The plan of the rest of this chapter is as follows; the next section briefly reviews the form
of predictors from the ARFIMA, which is the most basic long memory model. Section 2.3 reviews the SPE estimation methodology, which is based on using the LW as a first step and then
deriving the estimate of the short memory parameter(s) as a second step. Hence the complete estimator is referred to as the Local Whittle Two Step Estimator, or LW T SE. These estimates are
then used to replace the unknown parameters in the theoretical minimum MSE predictor. Section
2.4 discusses the practical issues involved in constructing predictors from a sample and reports the
results of a simulation study. The results indicate the difficulties in choosing the bandwidth for
the LW estimator. Several different LW estimators are considered based on bandwidth considerations. The predictors based on “optimal” bandwidths generally perform poorly in the simulation
designs. Their performance deteriorates with the degree of persistence of the short memory AR
components. While the MLE is expected to produce superior forecasts in terms of MSE, the extent
of the improvement is surprising.
Some additional results of particular interest concern the predictors when the estimation of
the model is mis-specified. This situation should conceptually be ideal for the LW T SE since it is
designed to be robust to ignorance of short run dynamics. However, the predictor based on MLE
from a mis-specified model is found to substantially outperform the predictor from the LW T SE for
all lead times up to 40 periods ahead. Further details are given in section 2.4. The chapter ends
with a short conclusions section.

2.2 Prediction from the ARFIMA Model
Several previous articles have examined issues concerning prediction from ARFIMA models. The
s step ahead predictor formed at forecast origin t is denoted by yt,s , and the prediction error and

10

MSE with known parameters are given by
et,s = yt+s − yt,s =

s−1

∑ ψ j εt+s− j

j=0

and
Var(et,s ) = σ 2

s−1

∑ ψ2
j

j=0

respectively. Some important results on the long run properties of predictors from ARFIMA models are due to Diebold and Lindner (1996), who showed that for I(d) where −1/2 < d < 1/2,
( )
( )
2
then Var en,s = O(1) and lims→∞ Var et,s = σε ∑∞ ψ 2 = Var(yt ). Moreover, for the non
j=0 j
stationary case, where 1/2 < d < 3/2, then the Var(et,s ) = O(s2d−1 ), which is undefined as the
forecast horizon gets large. Hence prediction from these models tends to be dominated at high forecast horizons by the hyperbolic decay that is evident in the theoretical optimal forecasts. At low
forecast horizons the impact of short memory components adds to the complexity of the forecasts.
The practical calculation of the optimum minimum MSE predictor is not as straightforward
as for ARMA models. The first proposed method was to use a truncated version of the infinite
autoregressive representation; see Granger and Joyeux (1980) and Geweke and Porter-Hudak
(1983). Also, Peiris (1987) and Peiris and Perera (1988) discussed computationally convenient
ways for calculating predictions from the autoregressive representation. Crato and Ray (1996)
have found that AIC is biased towards selecting low order AR models. However, Poskitt (2007)
has analyzed information criterion to determine the lag of autoregressive model.
An alternative methodology is to use the state-space representation, of Chan and Palma
(1998); although this can be computationally quite complicated. A further alternative predictor
is the partial moving average representation used in the context of regression models with ARMA
disturbances by Baillie (1980) and Yamamoto (1981). Then ARFIMA model can be expressed as
the partial moving average expansion
s−1

yt+s =

∑ ψ j εn+s− j + {1 − µ (L)π (L)}L−syt ,

j=0

11

(2.1)

where µ (L) = ∑s−1 ψ j L j . Equation (2.1) can also be written as
j=0
s−1

j=0

yt+s =

∞

j=0

∑ ψ j εt+s− j + ∑ τs j yt− j

(2.2)

where τs j is the weight placed on an observation j periods ago when forming the s step ahead
predictor. The prediction weights can be shown to be
{

τs j =

s−1

}

∑ πk+ j ψs−k+ j

+ πs+ j

(2.3)

k=1

where π j and ψ j are defined in (1.3) and (1.4) respectively. A brief derivation of the above results
is given in the Appendix 2A. The s step ahead predictor at time t is then given by
∞

yt,s =

∑ τs j yt− j .

(2.4)

j=0

We now consider the issues of parameter estimation before returning to the prediction problem
in practice when parameters are unknown and have to be replaced by their estimated values.

2.3 Estimation of ARFIMA Models
Given the substantial recent econometric literature on semi-parametric estimation of the long memory parameter, see, e.g., Phillips and Shimotsu (2006) and Phillips (2007), this study also considers an alternative two step estimation procedure. There is now a wide variety of semi-parametric
methods for the estimation of d and this study uses Local Whittle (LW ), which is proposed by Kunsch (1987) and Robinson (1995), and known to have desirable properties; see Robinson (1995),
Dalla, Giraitis, and Hidalgo (2005) and Phillips and Shimotsu (2006). The LW estimate of d is
obtained by minimizing the objective function
[
]
1 m 2d
2d m
ω j I(ω j ) −
ln
∑
∑ ln(ω j )
m j=1
m j=1

(2.5)

iω t 2
1
with respect to d, where I(ω j ) is the periodogram given by I(ω j ) = 2π T ∑T yt e j . An imj=1

portant issue concerns the choice of bandwidth, m. A relatively agnostic and often chosen value is

12

to choose m = T 0.5 . Methods for determining the optimal bandwidth in the sense of MSE reduction have been proposed by Henry and Robinson (1996), Delgado and Robinson (1996), Henry
(2001), Andrews and Guggenberger (2003), Andrews and Sun (2004), Phillips and Shimotsu
(2006) and the references therein. From Henry (2001), the optimal bandwidth, for −0.5 < d < 0.5,
is

(
mLW =

)
3 4/5 ∗ d −2/5 4/5
T
ν +
4π
12

(2.6)

′′
where ν ∗ = f ∗ (0)/2 f ∗ (0) and f is the spectral density function of the long memory process.

This study uses simulated data which allows ν ∗ to be analytically calculated. In particular, for the
ARFIMA(1, d, 0) model, it can be shown that ν ∗ = ϕ1 (1 − ϕ1 )−2 . For a known value of d, the
series can be fractionally filtered to obtain
t−p

ut = yt −

∑ πl (d)yt−l ,

l=1

where (1 − L)d yt = yt − ∑∞ πl (d)yt−l . Given the LW estimate of d denoted as dˆLW , then the
l=1
Feasible Fractionally Filtered (FFF) series is
t−p

ut = yt −
ˆ

ˆ
∑ πl (dˆLW )yt−l .

(2.7)

l=1

The ARMA(p, q) parameters of the original ARFIMA(p, d, q) process in equation (1.2) are then
estimated by minimizing the conditional sum of squares. Further discussion of this technique and
its properties can be found in Robinson (2006) and Baillie and Kapetanios (2009).
While there is a large literature on semi-parametric estimation of d, there is relatively little consideration of subsequent estimation of the remaining parameters in the model. The Local
Whittle Two Step Estimator (LW T SE), approach considered in this study is to fractionally filter
the observed series yt using the infinite AR representation of yt , given by
t−p

ut = yt −

∑ πl (d)yt−l ,

(2.8)

l=1

where d is replaced by its semi-parametric estimate. This generates residuals denoted by ut . The
ˆ
next stage is to estimate an ARMA model for the ut series, by minimizing the CSS. The small samˆ
ple properties of such an estimation strategy with the correct specification, as an alternative to the
13

one-step MLE, are considered in Baillie and Kapetanios (2009) which is concerned with a Monte
Carlo study. Baillie and Kapetanios (2009) also provide consistency and a rate of convergence of
second-step parameters in the LW T SE as
(
)
ˆ
β dˆLW − β0 (d0 ) = O p (m−1/2 )
(
)
ˆ
where β = (ϕ1 , ..., ϕ p , θ1 , ..., θq )′ , β0 (d0 ) denote the true parameter values, and β dˆLW be the
ˆ
LW T SE of β based on the fractionally filtered series ut .
The more direct approach to estimating the ARFIMA(p, d, q) process is to assume the innovations to be NID(0, σ 2 ), then the Gaussian likelihood is numerically maximized with respect to
the complete vector of parameters ϑ . Under these conditions, Fox and Taqqu (1986) have shown
the asymptotic distribution of the MLE to be
(
)
ˆ
T 1/2 ϑ − ϑ 0 → N{0, A(ϑ 0 )−1 }
where ϑ 0 denotes the true value of the vector of parameters, and where A(ϑ 0 ) is the information
matrix. Hosoya (1997) has shown, under the condition that non-Gaussian innovations satisfy mild
mixing and moment conditions, the asymptotic distribution of the MLE to be
(
)
ˆ
T 1/2 ϑ − ϑ 0 → N{0, A(ϑ 0 )−1 B(ϑ 0 )A(ϑ 0 )−1 }
where B(ϑ 0 ) is the outer product gradient, evaluated at the true parameter values ϑ 0 . Recently,
Baillie and Kapetanios (2008) have extended this analysis to include the estimation of long memory models with nonlinear autoregressive structures. The computation of such models is achieved
by the minimization of the conditional sum of squares (CSS).
It should be noted that the approximate time domain MLE approach has been successfully
applied in many extensions involving far more complicated models which include a long memory
parameter. This includes for example models with long memory and nonlinearity in the conditional
mean; see van Dijk, Frances, and Paap (2002) and Baillie and Kapetanios (2008); long memory
GARCH models of Baillie, Bollerslev, and Mikkelsen (1996), long memory stochastic volatility
models of Breidt, Crato, and de Lima (1998), long memory models with GARCH innovations,
multivariate long memory models, see Jensen (2009) and others.
14

2.4 Practical Issues and Simulation
Assuming the correct specification of the model, the practical predictors are formed by inserting
ˆ
parameter estimates into the minimum MSE predictor to obtain τs j . Hence the feasible predictor
with estimated parameters is

T −1

yt,s =
ˆ

ˆ
∑ τs j yt− j .

(2.9)

j=0

The purpose of this simulation is to assess importance of using LW T SE with agnostic bandwidth
choice, as compared to LW T SE with optimal bandwidth, as compared to AR(P) with P = ln(T )2 ,
as compared with MLE.
A natural approach for prediction with long memory is to use high order autoregressive
schemes. These methods can be based on an OLS projection on a lagged information set which is
repeated for each different forecast horizon; see Ray (1993), Crato and Ray (1996), Bhansali and
Kokoszka (2002), Doornik and Ooms (2004) and Barkoulas and Baum (2006). These studies
have produced rather mixed results for forecasting from long memory processes. For example,
Ray (1993) has noted the sensitivity of predictions to the order of approximation of the AR model
and also the sensitivity to the forecast horizon. Previous studies by Baillie and Kapetanios (2008)
and Baillie and Kapetanios (2009) have described the properties of MLE and the LW T SE for these
models.
The following simulation was designed to evaluate the relative performance of predictors
based on the LW T SE with agnostic bandwidth choice, the LW T SE with optimal bandwidth, the
AR(P), and compared with the benchmark MLE. A selection of the simulations that were performed are reported in the text and the reported results are indicative of the general findings.
Realizations of an ARFIMA(1, d, 0) process were generated for sample sizes of T = 200, and
T = 1, 000. The simulation designs are for values of the AR coefficient given by ϕ = 0.8, ϕ = 0.90,

ϕ = 0.95 and ϕ = 0.99; and for d = 0.2 and d = 0.4. In each simulation design, 10, 000 replications were generated with predictions generated up to s = 1, 2, 3, ....40 periods ahead from equation
(2.9). The bias and MSE of the different predictions were computed for each predictor based on

15

the different estimation methods, and also for each one of the 40 period forecast horizons. These
results are compared to the true model with known parameter values plugged into predictor formula. This study does not use the theoretical asymptotic results on prediction MSE which have
been derived for ARMA models by Baillie (1980) and Yamamoto (1981), and which have been
extended to ARFIMA models by Katayama (2008).
In this chapter we deal with the analogous issue concerning prediction. Tables 2.1 and 2.2
report the biases from the various methods for designs with a sample size of T = 200, while tables
2.3 and 2.4 report analogous quantities for T = 1, 000. As can be seen, the forecast biases are
extremely reasonable for virtually all methods and prediction horizons.
Tables 2.5 and 2.6 present the prediction MSE for the sample size of T = 200. The first column
gives the prediction lead time and the second column the theoretical prediction MSE derived form
knowledge of the true parameters in each design; see dot lines in Figures 1.1 and 1.2 for examples.
The MSE quantity was also computed from the predictor with the true theoretical values inserted
into the optimal prediction formula and reported in the third column of Tables 2.5 through 2.8. As
expected, these MSEs were extremely close to the true theoretical values. The rest of the table
gives the simulated MSEs from the MLE, AR(P) and LW based on the m = T 0.5 bandwidth and
also the LW based on the optimal bandwidth. In general the MLE compares well with the SPE
versions and the optimal bandwidth version of the LW is generally inferior to the simple LW . For
example, in Table 2.6, at 10 periods ahead, the MSE of MLE and LW with m = T 0.5 are 21.8
and 23.1, respectively. The performance of the optimal bandwidth LW deteriorates with increasing
persistence of the AR parameter. The prediction MSEs of the LW T SE markedly differ from those
of the true model for d = 0.4 and especially when ϕ = 0.9. In general the predictor based on
the MLE performs very well in the sense that its value is close to the MSE of the predictor with
known parameter values. The predictor based on the LW T SE is generally inferior with higher MSE
compared to that of the predictor based on the MLE. The predictor based on the AR(P) performs
poorly. For example, in Table 2.6, MSE at 40 period forecast from AR(P) is 42 percent higher
than theoretical MSE comparing to 2 percent from MLE. In general the relative performance of

16

the predictor based on the LW T SE deteriorates as the forecast lead time increases. Analogous
results for the larger sample size of T = 1, 000 are to be found in tables 2.7 and 2.8. In general the
performance of the predictors improves as the observations increases. We also consider the case of
ARFIMA(0, d, 1) process and ARFIMA(2, d, 1) process. The results in Tables 2.9 and 2.10 imply
that the performance of predictors from MLE are generally slightly better than LW T SE.
The above results are all predicated on the assumption that the order of the short memory component, i.e. an AR(1) process is known and that the unknown quantities are the parameter values.
This assumption is applied to both the LW T SE and the MLE and in this sense treats both methods
equally. In practice the order of say an ARMA(p, q) process for ut would have to be determined
before the parameter estimation method is implemented. In order to increase the realism of the
results a limited simulation was conducted where the order p of an autoregressive process was determined by Bayesian Information Criteria (BIC) for an ARFIMA(p, d, 0) model when using MLE
and also on the FFF series ut when using the LW T SE approach. Some interesting results were for
ˆ
when the short run dynamics of the ARFIMA model are mis-specified. This is a situation where
the MLE would be expected to potentially perform very poorly, and the LW to be superior since
it is robust to arbitrary short run dynamics. One simulation was based on the ARFIMA(2, d, 0)
process
(1 − L)d (1 − ϕ1 L − ϕ2 L2 )yt = εt
it can be shown after some algebra that

υ∗ =

(ϕ1 ϕ2 − ϕ1 − 4ϕ2 )
(1 − ϕ1 − ϕ2 )2

while the process
(1 − L)d (1 − 0.0800L − 0.9025L2 )yt = εt
has an autoregressive component with complex roots and which has a damping factor of
|−ϕ2 |1/2 = 0.98, which generates very slow decay it its autocorrelation function and impulse response weights with a sinusoidal pattern superimposed. In this exercise, we estimate
ARFIMA(1, d, 0) model for both the MLE and LW T SE. The results are presented in Tables 2.11
17

to 2.16. Again, the MLE, although based an a mis-specified model performs more satisfactorily
than the LW T SE for either agnostic or optimal bandwidth. The AR(P) performs very poorly comparing to the MLE and LW T SE. The results of prediction based on SPE are disappointing in
ARFIMA(1, d, 0) with high autoregressive coefficients and misspecified cases. These suggest the
need for further work to be done on the prediction issue in SPE.

2.5 Conclusion
This chapter considers the effects on multi step predication of using semi parametric Local Whittle
estimators compared to MLE for long memory ARFIMA models. We consider various representations for the minimum MSE predictor with known parameters. We then conduct a detailed
simulation study for when true parameters are replaced by estimates. The predictor based on MLE
is found to be superior in MSE sense to the predictor based on two step Local Whittle estimation.
The “optimal” bandwidth Local Whittle estimator gives rise to worse predictions that the Local
Whittle using an agnostic bandwidth of the square root of the sample size. Although the Local
Whittle is theoretically robust to short run dynamics, it nevertheless gives rise to inferior predictions to a predictor using MLE from a mis-specified model. This provides doubt on the usefulness
of the Local Whittle estimator for the issue of prediction in long memory ARFIMA models, and
suggests further work is required on how to form short and medium term predictions based on semi
parametric estimates of the long memory parameter.

18

APPENDICES

19

Appendix 2A: The prediction weight formula
Any stationary time series yt has a Wold decomposition of the form yt = ∑∞ ψ j εt− j . When
j=0
considering the the formation of the optimum minimmum MSE predictor, it is convenient to use
the partial moving average expansion:
yt+s = µ (L)εt+s + ∂ (L)yt ,

(2.10)

where µ (L) = ∑s−1 ψ j L j .
j=0
Let consider the case of yt is the ARFIMA(p, d, q) model where ϕ (L)(1 − L)d yt = θ (L)εt . It
follows that
yt+s = µ (L)εt+s + ∂ (L)yt
= µ (L)θ (L)−1 ϕ (L)(1 − L)d yt+s + ∂ (L)yt
= {µ (L)θ (L)−1 ϕ (L)(1 − L)d + ∂ (L)Ls }yt+s ,
where

{

∂ (L) =
Or by defining

φ ∗ (L) =

θ (L) − µ (L)ϕ (L)(1 − L)d
θ (L)

}
L−s .

}
{
d L−s ,
θ (L) − µ (L)ϕ (L)(1 − L)

the partial MA representation of the ARFIMA(p, d, q) model is
yt+s = µ (L)εt+s +
or

φ ∗ (L)
yt ,
θ (L)

s−1

∞

s−1

j=0

yt+s =

j=0

(2.11)

j=0

∑ ψ j εt+s− j + ∑ τs j yt− j = ∑ ψ j εt+s− j + τs(L)yt ,

(2.12)

where τs (L) = ∑∞ τs j L j . Note that τs j is the weight given to the (t − j) observation in the
j=0
calculation of the s step forecast.
Further, we consider this formula with the AR(1) model, yt = ϕ yt−1 + εt . It follows that

ψ j = ϕ j.

]
[
ϕ ∗ (L) = 1 − (1 + ϕ L + ϕ 2 L2 + ... + ϕ s−1 Ls−1 )(1 − ϕ L) L−s = ϕ s .
20

Hence yt+s = ∑s−1 ϕ j εt+s− j + ϕ s yt .
j=0
Next, we derive the explicit formula for the prediction weight.

Let consider the

ARFIMA(0, d, 0) model, (1 − L)d yt = εt . We can rewrite ARFIMA(0, d, 0) in AR(∞) and MA(∞)
representations as

∞

∑ π j L j ≡ π (L),

(1 − L)d = 1 −

j=1

and
(1 − L)−d = 1 +

∞

∑ ψ j L j ≡ ψ (L).

j=1

Since θ (L) = 1 and ϕ (L) = 1, then

τs

(L) = φ ∗ (L) =

{
}
d L−s = {1 − µ (L)π (L)} L−s
1 − µ (L)(1 − L)

(2.13)

By replacing π (L) and ψ (L) in equation (2.13), we can show that, in case of s = 1,

τ1, j = π1+ j

for

j = 0, 1, 2, ...,

(2.14)

in case of s = 2,

τ2, j = π1+ j ψ1+ j + π2+ j

for

j = 0, 1, 2, ...,

(2.15)

in case of s = 3,

τ3, j = π1+ j ψ2+ j + π2+ j ψ1+ j + π3+ j

for

j = 0, 1, 2

(2.16)

and so on. By the induction from equations (2.14) through (2.16), the prediction weights are
{
}

τs, j =

s−1

∑ πk+ j ψs−k+

+ πs+ j

(2.17)

k=1

for s = 1, 2, 3, ... and j = 0, 1, 2, ... . With the same argument, the formula in equation (2.17) is the
prediction weights for any ARFIMA(p, d, q) models.

21

Appendix 2B: Tables
Table 2.1: Biases of Different Predictors, ARFIMA(1, d, 0) with d = 0.4, ϕ = 0.8, and T = 200
Forecast
Horizon(s)
1
2
3
4
5
10
20
30
40

True
Parameter
value
-0.012
-0.009
-0.006
0.000
0.008
-0.019
-0.025
-0.028
-0.025

MLE

AR(P)

LW
m = T 0.5

LW
optimal m

-0.013
-0.010
-0.008
-0.002
0.007
-0.020
-0.025
-0.026
-0.024

-0.014
-0.008
-0.004
0.003
0.021
-0.012
-0.022
-0.018
-0.002

-0.011
-0.008
-0.005
0.001
0.010
-0.017
-0.024
-0.026
-0.024

-0.009
-0.007
-0.006
-0.001
0.007
-0.025
-0.035
-0.038
-0.036

Key: Number of replications is 10,000 and P = ln(T )2 ≈ 28 in the AR(P).
Table 2.2: Biases of Different Predictors, ARFIMA(1, d, 0) with d = 0.4, ϕ = 0.9, and T = 200
Forecast
Horizon(s)
1
2
3
4
5
10
20
30
40

True
Parameter
value
-0.008
0.003
0.013
0.031
0.032
0.054
0.045
0.043
0.004

MLE

AR(P)

LW
m = T 0.5

LW
optimal m

-0.008
0.002
0.012
0.029
0.030
0.047
0.029
0.023
-0.017

-0.005
0.009
0.016
0.038
0.044
0.089
0.071
0.068
0.021

-0.008
0.003
0.012
0.029
0.030
0.048
0.034
0.029
-0.012

-0.012
-0.004
0.003
0.016
0.013
0.009
-0.061
-0.156
-0.336

Key: See Table 2.1.

22

Table 2.3: Biases of Different Predictors, ARFIMA(1, d, 0) with d = 0.4, ϕ = 0.8, and T = 1, 000
Forecast
Horizon(s)
1
2
3
4
5
10
20
30
40

True
Parameter
value
-0.007
-0.005
-0.004
-0.004
-0.008
0.009
0.042
0.014
0.037

MLE

AR(P)

LW
m = T 0.5

LW
optimal m

-0.008
-0.006
-0.005
-0.006
-0.010
0.007
0.042
0.015
0.039

-0.007
-0.004
-0.006
-0.006
-0.012
0.008
0.036
0.000
0.023

-0.007
-0.005
-0.004
-0.005
-0.009
0.008
0.041
0.014
0.038

-0.007
-0.005
-0.003
-0.003
-0.007
0.011
0.044
0.017
0.040

Key: Number of replications is 10,000 and P = ln(T )2 ≈ 47 in the AR(P).

Table 2.4: Biases of Different Predictors, ARFIMA(1, d, 0) with d = 0.4, ϕ = 0.9, and T = 1, 000
Forecast
Horizon(s)
1
2
3
4
5
10
20
30
40

True
Parameter
value
-0.012
-0.015
-0.022
-0.006
-0.006
0.023
0.019
0.011
0.028

MLE

AR(P)

LW
m = T 0.5

LW
optimal m

-0.012
-0.016
-0.024
-0.008
-0.009
0.020
0.019
0.013
0.032

-0.011
-0.018
-0.024
-0.009
-0.008
0.031
0.037
0.031
0.045

-0.012
-0.015
-0.023
-0.007
-0.008
0.018
0.011
0.000
0.017

-0.011
-0.015
-0.023
-0.008
-0.010
0.012
0.002
-0.007
0.012

Key: See Table 2.3.

23

Table 2.5: MSEs of Different Predictors, ARFIMA(1, d, 0) with d = 0.4, ϕ = 0.8, and T = 200
Forecast
Horizon(s)

Theoretical
MSE

1
2
3
4
5
10
20
30
40

1.000
2.440
3.978
5.456
6.809
11.502
15.252
16.754
17.629

True
Parameter
value
1.008
2.452
4.027
5.575
6.954
11.611
15.356
17.076
18.189

MLE

AR(P)

LW
m = T 0.5

LW
optimal m

1.021
2.498
4.112
5.706
7.141
11.966
15.998
17.783
18.823

1.197
2.966
4.972
6.942
8.777
15.202
20.775
22.946
24.285

1.022
2.507
4.139
5.77
7.255
12.388
16.955
18.993
20.221

1.06
2.568
4.202
5.836
7.32
12.488
17.248
19.562
20.865

Key: See Table 2.1.

Table 2.6: MSEs of Different Predictors, ARFIMA(1, d, 0) with d = 0.4, ϕ = 0.9, and T = 200
Forecast
Horizon(s)

Theoretical
MSE

1
2
3
4
5
10
20
30
40

1.000
2.690
4.793
7.130
9.584
21.488
37.790
46.400
51.312

True
Parameter
value
0.995
2.687
4.801
7.116
9.466
20.982
37.072
46.141
50.366

MLE

AR(P)

LW
m = T 0.5

LW
optimal m

1.014
2.753
4.928
7.332
9.776
21.780
38.651
47.957
52.263

1.207
3.292
5.979
9.026
12.229
28.480
53.797
70.359
78.776

1.020
2.786
5.036
7.539
10.135
23.113
43.774
57.243
65.143

2.340
3.671
5.935
8.535
11.317
29.166
125.699
546.540
1973.282

Key: See Table 2.1.

24

Table 2.7: MSEs of Different Predictors, ARFIMA(1, d, 0) with d = 0.4, ϕ = 0.8, and T = 1, 000
Forecast
Horizon(s)

Theoretical
MSE

1
2
3
4
5
10
20
30
40

1.000
2.440
3.978
5.456
6.809
11.502
15.252
16.754
17.629

True
Parameter
value
0.997
2.436
3.989
5.518
6.899
11.556
15.365
16.724
17.663

MLE

AR(P)

LW
m = T 0.5

LW
optimal m

0.999
2.439
3.994
5.528
6.915
11.630
15.517
16.916
17.838

1.042
2.565
4.207
5.791
7.247
12.374
16.628
18.186
19.207

0.999
2.447
4.016
5.558
6.954
11.724
15.622
17.112
18.055

1.000
2.448
4.018
5.564
6.959
11.735
15.588
17.057
18.017

Key: See Table 2.3.

Table 2.8: MSEs of Different Predictors, ARFIMA(1, d, 0) with d = 0.4, ϕ = 0.9, and T = 1, 000
Forecast
Horizon(s)

Theoretical
MSE

1
2
3
4
5
10
20
30
40

1.000
2.690
4.793
7.130
9.584
21.488
37.790
46.400
51.312

True
Parameter
value
1.000
2.696
4.793
7.129
9.620
21.969
38.099
47.299
51.908

MLE

AR(P)

LW
m = T 0.5

LW
optimal m

1.004
2.706
4.819
7.175
9.680
22.114
38.282
47.565
52.241

1.054
2.838
5.087
7.603
10.314
23.566
41.372
51.854
57.663

1.015
2.747
4.897
7.324
9.899
22.821
40.101
50.257
55.751

1.022
2.748
4.887
7.279
9.828
22.545
39.466
49.067
54.084

Key: See Table 2.3.

25

Table 2.9: MSEs of Different Predictors, ARFIMA(0, d, 1) with d = 0.4, θ = 0.8, and T = 200
Forecast
Horizon(s)

Theoretical
MSE

1
2
3
4
5
10
20
30
40

1.000
2.440
2.800
3.001
3.138
3.494
3.781
3.927
4.023

TRUE
Parameter
Value
1.044
2.463
2.837
3.198
3.027
3.484
3.992
3.714
4.208

MLE

AR(P)

LW
m = T 0.5

1.060
2.498
2.861
3.215
3.038
3.506
4.028
3.733
4.214

1.213
2.822
3.370
3.788
3.638
4.102
4.597
4.355
5.047

1.131
2.749
3.116
3.476
3.276
3.667
4.174
3.888
4.342

Key: See Table 2.3.

Table 2.10: MSEs of Different Predictors, ARFIMA(2, d, 1) with d = 0.4, ϕ1 = −0.5, ϕ2 = −0.9,
θ = 0.6, and T = 200
Forecast
Horizon(s)

Theoretical
MSE

1
2
3
4
5
10
20
30
40

1.000
1.250
1.647
1.713
2.295
3.142
4.056
4.370
4.487

TRUE
Parameter
Value
0.975
1.258
1.621
1.702
2.256
3.173
4.030
4.442
4.566

Key: See Table 2.3.

26

MLE

AR(P)

LW
m = T 0.5

1.003
1.281
1.653
1.734
2.323
3.278
4.180
4.556
4.653

1.159
1.507
1.923
2.027
2.742
3.734
4.768
5.130
5.195

1.062
1.370
1.670
1.759
2.419
3.304
4.202
4.558
4.670

Table 2.11: Biases of Different Predictors, ARFIMA(2, d, 0) process with ϕ1 = 0.08,
ϕ2 = 0.9025, d = 0.2, and T = 400
Forecast
Horizon(s)
1
2
3
4
5
10
20
30
40

MLE

AR(P)

-0.009
-0.015
-0.01
0.001
-0.002
0.024
0.011
-0.02
-0.019

-0.009
-0.015
-0.01
-0.003
-0.003
0.011
0.002
-0.022
-0.011

LW
m = T 0.5
-0.011
-0.016
-0.013
-0.001
-0.006
0.019
0.002
-0.031
-0.033

Key: Number of replications is 10,000 and P = ln(T )2 ≈ 35 in the AR(P). The MLE and LW T SE
are estimated as ARFIMA(1, d, 0).

Table 2.12: Biases of Different Predictors, ARFIMA(2, d, 0) process with ϕ1 = 0.08,
ϕ2 = 0.9025, d = 0.4, and T = 400
Forecast
Horizon(s)
1
2
3
4
5
10
20
30
40

MLE

AR(P)

0.006
0.013
0.006
0.01
0.01
0.014
-0.007
-0.047
-0.032

0.003
0.013
0.003
0.007
0.009
0.017
0.004
-0.028
-0.011

Key: See Table 2.11.

27

LW
m = T 0.5
0.004
0.013
0.004
0.009
0.009
0.014
-0.007
-0.046
-0.031

Table 2.13: Biases of Different Predictors, ARFIMA(2, d, 0) process with ϕ1 = 0.08,
ϕ2 = 0.9025, d = 0.49, and T = 400
Forecast
Horizon(s)
1
2
3
4
5
10
20
30
40

MLE

AR(P)

0.005
-0.001
0.011
0.007
0.008
-0.002
-0.066
-0.068
-0.076

0.000
0.002
-0.003
0.002
-0.011
-0.026
-0.104
-0.116
-0.115

LW
m = T 0.5
0.002
0.000
0.008
0.008
0.005
-0.005
-0.077
-0.087
-0.102

Key: See Table 2.11.

Table 2.14: MSEs of Different Predictors, ARFIMA(2, d, 0) process with ϕ1 = 0.08, ϕ2 = 0.9025,
d = 0.2 and T = 400
Forecast
Horizon(s)
1
2
3
4
5
10
20
30
40

MLE

AR(P)

1.034
1.086
2.278
2.409
3.563
6.933
15.671
25.269
36.194

1.124
1.203
2.496
2.655
3.947
7.851
18.39
30.38
43.855

Key: See Table 2.11.

28

LW
m = T 0.5
1.09
1.15
2.4
2.555
3.755
7.215
15.803
24.699
34.329

Table 2.15: MSEs of Different Predictors, ARFIMA(2, d, 0) process with ϕ1 = 0.08, ϕ2 = 0.9025,
d = 0.4 and T = 400
Forecast
Horizon(s)
1
2
3
4
5
10
20
30
40

MLE

AR(P)

1.195
1.275
3.041
3.438
5.637
14.138
43.603
84.881
133.739

1.109
1.392
3.109
3.797
6.050
16.211
52.669
106.780
174.338

LW
m = T 0.5
1.180
1.442
3.267
3.963
6.272
15.832
46.090
85.967
131.450

Key: See Table 2.11.

Table 2.16: MSEs of Different Predictors, ARFIMA(2, d, 0) process with ϕ1 = 0.08, ϕ2 = 0.9025,
d = 0.49 and T = 400
Forecast
Horizon(s)
1
2
3
4
5
10
20
30
40

MLE

AR(P)

1.515
1.502
3.873
4.434
7.467
21.238
71.020
146.790
240.306

1.117
1.500
3.435
4.419
7.178
22.723
82.612
181.008
315.058

Key: See Table 2.11.

29

LW
m = T 0.5
1.350
1.812
4.199
5.382
8.520
24.537
77.451
152.963
242.127

Chapter 3

INFORMATION CRITERIA METHODS FOR SELECTION OF ARFIMA MODELS

3.1 Introduction
As previously stated, the ARFIMA model has become the standard basic model for representing
discrete time long memory processes. There are now many methods available for detecting the
presence of long memory, but surprisingly the issue of model selection in the ARFIMA(p, d, q)
model has not been satisfactorily resolved. Hence once it has been determined that the long memory parameter d is non zero, it then remains to determine the appropriate orders of the short memory ARMA structure; namely p and q.
It is first worth recollecting some results on the estimation of ARFIMA models that were summarized in the introduction. However, successful implementation of QMLE for ARFIMA models
requires the correct specification of the short memory component of the conditional mean. This
caveat can be avoided by the use of one of the many Semi Parametric Estimation (SPE) procedures,
where an estimate of the long memory parameter is obtained which does not require the specification of the short memory process. Following the work of Geweke and Porter-Hudak (1983), many
such SPEs have now been proposed, with versions of the Local Whittle (LW ) proving particularly
popular; see Phillips and Shimotsu (2006), Phillips (2007), etc. While this method appears attractive in terms of avoiding the specification of the short memory components of the process, it
does not however lead to a simple way of either specifying or estimating all the parameters of the
ARFIMA model.
This chapter proposes a solution to the above problem of model selection of ARFIMA models
by utilizing some standard Information Criteria, (IC). We show that model selection based on the
BIC is consistent in the sense of Sin and White (1996). The alternate strategy of using HQ is
also consistent, while the AIC is inconsistent. Our proposed method requires initial specification
30

of the maximum lags pmax and qmax of the ARFIMA(p, d, q) process which are to be entertained
by the investigator. The method then uses MLE or QMLE to estimate the possibly misspecified
(pmax + 1)(qmax + 1) models, and uses the BIC for model selection. We report the results of a
detailed simulation to assess the proposed methodology; and we find that the proposed MLE − IC
procedure generally works well in terms of the selection of the most appropriate model. The
method appears particularly effective for situations where the true data generating process is of
relatively low order with p and q less than or equal to two or three. For completeness we also
investigate the application of the IC based on a fractionally filtered series where an SPE such as
LW is used at an initial stage. This method works quite well for some designs, but is generally
inferior to the MLE based procedure on the original series. Hence we propose a more limited role
for SPE, since once the presence of long memory or fractional integration has been confirmed by
LW , or one of its many variants, the obvious strategy is to then revert immediately to using QMLE
with our suggested model selection strategy. We also suggest an adjustment of BIC which reflects
the sub optimal rate of convergence of parameters in second step of LW T SE.
We conclude that our proposed approach for selection of ARFIMA models works well for low
order models where p and q are less than or equal to 2; but there is mixed evidence for higher order
models. There are also detailed examples of the methodology applied to monthly CPI inflation
series.

3.2 Information Criteria for Model Selection
The successful implementation of a modeling strategy for using ARFIMA(p, d, q) models requires
an adequate model selection approach for the orders of p and q. Previous articles by Beran,
Bhansali, and Ocker (1998) and Crato and Ray (1996) have discussed this issue and have addressed less general formulations of the problem. There is a long literature on the implementation
of information criteria on weakly dependent processes; for example Shibata (1976) and Hannan
and Quinn (1979) have studied the AR(p) model and Hannan (1980) has investigated the stationary ARMA(p, q) model. The Bayesian (BIC) of Schwarz (1978) and the Hannan-Quinn (HQ)
31

Hannan and Quinn (1979) information criteria consistently estimate the lag order of linear models
with weakly dependent data, while the Akaike Information Criterion (AIC) is known to be inconsistent. Further, Sin and White (1996) and Kapetanios (2001) have shown that these properties
extend to nonlinear models for weakly dependent processes. Hidalgo (2002) has shown that similar results are valid for regressions involving long memory regressors. Recently Poskitt (2007)
has extended the optimality results of Shibata (1980) to stationary long memory processes.
Since none of the above results are directly relevant for the model considered in this study,
it is necessary to provide some new theory for lag order selection of p and q using information
criteria. In the following the conditional sum of squares objective function is denoted as
T

Q(y1 , ..., yT ; Θ p,q ) =

∑

εt (Θ p,q )2 ,

t=max(p,q)

where Θ p,q = (ϕ1 , ..., ϕ p , θ1 , ..., θq , d)′ and εt = εt (Θ p,q ) is defined as εt = θ (L)−1 ϕ (L)(1 − L)d yt .
Then, lag order selection, via the use information criteria, is carried out by maximizing
ICT (p, q) = −Q(y1 , ..., yT ; Θ p,q ) − cT (p, q)

(3.1)

over 0 ≤ p ≤ pmax and 0 ≤ q ≤ qmax , where cT (p, q) is a penalty term depending on p, q and T .
The following theorem provides a result on the properties of the estimated lag orders.
Theorem 3.1. Consider the ARFIMA(p, d, q) process

ϕ (L)(1 − L)d yt = θ (L)εt

(3.2)

where ϕ (L) and θ (L) are polynomials in the lag operator of orders p and q respectively, with
all their roots lying outside the unit circle. Then for some 0 ≤ p0 ≤ pmax and 0 ≤ q0 ≤ qmax ,
without loss of generality, it is assumed that ϕi ̸= 0, for i = 1, .., p0 and θi ̸= 0, for i = 1, .., q0 . The
estimates of the lag orders p0 and q0 , are denoted by p and q and are determined by minimizing
ˆ
ˆ
(3.1) over 0 ≤ p ≤ pmax and 0 ≤ q ≤ qmax . It is assumed that if p < p0 or q< q0 then there
[
]
exists a unique Θ∗ that minimizes E εt (Θ p,q )2 . Then, if cT (p, q) = o(T ) for all p, q, and
p,q
(
)
cT (p, q) − cT (p0 , q0 ) → ∞ when p > p0 and q > q0 , then limT →∞ Pr ( p, q) = (p0 , q0 ) = 1
ˆ ˆ
32

The conditions on cT (p, q) given in Theorem 3.1 are standard for information criteria consistency for weakly dependent data. They imply that the Bayesian and Hannan-Quinn information
criteria provide consistent estimates of p0 and q0 whereas the Akaike information criterion is not
consistent.

Proof of Theorem 3.1
To prove the Theorem 3.1 it is necessary to use the theoretical framework of Sin and White (1996),
and to show the two properties
({
lim Pr

T →∞

and that

({
lim Pr

T →∞

p < p0

} {
})
0
∪ q<q
=0

(3.3)

p > p0

} {
})
0
∩ q>q
=0

(3.4)

Together, (3.3) and (3.4) imply Theorem 3.1. It is first necessary to show that for all p and q,
ˆ
there exist some Θ∗ such that Θ p,q − Θ∗ = O p (T −1/2 ). The two separate cases of (i) p ≥
p,q
p,q
ˆ
p0 and q ≥ q0 and (ii) p < p0 or q < q0 are distinguished. For the first case, Θ p,q − Θ p,q =
O p (T −1/2 ) where Θ p,q = (ϕ1 , ..., ϕ p0 , 0, ..., 0, θ1 , ..., θq0 , 0, ..., 0, d)′ , follows by Theorems 2.1 and
2.2 of Hosoya (1997).

[
]
For the second case, note the assumption that E εt (Θ p,q )2 is uniquely minimized at some

Θ∗ . Then, on taking this assumption with the treatment of Hosoya (1997), who does not assume
p,q
ˆ
that the true model belongs to the class of models being estimated, it follows that Θ p,q − Θ∗ =
p,q
O p (T −1/2 ), which uses identical arguments to those in the proof of Corollary 1 of Baillie and
Kapetanios (2008).
It is now possible to start by showing (3.3). On defining Q∗ (Θ p,q ) = −Q(y1 , ..., yT ; Θ p,q )
T
and on taking a two term mean value expansion of Q∗ (Θ∗ ) around Θ p,q and noting that
p,q
T

33

ˆ
∇Q∗ (Θ p,q ) = 0, gives
T
(
)
ˆ
ˆ
ˆ
Q∗ (Θ∗ ) = Q∗ (Θ p,q ) + ∇Q∗ (Θ p,q )′ Θ p,q − Θ∗
p,q
p,q
T
T
T
)′
)
(
(
∗ (Θ ) Θ
ˆ p,q − Θ∗
¯ p,q ˆ p,q − Θ∗
+0.5 Θ
p,q ∇QT
p,q
(
)′
(
)
ˆ
ˆ
¯
ˆ
= Q∗ (Θ p,q ) + 0.5 Θ p,q − Θ∗
∇2 Q∗ (Θ p,q ) Θ p,q − Θ∗
p,q
p,q
T

T

¯
By the quadratic nature of Q∗ (Θ p,q ), it follows that ∇2 Q∗ (Θ p,q ) = O p (T ) and therefore that
T
T
ˆ
Q∗ (Θ p,q ) − Q∗ (Θ∗ ) = O p (1)
p,q
T
T
(
)
ˆ
Q∗ (Θ p,q ) = Q∗ (Θ∗ ) + o p (T ) = E Q∗ (Θ∗ ) + o p (T )
p,q
p,q
T
T
T

Then,

(3.5)

(3.6)

where the second equality follows by combining (2.8) of Robinson (2006) and Lemma 3.3 of
Hosoya (1997). Then, (3.3) holds if
(
)
lim Pr ICT (p, q) < ICT (p0 , q0 ) = 1

T →∞

(3.7)

when either p < p0 or q < q0 . However, by (3.6) and the fact that cT (p, q) = o(T ), ICT (p, q) <
ICT (p0 , q0 ) with probability approaching one, if
(

)

(

)

T −1 E Q∗ (Θ p0 ,q0 ) − T −1 E Q∗ (Θ∗ ) > 0
p,q
T
T
But this follows immediately by the identifiability condition that ϕi ̸= 0, i = 1, .., p0 and θi ̸= 0,
i = 1, .., q0 . Note that the weaker condition ϕ p0 ̸= 0, and θq0 ̸= 0, may not be sufficient as under
this condition the various sub models may not be strictly nested requiring more stringent conditions
for the penalty term. The next stage is to show (3.4). By the fact that Q∗ (Θ∗ ) = Q∗ (Θ∗ 0 0 ) for
p,q
T
T
p ,q
all pmax ≥ p ≥ p0 and qmax ≥ q ≥ q0 , and using (3.5), it follows that, for all pmax ≥ p ≥ p0 and
qmax ≥ q ≥ q0 ,
ˆ
ˆ
ICT (p0 , q0 ) − ICT (p, q) = Q∗ (Θ p0 ,q0 ) − Q∗ (Θ p,q ) + cT (p, q) − cT (p0 , q0 )
T
T
= cT (p, q) − cT (p0 , q0 ) + O p (1)
But, since cT (p, q) − cT (p0 , q0 ) → ∞ when p > p0 and q > q0 , (3.4) follows immediately.
34

3.3 Simulation Study
This section reports the results from a detailed Monte Carlo study to investigate the performance
of the various information criteria and estimation procedures. The model selection criteria we consider are Akaike Information Criterion (AIC), Baynesian Information Criterion (BIC), and HannanQuinn Information Criterion (HQ). ICs are defined as
ˆ
ln(σ 2 ) + kc(T )/T,
where c(T ) = 2 in the Akaike case, c(T ) = ln(T ) in the BIC case, c(T ) = 2α ln(ln(T )), α > 1 in
ˆ
the HQ case, and k = 2 + p + q is the number of parameters, σ 2 is the MLE estimator of the error
variance. The selected model is the one that has the minimum IC.
Tables 3.1 through 3.4 report the performance of the various MLE − IC applied to realizations
from an ARFIMA(1, d, 0) data generating process, and with a sample size of T = 400. Table 3.1
is for ϕ = 0.8 and d = 0.2; table 3.2 is for ϕ = 0.8 and d = 0.4; and table 3.3 is for ϕ = 0.8
and d = 0.8. It can be seen that the BIC and to a slightly lesser extent the HQ perform very well
for all the designs; and interestingly the third design with a long memory parameter in the non
stationary region has a very similar performance over the replications. In fact the BIC selects the
appropriate model on 89%, 91% and 88% of the occasions respectively. However, as the short
memory persistence increases with ϕ = 0.9 and d = 0.4 in table 3.4. then the BIC does deteriorate
and only selects the correct model on 59% occasions. The HQ is slightly worse and has success
rates of between 69% and 73%; while the inconsistent AIC is only successful 8% to 12% of the
time.
The results of applying the corresponding SPE −CSS − IC method to the fractionally filtered
series are also given in these tables. The performance of these corresponding IC are generally
slightly worse than for the MLE − IC and with a similar ranking in performance of the different
ICs. Hence there does not appear to be any reason for making use of the SPE and LW statistics for
model selection once long memory has been detected.
Table 3.5 reports the results for the ARFIMA(0, d, 1) model and the performance of the
35

MLE − BIC is excellent with an 80% success rate of selecting the correct model. interestingly, the
inclusion of the MA short memory parameterization significantly deteriorates the performance of
the SPE − BIC to only a 49% success rate. Table 3.6 describes the results for the ARFIMA(1, d, 1)
process, with ϕ = 0.5, θ = −0.7 and d = 0.4. The MLE − BIC has a correct selection rate of 63%.
The properties of the IC in higher order AR models are explored in Tables 3.7 through 3.8
which report simulation results for the IC when the data generating process is the ARFIMA(2, d, 0)
process of the form
(1 − 0.25L − 0.375L2 )(1 − L)0.4 yt = εt
where the short memory component has real roots of (4/3) and −2 and d = 0.4. The BIC is still the
best performing IC statistic and successfully selects the correct specification 78% of the time for a
sample size of T = 400 and 93% of the time for a sample size of T = 1, 000. The SPE −CSS −IC is
surprisingly slightly superior to the MLE − IC in these cases; which is an exception to the general
pattern of the simulation results.
A further ARFIMA(2, d, 0) process of the form
(1 − 1.00L + 0.50L2 )(1 − L)0.4 yt = εt
where the short memory component has complex conjugate roots of (1 + i) and (1 − i) is investigated in Table 3.9 and for a sample size of T = 400 the BIC only selects the correct specification
on 49% of the time. In this case the SPE −CSS − IC is substantially inferior with a success rate of
only 38%.
Tables 3.10 through 3.13 report results where the true model is an ARFIMA(2, d, 1); and it
can be seen that the performance of the BIC is very poor in some cases such as the process in table
3.10 where it is affected by strong first and second order AR coefficients of AR representation.
ARFIMA(2, d, 1) in Table 3.10 can be represented by [1 − (7/12)L1 − (29/72)L2 + (25/216)L3 +
...](1 − L)d yt = εt . So, high proportion of selecting ARFIMA(1, d, 0) or ARFIMA(2, d, 0) can
be expected. [Pukkila, Koreisha, and Kallinen (1990)] Hence, BIC tends to select the simpler
ARFIMA(1, d, 0) model 41% of the time. Another reason for selecting lower order model is near

36

cancellation of the AR and MA root. The success rates of the BIC are moderate for Tables 3.11 and
3.12 with 49% and 60%. However, the BIC selects the correct ARFIMA(2, d, 1) model on 92% of
the occasions in table 3.13.
The performance of the BIC is mixed for higher order AR short memory structures. The
ARFIMA(3, d, 0) model is analyzed in tables 3.14 and 3.15, and the ARFIMA(4, d, 0) model results
are reported in tables 3.16 through 3.19. The success rate of the BIC is very mixed and very much
depends on the parameter space of the data generating process. The model selection seems to work
well in the case of a generating process with strong coefficient at high order, such as ϕ3 = −0.92705
in case of table 3.15 and ϕ4 = −0.31 in case of tables 3.18 and 3.19. In table 3.18 it is excellent
with the BIC having 77% success rate and 83% in table 3.19. The ARFIMA(4, d, 1) model is
examined in table 3.20.

3.4 Modified Information Criteria for LW T SE
The previous results on applying the SPE − BIC and the SPE − HQIC to ARFIMA models has
made the standard assumption that all the parameters are estimated simultaneously in the estimation before the IC are calculated. However, the implementation of the LW T SE requires initial
estimation of d by a m1/2 consistent estimator and the short memory parameters to be based on the
second step procedure. Groen and Kapetanios (2010) have considered with problem in the issue
of estimating regressions with unknown factors, as in Bai and Ng (2002) and have conjectured the
possibility of modification of the IC. In our problem the LW T SE has a rate of convergence of the
short memory parameters maybe slower than MLE and depends on the first step SPE estimation
which is related to the convergence of dˆ to d. This is very similar to the two step regression type
problem in other areas of econometrics; see Baillie and Kapetanios (2009). One possibility is to
modify the information criteria as
ˆ
Modified IC = T ln(σ 2 ) + kc(T ) + kc(T )T δ

37

where, δ = 0.5α and the rate of convergence of the estimate of d is related to the bandwidth used
in the first step estimation, so in this case is m = T α , which is derived by Wright (1995) and where
m0.5 = T 0.5α , and where generally 0 < α < 0.8. Hence when α = 0.5, then δ = 0.25. We have
accordingly done some initial simulations with this modified SPE − BIC and have reported them
in tables 3.21 through 3.25.
To motivate this, consider the ARFIMA(p, d, q) model ϕ (L)(1 − L)d yt = θ (L)εt where ϕ (L)
and θ (L) are polynomials in the lag operator of orders p and q respectively, with all their roots lying
outside the unit circle, there are (1 + p + q) parameters in the vector β ′ = (d, ϕ1 , , ..., ϕ p , θ1 , ..., θq ).
When the coefficient vector β has superscript 0 then it refers to a model using the true set of parameters. Hats on β indicate estimated parameters in a model with estimated short memory parameters
and estimated d. Tildes on β indicate estimated parameters in a model with estimates short memory parameters, but the true d parameter. The penalty term for the generic set of variables is
0
denoted by CT and the penalty term for the true set of variables by CT . Finally, let e = (e1 , ..., eT )′ .

The feasible information criterion takes the following form
IC(β ,CT ) =

T
ln{σ 2 } +CT
2

We also introduce the infeasible criterion given by
IC(β ,CT ) =

T
ln{σ 2 } +CT
2

We decompose IC(β ,CT ) as follows



IC(β 1 ,CT ) = IC(β 2 ,CT ) +

σ2

1



T 

ln
2
2
σ2

Then, consistency requires that
ˆ
ˆ0 0
lim P{IC(β ,CT ) − IC(β ,CT ) < 0} = 0

T →∞

Then following the analysis of Groen and Kapetanios (2010) this becomes
[ ]
(
)
T
σ2
ln
= O p T 1−δ = o p (T )
2
σ2
38

(3.8)

(3.9)

0
as long as T → ∞. Since CT −CT = o(T ) and from Wright (1995),

(
)
1 T
′
(ut − ut )yt = O p T −δ , δ < 0.5
∑
T t=1

(3.10)

0
as long as CT,N −CT,N → −∞. But, (3.9) implies this is not enough for (3.8) to hold.

We consider ARFIMA(2, d, 0) process of the form (1 − 1.00L + 0.50L2 )(1 − L)0.4 yt = εt ,
because the performance of LW T SE is substantially worse than MLE. The results are shown in
Tables 3.21 and 3.22. In Table 3.22, the modified BIC selected the correct model on 87% which
is 15% higher than standard BIC. The result for ARFIMA(1, d, 0) with ϕ = 0.8 and d = 0.4 is
presented in Table 3.23. And we find that modified information criteria improve model selection
for LW T SE.

3.5 Inflation Example
An example of the above methodology is provided by the US CPI inflation series from January
1947 through July 2010, making T = 762 observations. The properties of inflation have been the
subject of much research in the past with some articles claiming to find evidence for long memory
and others examining the possibility of regime switching or some other type of non-linearity. The
balance of evidence would seem to suggest a combination of long memory and nonlinearity to be
the optimal model for explaining the inflation series. The purpose of the present study is mainly to
provide an example. First, we consider CPI inflation calculated from seasonally unadjusted CPI.
The application of the LW gave estimates of the long memory parameter of dLW = 0.451 from a
bandwidth of m = T 0.50 ; and dLW = 0.574 from m = T 0.60 ; and dLW = 0.392 from m = T 0.75 ; and
dLW = 0.380 from m = T 0.80 . Since the series is likely to be seasonal the initial specification of the
maximum lags was pmax = 15 and qmax = 4. While this choice was deliberately conservative for
the MA structure, the allowance made for seasonality still gave a total of 80 models to be estimated.
The application of the BIC suggests that the most appropriate model is the ARFIMA(0, d, 0), while
the BIC grid of values below is relatively flat in the neighborhood of (p, q) = (0, 0), (p, q) = (1, 0)

39

and (p, q) = (0, 1). Estimation of an ARFIMA(0, d, 0) model realized
(1 − L)0.332(0.046) {1000∆ ln(CPIt ) − 3.101(1.434)} = εt .
However, examination of standard diagnostics such as the autocorrelations of the residuals indicated the presence of substantial autocorrelations at the seasonal lags of 11, 12 and 13. After some experimentation the most appropriate model from the BIC was the ARFIMA(0, d, 0) −
SARMA(1, 0), see Table 3.26:
(1 − L)d (1 − ΦL12 ){1000∆ ln(CPIt ) − µ } = εt .
This model could not have been selected form the previous analysis since its choice depended
on the implementation of eleven zero restrictions on the AR lag polynomial of the standard
ARFIMA(p, d, q) model of the form φ1 = φ2 = .... = φ11 = 0. Hence a more appropriate application of the BIC methodology would be to search over the set of ARFIMA(p, d, q) − SARMA(P, Q)
models.
For seasonally adjusted series, see Table 3.27, the BIC for the ARFIMA(0, d, 0) is less
than the BIC from the seasonal ARFIMA(0, d, 0) − AR(12) model and the coefficient of the seasonal AR parameter is statistically insignificant. Hence an investigator would select the regular
ARFIMA(0, d, 0) model. The situation is reversed with the CPI series from seasonally unadjusted
data, where the coefficient of the seasonal AR parameter is statistically significantly different from
zero. Hence an investigator would select the seasonal ARFIMA(0, d, 0) − AR(12) model.

3.6 Conclusion
This chapter has examined estimation and lag order selection for fractionally integrated ARFIMA
models. We show the Bayesian (BIC) of and Hannan-Quinn (HQ) information criteria are consistent model selection procedures for the class of long memory ARFIMA models when the models
are estimated by QMLE. We compare and contrast this approach with a method that initially uses
a semi-parametric estimate of the long memory parameter. Simulation evidence indicates that the
40

BIC based on QMLE works very well for low order models but can have mixed properties for high
order models. The methodology is illustrated with analysis of series of inflation.

41

APPENDIX

42

Appendix 3A: Tables
Table 3.1: Percentage of Model Selected, DGP ARFIMA(1, d, 0), ϕ1 = 0.8, d = 0.2, T = 400
p
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
1
2
2
2
2
2
2
2
2
2
3
3
3
3
3
3
3
3
3
4
4
4
4
4

q
0
1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8
0
1
2
3
4

AIC
0
0
0
0
0
0
0
0
0
15
1
0
0
0
0
0
0
0
2
1
0
0
0
0
0
0
0
1
0
3
1
0
0
0
0
0
0
0
2
2
1

QMLE
BIC HQ
1
0
1
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
89
72
1
3
0
0
0
0
0
0
0
0
0
0
0
0
0
0
3
5
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
2
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
1
0
0

AIC
0
0
0
0
0
0
0
0
0
13
1
0
0
0
0
0
0
0
2
1
0
0
0
0
0
0
0
1
0
4
0
0
0
0
0
1
0
0
1
1
1

LW T SE
BIC
0
1
0
0
0
0
0
0
0
89
2
0
0
0
0
0
0
0
5
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0

HQ
0
0
0
0
0
0
0
0
0
70
3
1
0
0
0
0
0
0
7
1
1
0
0
0
0
0
0
1
0
3
1
0
0
0
0
0
0
0
0
1
0

p
4
4
4
4
5
5
5
5
5
5
5
5
5
6
6
6
6
6
6
6
6
6
7
7
7
7
7
7
7
7
7
8
8
8
8
8
8
8
8
8

q
5
6
7
8
0
1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8

AIC
1
0
1
1
0
0
1
1
4
2
2
1
1
0
0
1
1
3
4
2
3
2
0
0
1
0
2
2
6
4
3
0
0
1
0
1
2
5
7
7

QMLE
BIC HQ
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
1
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
1
1

AIC
1
1
1
1
0
0
1
1
3
3
2
1
1
0
0
0
1
3
3
3
3
2
0
0
1
0
2
3
6
5
5
0
0
0
0
1
2
5
8
6

LW T SE
BIC
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0

HQ
0
0
0
0
0
0
1
0
1
0
0
0
0
0
0
0
0
0
1
0
0
1
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
1
1

Key: Number of replications is 1,000 and numbers in bold show the percent of correct selections.

43

Table 3.2: Percentage of Model Selected, DGP ARFIMA(1, d, 0), ϕ1 = 0.8, d = 0.4, T = 400
p
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
1
2
2
2
2
2
2
2
2
2
3
3
3
3
3
3
3
3
3
4
4
4
4
4

q
0
1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8
0
1
2
3
4

AIC
0
0
0
0
0
0
0
0
0
12
1
1
0
0
0
0
0
0
2
1
0
0
0
1
0
0
0
1
0
3
1
0
1
1
0
1
0
0
1
1
1

QMLE
BIC HQ
1
0
1
1
1
1
0
0
0
0
0
0
0
0
0
0
0
0
91
73
2
3
0
1
0
0
0
0
0
0
0
0
0
0
0
0
3
6
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
2
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
1
0
0

LW T SE
AIC BIC
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
8
90
2
3
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
4
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
2
0
2
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
2
0
3
0
1
0

HQ
0
0
0
0
0
0
0
0
0
66
6
0
0
0
0
0
0
0
7
1
0
0
0
0
0
0
0
1
0
2
1
0
0
0
0
0
0
0
1
1
0

p
4
4
4
4
5
5
5
5
5
5
5
5
5
6
6
6
6
6
6
6
6
6
7
7
7
7
7
7
7
7
7
8
8
8
8
8
8
8
8
8

Key: See Table 3.1.

44

q
5
6
7
8
0
1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8

AIC
1
1
1
1
0
0
0
0
5
2
2
1
2
0
0
0
1
2
4
2
2
3
0
0
0
0
1
3
6
4
4
0
0
0
0
1
2
7
7
6

QMLE
BIC HQ
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
1

LW T SE
AIC BIC
1
0
1
0
1
0
1
0
0
0
0
0
1
0
1
0
5
0
3
0
3
0
2
0
2
0
0
0
0
0
0
0
0
0
3
0
4
0
3
0
4
0
2
0
0
0
0
0
0
0
0
0
1
0
2
0
7
0
4
0
5
0
0
0
0
0
0
0
0
0
2
0
1
0
5
0
7
1
4
0

HQ
0
0
0
0
0
0
0
1
2
1
1
0
0
0
0
0
0
1
1
0
0
1
0
0
0
0
0
0
1
0
1
0
0
0
0
0
0
0
1
0

Table 3.3: Percentage of Model Selected, DGP ARFIMA(1, d, 0), ϕ1 = 0.8, d = 0.8, T = 400
p
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
1
2
2
2
2
2
2
2
2
2
3
3
3
3
3
3
3
3
3
4
4
4
4
4

q
0
1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8
0
1
2
3
4

AIC
0
0
0
0
0
0
0
0
0
12
1
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
1
0
2
1
0
0
1
0
0
0
0
1
1
1

QMLE
BIC HQ
0
0
1
1
0
0
0
1
0
0
0
0
0
0
0
0
0
0
88
69
2
5
0
0
0
0
0
0
0
0
0
0
0
0
0
0
4
5
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
2
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
1
0
0

LW T SE
AIC BIC
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
11
88
1
2
0
0
0
0
0
0
0
0
0
0
0
0
0
0
2
4
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
2
0
1
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
1
0
2
0
1
0

HQ
0
1
0
1
0
0
0
0
0
66
4
0
0
0
0
0
0
0
7
2
0
0
0
0
0
0
0
1
0
3
1
0
0
0
0
0
1
0
1
1
1

p
4
4
4
4
5
5
5
5
5
5
5
5
5
6
6
6
6
6
6
6
6
6
7
7
7
7
7
7
7
7
7
8
8
8
8
8
8
8
8
8

Key: See Table 3.1.

45

q
5
6
7
8
0
1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8

AIC
1
1
1
1
0
0
1
1
4
2
2
2
2
0
0
1
1
2
4
3
3
2
0
0
1
1
2
3
6
5
3
0
0
1
0
1
2
5
4
7

QMLE
BIC HQ
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
1
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
1
1
1
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
1
1
1

LW T SE
AIC BIC
1
0
0
0
1
0
1
0
0
0
0
0
1
0
0
0
3
0
3
0
2
0
2
0
1
0
0
0
0
0
1
0
1
0
3
0
4
0
2
0
3
0
1
0
0
0
0
0
1
0
1
0
2
0
3
0
7
0
5
0
4
1
0
0
0
0
0
0
0
0
1
0
2
0
5
0
8
0
5
0

HQ
0
0
0
0
0
0
1
0
2
1
1
0
0
0
0
0
0
1
1
0
0
0
0
0
0
0
0
1
1
0
1
0
0
0
0
0
0
1
1
0

Table 3.4: Percentage of Model Selected, DGP ARFIMA(1, d, 0), ϕ1 = 0.9, d = 0.4, T = 400
p
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
1
2
2
2
2
2
2
2
2
2
3
3
3
3
3
3
3
3
3
4
4
4
4
4

q
0
1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8
0
1
2
3
4

AIC
0
0
0
0
0
0
0
0
0
7
4
0
0
0
0
0
0
0
3
0
0
0
0
0
0
0
0
1
0
2
0
0
1
0
0
0
0
0
2
2
1

QMLE
BIC
3
5
0
0
0
0
0
0
0
59
13
0
0
0
0
0
0
0
15
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0

HQ
1
2
1
1
0
0
0
0
0
43
14
0
0
0
0
0
0
0
15
1
0
0
0
0
0
0
0
3
0
2
0
0
0
0
0
0
0
0
1
1
0

LW T SE
AIC BIC
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
6
68
5
16
1
0
0
0
0
0
0
0
0
0
0
0
0
0
4
10
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
1
0
1
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
1
0
1
0

HQ
0
1
0
0
0
0
0
0
0
43
21
2
0
0
0
0
0
0
13
1
0
0
0
0
0
0
0
2
0
2
1
0
0
0
0
0
0
0
1
1
0

p
4
4
4
4
5
5
5
5
5
5
5
5
5
6
6
6
6
6
6
6
6
6
7
7
7
7
7
7
7
7
7
8
8
8
8
8
8
8
8
8

Key: See Table 3.1.

46

q
5
6
7
8
0
1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8

AIC
1
1
1
1
0
0
1
1
3
3
1
1
1
0
0
1
0
2
3
2
2
3
0
0
1
1
2
3
8
5
4
0
0
1
0
1
2
4
8
6

QMLE
BIC HQ
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
1
0
1
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
1
0
0
1
1
0
0
0
0
0
0
0
0
0
0
0
1
0
1
1
1
1
1

LW T SE
AIC BIC
1
0
1
0
1
0
1
0
0
0
0
0
1
0
1
0
3
0
3
0
2
0
2
0
1
0
0
0
0
0
1
0
1
0
3
0
3
0
3
0
2
0
2
0
0
0
0
0
0
0
0
0
2
0
2
0
6
0
5
0
5
0
0
0
0
0
0
0
0
0
1
0
2
0
6
0
5
0
8
0

HQ
0
1
0
0
0
0
0
0
1
1
0
0
0
0
0
0
0
1
1
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
1
1
1

Table 3.5: Percentage of Model Selected, DGP ARFIMA(0, d, 1), θ1 = 0.9, d = 0.4, T = 400
p
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
1
2
2
2
2
2
2
2
2
2
3
3
3
3
3
3
3
3
3
4
4
4
4
4

q
0
1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8
0
1
2
3
4

AIC
0
12
2
0
0
0
0
0
0
3
1
0
0
0
0
0
0
0
1
1
0
2
1
1
1
1
0
0
0
1
1
1
1
0
0
0
0
0
1
1
1

QMLE
BIC HQ
1
0
79
62
2
3
0
0
0
0
0
0
0
0
0
0
0
0
13
11
1
2
0
1
0
0
0
0
0
0
0
0
0
0
0
0
1
4
0
1
0
1
0
2
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
1
0
1
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
1

LW T SE
AIC BIC
0
2
5
49
1
6
1
0
0
0
0
0
0
0
0
0
0
0
2
26
1
2
1
5
0
0
0
0
0
0
0
0
0
0
0
0
2
5
1
3
0
0
1
0
1
0
1
0
0
0
0
0
0
0
1
1
0
0
1
0
1
0
1
0
1
0
1
0
1
0
0
0
0
0
0
0
0
0
1
0
2
0

HQ
0
34
6
1
0
0
0
0
0
17
3
8
0
0
0
0
0
0
6
3
1
1
1
0
0
0
0
2
0
1
1
1
0
0
0
0
0
0
1
1
1

p
4
4
4
4
5
5
5
5
5
5
5
5
5
6
6
6
6
6
6
6
6
6
7
7
7
7
7
7
7
7
7
8
8
8
8
8
8
8
8
8

Key: See Table 3.1.

47

q
5
6
7
8
0
1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8

AIC
3
1
2
2
0
0
1
1
1
3
3
2
1
0
0
1
0
1
2
3
5
4
0
0
0
0
1
2
3
5
5
0
0
0
1
0
1
3
5
5

QMLE
BIC HQ
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0

LW T SE
AIC BIC
2
0
2
0
1
0
1
0
0
0
0
0
1
0
1
0
2
0
2
0
3
0
3
0
2
0
0
0
0
0
0
0
0
0
2
0
3
0
2
0
4
0
4
0
0
0
0
0
0
0
0
0
1
0
3
0
5
0
6
0
7
0
0
0
0
0
0
0
1
0
1
0
1
0
4
0
5
0
5
0

HQ
1
0
0
0
0
0
0
1
1
1
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
1
1
1
1
0
0
0
0
0
0
0
0
1

Table 3.6: Percentage of Model Selected, DGP ARFIMA(1, d, 1), ϕ1 = 0.5, θ1 = −0.7, d = 0.4,
T = 400
p
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
1
2
2
2
2
2
2
2
2
2
3
3
3
3
3
3
3
3
3
4
4
4
4
4

q
0
1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8
0
1
2
3
4

AIC
0
0
1
1
0
0
0
0
0
0
9
1
0
0
0
0
0
0
0
1
0
0
1
1
1
0
0
0
0
0
2
1
2
2
1
1
0
0
0
1
1

QMLE
BIC HQ
0
0
5
1
8
6
1
2
0
1
0
0
0
0
0
0
0
0
0
0
63
49
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
2
0
0
0
0
0
2
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
3
0
1
0
1
1
1
0
0
1
1
0
0
0
0
0
0
0
1
0
0

AIC
0
0
0
1
0
0
0
0
0
0
10
1
1
0
0
0
0
0
0
1
0
0
1
1
0
1
0
0
0
0
2
1
1
1
1
1
0
0
0
1
2

LW T SE
BIC
0
3
8
1
0
0
0
0
0
0
67
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0

HQ
0
1
6
2
1
1
0
0
0
0
53
1
1
0
0
0
0
0
0
2
1
1
1
0
0
0
0
0
0
0
2
1
0
1
1
1
1
0
0
1
0

p
4
4
4
4
5
5
5
5
5
5
5
5
5
6
6
6
6
6
6
6
6
6
7
7
7
7
7
7
7
7
7
8
8
8
8
8
8
8
8
8

Key: See Table 3.1.

48

q
5
6
7
8
0
1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8

AIC
1
2
4
3
0
0
0
0
2
3
3
4
4
0
0
0
0
1
3
3
4
5
0
0
0
0
1
1
2
6
4
0
0
0
0
1
1
2
5
4

QMLE
BIC HQ
0
1
1
2
3
3
2
2
0
0
0
0
0
0
0
0
0
0
0
1
1
2
2
2
2
3
0
0
0
0
0
0
0
0
0
0
0
0
0
0
2
2
2
3
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
2
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1

AIC
2
2
4
4
0
0
0
1
1
3
4
4
4
0
0
0
1
0
3
3
3
5
0
0
0
0
0
2
2
4
4
0
0
0
1
1
2
3
5
6

LW T SE
BIC
1
1
2
2
0
0
0
0
0
0
1
2
2
0
0
0
0
0
0
0
1
2
0
0
0
0
0
0
0
0
2
0
0
0
0
0
1
0
0
1

HQ
1
1
2
2
0
0
0
0
0
1
2
3
3
0
0
0
0
0
1
1
1
3
0
0
0
0
0
1
0
1
2
0
0
0
0
0
1
0
1
1

Table 3.7: Percentage of Model Selected, DGP ARFIMA(2, d, 0), ϕ1 = 0.25, ϕ2 = 0.375, d = 0.4,
T = 400
p
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
1
2
2
2
2
2
2
2
2
2
3
3
3
3
3
3
3
3
3
4
4
4
4
4

q
0
1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8
0
1
2
3
4

AIC
0
0
2
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
10
1
0
0
0
0
0
0
0
1
0
0
0
1
1
0
0
1
0
0
2
1
1

QMLE
BIC HQ
1
0
0
0
11
9
0
0
0
1
0
0
0
0
0
0
0
0
2
0
1
1
1
3
0
0
0
0
0
0
0
0
0
0
0
0
80
64
0
3
0
0
0
0
0
1
0
0
0
0
0
0
0
0
2
3
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
2
0
1
0
1

AIC
0
0
1
0
1
0
0
0
0
0
0
2
0
0
0
0
0
0
10
1
0
0
0
0
1
0
0
0
0
0
0
1
1
0
1
1
0
0
2
1
1

LW T SE
BIC
0
0
8
0
0
0
0
0
0
1
0
2
0
0
0
0
0
0
83
1
0
0
0
0
0
0
0
2
0
0
0
0
0
0
0
0
0
0
0
0
0

HQ
0
0
6
0
0
1
0
0
0
0
0
5
0
0
0
0
0
0
66
1
1
0
0
0
0
0
0
2
1
0
0
1
0
0
0
0
0
0
2
1
0

p
4
4
4
4
5
5
5
5
5
5
5
5
5
6
6
6
6
6
6
6
6
6
7
7
7
7
7
7
7
7
7
8
8
8
8
8
8
8
8
8

Key: See Table 3.1.

49

q
5
6
7
8
0
1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8

AIC
1
1
1
2
0
0
1
2
1
1
3
2
2
0
0
1
1
4
4
3
2
3
0
0
1
1
3
3
3
4
5
0
0
1
1
2
4
5
4
7

QMLE
BIC HQ
0
0
0
0
0
0
0
1
0
0
0
0
0
1
0
1
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
1

AIC
1
2
1
1
0
0
1
2
1
1
2
2
1
0
0
1
1
5
4
3
1
3
0
0
0
1
2
3
3
3
7
0
0
1
0
2
3
5
6
5

LW T SE
BIC
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0

HQ
0
1
0
0
0
0
0
1
0
0
1
0
0
0
0
0
1
1
1
0
0
0
0
0
0
0
1
0
0
0
1
0
0
0
0
0
0
0
0
1

Table 3.8: Percentage of Model Selected, DGP ARFIMA(2, d, 0), ϕ1 = 0.25, ϕ2 = 0.375, d = 0.4,
T = 1, 000
p
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
1
2
2
2
2
2
2
2
2
2
3
3
3
3
3
3
3
3
3
4
4
4
4
4

q
0
1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8
0
1
2
3
4

AIC
0
0
2
0
1
1
0
0
0
0
0
1
0
1
0
0
0
0
31
7
1
0
0
0
0
0
0
6
1
0
0
0
0
0
1
0
1
0
1
1
0

QMLE
BIC HQ
0
0
0
0
5
4
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
2
0
0
0
0
0
0
0
0
0
0
0
0
93
80
1
4
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
5
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0

AIC
0
0
0
0
1
0
0
0
0
0
0
1
1
0
0
0
0
0
25
8
1
0
0
0
0
0
0
5
0
0
0
0
0
0
0
1
2
0
1
1
1

LW T SE
BIC
0
0
2
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
94
1
0
0
0
0
0
0
0
2
0
0
0
0
0
0
0
0
0
0
0
0
0

HQ
0
0
1
0
0
0
0
0
0
0
0
2
1
0
0
0
0
0
79
6
0
0
0
0
0
0
0
5
0
0
0
0
0
0
0
0
1
0
1
0
0

p
4
4
4
4
5
5
5
5
5
5
5
5
5
6
6
6
6
6
6
6
6
6
7
7
7
7
7
7
7
7
7
8
8
8
8
8
8
8
8
8

Key: See Table 3.1.

50

q
5
6
7
8
0
1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8

AIC
1
0
1
0
0
0
0
2
1
1
1
1
1
0
0
0
0
1
2
1
1
3
0
0
1
0
2
3
2
2
2
0
0
0
1
1
2
3
4
3

QMLE
BIC HQ
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0

AIC
1
1
0
1
1
0
1
1
1
1
1
1
2
0
0
1
1
2
1
1
2
2
0
0
0
1
1
3
2
2
4
0
0
0
1
2
1
5
5
3

LW T SE
BIC
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0

HQ
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0

Table 3.9: Percentage of Model Selected, DGP ARFIMA(2, d, 0), ϕ1 = 1, ϕ2 = −0.5, d = 0.4,
T = 400
p
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
1
2
2
2
2
2
2
2
2
2
3
3
3
3
3
3
3
3
3
4
4
4
4
4

q
0
1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8
0
1
2
3
4

AIC
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
6
1
0
0
0
1
1
0
0
0
0
0
0
0
2
3
2
2
0
0
2
0
0

QMLE
BIC HQ
0
0
0
0
0
0
1
1
0
0
0
0
0
0
0
0
1
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
49
37
1
1
0
0
0
0
0
0
0
1
0
0
0
0
0
0
1
1
0
0
0
0
0
0
0
0
1
2
2
3
2
2
1
1
0
1
0
0
0
1
0
0
0
0

AIC
0
0
0
0
0
0
1
0
1
0
0
0
0
0
0
0
0
0
4
1
1
0
0
1
1
1
0
0
0
0
0
0
2
2
2
1
0
0
1
0
1

LW T SE
BIC
0
0
0
1
0
0
1
0
1
0
0
1
0
0
0
0
0
0
38
5
1
0
0
0
1
0
0
4
1
0
0
0
2
2
2
1
0
0
0
0
0

HQ
0
0
0
1
0
1
1
0
1
0
0
0
0
0
0
0
0
0
26
5
1
0
0
0
1
0
0
4
1
0
0
0
2
2
2
1
1
0
1
0
0

p
4
4
4
4
5
5
5
5
5
5
5
5
5
6
6
6
6
6
6
6
6
6
7
7
7
7
7
7
7
7
7
8
8
8
8
8
8
8
8
8

Key: See Table 3.1.

51

q
5
6
7
8
0
1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8

AIC
3
5
5
4
0
0
1
1
0
2
4
5
3
0
0
1
1
2
2
3
4
4
0
0
0
0
2
2
2
4
4
0
0
0
0
1
2
4
4
4

QMLE
BIC HQ
2
2
5
5
4
5
2
3
0
0
0
0
0
1
0
0
0
0
1
1
3
3
5
5
2
2
0
0
0
0
0
0
0
0
0
1
0
0
2
2
3
3
2
3
0
0
0
0
0
0
0
0
0
0
0
0
0
1
3
3
2
3
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
3
3

AIC
3
5
5
3
0
0
1
1
1
2
4
5
3
0
0
0
1
2
2
2
4
4
0
0
0
0
1
3
2
4
5
0
0
0
0
1
2
3
4
4

LW T SE
BIC
3
4
5
2
0
0
0
0
0
1
3
4
2
0
0
0
0
0
0
2
3
3
0
0
0
0
0
0
1
2
3
0
0
0
0
0
0
0
1
1

HQ
3
5
5
2
0
0
1
1
0
1
4
4
2
0
0
0
0
1
0
2
3
3
0
0
0
0
0
0
1
2
3
0
0
0
0
0
0
0
1
2

Table 3.10: Percentage of Model Selected, DGP ARFIMA(2, d, 1), ϕ1 = 0.25, ϕ2 = 0.375
θ1 = 1/3, d = 0.4, T = 400
p
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
1
2
2
2
2
2
2
2
2
2
3
3
3
3
3
3
3
3
3
4
4
4
4
4

q
0
1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8
0
1
2
3
4

AIC
0
0
1
0
0
0
0
0
0
1
1
2
0
0
0
0
0
0
3
2
0
0
1
1
0
1
0
1
0
1
0
1
0
0
1
0
0
0
1
1
1

QMLE
BIC HQ
15
2
0
0
9
9
0
1
0
1
0
0
0
0
0
0
0
0
41
19
0
3
1
4
0
0
0
0
0
0
0
0
0
0
0
0
28
31
3
6
0
0
0
0
0
1
0
0
0
0
0
0
0
0
1
5
0
0
0
2
0
1
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
1
0
1

AIC
0
0
1
0
0
0
0
0
0
1
2
1
0
0
0
0
0
0
5
2
0
0
0
0
1
1
0
1
0
1
1
1
0
0
0
1
0
0
1
2
1

LW T SE
BIC
1
0
7
0
0
0
0
0
0
28
7
2
0
0
0
0
0
0
48
3
0
0
0
0
0
0
0
2
0
0
0
0
0
0
0
0
0
0
0
0
0

HQ
0
0
6
1
1
0
0
0
0
10
8
4
0
0
0
0
0
0
38
7
0
0
0
0
0
0
0
3
0
2
1
1
0
0
0
0
0
0
2
1
1

p
4
4
4
4
5
5
5
5
5
5
5
5
5
6
6
6
6
6
6
6
6
6
7
7
7
7
7
7
7
7
7
8
8
8
8
8
8
8
8
8

Key: See Table 3.1.

52

q
5
6
7
8
0
1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8

AIC
1
1
2
2
0
0
2
1
3
2
3
2
2
0
0
1
1
3
4
3
2
4
0
0
1
0
3
2
5
5
4
0
0
1
1
2
2
5
5
8

QMLE
BIC HQ
0
0
0
0
0
1
0
0
0
0
0
0
0
1
0
1
0
1
0
1
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
1
0
1
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0

AIC
0
1
1
1
0
0
1
2
3
2
3
2
2
0
0
1
1
3
4
3
2
3
0
0
1
0
3
3
4
4
5
0
0
0
0
2
2
5
6
7

LW T SE
BIC
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0

HQ
0
0
0
0
0
0
1
1
1
1
1
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
1
0
1

Table 3.11: Percentage of Model Selected, DGP ARFIMA(2, d, 1), ϕ1 = 1, ϕ2 = −0.5, θ1 = 1/3,
d = 0.4, T = 400
p
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
1
2
2
2
2
2
2
2
2
2
3
3
3
3
3
3
3
3
3
4
4
4
4
4

q
0
1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8
0
1
2
3
4

AIC
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
2
11
1
1
1
1
1
1
1
6
2
0
0
0
1
1
0
0
2
0
1
1
1

QMLE
BIC
0
0
0
3
1
0
0
0
0
0
0
1
0
0
0
0
0
0
28
49
1
0
0
0
0
0
0
16
1
0
0
0
0
0
0
0
0
0
0
0
0

HQ
0
0
0
2
2
1
0
0
0
0
0
0
0
0
0
0
0
0
12
45
2
1
1
0
1
0
0
15
2
0
0
0
0
0
0
0
3
0
0
1
0

AIC
0
0
0
1
1
0
1
0
0
0
0
0
0
0
0
0
0
0
1
9
2
1
0
1
1
0
1
7
2
0
0
0
1
1
1
0
2
0
2
1
1

LW T SE
BIC
0
0
0
1
0
0
0
0
0
0
0
1
0
0
0
0
0
0
16
45
2
0
0
0
0
0
0
31
1
0
0
0
0
0
0
0
1
0
0
0
0

HQ
0
0
0
1
1
1
0
0
0
0
0
0
0
0
0
0
0
0
7
38
4
1
0
1
0
0
0
25
2
1
0
0
0
0
0
0
3
1
1
1
0

p
4
4
4
4
5
5
5
5
5
5
5
5
5
6
6
6
6
6
6
6
6
6
7
7
7
7
7
7
7
7
7
8
8
8
8
8
8
8
8
8

Key: See Table 3.1.

53

q
5
6
7
8
0
1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8

AIC
1
1
2
1
1
0
1
1
1
1
1
2
2
1
0
1
1
1
3
2
2
3
0
0
1
1
2
4
3
4
2
0
0
1
1
1
3
4
6
7

QMLE
BIC HQ
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
1
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0

AIC
1
1
1
2
1
0
1
1
1
1
2
1
2
1
0
1
1
1
3
3
2
2
0
0
1
1
2
4
3
3
3
0
0
0
1
1
3
4
6
6

LW T SE
BIC
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0

HQ
0
0
0
0
1
0
1
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
1
1
0
0
0
0
0
0
0
0
0
0
0
0

Table 3.12: Percentage of Model Selected, DGP ARFIMA(2, d, 1), ϕ1 = 1.4, ϕ2 = −0.6
θ1 = −0.8, d = 0.4, T = 400
p
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
1
2
2
2
2
2
2
2
2
2
3
3
3
3
3
3
3
3
3
4
4
4
4
4

q
0
1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8
0
1
2
3
4

AIC
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
2
2
1
0
1
0
0
0
0
0
0
1
1
1
0
0
0
1
0
0
0
4
1

QMLE
BIC HQ
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
21
9
60
47
2
3
0
0
0
1
0
1
0
0
0
0
0
0
5
5
0
0
0
3
0
0
0
1
0
0
0
0
0
1
0
0
2
1
0
0
0
0
0
3
0
0

AIC
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
3
0
1
0
0
0
0
0
0
1
1
0
0
0
0
2
1
0
0
1
5
0

LW T SE
BIC
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
19
58
2
0
0
0
0
0
0
3
2
0
0
0
0
0
0
1
1
0
1
0
0

HQ
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
8
46
1
0
0
0
0
0
0
2
1
1
1
0
0
1
0
1
1
0
2
2
1

p
4
4
4
4
5
5
5
5
5
5
5
5
5
6
6
6
6
6
6
6
6
6
7
7
7
7
7
7
7
7
7
8
8
8
8
8
8
8
8
8

q
5
6
7
8
0
1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8

AIC
1
1
2
1
0
0
0
2
0
2
1
4
3
0
0
1
0
3
13
5
6
3
0
0
0
1
0
2
4
4
2
0
0
0
0
2
0
5
9
8

QMLE
BIC HQ
0
1
0
1
0
0
1
1
0
0
0
0
0
0
1
2
0
0
0
1
0
0
3
3
2
2
0
1
0
0
0
0
0
0
0
0
0
4
0
0
2
3
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
0
0
0
0
0
0
0
0
0
0
0
0
0
2
0
0
0
1

AIC
2
1
4
4
0
0
0
1
4
0
0
3
2
0
0
1
2
0
4
4
6
4
0
0
0
1
1
3
4
4
4
0
0
0
0
0
3
4
11
7

LW T SE
BIC
0
1
1
1
0
0
0
0
0
0
0
2
2
0
0
0
0
0
0
0
2
1
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
1

HQ
1
1
2
2
0
0
0
1
2
0
0
2
2
1
0
0
1
0
3
1
3
2
0
0
0
1
0
1
1
1
2
0
0
0
0
0
0
0
0
1

Key: Number of replications is 100 and numbers in bold show the percent of correct selections.

54

Table 3.13: Percentage of Model Selected, DGP ARFIMA(2, d, 1), ϕ1 = −0.5, ϕ2 = −0.9,
θ1 = −0.6, d = 0.4, T = 400
p
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
1
2
2
2
2
2
2
2
2
2
3
3
3
3
3
3
3
3
3
4
4
4
4
4

q
0
1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8
0
1
2
3
4

AIC
0
0
0
0
0
0
0
1
0
0
0
1
0
0
0
0
0
0
0
24
1
2
0
0
0
0
0
0
4
1
0
0
0
1
0
0
0
2
2
1
1

QMLE
BIC HQ
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
0
0
0
0
0
0
1
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
92
83
1
3
0
0
0
0
0
0
0
0
0
0
0
0
0
0
4
6
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
2
0
0
0
0
0
1
0
0

AIC
0
0
1
0
0
0
0
3
0
0
0
0
0
0
0
0
0
1
0
12
2
3
0
0
0
0
0
0
4
5
0
0
0
1
0
0
0
2
1
1
0

LW T SE
BIC
0
0
1
0
0
0
0
2
0
0
0
0
0
0
0
0
0
1
0
65
10
0
0
0
0
0
0
0
11
7
0
0
0
0
0
0
0
2
0
0
0

HQ
0
0
1
0
0
0
0
3
0
0
0
0
0
0
0
0
0
1
0
48
11
1
1
0
0
0
0
0
7
7
0
0
0
0
0
0
0
4
0
1
0

p
4
4
4
4
5
5
5
5
5
5
5
5
5
6
6
6
6
6
6
6
6
6
7
7
7
7
7
7
7
7
7
8
8
8
8
8
8
8
8
8

Key: See Table 3.12.

55

q
5
6
7
8
0
1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8

AIC
1
0
0
1
1
1
0
0
1
0
1
0
0
0
0
0
2
5
1
1
1
1
3
0
1
0
3
4
3
3
2
1
1
2
0
3
5
2
4
5

QMLE
BIC HQ
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0

AIC
0
1
1
0
2
1
0
1
5
1
2
0
2
1
0
0
0
1
3
2
1
0
0
0
0
0
3
2
7
6
1
1
0
1
1
2
1
4
3
8

LW T SE
BIC
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0

HQ
0
1
0
0
4
0
0
2
2
0
0
0
0
2
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
0
0
0
0
0
0
0
0
0
1

Table 3.14: Percentage of Model Selected, DGP ARFIMA(3, d, 0), ϕ1 = −13/20, ϕ2 = 3/5,
ϕ3 = 27/80, d = 0.4, T = 400
p
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
1
2
2
2
2
2
2
2
2
2
3
3
3
3
3
3
3
3
3
4
4
4
4
4

q
0
1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8
0
1
2
3
4

AIC
0
10
0
0
0
0
0
0
0
0
16
0
0
0
0
0
0
0
0
13
0
0
0
0
0
0
0
0
16
0
0
0
0
0
0
0
1
7
0
0
0

QMLE
BIC
0
10
0
0
0
0
0
0
0
0
17
1
0
0
0
0
0
0
0
13
0
0
0
0
0
0
0
5
16
0
0
0
0
0
0
0
0
7
0
0
0

HQ
0
10
0
0
0
0
0
0
0
0
17
1
0
0
0
0
0
0
0
13
0
0
0
0
0
0
0
4
16
0
0
0
0
0
0
0
1
7
0
0
0

AIC
0
10
0
0
0
0
0
0
0
0
15
0
0
0
0
1
0
0
0
10
0
0
0
0
0
0
0
3
13
0
0
0
0
0
0
0
0
4
0
0
0

LW T SE
BIC
0
10
0
0
0
0
0
0
0
0
16
1
0
0
0
0
0
0
0
10
1
0
0
0
0
0
0
12
13
0
0
0
0
0
0
0
1
4
0
0
0

HQ
0
10
0
0
0
0
0
0
0
0
15
1
0
0
0
1
0
0
0
10
1
0
0
0
0
0
0
10
13
0
0
0
0
0
0
0
1
4
0
0
0

p
4
4
4
4
5
5
5
5
5
5
5
5
5
6
6
6
6
6
6
6
6
6
7
7
7
7
7
7
7
7
7
8
8
8
8
8
8
8
8
8

Key: See Table 3.12.

56

q
5
6
7
8
0
1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8

AIC
0
0
0
0
0
16
1
0
0
0
0
0
0
0
7
0
1
0
0
0
0
0
0
5
0
0
0
0
0
0
0
0
4
0
1
0
0
0
0
2

QMLE
BIC HQ
0
0
0
0
0
0
0
0
0
0
16
16
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
7
7
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
4
4
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
4
4
0
0
0
0
0
0
0
0
0
0
0
0
0
0

AIC
0
0
0
1
0
16
0
0
0
2
0
0
0
0
4
0
1
0
0
0
3
0
0
7
0
0
0
0
0
0
0
0
6
0
0
1
0
0
0
3

LW T SE
BIC
0
0
0
0
0
15
0
0
0
0
0
0
0
0
4
0
0
0
0
0
0
0
0
7
0
0
0
0
0
0
0
0
6
0
0
0
0
0
0
0

HQ
0
0
0
0
0
16
1
0
0
0
0
0
0
0
4
0
0
0
0
0
0
0
0
7
0
0
0
0
0
0
0
0
6
0
0
0
0
0
0
0

Table 3.15: Percentage of Model Selected, DGP ARFIMA(3, d, 0), ϕ1 = −0.56163,
ϕ2 = 0.57986, ϕ3 = −0.92705, d = 0.4, T = 400
p
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
1
2
2
2
2
2
2
2
2
2
3
3
3
3
3
3
3
3
3
4
4
4
4
4

q
0
1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8
0
1
2
3
4

AIC
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
15
1
0
0
0
0
0
12
3
11
4
0
1
0
0
0
13
0
1
0
0

QMLE
BIC
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
19
0
28
0
0
0
0
0
0
38
1
1
0
0
0
0
0
0
12
0
0
0
0

HQ
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
7
0
20
1
0
0
0
0
0
39
1
10
1
0
0
0
0
0
15
1
0
0
0

AIC
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
2
1
0
0
0
0
0
0
0
3
17
2
0
0
0
0
0
0
6
3
0
0
0

LW T SE
BIC
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
7
1
0
0
0
0
0
0
0
40
42
0
0
0
0
0
0
0
10
0
0
0
0

HQ
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
5
1
1
1
0
0
0
0
0
16
45
4
0
0
0
0
0
0
13
2
0
0
0

p
4
4
4
4
5
5
5
5
5
5
5
5
5
6
6
6
6
6
6
6
6
6
7
7
7
7
7
7
7
7
7
8
8
8
8
8
8
8
8
8

Key: See Table 3.12.

57

q
5
6
7
8
0
1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8

AIC
0
0
0
0
3
0
4
1
0
2
1
0
3
1
0
1
1
0
2
0
1
0
0
0
2
0
3
2
2
1
1
0
0
1
0
0
3
0
2
2

QMLE
BIC HQ
0
0
0
0
0
0
0
0
0
2
0
0
0
0
0
0
0
0
0
0
1
1
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0

AIC
0
0
0
2
4
0
2
4
2
0
0
1
0
0
0
3
1
2
0
1
0
0
0
0
0
0
3
6
4
2
3
0
0
0
0
4
5
6
9
2

LW T SE
BIC
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0

HQ
1
0
0
0
4
1
0
0
0
1
0
0
0
0
0
1
0
1
0
0
0
0
0
0
0
0
2
0
1
0
0
0
0
0
0
0
0
0
0
0

Table 3.16: Percentage of Model Selected, DGP ARFIMA(4, d, 0), ϕ1 = 5/4, ϕ2 = −3/8,
ϕ3 = −1/4, ϕ4 = 3/16, d = 0.4, T = 400
p
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
1
2
2
2
2
2
2
2
2
2
3
3
3
3
3
3
3
3
3
4
4
4
4
4

q
0
1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8
0
1
2
3
4

AIC
0
0
0
1
1
0
1
1
1
0
0
1
1
0
0
0
0
0
0
0
2
0
0
1
1
1
0
1
0
0
0
1
1
3
4
3
2
0
0
0
1

QMLE
BIC HQ
0
0
0
0
0
0
4
2
3
4
0
0
0
1
1
1
1
1
0
0
0
0
14
8
1
2
0
0
0
0
0
0
0
0
0
0
12
4
0
0
2
5
0
0
0
0
0
0
0
1
0
0
0
0
10
6
0
1
0
1
0
0
0
1
0
1
2
3
3
3
2
2
8
10
0
0
0
0
0
0
0
0

AIC
0
0
0
0
1
0
1
1
1
0
0
1
1
0
0
0
0
0
0
0
1
0
0
0
1
1
0
0
0
0
0
1
1
3
3
2
4
0
1
1
1

LW T SE
BIC
0
0
0
3
3
0
0
1
1
0
0
15
1
0
0
0
0
0
12
0
3
0
0
0
1
1
0
7
1
0
0
0
1
2
3
2
10
0
0
0
0

HQ
0
0
0
1
3
1
1
1
1
0
0
8
2
0
0
0
0
0
4
0
5
0
0
1
1
1
0
4
2
1
0
1
1
2
3
2
13
0
0
1
0

p
4
4
4
4
5
5
5
5
5
5
5
5
5
6
6
6
6
6
6
6
6
6
7
7
7
7
7
7
7
7
7
8
8
8
8
8
8
8
8
8

Key: See Table 3.1.

58

q
5
6
7
8
0
1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8

AIC
2
4
6
4
0
0
1
1
0
1
4
7
5
0
0
1
0
1
1
3
5
4
0
0
0
0
1
1
2
3
4
0
0
0
0
1
1
2
2
3

QMLE
BIC HQ
1
1
3
4
5
5
3
3
0
1
0
0
0
0
0
0
0
0
0
0
3
4
5
5
3
4
0
0
0
0
0
0
0
0
0
0
0
0
1
1
3
4
3
4
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
2
3
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1

AIC
2
5
4
4
0
0
1
1
1
1
4
5
5
0
0
1
1
1
1
2
4
5
0
0
0
0
1
1
2
4
5
0
0
0
0
2
1
3
3
5

LW T SE
BIC
1
4
4
3
0
0
0
0
0
0
4
4
3
0
0
0
0
0
0
0
2
3
0
0
0
0
0
0
0
1
2
0
0
0
0
0
0
0
0
2

HQ
1
5
4
3
0
0
0
0
0
0
4
4
4
0
0
0
0
0
0
1
2
4
0
0
0
0
0
0
0
1
3
0
0
0
0
1
0
0
0
2

Table 3.17: Percentage of Model Selected, DGP ARFIMA(4, d, 0), ϕ1 = −9/20, ϕ2 = 73/100,
ϕ3 = 87/400, ϕ4 = 27/400, d = 0.4, T = 400
p
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
1
2
2
2
2
2
2
2
2
2
3
3
3
3
3
3
3
3
3
4
4
4
4
4

q
0
1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8
0
1
2
3
4

AIC
0
9
0
0
0
0
0
0
0
0
16
1
0
0
0
0
0
0
1
5
0
0
0
0
0
0
0
0
14
1
0
0
0
1
0
0
0
5
0
0
0

QMLE
BIC HQ
0
0
9
9
0
0
0
0
0
0
0
0
0
0
0
0
0
0
3
0
18
17
8
6
0
0
0
0
0
0
0
0
0
0
0
0
3
1
5
5
1
1
0
0
0
0
0
0
0
0
0
0
0
0
8
5
14
14
0
2
0
0
0
0
0
0
0
0
0
0
0
0
2
4
4
5
0
0
0
0
0
0

AIC
0
10
0
0
0
0
0
0
0
0
8
0
0
0
0
0
0
0
0
9
0
1
0
0
0
0
0
1
19
1
0
0
0
0
0
0
2
11
0
0
0

LW T SE
BIC
0
10
0
0
0
0
0
0
0
1
8
2
0
1
0
0
0
0
0
9
0
0
0
0
0
0
0
19
17
0
0
0
0
0
0
0
4
11
0
0
0

HQ
0
10
0
0
0
0
0
0
0
0
8
2
0
0
0
0
0
0
0
9
1
1
0
0
0
0
0
11
18
0
0
0
0
0
0
0
6
11
0
0
0

p
4
4
4
4
5
5
5
5
5
5
5
5
5
6
6
6
6
6
6
6
6
6
7
7
7
7
7
7
7
7
7
8
8
8
8
8
8
8
8
8

Key: See Table 3.12.

59

q
5
6
7
8
0
1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8

AIC
0
0
0
0
0
15
0
1
0
1
1
0
0
0
1
2
0
0
0
0
1
2
0
7
0
0
0
3
0
2
1
0
1
1
0
0
2
3
2
1

QMLE
BIC HQ
0
0
0
0
0
0
0
0
0
0
15
15
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
1
1
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
6
6
0
0
0
0
0
0
1
1
0
0
0
0
0
0
0
0
1
1
0
1
0
0
0
0
0
1
0
2
0
0
1
1

AIC
0
0
0
0
0
9
0
1
0
0
1
0
0
0
5
1
2
0
0
0
0
2
0
4
0
0
2
1
1
0
1
0
0
0
0
0
4
2
0
2

LW T SE
BIC
0
0
0
0
0
9
0
0
0
0
0
0
0
0
5
0
0
0
0
0
0
0
0
4
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0

HQ
0
0
0
0
0
9
0
1
0
0
0
0
0
0
5
2
1
0
0
0
0
0
0
4
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0

Table 3.18: Percentage of Model Selected, DGP ARFIMA(4, d, 0), ϕ1 = 1.350, ϕ2 = −0.7,
ϕ3 = 0.4, ϕ4 = −0.31, d = 0.4, T = 400
p
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
1
2
2
2
2
2
2
2
2
2
3
3
3
3
3
3
3
3
3
4
4
4
4
4

q
0
1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8
0
1
2
3
4

AIC
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
2
1
0
1
0
1
0
0
0
2
1
0
0
0
3
17
4
4
0
0

QMLE
BIC HQ
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
3
1
3
1
1
2
1
3
0
0
0
0
0
1
0
0
1
1
0
0
0
0
0
0
0
2
0
0
0
0
0
0
0
1
3
3
77
56
1
5
1
2
0
1
0
0

AIC
0
0
0
2
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
3
2
0
0
0
1
0
0
0
1
0
0
1
1
2
16
2
2
0
0

LW T SE
BIC
0
0
0
2
0
0
0
0
1
0
0
0
0
1
0
0
0
0
2
1
1
1
1
0
0
0
1
0
0
0
1
0
0
0
0
2
69
4
0
0
0

HQ
0
0
0
2
0
0
0
0
1
0
0
0
0
1
0
0
0
0
1
1
1
6
2
0
0
0
1
0
0
0
1
0
0
1
0
2
47
4
0
0
0

p
4
4
4
4
5
5
5
5
5
5
5
5
5
6
6
6
6
6
6
6
6
6
7
7
7
7
7
7
7
7
7
8
8
8
8
8
8
8
8
8

Key: See Table 3.12.

60

q
5
6
7
8
0
1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8

AIC
0
1
2
2
2
0
0
0
1
0
2
1
3
0
0
4
1
2
2
1
2
2
0
0
1
1
1
3
3
1
7
0
0
0
1
1
2
5
7
2

QMLE
BIC HQ
0
0
0
1
1
1
1
1
2
4
0
0
0
0
0
0
0
1
0
0
0
0
0
1
2
3
0
0
0
1
0
2
0
0
0
0
0
0
0
0
0
1
1
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
2
2
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
2
0
0

AIC
1
1
2
0
1
1
0
0
4
1
1
1
4
4
0
3
0
0
1
0
1
1
0
0
0
4
3
2
5
2
6
0
0
0
0
3
3
4
4
3

LW T SE
BIC
0
0
1
0
3
2
0
0
1
1
0
0
1
2
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1

HQ
0
1
1
0
3
1
0
0
3
1
0
0
3
5
0
2
0
0
1
0
0
1
0
0
0
1
0
0
1
0
3
0
0
0
0
1
0
0
0
1

Table 3.19: Percentage of Model Selected, DGP ARFIMA(4, d, 0), ϕ1 = 1.350, ϕ2 = −0.7,
ϕ3 = 0.4, ϕ4 = −0.31, d = 0.3, T = 400
p
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
1
2
2
2
2
2
2
2
2
2
3
3
3
3
3
3
3
3
3
4
4
4
4
4

q
0
1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8
0
1
2
3
4

AIC
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
4
0
0
0
0
0
0
0
0
1
1
0
1
0
0
22
1
1
0
0

QMLE
BIC HQ
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
4
0
1
1
2
1
1
5
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
2
0
0
0
0
0
1
0
0
0
0
83
66
1
3
0
1
0
0
0
0

AIC
0
0
1
0
0
1
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
2
1
0
0
0
1
0
0
0
1
0
0
1
0
0
15
1
1
1
0

LW T SE
BIC
0
0
1
0
0
1
0
0
1
0
0
0
0
0
0
0
0
0
4
2
1
1
1
0
0
0
1
0
0
0
0
0
0
0
0
0
68
2
0
0
0

HQ
0
0
1
0
0
1
0
0
1
0
0
0
0
0
0
0
0
1
1
0
1
4
3
1
0
0
1
0
0
0
1
0
0
1
0
0
50
3
1
0
0

p
4
4
4
4
5
5
5
5
5
5
5
5
5
6
6
6
6
6
6
6
6
6
7
7
7
7
7
7
7
7
7
8
8
8
8
8
8
8
8
8

Key: See Table 3.12.

61

q
5
6
7
8
0
1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8

AIC
1
1
2
2
3
1
1
0
2
5
0
4
2
0
0
2
1
5
1
1
1
4
0
0
3
3
2
2
2
1
4
0
0
2
0
1
2
6
0
2

QMLE
BIC HQ
0
0
0
1
1
1
1
2
2
3
0
2
0
1
0
0
0
0
0
0
0
0
0
0
1
1
0
1
0
0
0
1
0
0
0
2
0
0
0
0
0
0
1
1
0
0
0
0
0
1
1
2
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0

AIC
1
0
2
1
2
4
0
1
1
3
0
2
1
1
0
2
1
4
1
0
5
1
0
0
0
3
2
1
4
3
5
0
0
1
1
2
5
5
3
5

LW T SE
BIC
0
0
0
0
9
1
0
0
0
0
0
1
1
2
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
0

HQ
1
0
1
1
7
3
0
0
1
2
0
1
1
2
0
0
1
0
0
0
1
0
0
0
0
1
0
0
0
0
0
1
0
0
0
0
2
1
1
1

Table 3.20: Percentage of Model Selected, DGP ARFIMA(4, d, 1), ϕ1 = 5/4, ϕ2 = −3/8,
ϕ3 = −1/4, ϕ4 = 3/16, θ1 = 1/3,d = 0.4, T = 400
p
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
1
2
2
2
2
2
2
2
2
2
3
3
3
3
3
3
3
3
3
4
4
4
4
4

q
0
1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8
0
1
2
3
4

AIC
0
0
0
0
1
0
1
1
3
0
0
0
0
0
0
0
1
1
1
0
1
0
0
1
3
2
1
1
0
0
0
0
3
5
7
4
2
0
0
0
0

QMLE
BIC HQ
0
0
0
0
0
0
0
0
1
2
0
0
1
1
1
1
3
3
0
0
0
0
1
1
1
1
0
0
0
0
0
0
1
1
1
1
22
9
0
0
1
2
0
0
0
0
1
1
2
3
2
2
1
1
6
5
0
0
0
0
0
0
0
0
2
2
5
5
6
6
3
4
1
4
0
0
0
0
0
0
0
0

AIC
0
0
0
0
1
0
1
2
1
0
0
0
1
0
0
0
1
1
1
0
1
1
0
1
2
2
2
1
0
1
0
0
2
5
5
3
2
0
0
0
0

LW T SE
BIC
0
0
0
0
1
0
1
2
2
0
0
2
2
0
0
0
1
1
11
0
2
0
0
1
2
1
1
13
0
0
0
0
2
4
4
3
6
0
0
0
0

HQ
0
0
0
0
2
1
1
2
1
0
0
1
2
0
0
0
1
1
5
0
4
0
0
1
2
1
1
8
0
0
0
0
3
5
5
3
6
0
0
0
0

p
4
4
4
4
5
5
5
5
5
5
5
5
5
6
6
6
6
6
6
6
6
6
7
7
7
7
7
7
7
7
7
8
8
8
8
8
8
8
8
8

Key: See Table 3.1.

62

q
5
6
7
8
0
1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8

AIC
2
5
6
5
0
0
0
0
0
0
3
6
5
0
0
0
0
1
1
2
4
5
0
0
0
0
1
1
1
1
3
0
0
0
0
0
1
1
2
2

QMLE
BIC HQ
2
2
5
5
5
6
4
4
0
0
0
0
0
0
0
0
0
0
0
0
3
3
5
5
4
4
0
0
0
0
0
0
0
0
0
0
0
0
1
1
3
4
4
4
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
2
2
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1

AIC
2
4
6
5
0
0
0
1
0
2
3
5
6
0
0
1
0
1
1
1
3
5
0
0
0
0
1
1
1
2
3
0
0
0
0
1
1
2
2
3

LW T SE
BIC
2
4
6
4
0
0
0
0
0
1
3
4
4
0
0
0
0
0
0
1
2
3
0
0
0
0
0
0
0
1
2
0
0
0
0
0
0
0
0
1

HQ
2
4
6
5
0
0
0
0
0
1
3
5
5
0
0
1
0
0
0
1
2
4
0
0
0
0
0
0
0
2
2
0
0
0
0
0
0
0
0
1

Table 3.21: Modified BIC and HQ for LW T SE, Percentage of Model Selected, DGP
ARFIMA(2, d, 0) with ϕ1 = 1,ϕ2 = −0.5, d = 0.4 and T = 400
p
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
1
2
2
2
2
2
2
2
2
2
3
3
3
3
3
3
3
3
3
4
4
4
4
4

q
0
1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8
0
1
2
3
4

Std.
BIC
0
0
2
2
0
0
0
0
0
0
0
1
0
0
0
0
0
0
72
4
0
0
0
0
0
0
1
5
0
0
0
0
0
0
3
0
0
0
0
0
0

Std.
HQ
0
0
0
1
0
0
0
0
0
0
0
1
0
0
0
0
0
0
47
4
1
0
0
1
1
0
1
6
2
0
0
0
0
1
3
1
0
0
4
0
2

Mod.
BIC
0
0
2
0
0
0
0
0
0
0
2
1
0
0
0
0
0
0
87
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0

Mod.
HQ
0
0
2
0
0
0
0
0
0
0
2
1
0
0
0
0
0
0
82
3
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0

p
4
4
4
4
5
5
5
5
5
5
5
5
5
6
6
6
6
6
6
6
6
6
7
7
7
7
7
7
7
7
7
8
8
8
8
8
8
8
8
8

q
5
6
7
8
0
1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8

Std.
BIC
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
1
0
1
0
1
2
0
0
0
0
1
0
0
1
0
0
0
0
0
0
0
1
0
1

Std.
HQ
0
0
0
0
0
1
1
0
0
0
0
0
0
0
0
0
1
2
3
0
2
2
0
0
0
0
1
1
0
1
1
0
0
0
0
0
3
3
0
2

Mod.
BIC
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
1
0
1
2
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
1
0
1

Mod.
HQ
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
1
0
1
2
0
0
0
0
1
0
0
1
0
0
0
0
0
0
0
1
0
1

Key: Number of replication is 100. Std. and Mod. denote standard and modified.

63

Table 3.22: Modified BIC and HQ for LW T SE, Percentage of Model Selected, DGP
ARFIMA(2, d, 0) with ϕ1 = 1,ϕ2 = −0.5, d = 0.4 and T = 1, 000
p
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
1
2
2
2
2
2
2
2
2
2
3
3
3
3
3
3
3
3
3
4
4
4
4
4

q
0
1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8
0
1
2
3
4

Std.
BIC
0
2
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
64
10
1
0
0
0
0
0
0
4
3
0
0
0
0
0
0
1
0
0
0
0
0

Std.
HQ
0
2
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
54
12
1
0
2
0
0
0
0
3
4
0
0
0
0
0
0
1
0
0
0
0
0

Mod.
BIC
0
2
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
92
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0

Mod.
HQ
0
2
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
89
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0

Key: See Table 3.21

64

p
4
4
4
4
5
5
5
5
5
5
5
5
5
6
6
6
6
6
6
6
6
6
7
7
7
7
7
7
7
7
7
8
8
8
8
8
8
8
8
8

q
5
6
7
8
0
1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8

Std.
BIC
0
0
1
1
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
1
1
0
0
0
0
0
0
2
0
2
0
0
0
0
1
1
1
1
0
2

Std.
HQ
0
0
1
1
0
0
0
1
0
0
0
0
0
0
1
1
0
0
1
1
1
0
0
0
0
0
0
3
0
2
0
0
0
1
1
1
1
1
0
3

Mod.
BIC
0
0
1
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
1
0
1
0
0
0
0
0
0
1
0
0
0

Mod.
HQ
0
0
1
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
1
0
0
0
0
0
0
0
2
0
1
0
0
0
0
0
0
1
0
0
0

Table 3.23: Modified BIC and HQ for LW T SE, Percentage of Model Selected, DGP
ARFIMA(1, d, 0) with ϕ1 = 0.8, d = 0.4 and T = 400
p
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
1
2
2
2
2
2
2
2
2
2
3
3
3
3
3
3
3
3
3
4
4
4
4
4

q
0
1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8
0
1
2
3
4

Std.
BIC
0
2
0
0
0
0
0
0
0
66
2
0
0
0
0
0
0
0
4
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
1
0
0
1
0
0

Std.
HQ
0
1
1
0
0
0
0
0
0
38
5
0
0
0
0
0
1
0
5
3
0
0
0
0
0
0
1
0
1
4
0
0
0
0
0
1
0
0
1
0
0

Mod.
BIC
1
2
0
0
0
0
0
0
0
89
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
1
0
0

Mod.
HQ
0
2
0
0
0
0
0
0
0
81
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
1
0
0
1
0
0

Key: See Table 3.21

65

p
4
4
4
4
5
5
5
5
5
5
5
5
5
6
6
6
6
6
6
6
6
6
7
7
7
7
7
7
7
7
7
8
8
8
8
8
8
8
8
8

q
5
6
7
8
0
1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8

Std.
BIC
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
1
3
2
0
0
0
0
0
1
1
3
2
0
0
0
0
0
3
1
3
2

Std.
HQ
0
0
0
0
0
0
2
0
0
1
0
2
2
0
0
1
0
1
0
3
3
3
0
0
0
0
0
1
4
3
2
0
0
0
0
0
3
2
3
2

Mod.
BIC
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
2
1
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
1
0

Mod.
HQ
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
3
1
0
0
0
0
0
1
0
1
2
0
0
0
0
0
2
1
2
0

Table 3.24: Modified BIC and HQ for LW T SE, Percentage of Model Selected, DGP
ARFIMA(4, d, 0), ϕ1 = 5/4, ϕ2 = −3/8, ϕ3 = −1/4, ϕ4 = 3/16, d = 0.4, and T = 1, 000
p
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
1
2
2
2
2
2
2
2
2
2
3
3
3
3
3
3
3
3
3
4
4
4
4
4

q
0
1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8
0
1
2
3
4

Std.
SIC
0
6
0
0
4
0
0
0
0
0
3
3
0
0
0
1
0
0
2
1
9
0
0
0
2
0
0
4
1
0
0
0
0
0
0
2
25
2
0
0
0

Std.
HQ
0
6
0
0
4
0
1
0
0
0
3
0
1
0
0
1
0
0
0
0
8
0
0
0
2
0
0
0
1
0
0
0
0
0
0
2
22
3
2
0
0

Mod.
SIC
0
7
0
0
0
0
0
0
0
0
3
1
0
0
0
0
0
0
59
1
0
0
0
0
2
0
0
1
0
0
0
0
0
0
0
2
0
2
0
0
0

Mod.
HQ
0
7
0
2
0
0
0
0
0
0
3
13
0
0
0
0
0
0
35
1
1
0
0
0
2
0
0
4
0
0
0
0
0
0
0
2
2
2
0
0
0

Key: See Table 3.21

66

p
4
4
4
4
5
5
5
5
5
5
5
5
5
6
6
6
6
6
6
6
6
6
7
7
7
7
7
7
7
7
7
8
8
8
8
8
8
8
8
8

q
5
6
7
8
0
1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8

Std.
SIC
0
0
0
1
1
2
0
0
0
0
0
0
0
0
4
0
0
0
0
0
2
0
0
4
0
0
0
0
1
0
1
0
13
0
1
0
0
0
0
5

Std.
HQ
0
0
0
1
0
3
0
1
0
0
0
0
0
0
4
1
1
0
0
1
2
0
0
5
2
0
1
0
1
0
1
0
13
0
1
1
0
0
0
5

Mod.
SIC
0
0
0
0
0
2
0
0
0
0
0
0
0
0
3
0
0
0
0
0
0
0
0
4
0
0
0
0
0
0
0
0
11
0
1
0
0
0
0
1

Mod.
HQ
0
0
0
0
0
2
0
0
0
0
0
0
0
0
3
0
0
0
0
0
1
0
0
4
0
0
0
0
0
0
0
0
13
0
1
0
0
0
0
2

Table 3.25: Modified BIC and HQ for LW T SE, Percentage of Model Selected, DGP
ARFIMA(4, d, 0), ϕ1 = 1.350, ϕ2 = −0.7, ϕ3 = 0.4, ϕ4 = −0.31, d = 0.4, and T = 1, 000
p
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
1
2
2
2
2
2
2
2
2
2
3
3
3
3
3
3
3
3
3
4
4
4
4
4

q
0
1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8
0
1
2
3
4

Std.
SIC
0
39
0
0
0
0
0
0
0
0
8
0
0
0
0
0
0
0
0
3
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
8
1
0
0
0

Std.
HQ
0
39
0
0
0
0
0
0
0
0
8
0
0
0
0
0
0
0
0
3
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
6
2
0
0
0

Mod.
SIC
0
40
0
0
0
0
0
0
0
0
8
0
0
0
0
0
0
0
4
3
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
6
1
0
0
0

Mod.
HQ
0
40
0
0
0
0
0
0
0
0
8
0
0
0
0
0
0
0
0
3
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
9
1
0
0
0

Key: See Table 3.21

67

p
4
4
4
4
5
5
5
5
5
5
5
5
5
6
6
6
6
6
6
6
6
6
7
7
7
7
7
7
7
7
7
8
8
8
8
8
8
8
8
8

q
5
6
7
8
0
1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8

Std.
SIC
0
0
0
0
0
8
0
0
0
0
0
0
0
0
12
0
0
0
0
0
0
0
0
7
0
0
0
0
0
0
0
0
12
0
0
0
1
0
0
0

Std.
HQ
0
0
0
0
0
9
0
0
0
0
0
0
0
0
12
0
0
0
0
0
0
0
0
7
0
0
0
0
0
0
0
0
12
0
0
0
1
0
0
0

Mod.
SIC
0
0
0
0
0
8
0
0
0
0
0
0
0
0
12
0
0
0
0
0
0
0
0
6
0
0
0
0
0
0
0
0
11
0
0
0
0
0
0
0

Mod.
HQ
0
0
0
0
0
8
0
0
0
0
0
0
0
0
12
0
0
0
0
0
0
0
0
7
0
0
0
0
0
0
0
0
11
0
0
0
0
0
0
0

Table 3.26: Estimated ARFIMA Models for U.S. Inflation
(1 − ϕ L)(1 − L)d (1 − ΦL12 )(1000 × ∆ ln(CPIt ) − µt ) = (1 − θ L)εt
ARFIMA(0,d,0)

ϕ

-2003.73
3.101
(1.434)
0.332
(0.046)
-

ARFIMA(0,d,0)SARMA(1,0)
-1999.18
3.219
(1.482)
0.316
(0.047)
-

θ

-

-

ARFIMA(1,d,0)SARMA(1,0)
-1999.07
3.168
(1.371)
0.301
(0.057)
0.026
(0.072)
-

ϕ

-

σ2

11.26
(1.003)
-0.11
7.04
30.66
452.30
2.424
2.430
2.426

-0.111
(0.053)
11.127
(1.023)
-0.17
7.43
22.84
439.44
2.415
2.427
2.419

-0.111
(0.052)
11.123
(1.021)
-0.15
7.42
22.09
437.77
2.417
2.435
2.424

Log L
µ
d

m3
m4
Q(20)
Q2 (20)
AIC
BIC
HQIC

ARFIMA(0,d,1)SARMA(1,0)
-1999.06
3.164
(1.363)
0.3
(0.057)
-0.028
(0.074)
-0.111
(0.052)
11.123
(1.021)
-0.15
7.42
22.04
437.60
2.417
2.435
2.424

Key: Seasonally unadjusted inflation. Standard errors are reported in parentheses. Log L is the
maximized value of the log likelihood. Q(20) and Q2 (20) are the Ljung-Box Postmanteau
statistics calculated from the first 20 residual and squared residual autocorrelations respectively.
m3 and m4 are the sample skewness and kurtosis of the standardized residuals respectively.

68

Table 3.27: Estimated ARFIMA Models for U.S. Inflation
(1 − ϕ L)(1 − L)d (1 − ΦL12 )(1000 × ∆ ln(CPIt ) − µt ) = (1 − θ L)εt
ARFIMA(0,d,0)

ϕ

-1862.79
3.635
(1.982)
0.395
(0.055)
-

ARFIMA(0,d,0)SARMA(1,0)
-1859.89
3.857
(2.202)
0.391
(0.058)
-

θ

-

-

ARFIMA(1,d,0)SARMA(1,0)
-1858.55
3.485
(1.676)
0.333
(0.067)
0.094
-0.096
-

ϕ

-

σ2

7.778
-0.782
0.00
8.71
69.73
326.09
2.054
2.060
2.056

-0.088
(0.053)
7.72
-0.783
-0.02
8.83
65.36
341.57
2.049
2.061
2.054

-0.094
(0.054)
7.692
-0.782
0.09
8.88
56.67
326.40
2.048
2.066
2.055

Log L
µ
d

m3
m4
Q(20)
Q2 (20)
AIC
BIC
HQIC

Key: Seasonally adjusted inflation. See Table 3.26.

69

ARFIMA(0,d,1)SARMA(1,0)
-1858.12
3.427
(1.567)
0.325
(0.064)
-0.114
(0.100)
-0.096
(0.054)
7.684
-0.781
0.11
8.88
54.21
321.00
2.047
2.065
2.054

Chapter 4

METHODS FOR SELECTION OF ARFIMA MODELS: APPLICATIONS

4.1 Introduction
This chapter considers applications of the model selection procedure discussed in the previous
chapter. We find that Information Criteria (IC) are useful in ARFIMA model specification. Several
economic and finance time series are investigated, such as U.S. real output, G-7 monthly inflation,
interest rate and realized volatility of exchange rate. We consider both Maximum Likelihood
Estimation (MLE) and Local Whittle Two-Step Estimation (LW T SE) procedure. We find that the
selected model and estimation results vary with the IC and estimation procedure. The MLE − BIC,
which is our suggested criterion, generally selects parsimonious models.

4.2 Application to Real Output
Whether real Gross Domestic Product (GDP) is difference stationary or trend stationary is long
debate in macroeconomic research. Nelson and Plosser (1982) had argued that the presence of
unit root of GNP cannot be rejected. Campbell and Mankiw (1987) found strong persistence in
output shock. In contrast, Clark (1987) and Cochrane (1988) found small persistence on the Gross
National Product (GNP) series. An application of the long memory model contributes to this debate
by avoiding “knife-edge” unit root test distinction. The GPH estimator was applied by Diebold and
Rudebusch (1989) to the U.S. real GNP and Industrial Production (IP) index. They found evidence
of long memory in both measures of real output. Sowell (1992) used the ARFIMA model to test
whether the series should be modeled by trend-stationary model or difference-stationary model.
He estimated the ARFIMA model of first differenced GNP and found that GNP is consistent with
both models. Quarterly GDP and the monthly IP index of G-7 countries were studied by Cheung

70

and Lai (1992). They found that the U.S. and the U.K. GDP are fractional integrated. Using long
spans of real GNP and GNP per capita ranging from 1869 to 2001, Mayoral (2006) rejected unitroot and trend stationary hypothesis. She also found that the fractional integration (FI) model and
structural break (SB) model are more favorable than the linear ARIMA model. Using her proposed
technique for distinguishing between FI and SB, the FI model is more preferable than the SB
model and the persistence parameter is non-stationary but mean reversion.
The data considered in this section include quarterly data on real U.S. GDP ranging from the
first quarter of 1947 to the second quarter of 2010 and monthly data on the U.S. IP index ranging
from January 1947 to September 2010. Both series were seasonally adjusted and collected from
the Federal Reserve at St. Louis.1 The long memory models were estimated on the first differenced
series. The maximum lags were set at pmax = 8 and qmax = 8.
The model selection results from MLE and LW T SE are presented in Table 4.1.
ARFIMA(1, d, 0) and ARFIMA(0, d, 0) were selected by MLE − BIC for the GDP and IP index,
respectively. Table 4.2 presents the estimation results. For the GDP series, the MLE estimate of
d is -0.341. The LW estimates vary from -0.084 to 0.283. For the IP, we obtained a value of
d = 0.182. The long memory behaviors are consistent with existing literature.

4.3 Application to Inflation
The persistence of inflationary shocks is a central debate in macroeconomic literature. An understanding of the dynamics of inflation rates is crucial for central bankers in controlling the inflation.
In case of countries with high persistence of inflation, the inflationary shock will take a long time to
dissipate. Hassler and Wolters (1995) and Baillie, Chung, and Tieslau (1996) found that inflation
rates in major industrial countries are fractional integrated. The evidences of long memory were
extended to other industrial countries and emerging countries by Baum, Barkoulas, and Caglayan
(1999). The changing in persistence in U.S. inflation was considered by Kumar and Okimoto
1 http://research.stlouisfed.org/fred2

71

(2007). They found a decline of persistence in the last two decades. Hassler and Meller (2010)
provide a formal test for a break in the long memory parameter.
In this section, we focus on the inflation of G-7 countries, i.e. the U.S., Canada, the U.K,
France, Germany, Japan, and Italy. The monthly inflation series are calculated from Consumer
Price Index (CPI) obtained from the OECD statistical website.2 The data ranged from January
1995 to August 2010. The CPI series are seasonally unadjusted. Monthly inflation rates were
calculated by 1000∆(CPIt ).
In model identification, the maximum lags were set at pmax = 15 and qmax = 4. The results
from the model selection procedure are shown in Table 4.3. The results from estimating ARFIMA
models are given in Table 4.4. For the U.S., an ARFIMA(0, d, 0) model was found to be favorable
among 80 specifications, with the estimate of d being 0.365 and a robust asymptotic standard error
of 0.048. The LW estimation gave estimates of the long memory parameter of dLW = 0.665 from
a bandwidth of m = T 0.5 ; and dLW = 0.685 from m = T 0.6 ; and dLW = 0.368 from m = T 0.7 ; and
dLW = 0.361 from m = T 0.8 . For other G-7 countries, the long memory estimates from the MLE
range from 0.284 for the U.K. to 0.951 for Italy.

4.4 Application to Realized Volatility of Exchange Rate
Modeling the asset price return is crucial for financial economists and risk managers in risk management, portfolio management and other financial activities. In general, the compounded return
of asset price can be expected to be stationary or to have little serial correlation. On the contrary, it
is widely accepted that asset price volatility is more predictable. Ding, Granger, and Engle (1993)
found long memory feature of absolute return and power transformation of absolute return in the
stock market. The evidence of long memory behavior was found in either conditional variance
model or squared returns. Baillie, Bollerslev, and Mikkelsen (1996) applied the Fractional Generalized Autoregressive Conditional Heteroskedasticity (FIGARCH) model to the exchange rate and
2 http://stats.oecd.org

72

Bollerslev and Mikkelsen (1996) applied the exponential FIGARCH model to the stock price.3
The long memory stochastic volatility (LMSV ) model was introduced by Breidt, Crato, and de
Lima (1998) and Harvey (1998). They found the evidence of long memory stochastic volatility in
the exchange rate and stock price. The modified LMSV model using LW estimator was considered
by Hurvich and Ray (2003). Because of the availability of high frequency data in the financial
market, the “realized volatility”, which is the sum of squared of intraday returns, was introduced by
Andersen, Bollerslev, Diebold, and Labys (2001). They argued that their new volatility measure is
observable and able to implement to simple modeling techniques. Andersen, Bollerslev, Diebold,
and Labys (2001) found evidence of long memory behavior in DM/$ and Yen/$ realized volatility.
The ARFIMA models of realized volatility were considered in Andersen, Bollerslev, Diebold, and
Labys (2003) and Morana and Beltratti (2004).
The data in this section is the realized exchange rate volatility of Andersen, Bollerslev,
Diebold, and Labys (2003) for the Deutsche Mark (DM)/US Dollar and Japanese Yen/US Dollar
exchange rate. The daily realized volatility processes have been obtained by summing squared of
30-minute returns over 48 observations. The data cover the period from December 1, 1986 through
June 30, 1999, a total of 3,045 observations.4
The log of realized volatility is estimated by MLE and LW T SE with pmax = 8 and qmax = 8.
Table 4.5 presents the selected models for realized volatility series. MLE − BIC selects the most
parsimonious model, i.e. ARFIMA(0, d, 0) for both DM/$ and Yen/$. The estimates are presented
in Table 4.6. The log of realized volatilities of DM/$ and Yen/$ have long memory characteristic
with long memory parameters equal to 0.382 and 0.414, respectively. The LW estimates gave
similar long memory parameters to MLE.
3 Caporin (2003) studied the identification issue of FIGARCH models.
4 I am grateful to Prof. Francis X. Diebold for making daily series available in his website.

73

4.5 Application to Interest Rates
The study of persistence of interest rates is a crucial topic in macroeconomic and finance literature.
In conducting monetary policy, monetary authorities change short-term interest rates that will move
long-term interest rates and, eventually, affect the real economy. Earlier works on the long memory
property of interest rates were studied by Shea (1991) and Backus and Zin (1993). Using the
GPH estimator, Shea (1991) found non-stationary long memory in the interest rate term structure.
Backus and Zin (1993) applied ARFIMA model to 3-month coupon rate and found evidence
of long memory. Caporale and Gil-Alana (2010) implemented a semi parametric test for U.S.
interest rate levels and spreads. They found that spreads are nonstationary but mean-reverting.
Long memory properties were also found in other forms of interest rates, such as, U.S. real interest
rates Tsay (2000); corporate bond yields and spreads McCarthy, Pantalone, and Li (2009).
The interest rates considered in this study are the U.S. Treasury constant maturity rates with
monthly data spanning from January 1959 to September 2010. We apply the ARFIMA models to
different maturities, which are 3 and 6 months, and 1, 5, 7 and 10 years. Data were obtained from
the Board of Governors of the Federal Reserve System.5
The ARFIMA models of interest rates and interest rates spread are estimated with pmax = 15
and qmax = 4. Model selection results are presented in Tables 4.7 and 4.8. The parsimonious
models, e.g. ARFIMA(0, d, 1) and ARFIMA(2, d, 3), were chosen by MLE − BIC except for the
3-month maturity rate. The estimates of the ARFIMA models for interest rate levels are shown
in Table 4.9. The d parameter of 3-month maturity equals 0.408. The long memory parameters
of the other maturities fall into the non-stationary region and ranges between 0.834 and 0.905.
The estimates of interest spread between short- and long-term interest rates are shown in Table
4.10. The long memory parameter of the interest rate spread between the 10-year Treasury rate
and the 3-month Treasury rate (Y10-M3) is 0.805. The interest rate spread between the 10-year
Treasury rate and the 6-month Treasury rate (Y10-M6) is also high persistence with d parameter
in the non-stationary region. The results suggest a long memory feature in interest spreads.
5 http://www.federalreserve.gov/releases/h15/data.htm

74

4.6 Conclusion
This section examines the applications of ARFIMA model selections on a set of macroeconomic
and financial time series. We find that the MLE − BIC generally selects relatively more parsimonious models than MLE − AIC and MLE − HQ. We find the evidence of long memory properties
in every series we considered. We also find that the degree of persistence of MLE and LW T SE are
typically different.
Some of the estimated models appear to have substantial ARCH effects. It is unclear if BIC
remains valid in this situation and this is a topic to possibly be investigated in the future.

75

APPENDIX

76

Appendix 4A: Tables
Table 4.1: Model selection for U.S. Real Output
Estimation Selection
procedure Criteria
QMLE
AIC
QMLE
BIC
QMLE
HQ
LW T SE
AIC
LW T SE
BIC
LW T SE
HQ
Number of observations

Selected ARFIMA(p, d, q) model
100[ln(GDPt ) − ln(GDPt−1 )] 100[ln(IPt ) − ln(IPt−1 )]
(7,d,6)
(4,d,3)
(1,d,0)
(0,d,0)
(1,d,0)
(4,d,3)
(8,d,7)
(5,d,5)
(8,d,7)
(1,d,1)
(8,d,7)
(1,d,1)
254
765

Key: GDP denotes Real Quarterly U.S. GDP and ranges from first quarter of 1947 to second
quarter of 2010. IP denotes Index of Industrial Product and ranges from January 1947 to
September 2010. Both variables are first differenced and taken log.

77

Table 4.2: Estimated ARFIMA Models for U.S. Real Output

Model
Log L
ˆ
µ
dˆ
ˆ
ϕ1
ˆ
σ2
m3
m4
Q(20)
Q2 (20)
AIC
BIC
HQ
LW (m = T 0.5 )
LW (m = T 0.6 )
LW (m = T 0.7 )
LW (m = T 0.8 )

100[ln(GDPt ) − ln(GDPt−1 )]
(1,d,0)
-337.283
0.816
(0.027)
-0.341
(0.183)
0.703
(0.173)
0.842
(0.102)
0.26
4.68
23.60
34.09
-0.156
-0.128
-0.145
-0.084
-0.058
0.208
0.283

100[ln(IPt ) − ln(IPt−1 )]
(0,d,0)
-967.83
0.257
(0.099)
0.182
(0.045)
0.738
(0.053)
0.18
4.96
37.80
78.71
-0.302
-0.296
-0.299
0.066
0.115
0.24
0.261

Key: Standard errors are reported in parentheses. Log L is the maximized value of the log
likelihood. Q(20) and Q2 (20) are the Ljung-Box Postmanteau statistics calculated from the first
20 residual and squared residual autocorrelations respectively. m3 and m4 are the sample
skewness and kurtosis of the standardized residuals respectively. m is the choice of bandwidth.

78

Table 4.3: Model Selection for G-7 Monthly Inflation
Estimation
procedure
QMLE
QMLE
QMLE
LW T SE
LW T SE
LW T SE

Selection
Criteria
AIC
BIC
HQIC
AIC
BIC
HQIC

Canada
(4,d,2)
(4,d,2)
(4,d,2)
(9,d,1)
(9,d,1)
(9,d,1)

Order (p,d,q) of selected ARFIMA model
France Germany Italy
Japan U.K.
(14,d,3) (1,d,3) (10,d,4) (1,d,2) (2,d,1)
(3,d,3) (1,d,3) (1,d,1) (1,d,2) (2,d,1)
(12,d,0) (1,d,3) (10,d,4) (1,d,2) (2,d,1)
(15,d,3) (15,d,4) (10,d,4) (4,d,2,) (9,d,1)
(3,d,3) (5,d,4) (1,d,1) (4,d,2,) (9,d,1)
(12,d,0) (15,d,4) (10,d,4) (4,d,2,) (9,d,1)

Key: Number of observations is 668.

79

U.S.
(8,d,4)
(0,d,0)
(8,d,4)
(10,d,4)
(5,d,4)
(5,d,4)

Table 4.4: Estimated ARFIMA Models for G-7 Monthly Inflation

Model
Log L
ˆ
µ
dˆ
ˆ
ϕ1
ˆ
ϕ2
ˆ
ϕ3
ˆ
ϕ4
ˆ
θ1
ˆ
θ2
ˆ
θ3
ˆ
σ2
m3
m4
Q(20)
Q2 (20)
AIC
BIC
HQIC
LW (T 0.5 )
LW (T 0.6 )
LW (T 0.7 )
LW (T 0.8 )

Canada
(4,d,2)
-1841
2.419
(0.953)
0.365
(0.043)
-0.355
(0.064)
-1.054
(0.058)
-0.330
(0.063)
-0.056
(0.054)
-0.043
(0.017)
-0.972
(0.019)
14.625
(1.184)
0.55
5.37
48.83
24.94
2.704
2.751
2.722
0.856
0.692
0.386
0.368

France
(3,d,3)
-1739
1.855
(3.005)
0.564
(0.211)
-1.234
(0.227)
0.051
(0.396)
0.517
(0.257)
-

Germany
(1,d,3)
-1754
1.736
(1.397)
0.585
(0.101)
0.388
(0.128)
-

Italy
(1,d,1)
-1711
0.734
(1.774)
0.951
(0.257)
0.390
(0.178)
-

Japan
(1,d,2)
-2191
1.377
(2.214)
0.568
(0.085)
-0.163
(0.122)
-

-

-

-

-

-1.065
(0.268)
0.381
(0.495)
0.695
(0.312)
10.790
(1.259)
0.91
10.08
110.87
330.37
2.400
2.447
2.418
0.707
0.525
0.488
0.414

0.846
(0.117)
-0.019
(0.082)
0.023
(0.053)
11.295
(0.842)
0.62
4.71
119.42
56.51
2.439
2.473
2.452
0.791
0.608
0.259
0.230

Key: See Table 4.2.

80

U.S.
(0,d,0)
-1647
2.329
(0.881)
0.365
(0.048)
-

-

U.K.
(2,d,1)
-2052
3.885
(1.188)
0.284
(0.042)
-0.975
(0.051)
0.026
(0.051)
-

-

-

-

0.885
(0.071)
-

-0.995
(0.004)
-

-

-

0.334
(0.104)
0.339
(0.070)
-

-

-

9.917
(1.123)
0.14
9.54
57.17
275.14
2.303
2.324
2.311
0.924
0.649
0.504
0.453

41.775
(3.322)
0.54
5.22
142.99
187.16
3.744
3.771
3.755
0.593
0.648
0.344
0.201

27.592
(2.927)
1.35
8.50
259.27
83.17
3.330
3.357
3.340
0.722
0.656
0.417
0.283

8.174
(0.729)
-0.25
6.30
53.15
122.042
2.104
2.111
2.107
0.665
0.685
0.368
0.361

-

-

Table 4.5: Model Selection for Realized Volatility of Exchange Rates
Estimation
procedure
QMLE
QMLE
QMLE
LW T SE
LW T SE
LW T SE

Selection
Criteria
AIC
BIC
HQIC
AIC
BIC
HQIC

Selected ARFIMA(p, d, q) model
DM/$
JP Yen/$
(8,d,8)
(4,d,8)
(0,d,0)
(0,d,0)
(1,d,1)
(0,d,0)
(7,d,6)
(4,d,1)
(1,d,1)
(1,d,1)
(1,d,1)
(1,d,1)

Key: Number of observations is 3,045.

Table 4.6: Estimated ARFIMA Models for Realized Volatility of Exchange Rates

Model
Log L
ˆ
µ
dˆ
ˆ
σ2
m3
m4
Q(20)
Q2 (20)
AIC
BIC
HQIC
LW (m = T 0.5 )
LW (m = T 0.6 )
LW (m = T 0.7 )
LW (m = T 0.8 )

DM/$
(0,d,0)
-2727.495
-1.124
(0.156)
0.382
(0.013)
0.351
(0.009)
0.76
4.87
33.87
33.85
-1.046
-1.044
-1.045
0.555
0.589
0.500
0.476

Key: See Table 4.2.

81

JP/$
(0,d,0)
-2931.457
-1.271
(0.206)
0.414
(0.014)
0.402
(0.010)
0.70
4.47
15.56
76.26
-0.911
-0.910
-0.911
0.553
0.504
0.500
0.482

Table 4.7: Model Selection for Interest Rates
Estimation
procedure
QMLE
QMLE
QMLE
LW T SE
LW T SE
LW T SE

Selection
Criteria
AIC
BIC
HQ
AIC
BIC
HQ

3 months
(13,d,4)
(9,d,2)
(11,d,4)
(13,d,4)
(9,d,3)
(11,d,4)

Selected ARFIMA(p, d, q) model
6 months 1 year
5 years 7 years
(11,d,4) (12,d,4) (14,d,4) (12,d,4)
(2,d,3)
(2,d,3)
(0,d,1)
(0,d,1)
(11,d,4)
(9,d,2)
(6,d,4)
(0,d,1)
(14,d,4) (13,d,4) (14,d,4) (10,d,4)
(2,d,3)
(2,d,3)
(0,d,1)
(0,d,1)
(9,d,3)
(9,d,2)
(5,d,2) (10,d,4)

10 years
(10,d,4)
(0,d,1)
(0,d,1)
(9,d,4)
(0,d,2)
(0,d,2)

Key: Number of observations is 621.

Table 4.8: Model Selection for Interest Rate Spreads
Estimation
procedure
QMLE
QMLE
QMLE
LW T SE
LW T SE
LW T SE

Selection
Criteria
AIC
BIC
HQ
AIC
BIC
HQ

Selected ARFIMA(p, d, q) model
Y10-M3
Y10-M6
(13,d,4)
(10,d,4)
(0,d,1)
(0,d,1)
(14,d,0)
(5,d,2)
(12,d,4)
(3,d,4)
(1,d,2)
(3,d,4)
(12,d,4)
(3,d,4)

Key: Number of observations is 621. Y10-M3 is an interest rate spread between 10-year Treasury
rate and 3-month Treasury rate. Y10-M6 is an interest rate spread between 10-year Treasury rate
and 6-month Treasury rate.

82

Table 4.9: Estimated ARFIMA Models for U.S. Interest Rates
Model
Log L
ˆ
µ
dˆ
ˆ
ϕ1
ˆ
ϕ2
ˆ
ϕ3
ˆ
ϕ4
ˆ
ϕ5
ˆ
ϕ6
ˆ
ϕ7
ˆ
ϕ8
ˆ
ϕ9
ˆ
θ1
ˆ
θ2
ˆ
θ3
ˆ
σ2
m3
m4
Q(20)
Q2 (20)
AIC
BIC
HQIC
LW (T 0.5 )
LW (T 0.6 )
LW (T 0.7 )
LW (T 0.8 )

3 months
(9,d,2)
-296.58
2.817
(0.324)
0.408
(0.104)
-0.425
(0.097)
0.340
(0.102)
0.457
(0.070)
-0.115
(0.055)
0.257
(0.047)
-0.051
(0.050)
-0.146
(0.046)
0.061
(0.042)
0.353
(0.039)
-1.467
(0.060)
-0.807
(0.068)
0.152
(0.009)
-0.87
16.46
26.28
768.33
-1.844
-1.758
-1.811
0.794
1.003
0.884
0.958

6 months
(2,d,3)
-276.643
3.042
(0.313)
0.857
(0.040)
-0.892
(0.062)
-0.717
(0.053)
-

1 year
(2,d,3)
-285.458
3.365
(0.298)
0.834
(0.038)
-0.845
(0.052)
-0.695
(0.052)
-

5 years
(0,d,1)
-216.614
3.927
(0.301)
0.905
(0.038)
-

7 years
(0,d,1)
-144.877
4.047
(0.265)
0.905
(0.037)
-

10 years
(0,d,1)
-60.528
4.058
(0.238)
0.905
(0.036)
-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-1.502
(0.058)
-1.445
(0.051)
-0.580
(0.040)
0.143
(0.008)
-1.04
13.54
67.17
747.74
-1.928
-1.885
-1.911
0.820
1.002
0.895
0.971

-1.538
(0.046)
-1.493
(0.045)
-0.659
(0.035)
0.147
(0.008)
-0.58
10.94
51.01
877.71
-1.899
-1.856
-1.883
0.860
0.981
0.898
0.965

-0.531
(0.041)
-

-0.552
(0.040)
-

-0.509
(0.043)
-

-

-

-

0.118
(0.007)
-0.22
9.38
53.81
576.42
-2.134
-2.120
-2.128
0.938
0.959
0.920
0.982

0.093
(0.005)
0.03
7.57
43.22
502.16
-2.365
-2.351
-2.359
0.974
0.954
0.935
0.999

0.071
(0.004)
-0.02
7.93
40.07
416.31
-2.636
-2.622
-2.631
1.010
0.965
0.979
1.030

Key: See Table 4.2.
83

Table 4.10: Estimated ARFIMA models for U.S. Interest Rate Spreads

Model
Log L
ˆ
µ
dˆ
ˆ
θ1
ˆ
σ2
m3
m4
Q(20)
Q2 (20)
AIC
BIC
HQIC
LW (T 0.5 )
LW (T 0.6 )
LW (T 0.7 )
LW (T 0.8 )

Y10-M3
(0,d,1)
-221.182
1.186
(0.316)
0.805
(0.041)
-0.508
(0.046)
0.119
(0.007)
0.32
13.71
65.60
504.97
-2.119
-2.105
-2.114
0.415
0.690
0.848
0.884

Key: See Table 4.2.

84

Y10-M6
(0,d,1)
-116.9
0.972
(0.272)
0.856
(0.042)
-0.455
(0.047)
0.085
(0.005)
0.51
13.50
60.15
382.70
-2.455
-2.441
-2.449
0.421
0.751
0.888
0.926

Chapter 5

CONCLUSION

This dissertation considers two methodological issues in long memory modeling, multi-step forecasting and model selections. The comparisons between the parametric model, i.e. ARFIMA , and
the semi parametric model, e.g. LW , in term of forecasting and model selection are investigated.
Applications of model selection to real output, inflation rates, volatility, and interest rates are also
considered.
We consider the effects on multi step prediction of using LW T SE compared to MLE for
ARFIMA models in Chapter 2. From detailed simulations, we found that the predictor based on
MLE is generally superior in term of MSE sense to the predictor based on LW T SE. Even in case
of misspecification, the MLE still performs better than the LW T SE.
Chapter 3 considers model selection of the ARFIMA model. The Bayesian (BIC) and HannanQuinn (HQ) information criteria are consistent model selection procedures for the ARFIMA model
estimated by QMLE. Simulation results show that the BIC based on QMLE works very well for
low order models. We also suggest the modified information criteria for LW T SE.
In Chapter 4, we apply the model selection with MLE and LW to real output, inflation rates,
volatility, and interest rates series. We found that the BIC with MLE mostly select the parsimonious
model. The long memory parameters from the selected model substantiate long memory behavior
in these series.

85

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86

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