Linear Models and Time-Series Analysis E-Book

Marc S. Paolella

• Herausgeber: John Wiley & Sons
• Kategorie: Wissenschaft und neue Technologien
• Serie: Wiley Series in Probability and Statistics
• Sprache: Englisch

Seitenzahl: 1488

Veröffentlichungsjahr: 2018

Beschreibung

A comprehensive and timely edition on an emerging new trend in time series Linear Models and Time-Series Analysis: Regression, ANOVA, ARMA and GARCH sets a strong foundation, in terms of distribution theory, for the linear model (regression and ANOVA), univariate time series analysis (ARMAX and GARCH), and some multivariate models associated primarily with modeling financial asset returns (copula-based structures and the discrete mixed normal and Laplace). It builds on the author's previous book, Fundamental Statistical Inference: A Computational Approach, which introduced the major concepts of statistical inference. Attention is explicitly paid to application and numeric computation, with examples of Matlab code throughout. The code offers a framework for discussion and illustration of numerics, and shows the mapping from theory to computation. The topic of time series analysis is on firm footing, with numerous textbooks and research journals dedicated to it. With respect to the subject/technology, many chapters in Linear Models and Time-Series Analysis cover firmly entrenched topics (regression and ARMA). Several others are dedicated to very modern methods, as used in empirical finance, asset pricing, risk management, and portfolio optimization, in order to address the severe change in performance of many pension funds, and changes in how fund managers work. * Covers traditional time series analysis with new guidelines * Provides access to cutting edge topics that are at the forefront of financial econometrics and industry * Includes latest developments and topics such as financial returns data, notably also in a multivariate context * Written by a leading expert in time series analysis * Extensively classroom tested * Includes a tutorial on SAS * Supplemented with a companion website containing numerous Matlab programs * Solutions to most exercises are provided in the book Linear Models and Time-Series Analysis: Regression, ANOVA, ARMA and GARCH is suitable for advanced masters students in statistics and quantitative finance, as well as doctoral students in economics and finance. It is also useful for quantitative financial practitioners in large financial institutions and smaller finance outlets.

Leseprobe

Table of Contents

Cover

Preface

Part I: Linear Models: Regression and ANOVA

1 The Linear Model

1.1 Regression, Correlation, and Causality

1.2 Ordinary and Generalized Least Squares

1.3 The Geometric Approach to Least Squares

1.4 Linear Parameter Restrictions

1.5 Alternative Residual Calculation

1.6 Further Topics

1.7 Problems

1.A Appendix: Derivation of the BLUS Residual Vector

1.B Appendix: The Recursive Residuals

1.C Appendix: Solutions

2 Fixed Effects ANOVA Models

2.1 Introduction: Fixed, Random, and Mixed Effects Models

2.2 Two Sample ‐Tests for Differences in Means

2.3 The Two Sample ‐Test with Ignored Block Effects

2.4 One‐Way ANOVA with Fixed Effects

2.5 Two‐Way Balanced Fixed Effects ANOVA

3 Introduction to Random and Mixed Effects Models

3.1 One‐Factor Balanced Random Effects Model

3.2 Crossed Random Effects Models

3.3 Nested Random Effects Models

3.4 Problems

3.5 Appendix: Solutions

Part II: Time Series Analysis: ARMAX Processes

4 The AR(1) Model

4.1 Moments and Stationarity

4.2 Order of Integration and Long‐Run Variance

4.3 Least Squares and ML Estimation

4.4 Forecasting

4.5 Small Sample Distribution of the OLS and ML Point Estimators

4.6 Alternative Point Estimators of

4.7 Confidence Intervals for

4.8 Problems

5 Regression Extensions: AR(1) Errors and Time‐varying Parameters

5.1 The AR(1) Regression Model and the Likelihood

5.2 OLS Point and Interval Estimation of

5.3 Testing in the ARX(1) Model

5.4 Bias‐Adjusted Point Estimation

5.5 Unit Root Testing in the ARX(1) Model

5.6 Time‐Varying Parameter Regression

6 Autoregressive and Moving Average Processes

6.1 AR() Processes

6.2 Moving Average Processes

6.3 Problems

6.4 Appendix: Solutions

7 ARMA Processes

7.1 Basics of ARMA Models

7.2 Infinite AR and MA Representations

7.3 Initial Parameter Estimation

7.4 Likelihood‐Based Estimation

7.5 Forecasting

7.6 Bias‐Adjusted Point Estimation: Extension to the ARMAX model

7.7 Some ARIMAX Model Extensions

7.8 Problems

7.9 Appendix: Generalized Least Squares for ARMA Estimation

7.10 Appendix: Multivariate AR() Processes and Stationarity, and General Block Toeplitz Matrix Inversion

8 Correlograms

8.1 Theoretical and Sample Autocorrelation Function

8.2 Theoretical and Sample Partial Autocorrelation Function

8.3 Problems

8.4 Appendix: Solutions

9 ARMA Model Identification

9.1 Introduction

9.2 Visual Correlogram Analysis

9.3 Significance Tests

9.4 Penalty Criteria

9.5 Use of the Conditional SACF for Sequential Testing

9.6 Use of the Singular Value Decomposition

9.7 Further Methods: Pattern Identification

Part III: Modeling Financial Asset Returns

10 Univariate GARCH Modeling

10.1 Introduction

10.2 Gaussian GARCH and Estimation

10.3 Non‐Gaussian ARMA‐APARCH, QMLE, and Forecasting

10.4 Near‐Instantaneous Estimation of NCT‐APARCH(1,1)

10.5 ‐APARCH and Testing the IID Stable Hypothesis

10.6 Mixed Normal GARCH

11 Risk Prediction and Portfolio Optimization

11.1 Value at Risk and Expected Shortfall Prediction

11.2 MGARCH Constructs Via Univariate GARCH

11.3 Introducing Portfolio Optimization

12 Multivariate Distributions

12.1 Multivariate Student's

12.2 Multivariate Noncentral Student's

12.3 Jones Multivariate Distribution

12.4 Shaw and Lee Multivariate Distributions

12.5 The Meta‐Elliptical Distribution

12.6 MEST: Marginally Endowed Student's

12.7 Some Closing Remarks

12.A ES of Convolution of AFaK Margins

12.B Covariance Matrix for the FaK

13 Weighted Likelihood

13.1 Concept

13.2 Determination of Optimal Weighting

13.3 Density Forecasting and Backtest Overfitting

13.4 Portfolio Optimization Using (A)FaK

14 Multivariate Mixture Distributions

14.1 The Distribution

14.2 Model Diagnostics and Forecasting

14.3 MCD for Robustness and Estimation

14.4 Some Thoughts on Model Assumptions and Estimation

14.5 The Multivariate Laplace and Distributions

Part IVAppendices

Appendix ADistribution of Quadratic Forms

A.1 Distribution and Moments

A.2 Basic Distributional Results

A.3 Ratios of Quadratic Forms in Normal Variables

A.4 Problems

A.A Appendix: Solutions

Appendix B: Moments of Ratios of Quadratic Forms

B.1 For and

B.2 For

B.3 For

B.4 For

B.5 Useful Matrix Algebra Results

B.6 Saddlepoint Equivalence Result

Appendix CSome Useful Multivariate Distribution Theory

C.1 Student's Characteristic Function

C.2 Sphericity and Ellipticity

Appendix DIntroducing the SAS Programming Language

D.1 Introduction to SAS

D.2 Basic Data Handling

D.3 Advanced Data Handling

D.4 Generating Charts, Tables, and Graphs

D.5 The SAS Macro Processor

D.6 Problems

D.7 Appendix: Solutions

Bibliography

Index

End User License Agreement

List of Tables

Chapter 01

Table 1.1 Empirical mean squared error over the four regression parameters, based on replications.

Chapter 02

Table 2.1 The ANOVA table for the balanced one‐way ANOVA model. Mean squares denote the sums of squares divided by their associated degrees of freedom. Term in the expected mean square corresponding to the treatment effect is defined in (2.42 ).

Table 2.2 Empirical size of test with .

Table 2.3 The ANOVA table for the balanced two‐way ANOVA model without interaction effect, where “error df” is . Mean squares denote the sums of squares divided by their associated degrees of freedom. Table 2.4 is for the case with interaction, and also gives the expected mean squares.

Table 2.4 The ANOVA table for the balanced two‐way ANOVA model with interaction effect. Mean squares denote the sums of squares divided by their associated degrees of freedom. The expected mean squares are given in (2.65 ), (2.72 ), and (2.74 )

Chapter 03

Table 3.1 ANOVA table for the balanced one‐factor REM. The second column is specific to our model notation 3.1 , and is not necessary, but shown for further clarity.

Table 3.2 ANOVA table for the balanced two‐factor crossed REM.

Table 3.3 ANOVA table for the balanced two‐factor crossed additive (no interaction effect) random effects model, where the error degrees of freedom is .

Table 3.4 ANOVA table for the balanced three‐factor crossed REM.

Table 3.5 ANOVA table for the balanced two‐factor nested REM. Notation B(A) is short for “B within A”, indicating the hierarchy of the nested factor.

Table 3.6 ANOVA table for balanced two‐factor mixed nested REM.

Table 3.7 ANOVA table for the balanced three‐factor nested REM.

Table 3.8 ANOVA table for the balanced three‐factor mixed nested REM: Classes are fixed, sub‐ and subsubclasses are random.

Table 3.9 ANOVA table for the balanced three‐factor mixed nested REM: Classes and subclasses are fixed, subsubclasses are random.

Chapter 04

Table 4.1 Small‐sample behavior of AR(1) estimators (o.l.s.) and (m.l.e.) based on 1,000 simulated time series, near and on the stationarity border. The top panel gives the sample mean, the middle panel gives 1,000 times the variance, and the bottom panel gives the percentage of estimates exceeding 1.0.

Chapter 06

Table 6.1 Comparison of estimated standard errors for the AR(1) model. “Emp” is the empirically observed standard error of , calculated as the sample standard deviation of the based on simulation with 10,000 replications; is short for , which refers to use of the sample covariances to form the asymptotic variance–covariance matrix in 6.29 ; is short for and refers to use of to form the asymptotic variance–covariance matrix in 6.29 ; and “Hess” refers to use of the estimated Hessian matrix constructed from the quasi‐Newton method used to numerically maximize the likelihood. The values in parentheses indicate the sample standard error of the 10,000 estimates. Entries under “Known ” assume the process has zero mean, so that only and are estimated, and the are formed as in (8.6), i.e., without subtracting the mean from the data. For “ jointly estimated”, the model is extended to include an matrix consisting of a column of ones, and the are formed with mean subtraction, as in (8.10). Boldface entries indicate being closest to the empirically observed standard error of .

Chapter 12

Table 12.1 The estimated parameters using the set of 1,945 daily (log percentage) returns of Bank of America and Wal‐Mart Stores, as depicted in Figures 12.3 and 12.4 . “S‐L” refers to Shaw and Lee, “loglik” is the log‐likelihood evaluated at the obtained m.l.e., “std err Hess” refers to the approximate (asymptotic normal‐based) standard errors obtained as output from the optimization, and “std err NPB” and “std err PB” refer to use of the nonparametric and parametric bootstrap, respectively.

Table 12.2

Similar to Table 12.1 but for the FaK and AFaK distributions, as well as the MESTI, discussed in Section 12.6.2 .

Chapter 13

Table 13.1 The obtained average realized predictive log‐likelihood ( 13.3 ) (to four significant digits) for various models and weighted likelihood () and correlation shrinkage () parameter settings. Last column is the difference of ( 13.3 ) from that of the first entry.

Chapter 14

Table 14.1 Number of observations required to be removed until the likelihood ratio test comparing GA and normality does not reject at the 0.05 level.

List of Illustrations

b01

Figure A.1 True (via inversion formula) and second‐order s.p.a. density of the sample variance , for a sample of size 10 for with and corresponding to an AR(1) process with parameter . In the left panel, for , the two graphs are optically indistinguishable. The s.p.a. is about 14 times faster to compute.

Left:

relative percentage error (r.p.e.) incurred when using (A.35 ) with for the p.d.f. of (A.36 ) with .

Right:

r.p.e. using the exact expression (A.39 ). For both the c.d.f. in (A.35 ) and the p.d.f. in (A.39 ), numeric integration using Matlab's integration routine

quadgk

was used with default tolerance parameters for the absolute and relative error. It is well‐suited to these integrands because, paraphrasing from their documentation, “[it] may be most efficient for oscillatory integrands and any smooth integrand at high accuracies. It supports infinite intervals and can handle moderate singularities at the endpoints.”

Figure A.3

Left:

Exact density (A.40 ).

Right:

Discrepancy between exact density and use of the Geary and Pan methods. The graph is truncated from below.

Figure A.4 Exact density (solid), second‐order s.p.a. (A.45 ) (dashed) and normalized first‐order s.p.a. (A.43 ) (dash‐dot) of in (A.36 ) using .

Figure A.5 Saddlepoint (solid) and kernel density estimate (dashed) of , where , and such that and .

Chapter 02

Figure B.1

Top:

The exact density of in B.7 for (solid) and via the first‐order (dashed) and second‐order (dash‐dot) s.p.a.

Bottom:

Histograms of 100,000 simulated values.

Figure B.2 Exact mean (solid), and the mean plus and minus 1.96 times the exact standard deviation (dashed) of the Durbin–Watson test statistic B.7 under the null hypothesis of no autocorrelation, versus sample size, starting at .

Figure B.3 Exact mean (solid), and the mean times the square root of the exact variance (dashed) of the Durbin–Watson test statistic B.18 under the null hypothesis of no autocorrelation, versus sample size, starting at . The matrix consists of a column of ones and a time‐trend vector, .

Figure B.4 The exact density of in B.16 for the model (solid) and the first‐order (dashed) and second‐order (dash‐dot) s.p.a.

Figure B.5 For the three sample sizes (solid), (dashed) and (dash‐dot), the mean (left) and times the variance (right) of the Durbin–Watson statistic B.16 , as a function of the autoregressive parameter in B.14 , for the intercept model (top panels), intercept and time trend (middle panels), and intercept, time trend and cyclical, (bottom panels), where is the eigenvector in B.10 with .

Figure B.6 The mean of , and the mean plus and minus 1.96 times its standard deviation, as a function of , when using the mis‐specified model that erroneously assumes .

Figure B.7 The mean (left) and variance (right) (not multiplied by ), of the Durbin–Watson statistic, as a function of the autoregressive parameter . The true model is , , , , with , , and vector is the same as used in Example B.6, but the regression model is mis‐specified as . The solid lines are the exact values; the dashed lines were computed from B.60 and B.61 . The dash‐dot lines show the exact mean and variance when (so that the model would not be under‐specified). The arrows in the left plot indicate how to determine that value of such that the mean of would be precisely the same value if there were no autocorrelation () and if the model were not mis‐specified; it is .

Figure B.8 Same as Figure B.7 but using (top) and (bottom).

Figure B.9 The relative percentage error , as a function of parameter , based on the s.p.a. for the mean (left) and variance (right) of the Durbin–Watson statistic using the same parameter values as those in Figure B.7.

b03

Figure C.1 The unit sphere for with 200 (a), 1000 (b), and 3000 (c) random uniformly distributed points.

Figure C.2 Kernel density plots (truncated, so that the ‐axis is the same in each plot) of the distribution of H‐P test statistic for ellipticity. The ‐axis elements were divided by 1,000.

Top left:

Computed on the GARCH‐filtered log percentage returns of randomly drawn stocks out of 416 from the S&P500 index, and this done over random draws.

The remaining plots

show the same result but when restricting the stocks to be within the same of each of the 10 industry sectors that divide the stocks on the index.

b04

Figure D.1 PDV for input and calculation of variables.

Figure D.2 PDV illustrating branching via an

if

statement.

Figure D.3 PDV illustrating construction of two data sets.

Figure D.4 PDV using

merge

.

Figure D.5 Output from SAS

proc gplot

with overlaid data.

Figure D.6 Output from SAS

proc gchart

.

Figure D.7 Differences model for predicting consumption.

Figure D.8 Autoregressive model for predicting consumption.

Chapter 01

Figure 1.1 Scatterplot of age versus income overlaid with fitted regression curves.

Figure 1.2 Weight vector for an MA(1) model with and (top) and (bottom).

Figure 1.3 Percentage improvement for the two test groups as a function of number of sessions.

Figure 1.4 True and fitted piecewise regression.

Figure 1.5 Ellipsoid for intercept (horizontal axis) and slope (vertical axis) for the model in Example 14 , for (plus signs), (crosses) and (circles). The black dot is .

Figure 1.6 Ratio of lengths of Bonferroni to Scheffé confidence intervals. The top panel does not adjust for rank of , while the bottom panel does adjust.

Figure 1.7 Simulated relative percentage change between the recursive and BLUS residuals for a model with intercept and time trend, and 20 observations.

Chapter 02

Figure 2.1 Power of the test, given in (2.10 ) and (2.12 ), as a function of , using and .

Figure 2.2

Top:

Minimum required sample size as a function of , based on (2.13 ), using and .

Bottom::

Approximation to the sample size calculation computed using (2.17 ).

Figure 2.3 Size (solid, left axis) and power (dashed, right axis) for the two‐way model ignoring the effect of gender, with , and .

Figure 2.4

Top:

Power of the test as a function of , for fixed , , and , and three values of .

Bottom:

Similar, but is fixed at 16, and three values of are used. The middle dashed line is the same in both graphics.

Figure 2.5 Matlab output for the ANOVA example.

Figure 2.6 Matlab output from calling the function in Listing 2.3 as

x=anovacreate(6)

, and then running the built‐in Matlab function

p=anova1(x)

.

Figure 2.7 Histogram of 10,000 simulated ‐values of the one‐way ANOVA test with and under the null hypothesis, but with i.i.d. Student's data with 2 (top) and 4 (bottom) degrees of freedom.

Figure 2.8 Same as SAS Output 2.10, but having used Matlab's function

anovan

. Note that in the fourth placed after the decimal, the mean square for treatment B (“Phy Act” in Matlab; “Sport” in SAS) differs among the two outputs (by one digit), presumably indicating that different numeric algorithms are used for their respective computations. This, in turn, is most surely irrelevant given the overstated precision of the measurements (they are not accurate to all 14 digits maintained in the computer), and that the statistics and corresponding ‐values are the same to all digits shown in the two tables.

Figure 2.9 ANOVA table output from the Matlab code in Listing 2.8. Here,

Columns

refers to factor B, which is also clear because it has one degree of freedom, corresponding to use of . Similarly,

Rows

is for factor A, with treatments (and, thus, two degrees of freedom).

Figure 2.10 Histograms of ‐values corresponding to the simulation from the code in Listing 2.9.

Figure 2.11 ANOVA table output from the Matlab code in Listing 2.10. Compare to Figure 2.9 .

Figure 2.12 The default graphical output corresponding to the interaction effect

Treatment*Sport

from using the

means

statement in

proc anova

from SAS Listing 2.2.

Figure 2.13 Default graphical output from SAS'

proc glm

, showing the same data as in Figure 2.12 .

Chapter 03

Figure 3.1

Top:

Histograms of the m.l.e. of the three parameters, from left to right, , , and , of the one‐way REM, based on , , and replications. The vertical dashed line indicates the true value of the parameter in each graph.

Bottom:

Histograms of the approximate standard errors output from the BFGS algorithm, with the vertical dashed lines being the sample standard error of the m.l.e. point estimates of , , and .

Figure 3.2 Same as Figure 3.1 but for and .

Figure 3.3 Similar to the top panels in Figure 3.1 , namely histograms of the m.l.e. of the three parameters, from a) to c), , , and , of the one‐way REM, based on , , and replications, but such that 10 observations are missing, as shown in the code in Listing 3.5. These were obtained using the approximate m.l.e. method, and such that the obtained estimates for were multiplied by 1.0368. The vertical dashed line indicates the true value of the parameter in each graph.

Figure 3.4

Top:

Actual coverage probability as a function of tuning parameter for the confidence interval of the intraclass correlation coefficient for the one‐way unbalanced REM with , , 20 missing values, , and .

Bottom:

Same but having used .

Figure 3.5 Point estimates of the three variance components for the two‐way nested REM, based on the approximate m.l.e. with multiplicative factor adjustment for and use of 1,000 replications. True model parameters are those given in Listing 3.12, namely , , .

Chapter 04

Figure 4.1 Example of a simulated AR(1) process with and error bounds. The top panel is just a magnified view of the beginning of the series. The figures are plotted such that the observations, indicated as dots, are connected to enhance visibility, though note that the process is not continuous.

Figure 4.2 Example of AR(1) process with .

Figure 4.3

Top:

Example of random walk.

Bottom:

Same random walk but showing more observations.

Figure 4.4 Random walk without drift (top) and with drift (bottom) based on the same sequence.

Figure 4.5 The daily S&P 500 stock index over a 10 year period.

Figure 4.6 Variance of times as a function of .

Figure 4.7 The m.s.e. of one‐step ahead forecast for the AR(1) model as a function of autoregressive parameter , with . The solid line is the m.s.e. for , the dashed line is for , and the dash‐dot line is for .

Figure 4.8

Top:

True m.s.e. of ‐step ahead forecasts, , for , , and using exact m.l.e. (solid) and conditional m.l.e. (dotted), both restricted to yield stationary models. The dashed line is 4.34 using the true values of and , while the dash‐dot line is the same, but without the last term in 4.34 .

Bottom:

Same, but with . Note the change of the ‐axis.

Figure 4.9 Kernel density estimates of the distribution of the m.l.e. of in the AR(1) model using replications (values , , and ) and three sample sizes, as indicated.

Figure 4.10 Density of based on , , , and , for (a), (b), and (c) using simulation and kernel density estimation based on 10,000 replications (solid) and the saddlepoint approximation of 4.37 (dashed).

Figure 4.11

Left:

Performance comparison as a function of AR(1) parameter , of the least squares estimator, denoted as OLS, along with jackknife bias‐adjusted estimator 4.44 for several values of , as indicated, and based on 100,000 replications. The top graphs show the mean‐bias, the middle graphs show the median‐bias, and the bottom graphs show the m.s.e.

Right:

Same but concentrating on the area near the unit root.

Figure 4.12 The mean (solid), median (dashed), and mode (dash‐dot) of 4.37 , minus , versus , for .

Figure 4.13 Performance comparison of the least squares (OLS), exact maximum likelihood (MLE), mean‐bias‐adjusted (MBA), median‐unbiased (MED), and mode‐unbiased (MOD) estimators for parameter in the AR(1) model with . The top graphs show the mean‐bias, the middle graphs show the median‐bias, and the bottom graphs show the m.s.e.

Figure 4.14 Confidence intervals over replications when , for the indicated sample sizes.

Figure 4.15 Comparison of actual confidence interval coverage (top panel) and length (bottom panel) for parameter in the AR(1) model with , based on simulation with 20,000 replications and level of significance . “Exact” (solid) refers to 4.53 , “Asymp” (dashed) refers to c.i.s based on the m.l.e. and its asymptotic normal distribution. and “Asy Cut” (dash dot) is the same as “Asymp” but truncating the c.i.s to lie between and 1.

Figure 4.16 Similar to Figure 4.15 but for parameter . “Asy Cut” (dash dot) is the same as “Asymp” but truncating the c.i.s to lie above zero.

Chapter 05

Figure 5.1 Demonstration of computation of in 5.15 .

Figure 5.2 Power of the test for the null of based on the two‐sided equal‐tail confidence interval for , for significance level and and , for various matrices, as indicated, and based on simulated replications.

Figure 5.3 Power of the , , and tests, for significance level , two sample sizes and , and two matrices, as indicated. The lines corresponding to and are graphically indistinguishable in the top plot.

Figure 5.4

Top:

Limiting power corresponding to a significance level of , for several matrices, none of which contain a column of ones, as a function of sample size .

Bottom:

Same but showing the limiting power 5.29 , such that the matrices contain a column of ones.

Figure 5.5 Adjustment to corresponding to (a), (b), and (c), shown for the two indicated matrices (solid and dashed lines). The four sample sizes used are , , , and , moving from right to left within each plot.

Figure 5.6 Comparison of the m.s.e. for the least squares (OLS), mean bias‐adjusted (MBA), median unbiased (MED), and mode unbiased (MOD) estimators for parameter in the AR(1) model with . The top panels refer to use of , while the bottom panels are for . The right panels just focus on the range .

Figure 5.7 The performance of the various adjusted estimators, having used Cauchy innovations.

Figure 5.8

Top:

The power of the Dickey–Fuller unit root test, for two sample sizes and three matrices, as indicated.

Bottom:

Same, but based on the one‐sided c.i. of as in Section 4.7.

Figure 5.9

Top:

Approximate power of the test 5.32 computed analytically (solid line) and from simulation (dashed line) with significance level 0.05, for , , and , using .

Bottom:

Same, but based on .

Figure 5.10 Similar to the right panel of Figure 5.9 , but based on the test 5.35 from Shively (2001 ).

Figure 5.11 Top: Power of the test 5.32 computed from simulation from replications, using innovations, with significance level 0.05, , and . Bottom: Same, but for .

Figure 5.12

Top:

Actual size of the test 5.32 , for and nominal size , computed via simulation with replications, as a function of , the coefficient referring to the regressor capturing the break in the constant.

Bottom:

Same, but as a function of , the coefficient referring to the regressor capturing the break in the trend.

Figure 5.13 Twenty realizations of model 5.41 with , , and and as indicated in the titles. The thick solid line is the true mean of , obtained by setting .

Figure 5.14 Histograms of the least squares estimators for the HHRC model based on , , , , , and replications. For , about 25% of the estimates were zero.

Figure 5.15 Same as Figure 5.14 but based on the m.l.e.

Figure 5.16 Histograms of the least squares estimators for the HHRC model based on , , where is the eigenvector in (B.10) with , , , , and replications. For , about 33% of the estimates were zero.

Figure 5.17 Histograms of the m.l.e. for the HHRC model for , , , , but now based on , and replications, and assuming known and . The top panel further assumes is also known.

Figure 5.18

Top:

Twenty realizations of model 5.53 with , , , , and two values of .

Bottom:

Same, but having used .

Figure 5.19 Kernel density plots of based on replications, sample size , , , and (top) and (bottom).

Figure 5.20 Scatterplots showing the approximate standard error as a function of , where the values used correspond to the used in Figure 5.19 but having been truncated such that .

Figure 5.21 The power of the Nyblom and Mäkeläinen (1983 ) test 5.57 for three sample sizes and two models, as described in the text. Solid (dashed) lines were computed using the exact (s.p.a.) method; they are essentially indistinguishable, with use of the s.p.a. being about three times faster.

Figure 5.22 The power of the Shively (1988a ) test 5.60 and the value of such that the power of the test based on is 0.5 when the true equals , for three sample sizes and three models, as described in the text.

Figure 5.23 The power of the Shively (1988b ) test 5.62 and the value of such that the power of the test based on is 0.5 when the true equals , for three sample sizes and three models, as described in the text.

Chapter 06

Figure 6.1 Simulated unit‐root AR(4) process 6.12 (top) and part of the first difference series (bottom).

Figure 6.2 Contour plots of likelihoods of simulated AR(2) time series as functions of (horizontal axis) and (vertical axis). The left panel takes and observations, the middle panel takes , , and , and the right panel shows both these cases, but based on simulated series with observations.

Figure 6.3

Left:

Density of the exact m.l.e. (solid) and conditional m.l.e. (dashed) of parameter in the MA(1) model based on observations, for true (top), (bottom), and .

Right:

Same, but for .

Figure 6.4

Top:

Variance and covariance of estimated AR(2) parameters, as a function of , for constant and sample size . Solid lines show the asymptotic values from 6.30 .

Bottom:

Same, but for .

Figure 6.5 Two views of the allowed range of parameters of the AR(3) model such that the process is stationary.

Figure 6.6 Allowed range of and in the AR(3) model for a fixed value of , for several values of .

Figure 6.7 The bias (top) and m.s.e. (bottom) of the exact m.l.e. (solid line) and conditional m.l.e. (dashed line) of in the MA(1) model as a function of parameter , based on observations, , and 10,000 replications. The graphics were produced from the code in Listing 6.9.

Figure 6.8 The bias (top) and m.s.e. (bottom) of the moments‐method 6.60 (solid line) and long autoregression method 6.57 (dashed line) of estimating with closed‐form expressions, for the MA(1) model as a function of parameter , based on observations, , and 10,000 replications.

Chapter 07

Figure 7.1 Comparison of the conditional m.l.e. and the infinite AR representation method of Section 7.3.1 for estimation of ARMA(2,1) parameters (shown as vertical lines in the plots) for observations, based on replications.

Figure 7.2 Comparison of m.l.e. and o.l.s.‐based methods for estimation of ARMA(2,1) parameters (shown as vertical lines in the plots) for observations, based on replications.

Figure 7.3 Same as Figure 7.2 but using observations.

Figure 7.4 Similar to Figures 7.2 and 7.3 , but for the zero‐mean ARMA(1,3) model with , , , , and observations.

Figure 7.5 Histograms of fitted ARMA(1,1) parameters when the true model is ARMA(2,1) with , , , and , for and replications.

Figure 7.6 Simulated time series (solid line) with out‐of‐sample forecasts based on estimated parameters (circles) and based on the true parameters (crosses). The straight dashed (dash‐dot) line is the estimated (exact) regression term.

Figure 7.7 Inverse mean function for various values of (here denoted as ) and for the ARMAX(1,1) model with and observations.

Figure 7.8 Mean bias (top), median bias (middle), and m.s.e. (bottom) of the exact m.l.e. (dotted line) and the MA(1)‐modified median‐unbiased estimator with known (solid line), estimated (dashedline), and (dash‐dot line), based on a model with a constant, observations and normal innovations. The bottom graph is truncated.

Figure 7.9 Percentage reduction in m.s.e. compared to the exact m.l.e. of the modified mean‐adjusted (solid line) and median‐unbiased (dashed line) estimators: From bottom to top, , 0.8, 0.9, 1.

Figure 7.10 Similar to Figure 7.2 , comparing the o.l.s. and i.o.l.s. methods for estimation of ARMA(2,1) parameters for observations, based on replications.

Figure 7.11 Similar to Figure 7.10 , but comparing the o.l.s. and g.l.s. methods for estimation of the ARMA(2,1) parameters for observations, based on replications.

Chapter 08

Figure 8.1 TACF of the AR(1) process with (top) and (bottom).

Figure 8.2 TACF of the stationary AR(3) model with parameters (top left), (top right), (bottom left) and (bottom right).

Figure 8.3 The SACFs of four simulated AR(1) time series with and .

Figure 8.4 The SACFs of four simulated AR(1) time series with and .

Figure 8.5 Sample ACF for a simulated random walk with 200 observations.

Figure 8.6

Top:

The mean SACF for an MA(1) model with and , i.e., the mean of each is shown. Left is with known mean, middle is for an intercept term in the model, i.e., with an matrix consisting of a column of ones, and right is for an intercept and trend model, i.e., with an matrix consisting of a column of ones and the time‐trend vector .

Bottom:

Same but with .

Figure 8.7 Top panel is the standard deviation of , , for an MA(1) process with , intercept term, and four sample sizes , 30, 40, and 50 (top to bottom). The bottom panel is similar but for and , 0.3, 0.6, and .

Figure 8.8 and individual intervals for each , corresponding to an AR(2) model with , , and . Top (bottom) panel is based on ().

Figure 8.9 SACF of four simulated AR(2) time series with , , and (big solid circles) with overlaid bounds for the SACF based on the true parameters.

Figure 8.10 SACF of four simulated AR(2) time series with , , and (big solid circles) with overlaid bounds for the SACF based on the m.l.e. of the parameters.

Figure 8.11 Same as Figure 8.10, using the same four simulated time series, but based on the

wrong model

MA(2).

Figure 8.12 Same as Figures 8.10 and 8.11, using the same four simulated time series, but based on the

wrong model

MA(6).

Figure 8.13 Simulated joint density (based on 500,000 replications) of (‐axis) and (‐axis) with and .

Figure 8.14 Saddlepoint density (8.26) of (‐axis) and (‐axis) for and sample sizes (top) and (bottom).

Figure 8.15 The left graph is saddlepoint density (8.26) of (‐axis) and (‐axis) for , and corresponding to an AR(1) model with . The right graph is the corresponding density based on 1,000,000 simulations of the SACF . The code in Listing 8.7 was used for the simulation, but with simulated AR(1) time series instead of i.i.d. normal sequences, produced from the program

armasim

given in Listing 7.1. In both graphs, the dotted line to the right of the contour plot is the lower endpoint of the support of , based on (8.20). It might happen in the simulated p.d.f. that the density appears to go beyond the allowed support, but this is just an artifact of Matlab's plotting procedure; executing

r1=pair(:,1); r2=pair(:,2); sum(r2 <= (2*r1.*r1‐1))

yields precisely zero, i.e., all SACF pairs are in the support given in (8.17).

Figure 8.16 Density of based on (8.33) (solid), simulation (dashed), and asymptotic (dotted). Top panel is for ; bottom is for . The decline of the kernel density in the left tail is an artifact of the method; further simulation reveals that the true density indeed increases upwards.

Figure 8.17 The conditional p.d.f. of given (top) and (bottom) for an AR(2) model with constant, unknown mean () and parameters and . The solid line is the s.p.a. (8.34) and the dashed line is based on simulation. The Matlab code to generate the plots is developed in Problem 8.4.

Figure 8.18 TPACF of the stationary AR(3) model with parameters (top left), (top right), (bottom left), and (bottom right).

Figure 8.19 The SPACFs of the four simulated AR(1) time series with and as were used in Figure 8.3.

Figure 8.20 The theoretical ACF (top) and PACF (bottom) of the subset AR(5) model with , , and .

Figure 8.21 The theoretical ACF (top) and PACF (bottom) for the MA(1) model with .

Figure 8.22 Similar to Figure 8.15 but based on and an AR(2) model with and . The left graph is based on simulation, the middle graph is the SPA, and the right graph is the asymptotic distribution given in (8.56).

Chapter 09

Figure 9.1 Top panels are the theoretical correlograms corresponding to a stationary and invertible ARMA(2,2) model with parameters , , , , , and . The second row shows a realization of the process, with its sample correlograms plotted in the third row. The last row shows the theoretical ACF and PACF but based on the

estimated

ARMA(2,2) model of the data.

Figure 9.2 An informal graphical test for covariance stationarity is to compute the sample correlograms for non‐overlapping segments of the data.

Figure 9.3 The RSACF (left) and RSPACF (right) for models AR(3) (top), MA(2) (middle), and ARMA(1,1) (bottom).

Figure 9.4 The RSACF (left) and RSPACF (right) for models ARMA(1,2) (top), ARMA(2,2) (middle), and AR(5) (bottom).

Figure 9.5 Empirical distribution of , (left, middle, and right panels), where is the ratio of the m.l.e. of to its corresponding approximate standard error and is the Student's cdf with degrees of freedom. Rows from top to bottom correspond to , , , and , respectively.

Figure 9.6 Similar to the top row of Figure 9.5 but having used m.l.e. standard errors computed from bootstrap iterations.

Figure 9.7 Similar to Figure 9.6 but based on the bootstrapped ‐statistics under the null.

Figure 9.8 QQ plot of 300 simulated values of the statistic (denoted t stat) and the signed likelihood ratio statistic (denoted LRT stat) for testing ARMA(2,1) and ARMA(1,2) when the true model is an ARMA(1,1) with and , based on observations.

Figure 9.9 Same as Figure 9.8 but using observations.

Figure 9.10 Simulation‐based performance of the AICC and BIC criteria (9.3 ) for an AR(1) model as a function of parameter , with and .

Figure 9.11 Simulation‐based performance of AICC (left) and BIC (right) criteria in terms of percentage of under‐selection (top), correct selection (middle), and over‐selection (bottom) for an AR(2) model as a function of parameters (–axis) and (–axis), with and . Legend is, for , dots –, circles –, plus –, star –, square –, diamond –, pentagram –.

Figure 9.12 Same as Figure 9.11 but based on and .

Figure 9.13 Histograms of , based on replications with true data being iid normal, and taking . Top (bottom) panels are for ().

Figure 9.14 Bar plots corresponding to the chosen value of AR lag length , based on the CACF method, a true AR(1) process with parameter , , and , using and .

Figure 9.15 Performance of the various methods in the AR(1) case using , , , , and true AR(1) parameter denoted as in the graphics.

Figure 9.16 Histograms corresponding to the chosen value of AR lag length , based on the CACF, AICC, and BIC, using three values of tuning parameter (2, 6, and 10, from a to c), and replications. True model is Gaussian AR(1) with parameter , sample size , and known mean (no matrix). The CACF method uses .

Figure 9.17 Similar to Figure 9.16 , but for sample size and .

Figure 9.18 Based on the CACF method, bar plots corresponding to the chosen value of AR lag length , in percent, based on replications, using a true AR(4) process with parameters , , , and taking on the six values through , as indicated in the titles of the plots. The sample size is and is given in (9.11 ). The CACF tuning parameters are and as given in (9.10 ).

Figure 9.19 Similar to Figure 9.18 but based on .

Figure 9.20 Similar to Figure 9.10 : Performance of the three indicated AR order selection methods as a function of autoregressive parameter , for sample size , known mean (denoted by ), and , when the true AR order is and (falsely) assuming Gaussianity. The true innovation sequence consists of i.i.d. Student's realizations, with indicated in the titles (from top to bottom, , , , and ). Left (right) panels indicate the percentage of the replications that resulted in choosing ().

Figure 9.21 Same as Figure 9.20 but having used .

Figure 9.22 Histograms, based on replications and sample size , corresponding to the chosen value of AR lag length , for an AR(1) model with parameter , for the CACF, AICC, BIC, and SVD methods, using . The top panels correspond to a pure AR(1) model with no regressor matrix, while the bottom panels use , and the SVD method is applied to the ordinary least squares residuals based on this matrix. From left to right, the three values , , , as the tuning parameter of the SVD method, are used.

Figure 9.23

Top:

Similar to Figure 9.22 , but based on an AR(4) model with , , , and , no matrix, sample size , , and two sets of tuning parameters. The left panels use for the SVD and for the CACF, while right panels use and .

Bottom:

Same but .

Chapter 10

Figure 10.1 The returns , where is the exchange rate at time , of various Asian currencies versus the USD, from 1 January 1990 to 31 December 1998 (with the exception of the Indian rupee, which extends only until 2 July 1998).

Figure 10.2 Annualized volatility during the GFC based on a robustified GARCH model and marked with occurrences of several major events. This graphic, courtesy of Peter Hansen, is Figure 4 in Banulescu et al. (2016 ).

Figure 10.3 Realization of ( 10.1 ) from the code in Listing 10.1.

Figure 10.4 Results of m.l.e. estimation of a normal GARCH(1,1) process with observations, using replications. True parameters are indicated by vertical dashed lines.

Figure 10.5 The profile log‐likelihood (solid) in , for the returns on AT&T, and the local maxima of the log‐likelihood (circles). Left panels correspond to model ( 10.9 ) given below (and indicated with ), while the right panels correspond to model ( 10.2 ) (indicated with ).

Figure 10.6 Three realizations of a simulated ‐IGARCH process with and .

Figure 10.7 Similar to Figure 10.4 , but showing the m.l.e. of a fitted (mis‐specified) normal‐GARCH(1,1), when using a ‐GARCH(1,1) process as the d.g.p., with observations, using replications.

Figure 10.8 Similar to Figure 10.7 , but based on a ‐GARCH(1,1) process, and having used a sample size of observations.

Figure 10.9 The m.l.e. parameter estimates corresponding to the NCT‐GARCH model for the 30 assets of the DJIA, from January 1, 1993 until December 31, 2012, using non‐overlapping windows of length . The circle on the ‐axis indicates the median of the data. The thin vertical lines refer to the average parameter value for each of the 30 assets. (In the lower right panel, the NCT noncentrality parameter is denoted as .)

Figure 10.10 The m.l.e. parameter estimates corresponding to the NCT‐GARCH model for simulated NCT GARCH data, using length and replications. The circle on the ‐axis indicates the median of the data. (In the lower right panel, the NCT noncentrality parameter is denoted as .)

Figure 10.11 Illustration of the effect of varying the APARCH asymmetry parameter on the number of VaR violations (top) and the sum of the predicted log‐likelihood values (bottom). Results are out‐of‐sample ( forecasts) for the period January 4, 1993, to December 31, 2012, obtained from a rolling window exercise with window size . The data set under study is the 20‐year sequence of daily (percentage log) returns of the equally weighted portfolio of DJIA‐30 components (as of April 2013). The dashed lines refer to the NCT‐GARCH(1,1) model and the solid lines to the NCT‐APARCH(1,1) model with being varied. The dotted lines in the left panel depict the expected number of VaR violations at the 1% (lower lines; blue), 2.5% (middle lines; green) and 5% (upper lines; red) significance level, respectively.

Figure 10.12 Histograms of the ‐values of the three tests for stability, based on 957 data sets formed from 33 windows of data using , and 29 stocks comprising the DJIA.

Figure 10.13 Same as Figure 10.12 but having used the 100 largest market‐cap stocks from the S&P500 index, resulting in 4100 data sets and ‐values.

Figure 10.14 Histograms of ‐values of the GR‐L1 (left) and GR‐sup (right) tests, based on 5000 replications for sample size , of i.i.d. normal data (top row); data (second row); absolute value of i.i.d. data (third row); normal‐GARCH data with , , and (fourth row); and absolute value of same normal‐GARCH data (last row).

Figure 10.15

Left:

Sample autocorrelation function (SACF) in lag , , of the squared returns from the 20 years of daily NASDAQ Composite returns from January 1, 1990 to December 31, 2010, overlaid with their theoretical counterparts, given in ( 10.25 ), for the fitted plain GARCH(1,1) and various fitted MixN()‐GARCH(1,1) models.

Right:

Same but based on the 4000 daily NASDAQ Composite returns until March 16, 2011.

Figure 10.16

Left:

Estimate of the non‐parametric mixture weights according to ( 10.28 ) for the two‐component mixture GARCH(1,1) model based on about 10 years (2,500 data points) of NASDAQ Composite returns (April 6, 2011 to March 3, 2011). The chosen weighting function consists of linear functions, each with zero slope and estimated intercept. The first and last are very close to zero. Superimposed is a scaled histogram of the fitted innovations.

Middle:

Same but for (some of estimated weights are essentially zero); ignore the axis label with .

Right:

The fitted sigmoidal mixing weights based on ( 10.26 ) and ( 10.27 ) with , , and , i.e., .

Chapter 11

Figure 11.1 Examples of deviation plots for illustrating the unconditional coverage of VaR predictions. The ‐axis is the VaR level (the tail probability) in percent, with 1, 2.5, and 5 being commonly checked values. The ‐axis shows the deviation, so that a value of zero is ideal. Instead of showing tables of results for several VaR levels, such a graphic is more appealing and contains more information. The graphics are taken from Kuester et al. (2006 ), and pertain to VaR forecasts based on moving windows of 500 observations, from the log percentage returns of the daily closing prices of the NASDAQ composite index, from its inception on February 8, 1971, to June 22, 2001, yielding a total of 7,681 observations. The index itself is a market value‐weighted portfolio of more than 5,000 stocks. The various models depicted are described in detail in Kuester et al. (2006 ).

Figure 11.2 Center black lines are the returns on the equally weighted portfolio constructed from the 2,767 daily returns of components of the DJIA from January 2, 2001, to December 30, 2011 (based on the index composition as of June 8, 2009). Overlaid as colored lines are the associated one‐day‐ahead 1% (top) and 5% (bottom) VaR forecasts, using: (i) one of the non‐Gaussian GARCH COMFORT models (dashed red line), (ii) a non‐Gaussian but i.i.d. model (solid magenta line), and the Gaussian DCC model (solid blue line). Further overlaid are the VaR violations, depicted by + signs on the top and bottom of the graphs, using the same color as corresponds to the lines for the VaR predictions.

Figure 11.3 The (log percentage) returns on Merck & Co. Inc. for the dates indicated (top) and several filtered time series associated with the COMFORT model.

Figure 11.4 Similar to Figure 11.3 , but overlaying the results for all 30 series associated with the DJIA.

Figure 11.5 Scatterplot of the first two out of three portfolio weights, for different sampling schemes.

Figure 11.6

Left:

Cumulative return sequences of the DJIA data using the equally weighted allocation and the Markowitz iid long‐only framework (denoted Mark‐NS), based on moving windows of returns.

Right:

Circles indicate the average, over all the windows, of , where and refer to the analytic (optimized) and UCM‐based portfolio vectors, respectively. This was conducted times per sample size for , and otherwise times. Crosses indicate the average over the ‐values.

Figure 11.7 The same as the left panel of Figure 11.6 , i.e., comparison of cumulative returns for the stocks listed on the DJIA, but with other methods. From top to bottom, the first is the i.i.d. two‐component mixed normal distribution with parameters estimated via the MCD methodology, from Gambacciani and Paolella (2017 ) (in the color version, in purple). This is followed by eight runs of the UCM2 method based on 900 replications (black lines), the equally weighted method (red line), Markowitz (no short selling) based on the i.i.d. assumption (green line), and Markowitz (no short selling) but using the Gaussian DCC‐GARCH model for computing the expected returns and their covariance matrix (blue line).

Figure 11.8 Similar to Figure 11.7 but using (i) a modified version of the mixed normal MCD method such that a signal, based on information up to time , is used to determine if trading should take place at time or not, and (ii) a new variant of the COMFORT method discussed in Section 11.2.4 .

Figure 11.9

Top:

The ES span, , based on the UCM NCT‐APARCH(1,1) model of Section 11.3.4 (denoted in the title as ), shown as a histogram from 100,000 random portfolio replications drawn uniformly from the simplex via ( 11.46 ), for corresponding to the 252 trading days of years 2005 (left) and 2008 (right). Its minimum value is denoted by the (orange) line “ES span minimum”, while the minimum ES obtained by constrained optimization (

fmincon

; repeated 1,000 times because of non‐differentiability of the objective function) is indicated by the (red) line “min‐ES”. The (green) line “Markowitz” indicates the ES corresponding to the minimum variance portfolio allowing for short selling (i.e., negative portfolio weights). The short vertical (black) lines indicate the ES corresponding to putting a weight of one on a single asset (and the rest zero). The ‐axis was truncated on the right to improve readability, so that some (or all, in the case of the lower left panel) of the ES values corresponding to individual assets are not shown.

Bottom:

Same as top, but based on the i.i.d. discrete two‐component multivariate normal mixture model, as discussed in Chapter , fit via maximum likelihood with shrinkage (denoted in the title as ).

Chapter 12

Figure 12.1

Left:

The sorted values of the estimated tail thickness parameters (degrees of freedom) of the noncentral Student's distribution, for the 30 daily stock return series comprising the DJIA index, along with approximate 95% confidence intervals obtained via the non‐parametric bootstrap with replications.

Right:

The same as the left panel, but for the noncentrality parameter .

The ordering is the same as in the left panel

, thus allowing a comparison.

Figure 12.2 Similar to Figure 12.1 , but having used the NCT‐GARCH(1,1) model. The ‐axis of the left graphic is truncated for readability.

Figure 12.3 Daily percentage log returns on Bank of America (top) and Wal‐Mart (bottom).

Figure 12.4

Top:

Scatterplot of the returns on Bank of America and Wal‐Mart for the observations.

Bottom:

Scatterplot, now with truncated and equal axes, and omitting points near the center, with an overlaid contour plot of the fitted multivariate Student's density.

Figure 12.5 Bivariate contour plots of three MVNCT densities.

Figure 12.6

Top left:

The bivariate Jones (2002 ) distribution 12.9 for and degrees of freedom.

Top right:

Same, but its generalization 12.14 with and (and no noncentrality).

Bottom:

Similar to top, but additionally introduce asymmetry via noncentrality parameters and .

Figure 12.7 Contour plots of 12.16 (Shaw and Lee, 2008 , Eq. 4.10) and 12.19 (Shaw and Lee, 2008 , Eq. 4.18) for , , and . Compare to the top right panel of Figure 12.6 .

Figure 12.8 Examples of the bivariate FaK distribution 12.23 , each with zero location and unit scale, and degrees of freedom parameters given in the title (writing for parameter ).

Figure 12.9 Examples of the bivariate FaK distribution 12.23 based on simulation, each with zero location, unit scale, and zero correlation. The case with in the bottom left (in the graphics titles, having used instead of ) appears the most appropriate for financial returns data.

Figure 12.10 Examples of the bivariate AFaK (asymmetric FaK) distribution 12.25 , each with zero location and unit scale (and using instead of ).

Figure 12.11

Top:

Kernel density plots of based on 500 replications and observations using the two‐step method of estimating the parameters of the FaK model, with , indicated in the graphs, and two choices of , zero and .

Bottom:

Same, but having used full maximum likelihood estimation via the direct method, and plotted with a vertical offset because

they are otherwise graphically identical

; see the text for explanation.

Figure 12.12 Top row shows the usual MVT 12.3 with degrees of freedom, zero mean vector, , and two values of , zero (left) and 0.5 (right). The middle and last rows show the SMESTI distribution with and , , respectively (same and as first row).

Figure 12.13 Similar to the bottom four panels of Figure 12.12, but for the MESTI distribution, with and .

Figure 12.14 Estimated degrees of freedom parameter from a fitted NCT‐APARCH model using the fast estimation method discussed in Section 10.4, and based on the 10 years of daily returns data of the S&P500 from January 2004 to May 2014.

Figure 12.15 Same as Figure 12.14 but based on only the last 500 observations (two years of data).

Figure 12.16

Top:

Boxplots of the estimated degrees of freedom parameters based on division by financial sector, having used the entire 10 year data period.

Bottom:

Same, but based on only the last 500 observations (two years of data).

Figure 12.17 Similar to Figure 12.14 , but showing the estimated noncentrality (asymmetry) parameter from a fitted NCT‐APARCH model, based on the 10 years of daily returns data.

Figure 12.18 Same as Figure 12.17 but based on only the last 500 observations (two years of data). Note that some points might be missing due to the size of the ‐axis, chosen to be the same as in Figure 12.17 .

Figure 12.19 Comparison of five methods of estimating ES for a sequence of 100 rolling windows, using the equally weighted portfolio based on the constituents (as of April 2013) of the Dow Jones Industrial Average index (Wharton/CRSP), with starting dates August 8, 2012 to December 31, 2012. These are plotted as a function of time, and based on the equally weighted portfolio.

Figure 12.20

Upper left:

Percentage log returns of the equally weighted portfolio.

Mid and lower left:

Boxplots of 1% ES values obtained from simulations based on draws from the fitted copula for different non‐overlapping rolling windows of size , spanning January 4, 1993, to December 31, 2012. Timestamps denote the most recent date included in a data window. All values are obtained via the NCT estimator.

Upper right:

Boxplots of 1% ES values sorted in descending order by the average ES value, overlayed by the average of the estimated degrees of freedom parameters.

Mid right:

ES variances in log scale across rolling windows for different samples sizes , sorted by the average ES value per window.

Lower right:

Linear approximation of the above panel, overlayed by the linear approximation of the estimated degrees of freedom, based on .

Figure 12.21

Top:

Illustration of the discrepancy between the approximation of (obtained as the off‐diagonal term ) and the true value (obtained by bivariate numeric integration), as a function of , where , with , , varies along the ‐axis, and , with and specified in the legend of the plots.

Middle and bottom:

Same, but with using the AFaK distribution with but nonzero and .

Figure 12.22 Same as top two panels of Figure 12.21 but based on 12.53 and 12.54 (and using instead of in the legend).

Figure 12.23

Top:

Similar to Figure 12.22 but for , , a fixed value of of , and as a function of . The vertical dashed line indicates the case with , which agrees with the corresponding point in the bottom panel of Figure 12.22 (right‐most point of the dashed line).

Bottom:

Same, but for , . (Note that notation instead of is used in the titles, sparing the lazy author a re‐computation of the graphics.)

Chapter 13

Figure 13.1 (a) Estimates of using the weighted sample correlation estimator, as a function of weighting parameter , for bivariate normal data generated as , , where is a correlation matrix with single parameter , so that the correlation is varying linearly through time, from zero to 0.5. (b) Same as (a), again using the weighted sample correlation estimator, but for bivariate FaK data with and the same correlation structure. (c) Same as (b), but estimation is based on the weighted m.l.e.

Figure 13.2 Density forecast measure ( 13.3 ) over a grid of values from ( 13.5 ) and ‐values from ( 13.1 ), for the FaK model, using sample correlation.

Figure 13.3 Density forecasting performance measure ( 13.3 ) as a function of degrees of freedom shrinkage parameter , for the i.i.d. FaK model applied to the usual DJIA data, using the two forms of estimating correlation matrix , and with fixed and .

Figure 13.4 Cumulative returns of the equally weighted, Markowitz, and MVT models, the latter using the true parameter values and simulation based on samples to obtain the optimal portfolio. The thinner, dashed (red) line uses instead of (thicker, solid, red line). In all but the top left case, use of is at least as good as and in some cases, such as the last four panels, leads to substantially better results.

Figure 13.5 Similar to Figure 13.4 , but based on the FaK model, using the true parameter values. All plots were truncated in order to have the same ‐axis.

Figure 13.6 Performance comparison using the same four data sets as in Figure 13.5 , and having estimated the FaK parameters.

Figure 13.7 Similar to Figure 13.6 , with estimated parameters and using , but having used the alternative investment strategy based on choosing among the 10% of generated portfolios with the lowest ES the one with the highest expected return.

Figure 13.8 Cumulative returns on portfolios of the 30 stocks in the DJIA index, using the FaK model with the alternative investment strategy, the allocation, Markowitz (no short selling), and 400 random portfolios (showing only the most extreme ones to enhance graphic readability).

Chapter 14

Figure 14.1 Examples of scatterplots between pairs of stock return series (top) and their corresponding contour plots of the fitted distribution (bottom). In the scatterplots, the smaller (larger) dots correspond to the points assigned to the first (second) component, as determined by the approximate split discussed in Section 14.2.1 .

Figure 14.2 The estimated means (top), 30 variances (middle), and 435 correlations (bottom) for the first (left) and second (right) components of the normal mixture corresponding to the DJIA‐30 data set under study. Solid (dashed) vertical lines show the mean (median).

Figure 14.3 Estimation accuracy, as a function of prior strength parameter , measured as four divisions of from (14.9 ) ( is ignored), based on simulation with 10,000 replications and , of the parameters of the model, using as true parameters the m.l.e. of the DJIA‐30 data set.

Figure 14.4 Final values of returned from the EM algorithm based on the model, applied to the DJIA‐30 data set.

Figure 14.5

Left:

Final values of returned from the EM algorithm based on the model for a simulated set of multivariate normal data with , , using a mean and covariance equal to the sample mean and covariance from the DJIA‐30 data set.

Right:

Same, but having used a multivariate Laplace distribution with .

Figure 14.6 Truncated boxplots of the fitted GA parameters of the return series in the first (left) and second (right) component. Parameter has nothing to do with our use of for the dimension of the data, 30 in our case.

Figure 14.7 The first (left) and second (right) components of the McDonald's stock returns, with unrestricted and restricted GA densities, without and with outlier removal.

Figure 14.8

Left:

The traditional Mahalanobis distances computed for the DJIA‐30 returns, with 15% of the observations above the cutoff line.

Right:

Similar, but having used the robust Mahalanobis distance based on the mean vector and covariance matrix from the m.c.d. method, resulting in 33% above.

Figure 14.9

Left:

The traditional Mahalanobis distances computed for the observations in the first component, based on the EM‐split of the DJIA‐30 data.

Right:

Similar, but having used the robust Mahalanobis distance based on the mean vector and covariance matrix from the m.c.d. method.

Figure 14.10 Similar to Figure 14.9 but for the second component.

Figure 14.11

Left:

Plot of the standardized cusum where is given in (14.21 ), versus , for and several .

Right:

The normalized sum of the realized predictive log likelihood (14.22 ) as a function of prior strength hyper‐parameter , and based on estimation with a moving window of length (solid line) and (dashed line). For the latter, is still 250, and we use the convention in (14.23 ). The star shows the best obtained value, corresponding to a prior weight of , and is the same star in both panels of Figure 14.16 , while the top‐most horizontal line is the same line in the right panel of Figure 14.16 , showing the additional improvement from the methods discussed in Section 14.2.4 .

Figure 14.12 Both panels show the normalized sum of the realized predictive log‐likelihood ( 14.22 ) as a function of moving window size , , for three values of prior strength hyper‐parameter . The left uses , while the right uses . In the left panel, the plot for (the one at the top) has the same shape as the corresponding one in the right panel when the plot is magnified, with its maximum also at .

Figure 14.13 Similar to the right panel of Figure 14.11 , except that here the left panel shows the results for only assets (and four values of ), while the right is for . Observe that for all four window sizes, so that the density predictions were based on precisely the same data points, namely to .

Figure 14.14 The optimal value of (left) and the corresponding values of the attained normalized sum of the realized predictive log‐likelihood ( 14.22 ) (right), for various subsets of the DJ‐30 assets under study.

Figure 14.15

Left:

Overlays same plot in the right panel of Figure 14.14 , and additionally shows, as crosses, the result when taking to be from the regression line depicted in the left panel of Figure 14.14 , i.e., , where and . The resulting values based on and are virtually identical, except for the case.

Right:

Same as left, but based on a fixed value of .

Figure 14.16

Left:

Normalized sum of the realized predictive log‐likelihood, for and shrinkage hyper‐parameter , as a function of hyper‐parameter , which controls the shape of the weights used in the weighted likelihood calculation. The star at the bottom right of the left and right panels is the same star shown in the right panel of Figure 14.11 .

Right:

Normalized sum of the realized predictive log‐likelihood, for and shrinkage hyper‐parameter , as a function of , which dictates how many of the latest values of are averaged to form the value of . The big circle in both plots represents the same value. Both plots have the same ‐axis range, so that it is easy to see the improvement in using with weighted likelihood for applied just to the , compared to using with weighted likelihood applied to all model parameters, including . Finally, the horizontal line at the top of the graph is the result of taking to be .

Figure 14.17 Similar to Figure 14.16 but using assets instead of 30. Top is window size and (as ascertained from the left panel of Figure 14.13 ); bottom is window size and . For both window sizes, so that the results are directly comparable.

Figure 14.18

Left:

Overlays same plot in the right panel of Figure 14.14 , and additionally shows, as crosses, the result when taking to be from the regression line from the left panel of Figure 14.14 and additionally with (i) weighted likelihood, with and just applied to the , and (ii) a moving average of from ( 14.26 ), i.e., .

Right:

Same as left, except that instead of the distribution, we use the distribution (14.41 ), introduced in Section 14.5 .

Figure 14.19

Left:

The normalized sum of the realized predictive log‐likelihood ( 14.22 ) for the based on the 2,208 daily returns for returns on the SMI stocks as a function of shrinkage hyper‐parameter , for four values of weighted likelihood parameter (applied just to the ), and based on moving windows of length , with . The left uses a moving average of the estimated component weight from ( 14.25 ) with .

Right:

Similar, but uses ( 14.26 ).

Figure 14.20 Top panels parallel those in Figure 14.14 , but using the SMI‐20 data (with the two points corresponding to 15 assets refer to stocks 1 through 15, and 6 to 20, so that they do contain overlap). The bottom panels are similar to those in Figure 14.18 , showing the incremental improvement by using weighted likelihood and moving averages of (left) and by using the mixture Laplace distribution (right).

Figure 14.21

Left:

Similar to the right panel of Figure 14.11 , but such that the density forecasts were formed via ( 14.27 ). Note the difference in the ‐axis compared to the right panel of Figure 14.11 : Use of ( 14.27 ) and fitting two separate MVN distributions performs comparatively poorly.

Right:

Same, but based on the m.c.d. method of separation (14.29 ), for different values of tuning parameter , with overlaid graph from the right panel of Figure 14.11 , showing that use of the (prior‐augmented) m.l.e. via the EM algorithm results in nearly the same as with use of the m.c.d. split method. In fact, the latter, at all three cutoff values, is slightly better for .

Figure 14.22 Examples of the bivariate Laplace distribution; top left is normal, for comparison. The contour lines have the same height across plots.

Figure 14.23 Bivariate distributions as products of i.i.d. univariate Laplace with scale .

Figure 14.24

Left:

The normalized sum of the realized predictive log‐likelihood versus , based on the estimated via m.c.d. split and prior‐augmented m.l.e. via the EM algorithm for each separate Laplace component, using a moving window of length , and two sets of fixed