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# gjr

GJR conditional variance time series model

## Description

Use gjr to specify a univariate Glosten, Jagannathan, and Runkle (GJR) model. The gjr function returns a gjr object specifying the functional form of a GJR(P,Q) model, and stores its parameters values.

The key components of a gjr model include:

• The GARCH polynomial, which is composed of lagged conditional variances. The degree is denoted P.

• The ARCH polynomial, which is composed of the lagged squared innovations.

• The leverage polynomial, which is composed of lagged squared, negative innovations.

• Q is the maximum of the ARCH and leverage polynomial degrees.

P and Q are the maximal nonzero lags in the GARCH, and ARCH and leverage polynomials, respectively. Other model components include the innovation mean model offset, a conditional variance model constant, and innovations distribution.

All coefficients are unknown (NaN values) and estimable unless you specify their values using name-value pair argument syntax. To estimate models containing all or partially unknown parameter values given data, use estimate. For completely specified models (models in which all parameter values are known), simulate or forecast responses using simulate or forecast, respectively.

## Creation

### Syntax

Mdl = gjr
Mdl = gjr(P,Q)
Mdl = gjr(Name,Value)

### Description

example

Mdl = gjr returns a zero-degree conditional variance gjr object.

example

Mdl = gjr(P,Q) returns a gjr object that has a GARCH polynomial with a degree of P, and ARCH and leverage polynomials both with a degree of Q. All polynomials contain all consecutive lags from 1 through their degrees, and all coefficients are NaN values.

This shorthand syntax allows for easy model template creation in which you specify the polynomial degrees explicitly. The model template is suited for unrestricted parameter estimation, that is, estimation without any parameter equality constraints. However, after you create a model, you can alter property values using dot notation.

example

Mdl = gjr(Name,Value) sets properties or additional options using name-value pairs. You can specify multiple values. Enclose each name in single quotes. For example, 'ARCHLags',[1 4],'ARCH',{0.2 0.3} specifies the two ARCH coefficients in ARCH at lags 1 and 4.

This longhand syntax enables you to create more flexible models.

### Input Arguments

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The shorthand syntax provides an easy way for you to create model templates that are suitable for unrestricted parameter estimation. For example, to create a GJR(1,2) model containing unknown parameter values, enter:

Mdl = gjr(1,2);
To impose equality constraints on parameter values during estimation, set the appropriate property values using dot notation.

GARCH polynomial degree, specified as a nonnegative integer. In the GARCH polynomial and at time t, MATLAB® includes all consecutive conditional variance terms from lag t – 1 through lag tP.

You can specify this argument using the gjr(P,Q) shorthand syntax only.

If P > 0, then you must specify Q as a positive integer.

Example: gjr(1,1)

Data Types: double

ARCH polynomial degree, specified as a nonnegative integer. In the ARCH polynomial and at time t, MATLAB includes all consecutive squared innovation terms (for the ARCH polynomial) and squared, negative innovation terms (for the leverage polynomial) from lag t – 1 through lag tQ.

You can specify this argument using the gjr(P,Q) shorthand syntax only.

If P > 0, then you must specify Q as a positive integer.

Example: gjr(1,1)

Data Types: double

#### Name-Value Pair Arguments

Specify optional comma-separated pairs of Name,Value arguments. Name is the argument name and Value is the corresponding value. Name must appear inside quotes. You can specify several name and value pair arguments in any order as Name1,Value1,...,NameN,ValueN.

The longhand syntax enables you to create models in which some or all coefficients are known. During estimation, estimate imposes equality constraints on any known parameters.

Example: 'ARCHLags',[1 4],'ARCH',{NaN NaN} specifies an GJR(0,4) model and unknown, but nonzero ARCH coefficient matrices at lags 1 and 4.

GARCH polynomial lags, specified as the comma-separated pair consisting of 'GARCHLags' and a numeric vector of unique positive integers.

GARCHLags(j) is the lag corresponding to the coefficient GARCH{j}. The lengths of GARCHLags and GARCH must be equal.

Assuming all GARCH coefficients (specified by the GARCH property) are positive or NaN values, max(GARCHLags) determines the value of the P property.

Example: 'GARCHLags',[1 4]

Data Types: double

ARCH polynomial lags, specified as the comma-separated pair consisting of 'ARCHLags' and a numeric vector of unique positive integers.

ARCHLags(j) is the lag corresponding to the coefficient ARCH{j}. The lengths of ARCHLags and ARCH must be equal.

Assuming all ARCH and leverage coefficients (specified by the ARCH and Leverage properties) are positive or NaN values, max([ARCHLags LeverageLags]) determines the value of the Q property.

Example: 'ARCHLags',[1 4]

Data Types: double

Leverage polynomial lags, specified as the comma-separated pair consisting of 'LeverageLags' and a numeric vector of positive integers.

LeverageLags(j) is the lag corresponding to the coefficient Leverage{j}. The lengths of LeverageLags and Leverage must be equal.

Assuming all ARCH and leverage coefficients (specified by the ARCH and Leverage properties) are positive or NaN values, max([ARCHLags LeverageLags]) determines the value of the Q property.

Example: 'LeverageLags',1:4

Data Types: double

Conditional probability distribution of the innovation process, specified as the comma-separated pair consisting of 'Distribution' and a value in this table.

DistributionValueStructure Array
Gaussian'Gaussian'struct('Name','Gaussian')
Student’s t
 't' By default, DoF is NaN.
 struct('Name','t','DoF',DoF) DoF > 2 or DoF = NaN

The t distribution degrees of freedom parameter is estimable. That is, if you want estimate to estimate it along with all other unknown parameters, then set its value to NaN.

Example: 'Distribution',struct('Name','t','DoF',10)

Data Types: char | struct

## Properties

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You can set writable properties when you create an gjr object by using name-value pair argument syntax. Or, after you create a model, you can use dot notation. For example, to create a GJR(1,1) model with unknown coefficients, and then specify a t innovation distribution with unknown degrees of freedom, enter:

Mdl = gjr('GARCHLags',1,'ARCHLags',1);
Mdl.Distribution = "t";

This property is read-only.

GARCH polynomial degree, specified as a nonnegative integer. P is the maximum lag in the GARCH polynomial with a coefficient that is positive or NaN. Lags that are less than P can have coefficients equal to 0.

P specifies the minimum number of presample conditional variances required to initialize the model.

If you use name-value pair arguments to create the model, then MATLAB implements one of these alternatives (assuming the coefficient of the largest lag is positive or NaN):

• If you specify GARCH, then P is the number of elements therein.

• If you specify GARCHLags, then P is the largest lag therein. This alternative takes precedence over the others.

• Otherwise, P is 0.

Data Types: double

This property is read-only.

Maximal degree of ARCH and leverage polynomials, specified as a nonnegative integer. Q is the maximum lag among the ARCH and leverage polynomials in the model. In either polynomial, lags that are less than Q can have coefficients equal to 0.

Q specifies the minimum number of presample innovations required to initiate the model.

If you use name-value pair arguments to create the model, then MATLAB implements one of these alternatives (assuming the coefficients of the largest lags in the ARCH and leverage polynomials are positive or NaN):

• If you specify ARCH or Leverage, then Q is the maximum number of elements between the two specifications.

• If you specify ARCHLags or LeverageLags, then Q is the maximum element between the two specifications. This alternative takes precedence over the others.

• Otherwise, Q is 0.

Data Types: double

Conditional variance model constant, specified as a positive scalar or NaN value.

Data Types: double

GARCH polynomial coefficients, specified as a cell vector of positive scalars or NaN values.

• If you set the GARCHLags name-value pair argument, then the following conditions apply.

• The lengths of GARCH and GARCHLags are equal.

• GARCH{j} is the coefficient of lag GARCHLags(j).

• By default, GARCH is a numel(GARCHLags)-by-1 cell vector of NaN values.

• Otherwise, the following conditions apply.

• The length of GARCH is P.

• GARCH{j} is the coefficient of lag j.

• By default, GARCH is a P-by-1 cell vector of NaN values.

Data Types: cell

ARCH polynomial coefficients, specified as a cell vector of positive scalars or NaN values.

• If you set the ARCHLags name-value pair argument, then the following conditions apply.

• The lengths of ARCH and ARCHLags are equal.

• ARCH{j} is the coefficient of lag ARCHLags(j).

• By default, ARCH is a Q-by-1 cell vector of NaN values. For more details, see the Q property.

• Otherwise, the following conditions apply.

• The length of ARCH is Q.

• ARCH{j} is the coefficient of lag j.

• By default, ARCH is a Q-by-1 cell vector of NaN values.

Data Types: cell

Leverage polynomial coefficients, specified as a cell vector of numeric scalars or NaN values.

• If you set the LeverageLags name-value pair argument, then the following conditions apply.

• The lengths of Leverage and LeverageLags are equal.

• Leverage{j} is the coefficient of lag LeverageLags(j).

• By default, Leverage is a Q-by-1 cell vector of NaN values. For more details, see the Q property.

• Otherwise, the following conditions apply.

• The length of Leverage is Q.

• Leverage{j} is the coefficient of lag j.

• By default, Leverage is a Q-by-1 cell vector of NaN values.

Data Types: cell

This property is read-only.

The model unconditional variance, specified as a positive scalar.

The unconditional variance is

${\sigma }_{\epsilon }^{2}=\frac{\kappa }{\left(1-{\sum }_{i=1}^{P}{\gamma }_{i}-{\sum }_{j=1}^{Q}{\alpha }_{j}-\frac{1}{2}{\sum }_{j=1}^{Q}{\xi }_{j}\right)}.$

κ is the conditional variance model constant (Constant).

Data Types: double

Innovation mean model offset, or additive constant, specified as a numeric scalar or NaN value.

Data Types: double

Conditional probability distribution of the innovations process, specified as a structure array.

The Name field stores the name of the distribution, either "Gaussian" for the Gaussian distribution or "t" for the t distribution.

If Name is "t", then Distribution also contains the DoF field, which stores the t-distribution degrees of freedom.

By default, Distribution is struct('Name',"Gaussian"). When you create the object, if you specify that the underlying innovation process has a t distribution by using the Distribution name-value pair argument, then the DoF field is NaN by default.

Data Types: struct

Model description, specified as a string scalar or character vector. By default, this property describes the parametric form of the model, for example, "GJR(1,1) Conditional Variance Model (Gaussian Distribution)".

Data Types: string | char

### Note

• All NaN-valued model parameters, which include coefficients and the t-innovation-distribution degrees of freedom (if present), are estimable. That is, when you pass the resulting gjr object and data to estimate, MATLAB estimates all NaN-valued parameters. During estimation, estimate treats known parameters as equality constraints, that is,estimate holds any known parameters fixed at their values.

• All GARCH, ARCH, and Leverage coefficients are subject to a near-zero tolerance exclusion test. That is, the software:

1. Creates lag operator polynomials for each of the GARCH and ARCH components.

2. Compares each coefficient to the default lag operator zero tolerance, 1e-12.

3. Includes a coefficient in the model if its magnitude is greater than 1e-12, and excludes the coefficient otherwise. In other words, the software considers excluded coefficients to be sufficiently close to zero.

For details, see LagOp.

• Typically, the lags in the ARCH and leverage polynomials are the same, but their equality is not a requirement. Differing polynomials occur when:

• One of ARCH{Q} or Leverage{Q} meets the near-zero exclusion tolerance. In this case, MATLAB excludes the corresponding lag from the polynomial.

• You specify polynomials of differing lengths by setting the ARCHLags and LeverageLags name-value pair arguments, or by setting the ARCH or Leverage properties.

In either case, Q is the maximum lag between the two polynomials.

## Object Functions

 estimate Fit conditional variance model to data filter Filter disturbances through conditional variance model forecast Forecast conditional variances from conditional variance models infer Infer conditional variances of conditional variance models simulate Monte Carlo simulation of conditional variance models summarize Display estimation results of conditional variance model

## Examples

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Create a default gjr model object and specify its parameter values using dot notation.

Create a GJR(0,0) model.

Mdl = gjr
Mdl =
gjr with properties:

Description: "GJR(0,0) Conditional Variance Model (Gaussian Distribution)"
Distribution: Name = "Gaussian"
P: 0
Q: 0
Constant: NaN
GARCH: {}
ARCH: {}
Leverage: {}
Offset: 0

Mdl is a gjr model object. It contains an unknown constant, its offset is 0, and the innovation distribution is 'Gaussian'. The model does not have GARCH, ARCH, or leverage polynomials.

Specify two unknown ARCH and leverage coefficients for lags one and two using dot notation.

Mdl.ARCH = {NaN NaN};
Mdl.Leverage = {NaN NaN};
Mdl
Mdl =
gjr with properties:

Description: "GJR(0,2) Conditional Variance Model (Gaussian Distribution)"
Distribution: Name = "Gaussian"
P: 0
Q: 2
Constant: NaN
GARCH: {}
ARCH: {NaN NaN} at lags [1 2]
Leverage: {NaN NaN} at lags [1 2]
Offset: 0

The Q, ARCH, and Leverage properties update to 2, {NaN NaN}, and {NaN NaN}, respectively. The two ARCH and leverage coefficients are associated with lags 1 and 2.

Create a gjr model object using the shorthand notation gjr(P,Q), where P is the degree of the GARCH polynomial and Q is the degree of the ARCH and leverage polynomials.

Create an GJR(3,2) model.

Mdl = gjr(3,2)
Mdl =
gjr with properties:

Description: "GJR(3,2) Conditional Variance Model (Gaussian Distribution)"
Distribution: Name = "Gaussian"
P: 3
Q: 2
Constant: NaN
GARCH: {NaN NaN NaN} at lags [1 2 3]
ARCH: {NaN NaN} at lags [1 2]
Leverage: {NaN NaN} at lags [1 2]
Offset: 0

Mdl is a gjr model object. All properties of Mdl, except P, Q, and Distribution, are NaN values. By default, the software:

• Includes a conditional variance model constant

• Excludes a conditional mean model offset (i.e., the offset is 0)

• Includes all lag terms in the GARCH polynomial up to lags P

• Includes all lag terms in the ARCH and leverage polynomials up to lag Q

Mdl specifies only the functional form of a GJR model. Because it contains unknown parameter values, you can pass Mdl and time-series data to estimate to estimate the parameters.

Create a gjr model using name-value pair arguments.

Specify a GJR(1,1) model.

Mdl = gjr('GARCHLags',1,'ARCHLags',1,'LeverageLags',1)
Mdl =
gjr with properties:

Description: "GJR(1,1) Conditional Variance Model (Gaussian Distribution)"
Distribution: Name = "Gaussian"
P: 1
Q: 1
Constant: NaN
GARCH: {NaN} at lag [1]
ARCH: {NaN} at lag [1]
Leverage: {NaN} at lag [1]
Offset: 0

Mdl is a gjr model object. The software sets all parameters to NaN, except P, Q, Distribution, and Offset (which is 0 by default).

Since Mdl contains NaN values, Mdl is only appropriate for estimation only. Pass Mdl and time-series data to estimate.

Create a GJR(1,1) model with mean offset

where

and is an independent and identically distributed standard Gaussian process.

Mdl = gjr('Constant',0.0001,'GARCH',0.35,...
'ARCH',0.1,'Offset',0.5,'Leverage',{0.03 0 0.01})
Mdl =
gjr with properties:

Description: "GJR(1,3) Conditional Variance Model with Offset (Gaussian Distribution)"
Distribution: Name = "Gaussian"
P: 1
Q: 3
Constant: 0.0001
GARCH: {0.35} at lag [1]
ARCH: {0.1} at lag [1]
Leverage: {0.03 0.01} at lags [1 3]
Offset: 0.5

gjr assigns default values to any properties you do not specify with name-value pair arguments. An alternative way to specify the leverage component is 'Leverage',{0.03 0.01},'LeverageLags',[1 3].

Access the properties of a gjr model object using dot notation.

Create a gjr model object.

Mdl = gjr(3,2)
Mdl =
gjr with properties:

Description: "GJR(3,2) Conditional Variance Model (Gaussian Distribution)"
Distribution: Name = "Gaussian"
P: 3
Q: 2
Constant: NaN
GARCH: {NaN NaN NaN} at lags [1 2 3]
ARCH: {NaN NaN} at lags [1 2]
Leverage: {NaN NaN} at lags [1 2]
Offset: 0

Remove the second GARCH term from the model. That is, specify that the GARCH coefficient of the second lagged conditional variance is 0.

Mdl.GARCH{2} = 0
Mdl =
gjr with properties:

Description: "GJR(3,2) Conditional Variance Model (Gaussian Distribution)"
Distribution: Name = "Gaussian"
P: 3
Q: 2
Constant: NaN
GARCH: {NaN NaN} at lags [1 3]
ARCH: {NaN NaN} at lags [1 2]
Leverage: {NaN NaN} at lags [1 2]
Offset: 0

The GARCH polynomial has two unknown parameters corresponding to lags 1 and 3.

Display the distribution of the disturbances.

Mdl.Distribution
ans = struct with fields:
Name: "Gaussian"

The disturbances are Gaussian with mean 0 and variance 1.

Specify that the underlying disturbances have a t distribution with five degrees of freedom.

Mdl.Distribution = struct('Name','t','DoF',5)
Mdl =
gjr with properties:

Description: "GJR(3,2) Conditional Variance Model (t Distribution)"
Distribution: Name = "t", DoF = 5
P: 3
Q: 2
Constant: NaN
GARCH: {NaN NaN} at lags [1 3]
ARCH: {NaN NaN} at lags [1 2]
Leverage: {NaN NaN} at lags [1 2]
Offset: 0

Specify that the ARCH coefficients are 0.2 for the first lag and 0.1 for the second lag.

Mdl.ARCH = {0.2 0.1}
Mdl =
gjr with properties:

Description: "GJR(3,2) Conditional Variance Model (t Distribution)"
Distribution: Name = "t", DoF = 5
P: 3
Q: 2
Constant: NaN
GARCH: {NaN NaN} at lags [1 3]
ARCH: {0.2 0.1} at lags [1 2]
Leverage: {NaN NaN} at lags [1 2]
Offset: 0

To estimate the remaining parameters, you can pass Mdl and your data to estimate and use the specified parameters as equality constraints. Or, you can specify the rest of the parameter values, and then simulate or forecast conditional variances from the GARCH model by passing the fully specified model to simulate or forecast, respectively.

Fit a GJR model to an annual time series of stock price index returns from 1861-1970.

Load the Nelson-Plosser data set. Convert the yearly stock price indices (SP) to returns. Plot the returns.

load Data_NelsonPlosser;
sp = price2ret(DataTable.SP);

figure;
plot(dates(2:end),sp);
hold on;
plot([dates(2) dates(end)],[0 0],'r:'); % Plot y = 0
hold off;
title('Returns');
ylabel('Return (%)');
xlabel('Year');
axis tight;

The return series does not seem to have a conditional mean offset, and seems to exhibit volatility clustering. That is, the variability is smaller for earlier years than it is for later years. For this example, assume that an GJR(1,1) model is appropriate for this series.

Create a GJR(1,1) model. The conditional mean offset is zero by default. The software includes a conditional variance model constant by default.

Mdl = gjr('GARCHLags',1,'ARCHLags',1,'LeverageLags',1);

Fit the GJR(1,1) model to the data.

EstMdl = estimate(Mdl,sp);

GJR(1,1) Conditional Variance Model (Gaussian Distribution):

Value      StandardError    TStatistic     PValue
_________    _____________    __________    ________

Constant       0.0045728      0.0044199        1.0346       0.30086
GARCH{1}         0.55808           0.24        2.3253      0.020057
ARCH{1}          0.20461        0.17886         1.144       0.25263
Leverage{1}      0.18066        0.26802       0.67406       0.50027

EstMdl is a fully specified gjr model object. That is, it does not contain NaN values. You can assess the adequacy of the model by generating residuals using infer, and then analyzing them.

To simulate conditional variances or responses, pass EstMdl to simulate.

To forecast innovations, pass EstMdl to forecast.

Simulate conditional variance or response paths from a fully specified gjr model object. That is, simulate from an estimated gjr model or a known gjr model in which you specify all parameter values.

Load the Nelson-Plosser data set. Convert the yearly stock price indices to returns.

load Data_NelsonPlosser;
sp = price2ret(DataTable.SP);

Create a GJR(1,1) model. Fit the model to the return series.

Mdl = gjr(1,1);
EstMdl = estimate(Mdl,sp);

GJR(1,1) Conditional Variance Model (Gaussian Distribution):

Value      StandardError    TStatistic     PValue
_________    _____________    __________    ________

Constant       0.0045728      0.0044199        1.0346       0.30086
GARCH{1}         0.55808           0.24        2.3253      0.020057
ARCH{1}          0.20461        0.17886         1.144       0.25263
Leverage{1}      0.18066        0.26802       0.67406       0.50027

Simulate 100 paths of conditional variances and responses from the estimated GJR model.

numObs = numel(sp); % Sample size (T)
numPaths = 100;     % Number of paths to simulate
rng(1);             % For reproducibility
[VSim,YSim] = simulate(EstMdl,numObs,'NumPaths',numPaths);

VSim and YSim are T-by- numPaths matrices. Rows correspond to a sample period, and columns correspond to a simulated path.

Plot the average and the 97.5% and 2.5% percentiles of the simulated paths. Compare the simulation statistics to the original data.

dates = dates(2:end);
VSimBar = mean(VSim,2);
VSimCI = quantile(VSim,[0.025 0.975],2);
YSimBar = mean(YSim,2);
YSimCI = quantile(YSim,[0.025 0.975],2);

figure;
subplot(2,1,1);
h1 = plot(dates,VSim,'Color',0.8*ones(1,3));
hold on;
h2 = plot(dates,VSimBar,'k--','LineWidth',2);
h3 = plot(dates,VSimCI,'r--','LineWidth',2);
hold off;
title('Simulated Conditional Variances');
ylabel('Cond. var.');
xlabel('Year');
axis tight;

subplot(2,1,2);
h1 = plot(dates,YSim,'Color',0.8*ones(1,3));
hold on;
h2 = plot(dates,YSimBar,'k--','LineWidth',2);
h3 = plot(dates,YSimCI,'r--','LineWidth',2);
hold off;
title('Simulated Nominal Returns');
ylabel('Nominal return (%)');
xlabel('Year');
axis tight;
legend([h1(1) h2 h3(1)],{'Simulated path' 'Mean' 'Confidence bounds'},...
'FontSize',7,'Location','NorthWest');

Forecast conditional variances from a fully specified gjr model object. That is, forecast from an estimated gjr model or a known gjr model in which you specify all parameter values.

Load the Nelson-Plosser data set. Convert the yearly stock price indices (SP) to returns.

load Data_NelsonPlosser;
sp = price2ret(DataTable.SP);

Create a GJR(1,1) model and fit it to the return series.

Mdl = gjr('GARCHLags',1,'ARCHLags',1,'LeverageLags',1);
EstMdl = estimate(Mdl,sp);

GJR(1,1) Conditional Variance Model (Gaussian Distribution):

Value      StandardError    TStatistic     PValue
_________    _____________    __________    ________

Constant       0.0045728      0.0044199        1.0346       0.30086
GARCH{1}         0.55808           0.24        2.3253      0.020057
ARCH{1}          0.20461        0.17886         1.144       0.25263
Leverage{1}      0.18066        0.26802       0.67406       0.50027

Forecast the conditional variance of the nominal return series 10 years into the future using the estimated GJR model. Specify the entire return series as presample observations. The software infers presample conditional variances using the presample observations and the model.

numPeriods = 10;
vF = forecast(EstMdl,numPeriods,'Y0',sp);

Plot the forecasted conditional variances of the nominal returns. Compare the forecasts to the observed conditional variances.

v = infer(EstMdl,sp);
nV = size(v,1);
dates = dates((end - nV + 1):end);

figure;
plot(dates,v,'k:','LineWidth',2);
hold on;
plot(dates(end):dates(end) + 10,[v(end);vF],'r','LineWidth',2);
title('Forecasted Conditional Variances of Returns');
ylabel('Conditional variances');
xlabel('Year');
axis tight;
legend({'Estimation Sample Cond. Var.','Forecasted Cond. var.'},...
'Location','NorthWest');

expand all

## Tips

You can specify a gjr model as part of a composition of conditional mean and variance models. For details, see arima.

## References

[1] Glosten, L. R., R. Jagannathan, and D. E. Runkle. “On the Relation between the Expected Value and the Volatility of the Nominal Excess Return on Stocks.” The Journal of Finance. Vol. 48, No. 5, 1993, pp. 1779–1801.

[2] Tsay, R. S. Analysis of Financial Time Series. 3rd ed. Hoboken, NJ: John Wiley & Sons, Inc., 2010.

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