These are replications of the work in Chan and Maheu(2002). Standard GARCH models often have a hard time dealing with the occasional spikes (up or down) seen in many speculative market returnsthe GARCH recursion predicts a high variance will follow large residuals, but won't predict the initial "outlier" that triggers a high-volatility period. Chan and Maheu deal with this by combining a standard univariate GARCH model for the more systematic part of volatility with one of several forms of "jump" models which model the intermittent spikes as realizations of a Poisson process which, when it fires, injects a (presumably) high variance component into the process. The general form of the model (for a return series $$R$$) is

(1) $${R_t} = \mu + \sum\limits_{i = 1}^m {{{\phi _i}}{R_{t - i}}} + \sqrt {{h_t}} {z_t} + \sum\limits_{k = 1}^{{n_t}} {{Y_{t,k}}}$$

where

(2) $${z_t} \sim NID\left( {0,1} \right),{Y_{t,k}} \sim N\left( {{\theta _t},\delta _t^2} \right)$$

The mean is an AR(m) process, and the residual is a mixture of Normalsa mean zero process which will be governed by a GARCH recursion for $$h_t$$ and a set of independent (possibly non-zero mean) processes the number of which will be governed by a Poisson process. The use of a Poisson process (rather than a simpler binary process) allows for jumps of different magnitudes, as a process with a larger value of $$n_t$$ will add more variance to the process than a smaller one. With this,

(3) $${R_t}|{n_t},{h_t} \sim N\left( {\mu + \sum\limits_{i = 1}^p {{{\phi _i}}{R_{t - i}}} + {n_t}{\theta _t},{h_t} + {n_t}\delta _t^2} \right)$$

where $$h$$ is computed using the GARCH recursion

(4) $${h_t} = \omega + \sum\limits_{i = 1}^q {{\alpha _i}\varepsilon _{t - i}^2} + \sum\limits_{i = 1}^p {{\beta _i}{h_{t - i}}}$$

The likelihood element for $$R_t$$ will be the sum of the conditional probabilities in (3) over the values of $$n_t$$ from 0 to $$\infty$$ weighted by the Poisson probabilities of each $$n_t$$. However, the infinite sum isn't feasible, since the summands don't have a form which permits a limit sum. So this has to be approximated by a finite sum over (what one assumes) is a sufficiently large number of terms that the tail is negligible.

The Data

The data are 100 x the log difference in daily closing prices for the Dow Jones Industrial Average over the period from October 1, 1928 to January 11, 2000. This range includes both the stock market crashes in 1929 and 1987. This is an irregular time series data set which is handled using the mapped irregular date scheme introduced with Version 10 of RATS.

open data djia.txt

data(format=free,org=columns) 1 18940 year month day logret logrange close low high

*

* Data are daily with holiday skips.

*

cal(julian=%julianfromymd(year,month,day))

The authors do several calculations over ranges of certain years. With the mapped dates, the first entry in a year can be located by using year:1, and the last entry in a year can be located using (year+1):1-1 (entry before the first period in the next year). Thus, the following are used to locate the ends of the years 1950, 1969 and 1984:

compute end1950=1951:1-1

compute end1969=1970:1-1

compute end1984=1985:1-1

JUMPGARCH.SRC

This is a source file which has FUNCTIONs to do the calculations (for a specific entry T) of the log likelihood of different model types, sometimes producing values needed for future calculations as a side effect. There should be no need to edit this if you intend to use the models described in the paper.

The simpler JUMPGARCH function is for a fixed Poisson intensity. It takes as its arguments the current residual (U), the current GARCH variance (H), the Poisson parameter (LAMBDA), the squared variance of the Poisson jumps (DELTASQ) and the mean of the Poisson jumps (THETA). The calculation adds up the probability weighted Normal densities into JUMPGARCH and the (Poisson) probability weights into WT for all possible values of the Poisson process from 0 to KMAX. (KMAX is set to 20 at the top of the procedure file). Because the sum is truncated at KMAX, the probability weights won't add up to 1 (they should be close unless $$\lambda$$ is quite large, like 5 or more), so this normalizes by the actual sum. This is a minor technical fix to avoid numerical problems if a large value of $$\lambda$$ happens to get evaluated as part of the optimization procedure. The calculations are done with:

compute wt=0.0

compute jumpgarch=0.0

do k=0,kmax

compute jp =%poissonk(lambda,k)

compute wt =wt+jp

compute jumpgarch=jumpgarch+jp*exp(%logdensity(h+k*deltasq,u-k*theta))

end do k

compute jumpgarch=log(jumpgarch/wt)

JUMPGARCH.RPF

This handles the more basic calculations in the paper: it does summary statistics (for a partial sample), estimates a standard GARCH(1,1) model, and estimates a Poisson jump GARCH model with fixed intensity (Poisson mean) and fixed $$\theta$$ and $$\delta$$ over three sample ranges (1928-1950), (1951-1969) and (1978-1984). One of the reasons for this is that the fixed values seem unlikely to be adequate across the full sample. (Note that the main point of the paper is to extend the model to allow time-varying values for all of these, so this part of the analysis wouldn't generally be needed in a practical application of the more complicated "ARJI-GARCH" model.)

The mean is an AR(2), where 2 is the SBC-minimizing lag length (assuming homoscedasticity). This defines the FRML for the mean model and sets up the PARMSET for it:

linreg(define=meaneq) r

# constant r{1 2}

frml(lastreg,vector=meanb,parmset=meanparms) meanf

These set up series for the residuals (U), the GARCH model variance (H) and for the residuals for the jump intensity (XI).

set u = 0.0

set h = %sigmasq

*

set xi  = 0.0

This sets up a standard univariate GARCH(1,1) recursion and the PARMSET for its parameters:

nonlin(parmset=garchparms) omega alpha beta

frml hf = omega+alpha*u{1}^2+beta*h{1}

LAMBDA is the parameter for the constant intensity Poisson jump process, DELTA and THETA are the parameters governing the size of the jumps. These are set up into separate PARMSETS to make it easier to switch to a more complicated calculation of the Poisson intensity.

nonlin(parmset=poissonparms) lambda

nonlin(parmset=jumpparms) delta theta

and this is the log likelihood calculation for the model. This uses the more complicated ARJIGARCH function to evaluate the log likelihood (rather than JUMPGARCH) because it returns the value of XI (the Poisson residual).

frml logl = u=%eqnrvalue(meaneq,t,meanb),h=hf,arjigarch(u,h,lambda,delta^2,theta,xi)

The program does estimates over three subperiods: 1928-1950, 1951-1969 and 1970-1984. The process for doing a set of estimates is the same for each of those. First, a standard GARCH model is estimated to provide guess values. SSVAR is the implied steady-state variance from the model and is used to initialize the pre-sample values of H (the variance used in the GARCH recursion).

garch(p=1,q=1,equation=meaneq,resids=u,hseries=h) * end1950

*

compute ssvar=%beta(%nregmean+1)/(1-%beta(%nregmean+2)-%beta(%nregmean+3))

*

set u * %regstart()-1 = 0.0

set h * %regstart()-1 = ssvar

The guess values for most of the parameters are taken straight out of the GARCH output; first the mean model, then the GARCH parameters:

compute meanb=%xsubvec(%beta,1,%size(meanb))

compute omega=%beta(%nregmean+1),alpha=%beta(%nregmean+2),beta=%beta(%nregmean+3)

The Poisson jump model is initialized with an expected value of .1 jumps per period, with a zero mean and a variance of 10 x the steady state variance for the simple GARCH model.

compute theta=0.0,delta=sqrt(10.0*ssvar),lambda=.1

This does the estimation of the jump GARCH model. The PARMSET option uses the combined PARMSET's from the four parts of the model (mean model, GARCH model, Poisson intensity and jump parameters).

maximize(parmset=meanparms+garchparms+jumpparms+poissonparms,$pmethod=simplex,piters=2,method=bfgs,$

title="Constant Intensity Jump GARCH, 1928-1950") logl gstart end1950

It's hardly a surprise that the log likelihood for the jump GARCH is quite a bit better than the simple GARCH model since it has a "model" for isolated outliers and GARCH does not. The main point of running the jump GARCH models (which were not novel to this paper) is to show that the estimates for the jump model parameters differ greatly from one subperiod to another, with lambda estimates going from .11 to 1.67 to .020.

Instead, the authors recommend a model where the jump model parameters adapt, labeling the new model ARJI-GARCH (AutoRegressive Jump Intensity). This has the jump intensity evolving according to

$${\lambda _t} = {\lambda _0} + \rho {\lambda _{t - 1}} + \gamma {\xi _{t - 1}}$$

where

$${\xi _t} = E[{n_t}|t] - {\lambda _t}$$

is the "residual" in the jump intensity: the best guess as to the number of jumps given the data at t minus the expected number of jumps predicted by the model given data only through t-1. Because $$\lambda$$ has to be positive and $$\xi$$ can be negative, they show that a sufficient condition for $$\lambda$$ to stay in bounds is for $$\rho \ge \gamma$$, which isn't binding in this example, but might be in other applications.

ARJIGARCH.RPF

The formula to generate the recursion for the ARJI is

declare real lambda0 rho gamma

frml lambdaf = lambda0+rho*lambda_t{1}+gamma*xi_t{1}

The second program file (ARJIGARCH.RPF) first estimates a constant intensity model over the data through 1984 (JUMPGARCH.RPF only does subsamples) which uses the same setup described above. The "simple" ARJI-GARCH model uses much of the same setup, but changes the PARMSET for the Poisson intensity to include the RHO and GAMMA and adds a calculation of the time-varying LAMBDA to the log likelihood (the LAMBDA_T=LAMBDAF clause in the calculation):

nonlin(parmset=poissonparms) lambda0 rho gamma

compute lambda0=.05,rho=.5,gamma=.05

frml logl = u=%eqnrvalue(meaneq,t,meanb),h=hf,lambda_t=lambdaf,deltasq_t=zeta0^2,theta_t=eta0,\$

ARJIgarch(u,h,lambda_t,deltasq_t,theta_t,xi_t)

Note that the guess values have RHO well above GAMMA which is enough in this case to avoid any problems with LAMBDA going negative. In practice, you might have to add the constraint RHO>=GAMMA to POISSONPARMS to prevent problems. The guess value for LAMBDA0 will produce a steady-state value of .1 ($${\lambda _0}/(1 - \rho )$$), so is similar to what we used before.

The estimates show the ARJI model to fit better than the constant intensity model over the full sample and it is definitely the preferred model under the SBC. The authors generate a graph showing how the probability of jumps during two periods of instability in the marketthe constant intensity model puts almost all the mass at 0, while the others put the majority on somewhere between 1 and 3.

The paper then extends the model by allowing different time-varying calculations for the mean and variance of the jumps which are fairly straightforward extensions requiring additional FRML's for the evolution of those values.

Output (JUMPGARCH.RPF)

Statistics on Series R

Irregular Data From 1928:10:01 To 1984:12:31

Observations                 15149

Sample Mean               0.010703      Variance                   1.247249

Standard Error            1.116803      SE of Sample Mean          0.009074

t-Statistic (Mean=0)      1.179572      Signif Level (Mean=0)      0.238189

Skewness                  0.056557      Signif Level (Sk=0)        0.004489

Kurtosis (excess)        15.525326      Signif Level (Ku=0)        0.000000

Jarque-Bera          152151.840766      Signif Level (JB=0)        0.000000

West-Cho Modified Q Test, Series R

Q(15)    24.63

Signif. 0.0551

Statistics on Series ABSR

Irregular Data From 1928:10:01 To 1984:12:31

Observations                 15149

Sample Mean               0.713042      Variance                   0.738901

Standard Error            0.859594      SE of Sample Mean          0.006984

t-Statistic (Mean=0)    102.097277      Signif Level (Mean=0)      0.000000

Skewness                  4.125962      Signif Level (Sk=0)        0.000000

Kurtosis (excess)        31.494966      Signif Level (Ku=0)        0.000000

Jarque-Bera          669097.958061      Signif Level (JB=0)        0.000000

West-Cho Modified Q Test, Series ABSR

Q(15)   3387.43

Signif.  0.0000

Statistics on Series RSQ

Irregular Data From 1928:10:01 To 1984:12:31

Observations                    15149

Sample Mean                  1.247282      Variance                  27.256501

Standard Error               5.220776      SE of Sample Mean          0.042417

t-Statistic (Mean=0)        29.405019      Signif Level (Mean=0)      0.000000

Skewness                    18.393694      Signif Level (Sk=0)        0.000000

Kurtosis (excess)          524.685447      Signif Level (Ku=0)        0.000000

Jarque-Bera          174622605.319893      Signif Level (JB=0)        0.000000

West-Cho Modified Q Test, Series RSQ

Q(15)   469.57

Signif. 0.0000

Linear Regression - Estimation by Least Squares

Dependent Variable R

Irregular Data From 1928:10:03 To 1999:12:31

Usable Observations                     18938

Degrees of Freedom                      18935

Centered R^2                        0.0043336

R-Bar^2                             0.0042285

Uncentered R^2                      0.0046753

Mean of Dependent Variable       0.0204720350

Std Error of Dependent Variable  1.1050085467

Standard Error of Estimate       1.1026698192

Sum of Squared Residuals         23022.701627

Regression F(2,18935)                 41.2074

Significance Level of F             0.0000000

Log Likelihood                    -28721.2509

Durbin-Watson Statistic                1.9998

Variable                        Coeff      Std Error      T-Stat      Signif

************************************************************************************

1.  Constant                      0.020140484  0.008015291      2.51276  0.01198739

2.  R{1}                          0.055207015  0.007261619      7.60258  0.00000000

3.  R{2}                         -0.038910488  0.007261625     -5.35837  0.00000008

GARCH Model - Estimation by BFGS

Convergence in    23 Iterations. Final criterion was  0.0000081 <=  0.0000100

Dependent Variable R

Irregular Data From 1928:10:03 To 1950:12:30

Usable Observations                      6550

Log Likelihood                     -9406.4477

Variable                        Coeff      Std Error      T-Stat      Signif

************************************************************************************

1.  Constant                      0.035677146  0.009955540      3.58365  0.00033883

2.  R{1}                          0.108016517  0.013985353      7.72355  0.00000000

3.  R{2}                         -0.031298660  0.013470313     -2.32353  0.02015077

4.  C                             0.010724611  0.001597228      6.71452  0.00000000

5.  A                             0.099929699  0.007562278     13.21423  0.00000000

6.  B                             0.898277050  0.007169563    125.29034  0.00000000

Constant Intensity Jump GARCH, 1928-1950 - Estimation by BFGS

Convergence in    25 Iterations. Final criterion was  0.0000011 <=  0.0000100

Irregular Data From 1928:10:03 To 1950:12:30

Usable Observations                      6550

Function Value                     -9153.7843

Variable                        Coeff      Std Error      T-Stat      Signif

************************************************************************************

1.  MEANB(1)=Constant             0.075364609  0.010055696      7.49472  0.00000000

2.  MEANB(2)=R{1}                 0.094288867  0.013066505      7.21607  0.00000000

3.  MEANB(3)=R{2}                -0.062559854  0.012883249     -4.85591  0.00000120

4.  OMEGA                         0.000935896  0.000925377      1.01137  0.31184036

5.  ALPHA                         0.061105581  0.005867058     10.41503  0.00000000

6.  BETA                          0.919413167  0.006917522    132.91078  0.00000000

7.  DELTA                        -1.453061700  0.137892512    -10.53764  0.00000000

8.  THETA                        -0.643078030  0.114071863     -5.63748  0.00000002

9.  LAMBDA                        0.112189630  0.019650941      5.70912  0.00000001

Q Test on Standardized Squared Residuals

Q(15)    51.46

Signif. 0.0000

Q Test on Jump Intensity Residuals

Q(15)    22.72

Signif. 0.0904

GARCH Model - Estimation by BFGS

Convergence in    21 Iterations. Final criterion was  0.0000000 <=  0.0000100

Dependent Variable R

Irregular Data From 1951:01:02 To 1969:12:31

Usable Observations                      4806

Log Likelihood                     -4367.1764

Variable                        Coeff      Std Error      T-Stat      Signif

************************************************************************************

1.  Constant                      0.038851150  0.008915464      4.35773  0.00001314

2.  R{1}                          0.192880542  0.015355761     12.56079  0.00000000

3.  R{2}                         -0.077274531  0.015498505     -4.98593  0.00000062

4.  C                             0.036908249  0.006790743      5.43508  0.00000005

5.  A                             0.120245314  0.014668999      8.19724  0.00000000

6.  B                             0.787904897  0.028768073     27.38817  0.00000000

Constant Intensity Jump GARCH, 1951-1969 - Estimation by BFGS

Convergence in    47 Iterations. Final criterion was  0.0000095 <=  0.0000100

Irregular Data From 1951:01:02 To 1969:12:31

Usable Observations                      4806

Function Value                     -4218.7901

Variable                        Coeff      Std Error      T-Stat      Signif

************************************************************************************

1.  MEANB(1)=Constant             0.212737185  0.032608681      6.52394  0.00000000

2.  MEANB(2)=R{1}                 0.184285775  0.015091772     12.21101  0.00000000

3.  MEANB(3)=R{2}                -0.081914275  0.015226597     -5.37968  0.00000007

4.  OMEGA                         0.000000000  0.000000000      0.00000  0.00000000

5.  ALPHA                         0.110824312  0.013167298      8.41663  0.00000000

6.  BETA                          0.802516504  0.025030910     32.06102  0.00000000

7.  DELTA                         0.272107221  0.034486582      7.89023  0.00000000

8.  THETA                        -0.112731030  0.018082583     -6.23423  0.00000000

9.  LAMBDA                        1.666095192  0.402941904      4.13483  0.00003552

Q Test on Standardized Squared Residuals

Q(15)    26.83

Signif. 0.0302

Q Test on Jump Intensity Residuals

Q(15)    23.91

Signif. 0.0666

GARCH Model - Estimation by BFGS

Convergence in    20 Iterations. Final criterion was  0.0000057 <=  0.0000100

Dependent Variable R

Irregular Data From 1970:01:02 To 1984:12:31

Usable Observations                      3791

Log Likelihood                     -4837.4740

Variable                        Coeff      Std Error      T-Stat      Signif

************************************************************************************

1.  Constant                      0.018516225  0.015037639      1.23133  0.21820124

2.  R{1}                          0.160195579  0.016767079      9.55417  0.00000000

3.  R{2}                         -0.027080931  0.016663934     -1.62512  0.10413650

4.  C                             0.008952538  0.002126896      4.20920  0.00002563

5.  A                             0.049328981  0.005603675      8.80297  0.00000000

6.  B                             0.940243790  0.006834756    137.56801  0.00000000

Constant Intensity Jump GARCH, 1970-1984 - Estimation by BFGS

Convergence in    22 Iterations. Final criterion was  0.0000072 <=  0.0000100

Irregular Data From 1970:01:02 To 1984:12:31

Usable Observations                      3791

Function Value                     -4816.2804

Variable                        Coeff      Std Error      T-Stat      Signif

************************************************************************************

1.  MEANB(1)=Constant            -0.002294792  0.013848382     -0.16571  0.86838653

2.  MEANB(2)=R{1}                 0.162630679  0.017083748      9.51961  0.00000000

3.  MEANB(3)=R{2}                -0.029561443  0.017033039     -1.73554  0.08264600

4.  OMEGA                         0.007285182  0.002380861      3.05989  0.00221415

5.  ALPHA                         0.040307222  0.005574039      7.23124  0.00000000

6.  BETA                          0.947085857  0.007327168    129.25673  0.00000000

7.  DELTA                         1.429793224  0.284569463      5.02441  0.00000050

8.  THETA                         0.990834860  0.778600896      1.27258  0.20316578

9.  LAMBDA                        0.019507438  0.015908158      1.22625  0.22010320

Q Test on Standardized Squared Residuals

Q(15)    19.01

Signif. 0.2133

Q Test on Jump Intensity Residuals

Q(15)    30.18

Signif. 0.0113

Output (ARJIGARCH.RPF)

Linear Regression - Estimation by Least Squares

Dependent Variable R

Irregular Data From 1928:10:03 To 1999:12:31

Usable Observations                     18938

Degrees of Freedom                      18935

Centered R^2                        0.0043336

R-Bar^2                             0.0042285

Uncentered R^2                      0.0046753

Mean of Dependent Variable       0.0204720350

Std Error of Dependent Variable  1.1050085467

Standard Error of Estimate       1.1026698192

Sum of Squared Residuals         23022.701627

Regression F(2,18935)                 41.2074

Significance Level of F             0.0000000

Log Likelihood                    -28721.2509

Durbin-Watson Statistic                1.9998

Variable                        Coeff      Std Error      T-Stat      Signif

************************************************************************************

1.  Constant                      0.020140484  0.008015291      2.51276  0.01198739

2.  R{1}                          0.055207015  0.007261619      7.60258  0.00000000

3.  R{2}                         -0.038910488  0.007261625     -5.35837  0.00000008

GARCH Model - Estimation by BFGS

Convergence in    22 Iterations. Final criterion was  0.0000017 <=  0.0000100

Dependent Variable R

Irregular Data From 1928:10:03 To 1984:12:31

Usable Observations                     15147

Log Likelihood                    -18671.5083

Variable                        Coeff      Std Error      T-Stat      Signif

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1.  Constant                      0.030382787  0.005636423      5.39044  0.00000007

2.  R{1}                          0.144533761  0.009006106     16.04842  0.00000000

3.  R{2}                         -0.039939752  0.008708777     -4.58615  0.00000451

4.  C                             0.008026994  0.000926460      8.66415  0.00000000

5.  A                             0.078417793  0.004304920     18.21585  0.00000000

6.  B                             0.914636358  0.004558589    200.64023  0.00000000

Constant Intensity Jump GARCH, Full Sample - Estimation by BFGS

Convergence in    25 Iterations. Final criterion was  0.0000007 <=  0.0000100

Irregular Data From 1928:10:04 To 1984:12:31

Usable Observations                     15146

Function Value                    -18314.5453

Variable                        Coeff      Std Error      T-Stat      Signif

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1.  MEANB(1)=Constant             0.069233472  0.008117934      8.52846  0.00000000

2.  MEANB(2)=R{1}                 0.138895120  0.008358254     16.61772  0.00000000

3.  MEANB(3)=R{2}                -0.056147005  0.008362669     -6.71401  0.00000000

4.  OMEGA                         0.000525046  0.000822140      0.63863  0.52306105

5.  ALPHA                         0.066570823  0.003980262     16.72524  0.00000000

6.  BETA                          0.919270528  0.004554112    201.85503  0.00000000

7.  ZETA0                         0.874106660  0.096271642      9.07959  0.00000000

8.  ETA0                         -0.322133747  0.057802714     -5.57299  0.00000003

9.  LAMBDA0                       0.151357539  0.038125319      3.97000  0.00007187

Q Test on Standardized Squared Residuals

Q(15)    18.24

Signif. 0.2500

Q Test on Jump Intensity Residuals

Q(15)    30.97

Signif. 0.0089

ARJI-GARCH - Estimation by BFGS

Convergence in    33 Iterations. Final criterion was  0.0000020 <=  0.0000100

Irregular Data From 1928:10:04 To 1984:12:31

Usable Observations                     15146

Function Value                    -18274.8689

Variable                        Coeff      Std Error      T-Stat      Signif

************************************************************************************

1.  MEANB(1)=Constant             0.057051646  0.007071514      8.06781  0.00000000

2.  MEANB(2)=R{1}                 0.133111046  0.008440042     15.77137  0.00000000

3.  MEANB(3)=R{2}                -0.058052831  0.008361637     -6.94276  0.00000000

4.  OMEGA                         0.002989075  0.000603580      4.95224  0.00000073

5.  ALPHA                         0.034646341  0.004149190      8.35014  0.00000000

6.  BETA                          0.949480037  0.004643237    204.48666  0.00000000

7.  ZETA0                         1.176797434  0.106115207     11.08981  0.00000000

8.  ETA0                         -0.395222062  0.062259453     -6.34798  0.00000000

9.  LAMBDA0                       0.013127653  0.003175585      4.13393  0.00003566

10. RHO                           0.915942675  0.019418309     47.16902  0.00000000

11. GAMMA                         0.491712724  0.072537592      6.77873  0.00000000

Q Test on Standardized Squared Residuals

Q(15)     6.87

Signif. 0.9612

Q Test on Jump Intensity Residuals

Q(15)     9.83

Signif. 0.8303

ARJI-R{1}^2 GARCH - Estimation by BFGS

Convergence in    30 Iterations. Final criterion was  0.0000000 <=  0.0000100

Irregular Data From 1928:10:04 To 1984:12:31

Usable Observations                     15146

Function Value                    -18231.1754

Variable                        Coeff      Std Error      T-Stat      Signif

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1.  MEANB(1)=Constant             0.071364949  0.007513916      9.49770  0.00000000

2.  MEANB(2)=R{1}                 0.148592120  0.008700510     17.07855  0.00000000

3.  MEANB(3)=R{2}                -0.063810012  0.008286581     -7.70040  0.00000000

4.  OMEGA                         0.001968097  0.000499047      3.94371  0.00008023

5.  ALPHA                         0.029297868  0.003394138      8.63190  0.00000000

6.  BETA                          0.956417431  0.003920155    243.97441  0.00000000

7.  ZETA0                         0.872576426  0.070260871     12.41909  0.00000000

8.  ZETA1                         0.133112642  0.046988267      2.83289  0.00461291

9.  ETA0                         -0.520041723  0.076046190     -6.83850  0.00000000

10. ETA1                          0.032000592  0.049567878      0.64559  0.51854406

11. ETA2                         -0.206286465  0.044019300     -4.68627  0.00000278

12. LAMBDA0                       0.018853971  0.004420945      4.26469  0.00002002

13. RHO                           0.910480072  0.019283446     47.21563  0.00000000

14. GAMMA                         0.522667477  0.082299398      6.35081  0.00000000

Q Test on Standardized Squared Residuals

Q(15)    14.62

Signif. 0.4792

Q Test on Jump Intensity Residuals

Q(15)    15.52

Signif. 0.4147

ARJI-h GARCH - Estimation by BFGS

Convergence in    43 Iterations. Final criterion was  0.0000006 <=  0.0000100

Irregular Data From 1928:10:04 To 1984:12:31

Usable Observations                     15146

Function Value                    -18141.8883

Variable                        Coeff      Std Error      T-Stat      Signif

************************************************************************************

1.  MEANB(1)=Constant             0.109621552  0.009990126     10.97299  0.00000000

2.  MEANB(2)=R{1}                 0.120534261  0.012055362      9.99839  0.00000000

3.  MEANB(3)=R{2}                -0.067357788  0.007845604     -8.58542  0.00000000

4.  OMEGA                         0.001522318  0.000321873      4.72956  0.00000225

5.  ALPHA                         0.019922410  0.002798674      7.11852  0.00000000

6.  BETA                          0.959969514  0.004374653    219.43902  0.00000000

7.  ZETA0                        -0.000000600  0.135081331 -4.43961e-06  0.99999646

8.  ZETA1                         1.599799015  0.124202783     12.88054  0.00000000

9.  ETA0                         -0.292333568  0.038904292     -7.51417  0.00000000

10. ETA1                          0.116401917  0.035149165      3.31166  0.00092746

11. ETA2                         -0.080473686  0.029113105     -2.76417  0.00570671

12. LAMBDA0                       0.069955714  0.015266708      4.58224  0.00000460

13. RHO                           0.842563777  0.025377026     33.20183  0.00000000

14. GAMMA                         0.443871269  0.060380695      7.35121  0.00000000

Q Test on Standardized Squared Residuals

Q(15)     8.01

Signif. 0.9233

Q Test on Jump Intensity Residuals

Q(15)    14.69

Signif. 0.4736

Graphs (All out of ARJIGARCH.RPF)