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 returns⎯the 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 Normals⎯a 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 market⎯the 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
************************************************************************************
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
************************************************************************************
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
************************************************************************************
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)
