GARCH – Differences in GARCH Estimates Between rugarch (R) and EViews

eviewsgarchrtime seriesvolatility-forecasting

I modelled a stock's volatility using the "rugarch" package in R and Eviews.
The estimated model is GARCH(1,1).

Data is as below:

> dput(datax)
c(0.00240428226573286, 0.00718664351112785, 0.00417663958775449, 
-0.0124234291416307, 0.00615240249156912, 0.0096846888172486, 
0.0106526433200909, -0.00786660798829253, -0.0122874870756498, 
-0.000314141256930967, 0.000471174886371273, -0.0208884504520821, 
-0.0149969692551366, 0.0241492647161508, 0.00419227605454964, 
0.0178426729434715, 0.00339145325161994, 0.00518480259013288, 
0.0144432753009873, -0.000454914348644309, -0.0129016560881787, 
0.0104447845272464, 0.0167547608104748, -0.00405921117604713, 
-0.0300729637845212, -0.00822872240789607, 0.00278348586175703, 
-0.00943594943234238, -5.99101840794702e-05, 0.000996016229104058, 
-0.000829404324086624, 0.0258218725118393, 0.00877055916031999, 
-0.00588618984169464, 0.0254017935654574, 0.00805703215794296, 
-0.0191565531978934, 0.0152034393746021, -0.00363509820161312, 
0.0117471147043791, 0.00185834076893698, 0.0109010059113128, 
0.000525595380350907, 0.00471136142307849, 0.00378484394178535, 
0.00256537092911024, 0.0134933997293825, 0.00363203707933835, 
0.00448837964129467, 0.00916296013641471, -0.0135706087748861, 
0.00426982136304233, 0.0249833876507619, 0.019064654422376, 0.00552211291815752, 
-0.0198178211588615, -0.0170519265736608, 0.0120525451282543, 
-2.37224843004924e-05, -0.000146280600871407, 0.00477158577627002, 
0.014383729883134, 0.00421564947003716, -0.0109717193626331, 
0.0182095942293206, -0.0108949339087712, -0.0176664501445263, 
-0.00426136213898864, 0.0163439380148436, 0.0178444666710593, 
-0.00319282678270305, 0.0233384597132087, 0.00102961481001707, 
-0.00247084887334204, 0.00896252965220334, -0.00107396173656049, 
-0.00815784718500012, 0.00140858905036012, 0.00184093926106854, 
-0.00841584807540308, 0.0075738167721191, -0.00446712175714303, 
0.00121321378245653, -0.00175673411449218, -0.0177111606464013, 
-0.0334792540725228, -0.021030442634931, -0.00865103680757429, 
-0.00421196940949287, 0.00881717323241915, -0.00329042016708669, 
0.00285661638368673, -0.0091984467251045, -0.00635158904182198, 
-0.0104482543279794, 0.00359730221733479, -0.000431138620978544, 
-0.0115382875664558, 0.00533047803197206, 0.0341741617915705, 
-0.0104470816276567, -0.00732144270561363, -8.62463751900577e-05, 
0.00646817036545677, -0.00936218069462136, -0.00992630520256377, 
-0.00995174579671421, 0.0241883533000227, 0.00578491104143275, 
-0.00555291698752569, 0.00859969423753348, -0.0143210399197393, 
-0.0106656279629131, -0.00460249153100456, -0.010173846679443, 
0.00628725502787653, -0.0207505962948851, 0.0101111964328382, 
0.0273637272462803, 0.000106801184665883, -0.00340971899708542, 
0.00930603680385644, -0.0342227015274759, -0.00272057238669277, 
0.0232516656596928, -0.00282957942797246, 0.00137068445465083, 
0.0146662205383272, 0.00557236352191204, -0.00470848819449188, 
0.01545895026171, 0.0237779175036614, 0.0022179786175851, 0.0154723164160355, 
0.00284859279265781, -0.0734795439085705, -0.0101844065754371, 
-0.0168785371789415, -0.0451668472504849, 0.00200162750463484, 
0.0333872319914033, -0.00655444963038754, 0.0186375681927853, 
0.00224002580098137, 0.00214365086334034, 0.01717336389666, -0.0119039720455039, 
-0.0166511258509399, 0.0208862527198921, -0.000787824156615713, 
0.0222754484996237, 0.00954288703328032, -0.00727556935841456, 
0.0137326236782958, -0.0102379006489173, 0.00311433870608724, 
-0.0098021206176373, 0.00565514504945241, -0.00226609648997211, 
0.00223756041797607, -0.00246408074099946, -0.0079808840138309, 
-0.0158660954154453, 0.00881042570067692, 0.00428512104701007, 
-0.0130623086945807, -0.00198847471210328, -0.00151270842662043, 
0.0135073344736334, 0.0117315016530082, 0.00260857338333409, 
-0.00451831385594481, 0.00257655670181478, -0.0101997675299845, 
-0.0135002265633961, 0.0214784602834559, -0.00461067140901328, 
0.00776184432271698, 0.0238427144319235, -0.000495135212737807, 
-0.0387403757953813, 0.00565275629502793, 0.00667937353452963, 
-0.00776691741432067, -0.00766349350523576, 0.00958945509957054, 
-0.00217288868014798, 0.0102753819264656, 0.000527389159218572, 
0.00104278741483199, -0.00180603147915903, -0.0024659310273023, 
-0.00235092677265314, -0.00843919357593137, 0.00963829977017916, 
-0.00150498322541637, 0.0115911140449825, 0.0110373532490922, 
0.000537162615545483, -0.00517091804718639, 0.013716249701627, 
-0.00626535235492476, -0.00044537835702485, -0.00833167889318887, 
-0.00517343387401326, 0.00259197243534537, -0.0142035782174386, 
-0.00332432066398702, -0.00637259964301151, -0.0319897877349788, 
0.0188708261874773, 0.00902118748854797, -0.00208183811690787, 
0.00198454105666279, -0.0156560254475639, -0.0100950585869821, 
0.00978093861225737, -0.00522315899479864, 0.00503232384557784, 
0.00666876157161944, -0.00126191845920864, 0.00354688366125622, 
-0.0102764670148119, -0.0113092429587738, 0.00229380810354662, 
0.00839328069543654, -0.0105198364905359, -0.002837775658179, 
-0.0201399087459038, 0.0119401772698691, 0.00284045488011309, 
0.0246084027100704, 0.00788890594633074, -0.00133535072794366, 
-0.00266444216114259, 0.00674294083180094, 0.00986258515676042, 
-0.00148717060305792, 0.0103228516356264, -0.00114563042290783, 
-0.00558149616106718, 0.00839029408001757, -0.00242454214368415, 
-0.00277027874972191, -0.00560435091364653, 0.000731425659337148, 
-0.00428107774040676, 0.0109993438147029, 0.0037087621145826, 
0.00388281880721841, -0.00492902801425465, -0.0147212663223222, 
-0.0062137466061678, 0.00318246089141461, 0.00938513022545173, 
0.00372645244357095, -9.69066555711606e-06, 0.0035197962925686, 
0.0406780148963204, 0.0077983167274418, 0.00229569544477393, 
0.00793643833981328, 0.00504391169459417, -0.00580243023076754, 
0.00927432095852687, -0.000232971205631927, 0.0138722766791695, 
-0.0129039060692566, 0.00836494753892758, 1.01399352825382e-05, 
0.0283457779811229, 0.00067442071407342, 0.00637900121597035, 
0.00626980084182271, 0.0113243798290323, -0.0117401689487977, 
0.00135979977779499, 0.00879045569253378, 0.00656352401512272, 
-0.0153928479424028, 0.0125530726116168, -0.00561643658804734, 
-0.00227591872884325, 0.0034633081250135, 0.00727107400641813, 
-0.00273647607013316, 0.00425203735149005, -0.00488867171599416, 
0.00683394561459849, -0.00992043957091049, -0.00560198247430499, 
-0.00327635391489345, 0.0208371203446358, 0.00684650777054152, 
-0.00235817540968775, 0.0146372216938975, -0.00254461570527909, 
-0.0147392682797047, -0.00540476259961409, 0.00681741066701314, 
-0.00202936679798782, -0.00328393800144688, 0.0034608210234186, 
0.00915650804831181, 0.0024681397557007, 0.00452850517684666, 
-0.00325770029997052, -0.00883931780571601, -0.000500965633410289, 
0.00686775854728872, -0.00763740444788219, 0.00541127306787104, 
-0.0101645080291242, 0.000140351294030339, -0.00375751614814845, 
-0.00312954762505058, -0.000642140719694595, 0.0047835723077192, 
-0.00403350849344264, -0.00205117718248182, 0.0305259473841222, 
-0.00368893454833596, 0.000524259091893242, -0.0119619058683504, 
0.00214533859236532, 0.00653076907380878, 0.00791071061486903, 
-0.0062537972532688, 0.0135117597884715, 0.00416939885856848, 
0.0148088851181498, 0.00883162254853787, -0.00119022679055902, 
-0.00254082633103003, 0.00394659516152096, -0.003168068545504, 
-0.00524040660809355, -0.00882022385438397, 0.00951940493577297, 
-0.00101927410289804, 0.015761701773787, 0.00909395368709731, 
-0.0112922617960063, -0.00123833318798283, 0.00620620396185423, 
0.00598439589524169, -0.00455009463326661, -0.00605233754141565, 
0.0130798753275556, 0.0135739452716361, 0.00608364063475264, 
-0.00613010218358134, -0.00184034344641404, 0.00197347969190886, 
-0.00387874641259245, 0.00199036225790472, -0.00180383171416842, 
0.0153096987521142, -0.00686017554850871, 0.0014203900944505, 
-0.00729808882639027, 0.00369152733962785, 0.00980434412762321, 
0.00503305294462741, -0.00143341801999597, 0.00338400714536036, 
-0.00906640410340387, -0.00552950392268947, 0.0115387367679265, 
-0.000633026777244083, 0.00121665545139571, 0.00683871348798348, 
-0.00434128430549308, 0.00977794561054779, -0.00425650993954818, 
0.00249283999941774, 0.000815176235308357, 0.00679613674310175, 
-0.00458861771460839, -0.001166401766314, -0.00540718042119614, 
0.0100685043595448, 0.0204185872102673, 0.00605956410345243, 
0.00385001917730676, 0.00922236514154662, 0.00985160106128902, 
-0.00470606734079837, 0.01594519327646, -0.00636892362420838, 
0.00100412807768002, -0.00123875407891383, 0.00308910806569429, 
0.00154485396972071, 0.0109979003939937, -0.00640462572168055, 
-0.0015637144202536, -0.0129542930903757, 0.0035548530292111, 
0.00588116908225444, 0.0129026905494971, 0.0113209668876699, 
-0.00129441124807883, -0.00846832936736064, -0.00844436119602499, 
-0.00779763940020217, 0.023781763109044, -0.0242478267815525, 
-0.000479662645538781, -0.000343118163115719, 0.00352384560039809, 
0.0130894063298506, -0.000188021398532356, 0.00329381722090716, 
0.0018447861748303, 0.0054929799543082, 0.00531453264371429, 
0.000753024431418226, -0.00374371676477558, -0.0103937514181691, 
0.0067629682572683, 0.0011958712688962, -0.0118359004134643, 
0.00923609281688798, -0.00300438045761275, -0.00896634115784245, 
0.000819686759950145, -0.00465327468340249, -0.0112668808388143, 
-0.0152929145318392, 0.00386127972024042, -0.0126357426677117, 
0.0011690144781813, -0.0179534149314371, 0.0160931118496812, 
-0.0264315783876601, 0.0140562888877458, 0.00249690206283404)

The R code is:

library(rugarch)

datax<-as.data.frame(datax)
model11<-ugarchspec(variance.model=list(model="sGARCH",
                                         garchOrder = c(1,1),
                                         external.regressors =NULL),
                     mean.model=list(armaOrder=c(0,0), include.mean=FALSE),
                     distribution.model = "norm")

fit11<-ugarchfit(data=datax,spec=model11)

The estimated coefficients with "rugarch" are without intercept in mean equation:

> fit11

*---------------------------------*
*          GARCH Model Fit        *
*---------------------------------*

Conditional Variance Dynamics   
-----------------------------------
GARCH Model : sGARCH(1,1)
Mean Model  : ARFIMA(0,0,0)
Distribution    : norm 

Optimal Parameters
------------------------------------
        Estimate  Std. Error  t value Pr(>|t|)
omega   0.000001    0.000002  0.63044 0.528405
alpha1  0.025113    0.014814  1.69521 0.090035
beta1   0.963648    0.016994 56.70583 0.000000

Robust Standard Errors:
        Estimate  Std. Error  t value Pr(>|t|)
omega   0.000001    0.000022 0.068042  0.94575
alpha1  0.025113    0.112516 0.223197  0.82338
beta1   0.963648    0.138717 6.946883  0.00000

The estimated coefficients with "rugarch" are with intercept in mean equation:

> fit11

*---------------------------------*
*          GARCH Model Fit        *
*---------------------------------*

Conditional Variance Dynamics   
-----------------------------------
GARCH Model : sGARCH(1,1)
Mean Model  : ARFIMA(0,0,0)
Distribution    : norm 

Optimal Parameters
------------------------------------
        Estimate  Std. Error  t value Pr(>|t|)
mu      0.001090    0.000531  2.05445 0.039932
omega   0.000001    0.000004  0.30336 0.761618
alpha1  0.026484    0.030789  0.86018 0.389691
beta1   0.964029    0.033893 28.44323 0.000000

Robust Standard Errors:
        Estimate  Std. Error  t value Pr(>|t|)
mu      0.001090    0.001997 0.545712  0.58526
omega   0.000001    0.000076 0.016534  0.98681
alpha1  0.026484    0.542671 0.048803  0.96108
beta1   0.964029    0.603895 1.596352  0.11041

Using the same data I estimated GARCH(1,1) model with EViews. The results are:

Dependent Variable: RETURN              
Method: ML ARCH - Normal distribution (BFGS / Marquardt steps)              
Date: 10/30/17   Time: 19:49                
Sample: 1 438               
Included observations: 438              
Convergence achieved after 22 iterations                
Coefficient covariance computed using outer product of gradients                
Presample variance: backcast (parameter = 0.7)              
GARCH = C(1) + C(2)*RESID(-1)^2 + C(3)*GARCH(-1)                

Variable    Coefficient Std. Error  z-Statistic Prob.  

C           6.30E-06    3.63E-06    1.738057    0.0822
RESID(-1)^2 0.042247    0.017263    2.447164    0.0144
GARCH(-1)   0.912332    0.039907    22.86162    0.0000

R-squared   -0.005593               Mean dependent var      0.000863
Adjusted R-squared  -0.003297       S.D. dependent var      0.011552
S.E. of regression  0.011571        Akaike info criterion   -6.108230
Sum squared resid   0.058645        Schwarz criterion       -6.080269
Log likelihood  1340.702            Hannan-Quinn criter.    -6.097197
Durbin-Watson stat  1.965053

So, there is some slight differences between the estimates:

  • "rugarch" estimates the GARCH coefficient as 0.963 while EViews 0.912.
  • "rugarch" estimates the error coefficient as 0.025 while EViews 0.042.

And also the estimated standard errors are different.

Are those differences natural? Or am I doing something wrong?

Best Answer

The behavior that you see is due to the presample variance option in EViews.

rugarch uses the variance of all data points and EViews uses backcasting using a parameter of 0.7. More precisely, EViews uses this formula for initialization of the variance:

$\sigma_0^2 = \lambda \hat\sigma^2 + (1-\lambda) \sum_{t=0}^T \lambda^{T-t-1} \cdot \varepsilon_{T-t}^2$

where $\lambda$ is the backcast parameter (default in EViews: 0.7, default in rugarch, fGarch, and gretl: 1.0) and $\hat\sigma^2$ is the unconditional variance of all residuals $\varepsilon_1, \ldots, \varepsilon_T$.

This is the explanation in the EViews manual regarding this choice of the variance initialization (whatever outperform means for them):

Our experience has been that GARCH models initialized using backcast 
exponential smoothing often outperform models initialized 
using the unconditional variance.

Using the same approach as rugarch, we get:

Dependent Variable: RETURN              
Method: ML - ARCH (Marquardt) - Normal distribution             
Date: 10/30/17   Time: 20:26                
Sample: 1 438               
Included observations: 438              
Convergence achieved after 11 iterations                
Presample variance: unconditional               
GARCH = C(2) + C(3)*RESID(-1)^2 + C(4)*GARCH(-1)                

Variable    Coefficient Std. Error  z-Statistic Prob.  

C   0.001093    0.000547    1.996164    0.0459

    Variance Equation           

C           1.33E-06    9.92E-07    1.345676    0.1784
RESID(-1)^2 0.026633    0.007118    3.741463    0.0002
GARCH(-1)   0.963334    0.011717    82.21705    0.0000

which is quite close to the estimates of rugarch. However, we still see some differences in the standard errors: they are lower for rugarch resulting in differences regarding hypothesis testing of the GARCH parameters.

For comparison, these are the estimates using the fGarch library for R:

        Estimate  Std. Error  t value Pr(>|t|)    
mu     1.088e-03   5.198e-04    2.094  0.03627 *  
omega  1.333e-06   1.060e-06    1.257  0.20864    
alpha1 2.663e-02   9.728e-03    2.738  0.00618 ** 
beta1  9.633e-01   1.283e-02   75.078  < 2e-16 ***

and gretl:

             coefficient   std. error      z      p-value
  -------------------------------------------------------
  const      0.00108849    0.000519879    2.094   0.0363  **

  alpha(0)   1.33330e-06   1.12351e-06    1.187   0.2353 
  alpha(1)   0.0266348     0.0101192      2.632   0.0085  ***
  beta(1)    0.963349      0.0138129     69.74    0.0000  ***

which seem to be more similar to EViews than to rugarch.

What astonishes me most is the drop in the standard errors of the rugarch estimates: e.g. from 57 to 28, or from 1.7 to 0.9 - they appear as if they were halved.

Maybe this finding is worth posting on the R-SIG-Finance mailing list as we don't observe such a drop in the estimated standard errors for the other packages / programs.

fGarch without mean:

        Estimate  Std. Error  t value Pr(>|t|)    
omega  1.488e-06   1.219e-06    1.221  0.22227    
alpha1 2.517e-02   9.602e-03    2.621  0.00875 ** 
beta1  9.635e-01   1.424e-02   67.681  < 2e-16 ***

gretl without mean:

             coefficient   std. error      z      p-value
  -------------------------------------------------------
  alpha(0)   1.48801e-06   1.31591e-06    1.131   0.2581 
  alpha(1)   0.0251710     0.0101023      2.492   0.0127  **
  beta(1)    0.963509      0.0156173     61.69    0.0000  ***

EViews without mean:

Variable    Coefficient Std. Error  z-Statistic Prob.  

    Variance Equation           

C           1.49E-06    1.13E-06    1.314825    0.1886
RESID(-1)^2 0.025179    0.006580    3.826449    0.0001
GARCH(-1)   0.963505    0.012718    75.75811    0.0000
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