GARCH estimates differ in rugarch (R) vs. EViews

Question

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, 
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0.00121321378245653, -0.00175673411449218, -0.0177111606464013, 
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0.00285661638368673, -0.0091984467251045, -0.00635158904182198, 
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0.00646817036545677, -0.00936218069462136, -0.00992630520256377, 
-0.00995174579671421, 0.0241883533000227, 0.00578491104143275, 
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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, 
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-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, 
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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?

Answer 1

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