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 Quantlet: bigarch Description: estimates the BEKK (Baba, Engle, Kraft, Kroner) volatility representation for a bivariate conditionally heteroscedastic time series and evaluates the maximum of the quasi log likelihood function in a GARCH(1,1) frame of the following form: S_t=C_(0)^T*C_(0)+A_(11)^T*e_(t-1)*e_(t-1)^T*A_(11)+G_(11)^T* S_(t-1)*G_(11)

Reference(s):
Engle, R. F. and Kroner, K. F. (1995). Multivariate Simultaneous Generalized GARCH. Econometric Theory: 11, pp. 122-150. Fengler, M. and Herwartz, H. (2001). Multivariate Volatility Models. XploRe Finance Guide.

 Usage: {th,liks}=bigarch(theta,et) Input: theta columns of 2x2 ARCH and GARCH parameter matrices et T x 2 matrix containing the time series data Output: th elements of the upper triangular 2x2 matrix of the deterministic variance components and estimated columns of 2x2 ARCH and GARCH parameter matrices liks evaluated log likelihood function

Example:
```library("times")
theta=#(0.5, 0.01, 0.01, 0.5, 0.6, 0.01, 0.01, 0.7)
{th, liks} = bigarch(theta, et)
th
liks

```
Result:
```Contents of th
[ 1,]  0.66929
[ 2,]  0.10183
[ 3,]  0.026132
[ 4,]  0.37569
[ 5,]  0.20174
[ 6,]  -0.2142
[ 7,]  0.28792
[ 8,] -0.011575
[ 9,] -0.02021
[10,]  0.47874
[11,]  0.72634

Contents of liks
[1,]   1035.7
```
Example:
```library("xplore")
library("stats")
library("times")
dat1=tdiff(log(et1[,1]))
dat2=tdiff(log(et1[,2]))
dat=dat1~dat2
a=1:dim(dat1)
b=1:dim(dat2)
dat1gr=a~dat1
dat2gr=b~dat2
fxrate="DEM/GBP"|"DEM/USD"
; summary statistics
summarize(dat1~dat2, fxrate)
disp=createdisplay(2,1)
show(disp,1,1,dat1gr)
show(disp,2,1,dat2gr)
setgopt(disp,1,1,"xlabel", "Time", "title", "Foreign exchange rate returns: DEM/USD","yvalue",0|1)
setgopt(disp,2,1,"xlabel", "Time", "title", "Foreign exchange rate returns: DEM/GBP","yvalue",0|1)
; fix starting values
theta=#(0.28,-0.06,-0.05, 0.2, 0.9, 0.03, 0.02, 0.9)
;call bigarch
{coeff, maxlik} = bigarch(theta, dat)
coeff
maxlik

```
Result:
```Contents of summ
[1,] "         "
[2,] "             Minimum     Maximum        Mean      Median   Std.Error"
[3,] "        -------------------------------------------------------------"
[4,] "DEM/GBP    -0.040125    0.031874 -4.7184e-06           0   0.0070936"
[5,] "DEM/USD    -0.046682    0.038665  0.00011003           0   0.0069721"
[6,] " "

Contents of coeff
[ 1,]  0.0011516
[ 2,]  0.00031009
[ 3,]  0.00075685
[ 4,]  0.28185
[ 5,] -0.057194
[ 6,] -0.050449
[ 7,]  0.29344
[ 8,]  0.93878
[ 9,]  0.025117
[10,]  0.027503
[11,]   0.9391

Contents of maxlik
[1,]   -28599

According to the model setup outlined above the results can be
expressed by the following matrices:

C_(0) =
|0.0011519  0.0031009|
|     0     0.0075685|

A_(11)=
|0.28185    -0.050449|
|-0.057194   0.29344 |

G_(11)=
|0.93878     0.025117|
|0.027503    0.9391  |
```

Author: H. Herwatz, F. Noetzel G. Aydinli, W. Haerdle, 20010419 license MD*Tech
(C) MD*TECH Method and Data Technologies, 05.02.2006