This is the replication from Sadorsky(2012). This fits a variety of GARCH models to a combination of returns to oil prices, "clean energy" and technology stocks and analyzes the effect of implied pair-wise hedging strategies. Note that the time-varying pairwise hedge ratio and portfolio weight calculations are common to all types of GARCH models, as they are functions only of the time-varying covariance matrices.
This is discussed in greater detail as part of the ARCH/GARCH and Volatility Models e-course.
The data set is 10 years of daily data over the 2001-2010 period, which includes the run up in oil prices and sharp declines in the stock prices in 2008. The mean model chosen is a one lag VAR on the returns:
system(model=var1)
variables dleco dloil dlpse
lags 1
det constant
end(system)
The author fits four different multivariate GARCH models to this: a standard BEKK, and three that use the VARMA model for the individual variances, one using a diagonal correlation model (assuming zero correlations between the residuals), one with CC (constant correlation) and one with DCC. Several of these models prove somewhat difficult to fit to this data set. The ones with MV=DIAG and MV=DCC both fail to converge with standard guess values. In this case (and this will be specific to this data set) we found that starting the estimation at entry 250, and feeding those estimates into a full sample estimate (using the INITIAL option on GARCH) provided convergence. (Starting at 250 avoids some large isolated outliers in two of the series). This is how that is done for the DCC model:
garch(model=var1,p=1,q=1,mv=dcc,variances=varma,method=bfgs,$
iters=50,pmethod=simplex,piters=10,noprint) 250 *
garch(model=var1,p=1,q=1,mv=dcc,variances=varma,method=bfgs,iters=500,$
initial=%beta,pmethod=simplex,piters=10,robusterrors,rvectors=rdcc,hmatrices=hdcc)
The first only does 50 iterations, because there is no reason to iterate to convergence when it's being used just for guess values.
All the model estimates save the (non-standardized) residuals (with the RVECTORS option, here into RDCC) and the time-variance covariance matrix estimates (with HMATRICES, here into HDCC).
The paper does a set of univariate diagnostics on each model, showing the Q on the (univariate) standardized residuals (testing for residual autocorrelation) and on their squares (testing for residual ARCH effects). With a multivariate model, you would generally prefer to do tests on jointly standardized residuals, but CC and DCC models only work with a limited amount of relationship among the series, so the univariate diagnostics will probably be adequate.
Among the four models fit, the DCC-VARMA variances model has the best log likelihood and best univariate diagnostics, so that is chosen for further analysis.
Portfolio Calculations
For the chosen DCC-VARMA variances model, the paper computes several portfolio optimization strategies using the (time-varying) conditional covariance matrices produced by the GARCH instruction. These are all done pairwise, assuming mean zero returns, so they are variance-minimizing strategies. One of these has the portfolio weights assuming a "no short" strategy, thus, they are forced to be bounded between 0 and 1 for each asset. The value at entry t for the asset pair i,j (using covariance matrix HDCC(T)) is:
%min(1.0,%max(0.0,(hdcc(t)(j,j)-hdcc(t)(i,j))/(hdcc(t)(i,i)-2.0*hdcc(t)(i,j)+hdcc(t)(j,j))))
where the optimizing weight on asset i is
\(\left( {{\sigma _{jj}} - {\sigma _{ij}}} \right)/({\sigma _{jj}} - 2{\sigma _{ij}} + {\sigma _{jj}})\)
which can be negative if the correlation between the assets is high positive and the variance of i is quite a bit higher than that of j (and can be greater than 1 if the correlation is high and the variance of i is quite a bit lower), so the calculation gets truncated to the [0,1] range.
The second portfolio calculation assumes a strategy of using one asset to hedge a unit position in the other. The short position for hedging i using j is
set hedges(i,j) = hdcc(t)(i,j)/hdcc(t)(j,j)
Both portfolio calculations give time-varying results. The hedge ratios are presented both graphically and in table form as statistics across the sample. The graphs of the hedge ratios are done with
spgraph(vfields=%nvar,hfields=%nvar-1,fillby=rows,$
footer="Fig 4. Time-varying hedge ratios computed from DCC model")
do i=1,%nvar
do j=1,%nvar
if i==j
next
graph(header="Hedge of "+labels(i)+" with "+labels(j))
# hedges(i,j)
end do j
end do i
spgraph(done)
which will do (in this case), a 3 x 2 array of graphs showing all pairs, while the table is done with
report(use=wreport,action=define,title="Table 6. Hedge ratio (long/short) summary statistics")
report(use=wreport,atrow=1,atcol=2,align=center) "Mean" "St Dev" "Min" "Max"
do i=1,%nvar
do j=1,%nvar
if j==i
next
stats(noprint,fract) hedges(i,j)
report(use=wreport,row=new,atcol=1) labels(i)+"/"+labels(j) %mean sqrt(%variance) %minimum %maximum
end do j
end do i
report(use=wreport,action=format,atcol=2,tocol=5,picture="*.##",align=decimal)
report(use=wreport,action=show)
A similar table is constructed with the portfolio weights, though because those have to add to one between the two elements of a pair, only one direction is shown.
Output
Statistics on Series DLECO
Observations 2609
Sample Mean -0.023126 Variance 4.808126
Standard Error 2.192744 SE of Sample Mean 0.042929
t-Statistic (Mean=0) -0.538703 Signif Level (Mean=0) 0.590138
Skewness -0.233807 Signif Level (Sk=0) 0.000001
Kurtosis (excess) 4.600006 Signif Level (Ku=0) 0.000000
Jarque-Bera 2324.044865 Signif Level (JB=0) 0.000000
Minimum -14.467302 Maximum 14.519496
01-%ile -6.300807 99-%ile 6.119683
05-%ile -3.431977 95-%ile 3.161663
10-%ile -2.462392 90-%ile 2.204196
25-%ile -1.070449 75-%ile 1.144094
Median 0.000000
Statistics on Series DLOIL
Observations 2609
Sample Mean 0.047015 Variance 6.269467
Standard Error 2.503890 SE of Sample Mean 0.049021
t-Statistic (Mean=0) 0.959090 Signif Level (Mean=0) 0.337602
Skewness -0.144539 Signif Level (Sk=0) 0.002593
Kurtosis (excess) 4.598233 Signif Level (Ku=0) 0.000000
Jarque-Bera 2307.585469 Signif Level (JB=0) 0.000000
Minimum -16.544513 Maximum 16.409725
01-%ile -7.089327 99-%ile 6.293575
05-%ile -3.850403 95-%ile 3.771591
10-%ile -2.762245 90-%ile 2.830286
25-%ile -1.282170 75-%ile 1.443507
Median 0.000000
Statistics on Series DLPSE
Observations 2609
Sample Mean 0.011012 Variance 2.652916
Standard Error 1.628778 SE of Sample Mean 0.031888
t-Statistic (Mean=0) 0.345344 Signif Level (Mean=0) 0.729863
Skewness 0.160638 Signif Level (Sk=0) 0.000815
Kurtosis (excess) 3.821450 Signif Level (Ku=0) 0.000000
Jarque-Bera 1598.740511 Signif Level (JB=0) 0.000000
Minimum -8.120620 Maximum 10.841189
01-%ile -4.482048 99-%ile 4.497959
05-%ile -2.713011 95-%ile 2.608114
10-%ile -1.883214 90-%ile 1.685204
25-%ile -0.738329 75-%ile 0.791930
Median 0.029164
Covariance\Correlation Matrix
DLECO DLOIL DLPSE
DLECO 4.806282759 0.23752 0.76512
DLOIL 1.303597467 6.267063950 0.11272
DLPSE 2.731570875 0.459537833 2.651899319
Covariance\Correlation Matrix
DLECOSQ DLOILSQ DLPSESQ
DLECOSQ 152.4438821 0.27812 0.60921
DLOILSQ 55.1928376 258.3457627 0.17157
DLPSESQ 48.1065179 17.6366691 40.9039365
MV-GARCH, BEKK - Estimation by BFGS
Convergence in 66 Iterations. Final criterion was 0.0000075 <= 0.0000100
With Heteroscedasticity/Misspecification Adjusted Standard Errors
Usable Observations 2608
Log Likelihood -14266.9233
Variable Coeff Std Error T-Stat Signif
************************************************************************************
Mean Model(DLECO)
1. DLECO{1} 0.019573414 0.016165336 1.21083 0.22596198
2. DLOIL{1} 0.019096179 0.011376907 1.67850 0.09324884
3. DLPSE{1} 0.067872021 0.019518978 3.47723 0.00050662
4. Constant 0.032041974 0.018403893 1.74104 0.08167599
Mean Model(DLOIL)
5. DLECO{1} -0.011577350 0.017873402 -0.64774 0.51715198
6. DLOIL{1} -0.021110504 0.016192621 -1.30371 0.19233201
7. DLPSE{1} 0.060495454 0.022146678 2.73158 0.00630312
8. Constant 0.085971430 0.045071723 1.90744 0.05646414
Mean Model(DLPSE)
9. DLECO{1} -0.017909124 0.010088659 -1.77517 0.07586918
10. DLOIL{1} 0.005639983 0.007638076 0.73840 0.46026919
11. DLPSE{1} -0.019401489 0.015054125 -1.28878 0.19747379
12. Constant 0.062413352 0.014230776 4.38580 0.00001156
13. C(1,1) 0.265049367 0.046804583 5.66289 0.00000001
14. C(2,1) 0.023039362 0.071797341 0.32089 0.74829045
15. C(2,2) 0.192431324 0.043517339 4.42195 0.00000978
16. C(3,1) 0.081814877 0.018318400 4.46627 0.00000796
17. C(3,2) 0.015920962 0.017930156 0.88794 0.37457125
18. C(3,3) -0.045230593 0.017068857 -2.64989 0.00805180
19. A(1,1) 0.320762056 0.043253900 7.41580 0.00000000
20. A(1,2) -0.007085789 0.043865147 -0.16154 0.87167147
21. A(1,3) 0.062299270 0.018392287 3.38725 0.00070597
22. A(2,1) -0.021366507 0.022823763 -0.93615 0.34919503
23. A(2,2) 0.186518807 0.021765517 8.56946 0.00000000
24. A(2,3) 0.004041510 0.011079484 0.36477 0.71527994
25. A(3,1) -0.131142790 0.047582869 -2.75609 0.00584964
26. A(3,2) 0.062099397 0.048235763 1.28741 0.19795002
27. A(3,3) 0.127785901 0.024826570 5.14714 0.00000026
28. B(1,1) 0.934232198 0.015756523 59.29177 0.00000000
29. B(1,2) -0.000401384 0.014700778 -0.02730 0.97821760
30. B(1,3) -0.020025915 0.006109288 -3.27795 0.00104565
31. B(2,1) 0.009030640 0.007335344 1.23111 0.21828046
32. B(2,2) 0.978721442 0.005522196 177.23411 0.00000000
33. B(2,3) 0.001533730 0.002911686 0.52675 0.59836729
34. B(3,1) 0.039372435 0.012753793 3.08712 0.00202109
35. B(3,2) -0.015055391 0.013812903 -1.08995 0.27573462
36. B(3,3) 0.997043089 0.005832852 170.93577 0.00000000
MV-Diagonal GARCH with VARMA Variances - Estimation by BFGS
Convergence in 83 Iterations. Final criterion was 0.0000059 <= 0.0000100
With Heteroscedasticity/Misspecification Adjusted Standard Errors
Usable Observations 2608
Log Likelihood -15552.1636
Variable Coeff Std Error T-Stat Signif
************************************************************************************
Mean Model(DLECO)
1. DLECO{1} 0.030481095 0.028775942 1.05926 0.28948310
2. DLOIL{1} 0.022755014 0.015499767 1.46809 0.14208047
3. DLPSE{1} 0.073888207 0.035338731 2.09086 0.03654095
4. Constant 0.062999845 0.033642900 1.87260 0.06112303
Mean Model(DLOIL)
5. DLECO{1} -0.009020683 0.029647462 -0.30426 0.76092605
6. DLOIL{1} -0.021887210 0.019119893 -1.14473 0.25231897
7. DLPSE{1} 0.058234330 0.037108880 1.56928 0.11658204
8. Constant 0.100480794 0.042033491 2.39049 0.01682574
Mean Model(DLPSE)
9. DLECO{1} -0.013208908 0.019716507 -0.66994 0.50289501
10. DLOIL{1} -0.003394763 0.010338702 -0.32835 0.74264341
11. DLPSE{1} -0.019496539 0.028937419 -0.67375 0.50047129
12. Constant 0.047274494 0.020382957 2.31931 0.02037798
13. C(1) 0.043627286 0.020167127 2.16329 0.03051910
14. C(2) 0.074197223 0.028661135 2.58877 0.00963181
15. C(3) 0.012666759 0.005366026 2.36055 0.01824799
16. A(1,1) 0.080909044 0.014246091 5.67939 0.00000001
17. A(1,2) -0.000774837 0.005455598 -0.14203 0.88705938
18. A(1,3) 0.005287484 0.017848635 0.29624 0.76704659
19. A(2,1) 0.020036528 0.011906194 1.68287 0.09240105
20. A(2,2) 0.046762537 0.012411699 3.76762 0.00016481
21. A(2,3) -0.005266841 0.011511899 -0.45751 0.64730250
22. A(3,1) 0.015558650 0.004676212 3.32719 0.00087726
23. A(3,2) 0.000359055 0.001713039 0.20960 0.83397894
24. A(3,3) 0.024609894 0.010999392 2.23739 0.02526107
25. B(1,1) 0.893870053 0.017966138 49.75304 0.00000000
26. B(1,2) 0.005828621 0.009470009 0.61548 0.53823639
27. B(1,3) 0.007265283 0.019517978 0.37224 0.70971757
28. B(2,1) -0.024855421 0.016799998 -1.47949 0.13900953
29. B(2,2) 0.937784819 0.017114673 54.79420 0.00000000
30. B(2,3) 0.017666196 0.015281370 1.15606 0.24765621
31. B(3,1) -0.019636762 0.005962557 -3.29335 0.00099003
32. B(3,2) 0.000323408 0.002746993 0.11773 0.90628039
33. B(3,3) 0.974364982 0.013545753 71.93140 0.00000000
MV-CC GARCH with VARMA Variances - Estimation by BFGS
Convergence in 108 Iterations. Final criterion was 0.0000077 <= 0.0000100
With Heteroscedasticity/Misspecification Adjusted Standard Errors
Usable Observations 2608
Log Likelihood -14351.3657
Variable Coeff Std Error T-Stat Signif
************************************************************************************
Mean Model(DLECO)
1. DLECO{1} 0.008120563 0.028093415 0.28906 0.77253871
2. DLOIL{1} 0.022236180 0.013562402 1.63955 0.10109964
3. DLPSE{1} 0.090480364 0.037033431 2.44321 0.01455736
4. Constant 0.049047637 0.027307654 1.79611 0.07247652
Mean Model(DLOIL)
5. DLECO{1} -0.017880464 0.022362459 -0.79958 0.42395705
6. DLOIL{1} -0.014809566 0.022218998 -0.66653 0.50507419
7. DLPSE{1} 0.058604631 0.026386409 2.22102 0.02634990
8. Constant 0.113425743 0.044387899 2.55533 0.01060871
Mean Model(DLPSE)
9. DLECO{1} -0.030288555 0.019681950 -1.53890 0.12382869
10. DLOIL{1} 0.002492577 0.010029190 0.24853 0.80372263
11. DLPSE{1} -0.010741740 0.029875899 -0.35955 0.71918719
12. Constant 0.053292898 0.020544372 2.59404 0.00948558
13. C(1) 0.139104179 0.007197003 19.32807 0.00000000
14. C(2) 0.100069478 0.018494057 5.41090 0.00000006
15. C(3) 0.040684130 0.003300767 12.32566 0.00000000
16. A(1,1) 0.086445848 0.005808065 14.88376 0.00000000
17. A(1,2) 0.008765425 0.004174545 2.09973 0.03575243
18. A(1,3) 0.011142408 0.021051970 0.52928 0.59661051
19. A(2,1) 0.027480313 0.011991635 2.29162 0.02192737
20. A(2,2) 0.052357605 0.012484207 4.19391 0.00002742
21. A(2,3) -0.005371610 0.011771817 -0.45631 0.64816632
22. A(3,1) 0.020769259 0.000670237 30.98792 0.00000000
23. A(3,2) 0.001133823 0.001189635 0.95308 0.34054723
24. A(3,3) 0.020774603 0.006266982 3.31493 0.00091666
25. B(1,1) 0.845396489 0.009769298 86.53606 0.00000000
26. B(1,2) -0.002342899 0.001591898 -1.47176 0.14108458
27. B(1,3) 0.019770958 0.023049327 0.85777 0.39102100
28. B(2,1) -0.049525974 0.022173025 -2.23361 0.02550850
29. B(2,2) 0.933207709 0.013703176 68.10156 0.00000000
30. B(2,3) 0.031718263 0.019276139 1.64547 0.09987342
31. B(3,1) -0.038807523 0.000501056 -77.45154 0.00000000
32. B(3,2) 0.000487241 0.000606292 0.80364 0.42160412
33. B(3,3) 0.984985696 0.007030850 140.09482 0.00000000
34. R(2,1) 0.213325919 0.016222517 13.14999 0.00000000
35. R(3,1) 0.761037385 0.007535435 100.99448 0.00000000
36. R(3,2) 0.078359998 0.018628722 4.20641 0.00002595
MV-DCC GARCH with VARMA Variances - Estimation by BFGS
Convergence in 114 Iterations. Final criterion was 0.0000039 <= 0.0000100
With Heteroscedasticity/Misspecification Adjusted Standard Errors
Usable Observations 2608
Log Likelihood -14234.9849
Variable Coeff Std Error T-Stat Signif
************************************************************************************
Mean Model(DLECO)
1. DLECO{1} 0.012758570 0.026159387 0.48772 0.62574511
2. DLOIL{1} 0.020603364 0.012750235 1.61592 0.10611153
3. DLPSE{1} 0.074930315 0.031732702 2.36130 0.01821117
4. Constant 0.036784075 0.029473222 1.24805 0.21201249
Mean Model(DLOIL)
5. DLECO{1} -0.015225738 0.029651275 -0.51349 0.60760612
6. DLOIL{1} -0.018606639 0.019460378 -0.95613 0.33900688
7. DLPSE{1} 0.059148634 0.033671345 1.75665 0.07897816
8. Constant 0.099333449 0.037088568 2.67828 0.00740021
Mean Model(DLPSE)
9. DLECO{1} -0.025503494 0.017653414 -1.44468 0.14854843
10. DLOIL{1} 0.005030439 0.009221584 0.54551 0.58540483
11. DLPSE{1} -0.013944973 0.026674436 -0.52278 0.60112447
12. Constant 0.059990205 0.019821848 3.02647 0.00247428
13. C(1) 0.059583205 0.021695872 2.74629 0.00602730
14. C(2) 0.082571929 0.030024774 2.75013 0.00595722
15. C(3) 0.018331797 0.005831619 3.14352 0.00166930
16. A(1,1) 0.080523282 0.015875436 5.07219 0.00000039
17. A(1,2) 0.004815119 0.006409883 0.75120 0.45253098
18. A(1,3) 0.002119574 0.018127149 0.11693 0.90691699
19. A(2,1) 0.022571933 0.009509899 2.37352 0.01761945
20. A(2,2) 0.056075358 0.013725253 4.08556 0.00004397
21. A(2,3) -0.004190445 0.013550611 -0.30924 0.75713594
22. A(3,1) 0.015560864 0.004482814 3.47123 0.00051809
23. A(3,2) 0.001559431 0.002201553 0.70833 0.47873912
24. A(3,3) 0.025640971 0.006826337 3.75618 0.00017252
25. B(1,1) 0.888034378 0.023331546 38.06153 0.00000000
26. B(1,2) 0.000527860 0.011287468 0.04677 0.96270042
27. B(1,3) 0.015149804 0.022642399 0.66909 0.50343804
28. B(2,1) -0.030110264 0.014223161 -2.11699 0.03426085
29. B(2,2) 0.927631722 0.018170822 51.05062 0.00000000
30. B(2,3) 0.021181078 0.018152107 1.16687 0.24326446
31. B(3,1) -0.022309075 0.006245723 -3.57190 0.00035441
32. B(3,2) -0.000727499 0.003439264 -0.21153 0.83247558
33. B(3,3) 0.975736155 0.008466516 115.24648 0.00000000
34. DCC(A) 0.018933853 0.003466412 5.46209 0.00000005
35. DCC(B) 0.977636695 0.004348559 224.81855 0.00000000
Univariate Diagnostics for BEKK
ECO
Q(20)r 16.57722 0.68023
Q(20)r^2 34.73182 0.02158
OIL
Q(20)r 10.04333 0.96738
Q(20)r^2 26.82265 0.14033
PSE
Q(20)r 19.13459 0.51309
Q(20)r^2 43.71686 0.00164
Univariate Diagnostics for Diag
ECO
Q(20)r 13.11870 0.87223
Q(20)r^2 19.99362 0.45833
OIL
Q(20)r 10.31916 0.96201
Q(20)r^2 18.11960 0.57953
PSE
Q(20)r 18.47843 0.55592
Q(20)r^2 30.89500 0.05659
Univariate Diagnostics for CCC
ECO
Q(20)r 14.99150 0.77689
Q(20)r^2 32.17422 0.04148
OIL
Q(20)r 9.96097 0.96887
Q(20)r^2 16.22758 0.70241
PSE
Q(20)r 19.76412 0.47277
Q(20)r^2 46.24004 7.47071e-04
Univariate Diagnostics for DCC
ECO
Q(20)r 15.82057 0.72769
Q(20)r^2 20.73236 0.41303
OIL
Q(20)r 10.08686 0.96657
Q(20)r^2 17.20482 0.63964
PSE
Q(20)r 18.45552 0.55743
Q(20)r^2 31.73149 0.04624
Mean St Dev Min Max
ECO/OIL 0.19 0.18 -0.20 0.74
ECO/PSE 1.10 0.30 0.58 1.93
OIL/ECO 0.23 0.22 -0.49 0.80
OIL/PSE 0.13 0.37 -0.80 1.12
PSE/ECO 0.57 0.16 0.26 1.14
PSE/OIL 0.06 0.13 -0.16 0.41
Mean St Dev Min Max
ECO/OIL 0.60 0.14 0.20 0.89
ECO/PSE 0.17 0.27 0.00 1.00
OIL/PSE 0.26 0.15 0.00 0.71
Graphs
Original data (end of day prices)
Squared returns
Time-varying conditional correlations
