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Financial Econometrics Tutorial Exercise 9   Solutions Question 1 AR(8) model is as follows: ARIMA regression Sample:  2 – 3907                               Number of obs      =      3906                                                 Wald chi2(8)       =    229.32 Log likelihood =  11708.15                      Prob > chi2        =    0.0000 ——————————————————————————              |                 OPG      D.lftse |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval] ————-+—————————————————————- lftse        |        _cons |  -.0000123    .000184    -0.07   0.947    -.0003729    .0003483 ————-+—————————————————————- ARMA         |           ar |          L1. |  -.0450127    .010497    -4.29   0.000    -.0655865   -.0244389          L2. |  -.0500389   .0090027    -5.56   0.000    -.0676839    -.032394          L3. |  -.0831168   .0091547    -9.08   0.000    -.1010596   -.0651739          L4. |   .0544616   .0092167     5.91   0.000     .0363972    .0725259          L5. |  -.0508978   .0091385    -5.57   0.000    -.0688089   -.0329866          L6. |  -.0361694   .0099771    -3.63   0.000    -.0557241   -.0166147          L7. |   .0298347   .0102067     2.92   0.003     .0098299    .0498395          L8. |   .0258334   .0106135     2.43   0.015     .0050313    .0466356 ————-+—————————————————————-       /sigma |   .0120773   .0000733   164.87   0.000     .0119337    .0122208 —————————————————————————— Note: The test of the variance against zero is one sided, and the two-sided confidence interval is truncated at zero. LM test for GARCH effects is based on the following auxiliary regression:       Source |       SS       df       MS              Number of obs =    3894 ————-+——————————           F( 12,  3881) =   92.89        Model |  .000144178    12  .000012015           Prob > F      =  0.0000     Residual |  .000501973  3881  1.2934e-07           R-squared     =  0.2231 ————-+——————————           Adj R-squared =  0.2207        Total |  .000646152  3893  1.6598e-07           Root MSE      =  .00036 ——————————————————————————           e2 |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval] ————-+—————————————————————-           e2 |          L1. |   .0106259   .0160427     0.66   0.508    -.0208271    .0420789          L2. |   .1376461   .0160188     8.59   0.000     .1062401    .1690522          L3. |   .1900582   .0160727    11.82   0.000     .1585465      .22157          L4. |   .1341395   .0163377     8.21   0.000     .1021082    .1661708          L5. |   .1246511   .0164634     7.57   0.000     .0923734    .1569288          L6. |  -.0087989   .0165814    -0.53   0.596    -.0413081    .0237103          L7. |  -.0185014   .0165813    -1.12   0.265    -.0510103    .0140075          L8. |  -.0442693   .0164625    -2.69   0.007    -.0765453   -.0119934          L9. |   .0519385   .0163379     3.18   0.001     .0199068    .0839701         L10. |   .1093726   .0160733     6.80   0.000     .0778596    .1408856         L11. |   .0551023   .0159913     3.45   0.001     .0237501    .0864546         L12. |   .0319695   .0160181     2.00   0.046     .0005648    .0633742              |        _cons |   .0000327   6.82e-06     4.79   0.000     .0000193    .0000461 —————————————————————————— The null hypothesis is:  H0:d1=…=d12=0. The test statistic is:       t = TR2 ~ c2(12)  if H0 is true. T = 3894         R2 = 0.2231     Þ        t = 3894´0.2231 = 868.9 Decision rule:  Accept H0 if t £                         Reject H0 if t >  from c2(12) is 21.03       Þ        Decision is reject H0 Þ        ARCH or GARCH effects are present. Question 2 The three estimations required in this question are shown below. 1.         AR(8)-ARCH(1) model: ARIMA regression Sample:  2 – 3907                               Number of obs      =      3906                                                 Wald chi2(8)       =    229.32 Log likelihood =  11708.15                      Prob > chi2        =    0.0000 ——————————————————————————              |                 OPG      D.lftse |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval] ————-+—————————————————————- lftse        |        _cons |  -.0000123    .000184    -0.07   0.947    -.0003729    .0003483 ————-+—————————————————————- ARMA         |           ar |          L1. |  -.0450127    .010497    -4.29   0.000    -.0655865   -.0244389          L2. |  -.0500389   .0090027    -5.56   0.000    -.0676839    -.032394          L3. |  -.0831168   .0091547    -9.08   0.000    -.1010596   -.0651739          L4. |   .0544616   .0092167     5.91   0.000     .0363972    .0725259          L5. |  -.0508978   .0091385    -5.57   0.000    -.0688089   -.0329866          L6. |  -.0361694   .0099771    -3.63   0.000    -.0557241   -.0166147          L7. |   .0298347   .0102067     2.92   0.003     .0098299    .0498395          L8. |   .0258334   .0106135     2.43   0.015     .0050313    .0466356 ————-+—————————————————————-       /sigma |   .0120773   .0000733   164.87   0.000     .0119337    .0122208 —————————————————————————— Note: The test of the variance against zero is one sided, and the two-sided confidence interval is truncated at zero. 2.         AR(8)-ARCH(13) model: ARCH family regression Sample: 10 – 3907                                  Number of obs   =      3898 Distribution: Gaussian                             Wald chi2(8)    =     20.30 Log likelihood =  12499.32                         Prob > chi2     =    0.0093 ——————————————————————————              |                 OPG      D.lftse |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval] ————-+—————————————————————- lftse        |        lftse |          LD. |  -.0507088   .0164901    -3.08   0.002    -.0830289   -.0183887         L2D. |  -.0310939   .0172248    -1.81   0.071     -.064854    .0026662         L3D. |  -.0343304   .0169922    -2.02   0.043    -.0676346   -.0010263         L4D. |   .0132373   .0166647     0.79   0.427    -.0194249    .0458995         L5D. |  -.0271206   .0158592    -1.71   0.087    -.0582041    .0039628         L6D. |   -.009055   .0161069    -0.56   0.574     -.040624    .0225139         L7D. |   .0121692   .0159335     0.76   0.445    -.0190599    .0433984         L8D. |  -.0083213   .0164358    -0.51   0.613    -.0405348    .0238923              |        _cons |   .0004391   .0001313     3.34   0.001     .0001817    .0006964 ————-+—————————————————————- ARCH         |         arch |          L1. |   .0644141   .0148145     4.35   0.000     .0353783    .0934499          L2. |   .1306674   .0202931     6.44   0.000     .0908936    .1704412          L3. |   .1210375   .0202321     5.98   0.000     .0813833    .1606916          L4. |   .1117587   .0172343     6.48   0.000       .07798    .1455373          L5. |   .0832166   .0186993     4.45   0.000     .0465666    .1198666          L6. |   .0809189   .0182995     4.42   0.000     .0450525    .1167853          L7. |   .0378929   .0134227     2.82   0.005     .0115849    .0642008          L8. |   .0602592   .0159361     3.78   0.000     .0290251    .0914933          L9. |   .0291812   .0147068     1.98   0.047     .0003565     .058006         L10. |   .0335833     .01459     2.30   0.021     .0049873    .0621793         L11. |   .0703043   .0153343     4.58   0.000     .0402497    .1003589         L12. |   .0405259   .0134845     3.01   0.003     .0140968    .0669551         L13. |    .035576   .0149039     2.39   0.017     .0063649    .0647871              |        _cons |   .0000172   1.64e-06    10.44   0.000     .0000139    .0000204 —————————————————————————— 3.         AR(3)-ARCH(12) model: ARCH family regression Sample: 5 – 3907                                   Number of obs   =      3903 Distribution: Gaussian                             Wald chi2(3)    =     16.64 Log likelihood =  12508.43                         Prob > chi2     =    0.0008 ——————————————————————————              |                 OPG      D.lftse |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval] ————-+—————————————————————- lftse        |        lftse |          LD. |  -.0506947   .0163397    -3.10   0.002    -.0827199   -.0186695         L2D. |  -.0297629   .0171462    -1.74   0.083     -.063369    .0038431         L3D. |  -.0358244   .0169617    -2.11   0.035    -.0690687   -.0025801              |        _cons |   .0004239   .0001304     3.25   0.001     .0001684    .0006794 ————-+—————————————————————- ARCH         |         arch |          L1. |   .0628625   .0145925     4.31   0.000     .0342617    .0914633          L2. |   .1328414   .0202547     6.56   0.000      .093143    .1725398          L3. |   .1238245    .020145     6.15   0.000      .084341     .163308          L4. |   .1120446   .0169238     6.62   0.000     .0788745    .1452147          L5. |   .0877618   .0188854     4.65   0.000      .050747    .1247765          L6. |   .0850323    .018756     4.53   0.000     .0482712    .1217934          L7. |   .0455209   .0138337     3.29   0.001     .0184075    .0726344          L8. |   .0602277   .0154362     3.90   0.000     .0299734     .090482          L9. |   .0351049    .014796     2.37   0.018     .0061052    .0641046         L10. |   .0355681   .0147577     2.41   0.016     .0066435    .0644926         L11. |   .0713004   .0153886     4.63   0.000     .0411393    .1014615         L12. |   .0415666   .0136439     3.05   0.002      .014825    .0683082              |        _cons |    .000018   1.64e-06    10.98   0.000     .0000148    .0000212 —————————————————————————— Question 3 The AR(3)-GARCH(1,1) model is as follows: ARCH family regression Sample: 5 – 3907                                   Number of obs   =      3903 Distribution: Gaussian                             Wald chi2(3)    =     16.16 Log likelihood =   12515.7                         Prob > chi2     =    0.0011 ——————————————————————————              |                 OPG      D.lftse |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval] ————-+—————————————————————- lftse        |        lftse |          LD. |   -.053993   .0174066    -3.10   0.002    -.0881093   -.0198767         L2D. |  -.0295708    .016792    -1.76   0.078    -.0624824    .0033409         L3D. |  -.0330701   .0164831    -2.01   0.045    -.0653765   -.0007638              |        _cons |   .0004082    .000131     3.12   0.002     .0001514    .0006651 ————-+—————————————————————- ARCH         |         arch |          L1. |   .1036631   .0076451    13.56   0.000     .0886789    .1186472              |        garch |          L1. |    .888038   .0079332   111.94   0.000     .8724892    .9035868              |        _cons |   1.38e-06   2.42e-07     5.72   0.000     9.09e-07    1.86e-06 —————————————————————————— The test to determine whether a fitted model containing an ARCH or GARCH component adequately describes the time-path of the conditional variance, are based on the property that ut can be written vtst, where vt = ut/st ~ N(0,1). We can estimate vt using A Ljung-Box test can be used to determine whether  are white noise. The null hypthesis is: H0:rk=0 for k=1…12, where rk is the ACF (autocorrelation function) for vt. The test statistic is: t = ~ c2(12)  if H0 is true. Portmanteau test for white noise —————————————  Portmanteau (Q) statistic =     5.6995  Prob > chi2(13)           =     0.9564 t = 5.70            from c2(13) is 22.362     Þ        Decision is accept H0  Þ        are white noise, so the fitted model adequately describes the time-path of the conditional variance. Question 4 The GJR threshold GARCH (or TARCH) model is as follows: ARCH family regression Sample: 5 – 3907                                   Number of obs   =      3903 Distribution: Gaussian                             Wald chi2(3)    =      9.83 Log pseudolikelihood =  12591.65                   Prob > chi2     =    0.0200 ——————————————————————————              |             Semirobust      D.lftse |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval] ————-+—————————————————————- lftse        |        lftse |          LD. |  -.0461048   .0165137    -2.79   0.005     -.078471   -.0137386         L2D. |  -.0215618   .0176625    -1.22   0.222    -.0561797    .0130561         L3D. |  -.0204926   .0171727    -1.19   0.233    -.0541505    .0131654              |        _cons |   .0000192   .0001339     0.14   0.886    -.0002431    .0002816 ————-+—————————————————————- ARCH         |         arch |          L1. |    .145324   .0204361     7.11   0.000       .10527     .185378              |        tarch |          L1. |   -.152914   .0217008    -7.05   0.000    -.1954467   -.1103813              |        garch |          L1. |   .9147153   .0122113    74.91   0.000     .8907817     .938649              |        _cons |   1.63e-06   3.60e-07     4.53   0.000     9.27e-07    2.34e-06 —————————————————————————— This model suggests the ARCH coefficient for a negative shock (=0.1453) is much larger than the ARCH coefficient for a positive shock (=0.1453–0.1529=–0.0024, not significantly different from zero). Therefore the asymmetry is large and highly significant.

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