4. (14 Points)
a.) Given that rt is a GARCH(1,1) process:
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please, represent rt2 as an ARMA(p,q) process and provide expressions for autoregressive and moving average coefficients. Also comment shortly on the properties of resulting error term. (7 points)
b.) For a series of an emerging market stock return (denoted further as ret_ban) you’ve estimated the (G)ARCH model, which estimates and diagnostic checks are provided below (Tables 1-4). Please, write down the formal specification; point out problems, evident from diagnostic checks and possible ways of enhancing the model. (7 points)
Dependent Variable: RET_BAN |
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Method: ML - ARCH (Marquardt) - Normal distribution |
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Sample (adjusted): 8/01/2000 2/13/2006 |
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Included observations: 1445 after adjustments |
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Convergence achieved after 8 iterations |
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Presample variance: backcast (parameter = 0.7) |
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GARCH = C(2) + C(3)*RESID(-1)^2 + C(4)*RESID(-2)^2 + C(5)*RESID(-3)^2 |
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Variable |
Coefficient |
Std. Error |
z-Statistic |
Prob. |
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C |
0.601556 |
0.044381 |
13.55435 |
0.0000 |
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Variance Equation |
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C |
1.693717 |
0.146653 |
11.54911 |
0.0000 |
RESID(-1)^2 |
0.316590 |
0.041872 |
7.560958 |
0.0000 |
RESID(-2)^2 |
0.199492 |
0.044408 |
4.492297 |
0.0000 |
RESID(-3)^2 |
0.120735 |
0.030461 |
3.963575 |
0.0001 |
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R-squared |
-0.001403 |
Mean dependent var |
0.681075 |
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Adjusted R-squared |
-0.004185 |
S.D. dependent var |
2.123340 |
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S.E. of regression |
2.127779 |
Akaike info criterion |
4.145861 |
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Sum squared resid |
6519.516 |
Schwarz criterion |
4.164116 |
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Log likelihood |
-2990.384 |
Hannan-Quinn criter. |
4.152674 |
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Durbin-Watson stat |
1.852277 |
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Table 1.
Table 2.
Table 3
Heteroskedasticity Test: ARCH |
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F-statistic |
2.295783 |
Prob. F(10,1424) |
0.0114 |
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Obs*R-squared |
22.76811 |
Prob. Chi-Square(10) |
0.0116 |
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Test Equation: |
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Dependent Variable: WGT_RESID^2 |
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Method: Least Squares |
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Sample (adjusted): 8/15/2000 2/13/2006 |
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Included observations: 1435 after adjustments |
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Variable |
Coefficient |
Std. Error |
t-Statistic |
Prob. |
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C |
0.777011 |
0.088337 |
8.796017 |
0.0000 |
WGT_RESID^2(-1) |
-0.003260 |
0.026472 |
-0.123162 |
0.9020 |
WGT_RESID^2(-2) |
-0.009755 |
0.026413 |
-0.369325 |
0.7119 |
WGT_RESID^2(-3) |
-0.043299 |
0.026377 |
-1.641555 |
0.1009 |
WGT_RESID^2(-4) |
0.051612 |
0.026389 |
1.955842 |
0.0507 |
WGT_RESID^2(-5) |
0.001045 |
0.026413 |
0.039569 |
0.9684 |
WGT_RESID^2(-6) |
0.031191 |
0.026411 |
1.180996 |
0.2378 |
WGT_RESID^2(-7) |
0.029046 |
0.026389 |
1.100654 |
0.2712 |
WGT_RESID^2(-8) |
0.055072 |
0.026379 |
2.087776 |
0.0370 |
WGT_RESID^2(-9) |
0.067259 |
0.026419 |
2.545874 |
0.0110 |
WGT_RESID^2(-10) |
0.046030 |
0.026478 |
1.738431 |
0.0824 |
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R-squared |
0.015866 |
Mean dependent var |
1.002920 |
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S.E. of regression |
1.405708 |
Akaike info criterion |
3.526595 |
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F-statistic |
2.295783 |
Durbin-Watson stat |
2.000195 |
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Prob(F-statistic) |
0.011358 |
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Table 4.