Applications of state space models in finance

90 6 Time-varying market beta risk of pan-European sectors
Table 6.3: Comparison of different GARCH(1,1) specifications.
To decide whether (i) the innovations zt in (6.7) should be modeled by a normal or a standardized Student-t distribution, (ii) asymmetric
effects should be explicitly taken into account, and (iii) a nonzero constant should be included in the mean equation in (6.6), eight different
GARCH(1,1) specifications have been analyzed. This table reports the estimated values of the corresponding log-likelihood functions and the
Bayesian information criteria.
Sector symmetric GARCH asymmetric GARCH
[zt ∼ N, µ �= 0] [zt ∼ N, µ = 0] [zt ∼ t, µ �= 0] [zt ∼ t, µ = 0] [zt ∼ N, µ �= 0] [zt ∼ N, µ = 0] [zt ∼ t, µ �= 0] [zt ∼ t, µ = 0]
log L BIC log L BIC log L BIC log L BIC log L BIC log L BIC log L BIC log L BIC
Broad 2215.9 −4.91 2209.6 −4.90 2228.5 −4.93 2220.1 −4.92 2226.2 −4.93 2223.5 −4.93 2234.6 −4.94 2229.5 −4.93
Automobiles 1871.9 −4.14 1870.7 −4.15 1892.5 −4.18 1890.6 −4.19 1879.9 −4.15 1879.7 −4.16 1895.0 −4.18 1893.9 −4.18
Banks 2112.8 −4.68 2108.0 −4.68 2132.4 −4.72 2124.1 −4.71 2119.3 −4.69 2116.7 −4.69 2136.0 −4.72 2129.9 −4.71
Basics 1988.2 −4.40 1985.5 −4.40 2013.8 −4.45 2009.6 −4.45 1995.7 −4.41 1993.8 −4.42 2017.7 −4.45 2014.6 −4.45
Chemicals 2110.1 −4.67 2106.3 −4.67 2117.5 −4.68 2112.2 −4.68 2120.7 −4.69 2119.3 −4.69 2125.5 −4.69 2122.8 −4.70
Construction 2112.9 −4.68 2109.8 −4.68 2129.7 −4.71 2124.3 −4.71 2126.0 −4.70 2124.6 −4.71 2137.6 −4.72 2134.5 −4.72
Financials 2144.4 −4.75 2137.4 −4.74 2175.7 −4.81 2168.1 −4.80 2156.2 −4.77 2152.0 −4.77 2181.6 −4.82 2175.3 −4.81
Food 2240.5 −4.97 2238.1 −4.97 2257.3 −5.00 2252.1 −4.99 2247.5 −4.97 2246.3 −4.98 2261.8 −5.00 2258.0 −5.00
Healthcare 2083.3 −4.61 2076.7 −4.61 2090.4 −4.62 2083.8 −4.62 2088.4 −4.62 2084.6 −4.62 2093.1 −4.62 2088.4 −4.62
Industrials 2144.2 −4.75 2137.8 −4.74 2158.9 −4.78 2150.9 −4.77 2154.8 −4.77 2151.3 −4.77 2164.3 −4.78 2158.9 −4.78
Insurance 1973.8 −4.37 1971.3 −4.37 2006.3 −4.44 2003.1 −4.44 1981.3 −4.38 1980.4 −4.39 2008.6 −4.43 2006.2 −4.44
Media 1927.4 −4.27 1925.1 −4.27 1941.8 −4.29 1937.5 −4.29 1930.7 −4.27 1929.8 −4.27 1943.5 −4.29 1940.7 −4.29
Oil & Gas 2048.7 −4.54 2044.0 −4.53 2056.8 −4.55 2052.6 −4.55 2061.0 −4.56 2059.4 −4.56 2066.7 −4.56 2064.9 −4.57
Personal 1570.8 −4.56 1567.7 −4.56 1577.9 −4.57 1574.2 −4.57 1582.9 −4.59 1582.3 −4.60 1587.4 −4.59 1586.2 −4.60
Retail 1466.1 −4.25 1464.6 −4.26 1521.2 −4.41 1517.6 −4.41 1478.8 −4.28 1478.5 −4.29 1526.2 −4.41 1524.1 −4.42
Technology 1780.0 −3.94 1776.9 −3.94 1793.5 −3.96 1788.3 −3.96 1781.8 −3.93 1779.5 −3.94 1794.3 −3.96 1790.2 −3.95
Telecom 1863.2 −4.12 1858.5 −4.12 1866.1 −4.12 1861.0 −4.12 1864.9 −4.12 1861.3 −4.12 1867.4 −4.12 1863.3 −4.12
Travel 1640.8 −4.77 1637.4 −4.77 1652.3 −4.79 1649.7 −4.79 1653.5 −4.79 1652.9 −4.80 FTC a FTC FTC FTC
Utilities 2270.9 −5.03 2265.7 −5.03 2284.4 −5.06 2276.1 −5.04 2273.4 −5.03 2269.7 −5.03 2285.6 −5.05 2278.6 −5.04
Mean BIC −4.55 −4.55 −4.58 −4.58 −4.56 −4.56 −4.57 −4.57
a Failed to converge.