Empirical results Directory UMM :Data Elmu:jurnal:M:Multinational Financial Management:

Observations are sampled at weekly intervals. The weekly data ranges from November 6, 1987 to August 28, 1998, which is a 565-data-point series. However, this paper works with rates of return and use the first difference of information variables, and finally all the instruments are used with a one-week lag, relative to the excess return series; that leaves 562 observations expanding from November 27, 1987 to August 28, 1998. Table 1 describes the variables and their symbols used in this paper. All the data are extracted from DATASTREAM. Summary statistics for bank stock returns, risk factors, and instruments used in this paper are presented in Table 2. The mean excess returns for three bank stock portfolios are 0.2551 for Money Center bank, 0.2372 for Large bank, and 0.2452 for Regional bank. These mean excess returns are all greater than 0.1065, the mean excess return for MSCI world equity index. However, their standard deviations are also greater than that of MSCI world equity index, indicating that investors are compensated for a higher risk premium when holding bank stocks. The positive change in the exchange rate reflects the depreciation of the US dollar against the currencies of ten industrialized countries. The coefficients of skewness and excess kurtosis reveal nonnormality in the data. The last two columns in Table 2 report the Ljung-Box portmanteau test statistics for indepen- dence in the return and squared return series up to 24 lags, denoted by Q 24 and Q 2 24 respectively. The Ljung-Box portmanteau test statistics for independence in the standardized residuals are calculated using autocorrelations up to 24 lags, and they follow a x 2 distribution with 24 degrees of freedom. 6 The hypothesis of linear independence is rejected at 5 level for Money Center bank and 1 level for Regional bank. Independence of the squared return series is rejected at 1 level for all three bank portfolio returns, MSCI world equity returns, and the exchange rate changes. Clearly, the nonlinear dependencies are much prevalent than the linear dependencies found in the data and it is consistent with the volatility clustering observed in most financial data: Large small changes in prices tend to be followed by large small changes of either sign. The GARCH model used in this study is well known to capture this property.

6. Empirical results

6 . 1 . Unconditional test of three-factor model : NLSUR 6ia GMM Following Ferson and Harvey 1994, this paper first estimates and tests the unconditional three-factor asset pricing model Eq. 11 where the expected factor risk premia are assumed to be time-invariant as a restricted nonlinear seemingly unrelated regression model NLSUR. The NLSUR via Hansen’s 1982 generalized method of moments GMM, which is valid under weak statistical assumptions, is implemented. To apply the GMM technique, the data used in the estimation must 6 The formula for the Ljung-Box statistic is LBk = TT + 2 k j = 1 r j 2 T − j, where r j is the jth lag autocorrelation, k is the number of autocorrelations, and T is the sample size. Ljung and Box, 1978 Table 1 Variable definitions and symbols Name Symbols Money center Bank America New BAC CMB Chase Manhattan CHM. Banking LN CHNY Citigroup CCI First Chicago NBD FCN Morgan JP CO.PF.A MORG Bank of New York BK Larger bank Wells Fargo WFC First Union FTU FLT Fleet Finl.GP Mellon Bank MEL NCBM Nat.City Bancorp. NCC Nat.City NOB Norwest PNC Bank PNC RNB Republic NY. WB Wachovia Corp. Amsouth Banc. ASO Regional bank Comerica CMA CF Crestar Finl. First Security FSCO Firstar FSR KEY Keycorp Merc.Bancorp MTL Nthn.Trust NTRS Riggs Natl RIGS Signet Banking SBK Star Banc STB US Bancorp Del USB Valley Nat.Bk. VLY Union Planters UPC Information 6ariables Euro-currency LDN US 7 day ECUSD7D SP 500 composite-dividend yield SPDY FRTCM10 US treasury constant maturities 10 year US corporate bond moody’s S’ND AAA FRCBAAA FRCBBAA US corporate bond moody’s S’ND BAA Risk factors MSWRLD MSCI world US US treasury constant maturities 10 year FRTCM10 USTRDW US Index FED-trade weighted Table 2 Summary statistics of bank stock returns, risk factors and instruments a Mean S.D. Skewness Kurtosis Q24 Q 2 24 Indi6idual banks Money center − 0.1377 2.6666 BAC 28.2540 0.2961 95.7098 4.2006 CMB 4.6039 − 0.1748 1.9237 49.5169 281.8171 0.2347 3.8718 0.1329 17.1957 365.9468 51.3717 0.6023 CHNY 0.1498 3.2391 25.1967 4.4007 14.0356 CCI 0.3874 3.4208 0.2565 0.07445 0.6309 39.0412 107.7093 FCN MORG − 1.2863 0.0555 10.9677 20.4297 13.9133 1.5067 0.02055 1.1759 37.2687 142.5659 4.1628 WFC 0.3206 Large − 0.1884 2.1950 BK 32.6233 0.3053 126.8239 4.0567 0.1976 1.1985 30.9891 3.6531 93.3796 0.2745 FTU 3.9844 0.1719 0.8171 6.0529 33.8288 105.9370 FLT − 0.0135 0.8889 26.3407 MEL 12.6556 0.2654 3.6839 − 0.4992 26.0323 47.1987 4.4375 128.7636 0.2022 NCBM 2.9674 0.2334 − 0.1610 0.9782 22.4665 107.1514 NCC NOB 0.0359 0.3616 0.6903 35.5935 76.8951 3.4668 − 0.01301 1.8859 45.6492 3.6239 230.4390 PNC 0.1340 2.8750 0.1368 − 0.5059 4.0154 21.0266 43.9046 RNB WB − 0.0961 0.2614 0.8838 22.0680 54.9341 2.9199 0.5742 1.74571 25.4122 23.6111 2.8732 ASO 0.2627 Regional − 0.0038 1.8839 CMA 25.9542 0.2998 22.8203 3.0293 0.2584 2.4376 19.4834 3.9932 264.9610 CF 0.2469 3.8188 0.2806 − 0.2368 1.8083 56.9787 63.9084 FSCO 1.5854 14.8356 34.5842 FSR 5.6308 0.3341 3.3272 4.8841 35.9988 21.0147 3.2253 11.5172 0.2000 KEY 3.4866 0.2626 − 0.0276 1.4890 13.3112 16.0097 MTL 1.3848 11.3948 35.6350 NTRS 68.2217 0.3322 3.1791 − 0.1265 1.6827 65.7222 5.7595 237.0455 RIGS − 0.0468 5.5220 0.2323 − 2.3421 39.8593 40.1195 0.7907 SBK 0.6384 5.5347 13.1720 STB 55.6055 0.3611 3.5700 − 0.0826 2.2598 18.8591 3.5886 97.2846 0.2687 USB 3.6315 0.2046 0.4646 4.5719 26.2556 25.5229 VLY UPC 3.8087 0.3365 1.6575 25.5378 41.6411 0.2114 Bank portfolios − 0.2883 Money center 1.1116 0.2551 41.3042 166.5553 2.9465 0.2372 2.3308 − 0.0938 0.7425 30.9497 108.1141 Large 2.2828 0.2452 − 0.1537 1.5120 42.7845 139.8178 Regional Risk factors 0.5084 F W 11.2245 0.1065 15.5908 259.7811 1.8389 F INT 0.1480 − 0.00012 0.20301 31.5452 26.6112 0.00248 0.1401 0.8599 21.8488 54.8806 1.1725 0.0175 F FX Instruments 0.2343 − 0.6946 11938 10325 SPDIV − 0.0551 0.0305 − 1.0056 63.6573 115.3779 0.00747 135.3716 − 0.00008 D USTP Table 2 Continued Q 2 24 Skewness Kurtosis Q24 Mean S.D. − 0.000018 0.000667 − 0.2520 44.1744 182.2403 2.9766 D USDP a ‘Money Center’ is the equally weighted average of excess returns on 7 individual money center banks. ‘Large’ is the equally weighted average of excess returns on 11 individual large banks. ‘Regional’ is the equally weighted average of excess returns on 13 individual regional banks. F W is the excess return on the Morgan Stanley world stock total return index in US . F INT is the log first difference of FRTCM10. F FX is the log first difference of USTRDW. SPDIV is the first difference of SP500 dividend yield in excess of ECUSD7D. DUSTP is the first difference of FRTCM10 in excess of ECUSD7D. DUSDP is the first difference of FRCBBAA in excess of FRCBAAA. Q24 and Q 2 24 are the Ljung-Box test statistics of order 24 for serial correlation in the standardized residuals and standardized residuals squared. Denotes statistically significant at the 5 level. Denotes statistically significant at the 1 level. be stationary. As a result, unit root tests are conducted to check if the variables used in the estimation are stationary. The test results not shown here indicate that the null hypothesis of unit root nonstationarity is strongly rejected. The GMM estimator is robust in the sense that one can avoid the usual assumption of homoskedasticity and normality, which are unlikely to hold in these data. 7 The advantage of this approach over the traditional Fama-McBeth approach is that the parameters l, b can be estimated jointly and it explicitly allows for contempora- neous correlations across the N assets, and thus is more efficient. A vector of ones and the contemporaneous values of the factor risk s, F k,t , are used as the instruments in the GMM. The orthogonality conditions therefore imply Eo it F k,t = 0 and Eo it = 0, for all i = 1, ······ ,N and k = 1, ······ ,K. The estimation results are presented in Table 3. The unconditional three-factor model is not rejected at 5 level based on the test of overidentifying restrictions x 28 2 = 11.1657 with a P-value of 0.99 8 . The estimates of factor loadings b’s indicates that almost all banks 27 out of 31 have significant negative factor loadings on interest rate risk at 1 level, 14 banks are sensitive to exchange rate risk, and 20 banks are sensitive to the world market risk. The negative factor loadings on interest rate risk indicate that banks are hurt by unexpected increases in the interest rates. As far as the factor risk premia are concerned, the interest rate risk premium is the only significant factor premium at 5 level with a point estimate of − 0.0006. This negative interest rate risk premium is consistent with previous work of Sweeney and Wagra 1986 and Choi et al. 1992 who use similar approach. 7 An alternative approach used by previous researchers see, e.g., Gibbons, 1982; McElory and Burmeister, 1988; Jorion, 1991; Prasad and Rajan, 1995; Choi and Rajan, 1997; and Choi et al., 1998 is the iterated non-linear seemingly unrelated regression method, which is asymptotically equivalent to maximum-likelihood estimation under the assumption of normality. 8 The total number of moments is 31 × 4 = 124 and the total number of parameters to be estimated is 31 × 3 + 3 = 96; therefore we have 28 degree of freedoms for testing overidentifying restrictions. Table 3 Unconditional three-factor asset pricing model: NLSUR via GMM estimation a b i,W b i,INT b I,FX R it − 355.2634 −4.8714 0.4171 2.8459 0.3219 3.4618 BAC 0.3518 2.1863 0.2257 −5.1180 2.2131 CMB − 408.7090 0.2947 CHNY 2.1192 − 59.6767 − 0.0020 −0.0233 −0.8671 0.1044 0.6772 0.2182 −6.1226 2.2324 CCI − 469.1484 −6.2034 − 366.6609 0.2587 2.1773 0.2274 3.0163 FCN −5.5951 − 145.5067 0.0826 1.5770 0.1032 3.1078 MORG 0.3258 2.2371 0.3475 −5.0166 3.7614 WFC − 363.2796 − 392.0966 −5.5666 0.1305 0.9198 0.2478 2.7552 BK 0.2225 1.7362 0.1764 −5.1118 2.1723 − 324.9226 FTU −4.7744 − 329.8369 0.2923 2.1000 0.2565 2.9069 FLT −5.7802 − 369.6595 0.3316 2.5809 0.1069 1.3117 MEL 0.0526 0.3305 0.1926 −0.3427 1.9128 NCBM − 26.9364 −5.4521 − 280.3459 0.2181 2.1119 0.2778 4.2394 NCC 0.1140 0.9584 0.2660 −7.0917 3.5202 − 421.0542 NOB 0.2746 2.1681 0.0421 PNC 0.5235 − 363.9035 −5.7714 0.0954 0.9517 0.1874 −5.4671 2.9483 RNB − 272.1622 WB −6.9453 0.2434 2.4114 0.1136 1.7723 − 349.7096 0.1204 1.1895 0.0661 −5.1460 1.0294 ASO − 258.4112 −7.0990 − 367.5825 0.2752 2.6534 0.2420 3.6723 CMA −4.7606 − 330.9412 0.1855 1.3220 0.2136 2.4033 CF 0.1604 1.1729 0.0499 −3.0319 0.5763 FSCO − 204.9110 −5.1625 − 300.0107 0.1967 1.6786 0.1525 2.0539 FSR 0.3588 3.2112 0.1685 −5.7198 2.3753 − 318.4414 KEY 0.2195 1.7654 0.1230 MTL 1.5621 − 210.9960 −3.4297 0.1989 1.7876 0.2227 −5.2820 3.1571 NTRS − 291.8416 0.3830 1.8568 0.1025 0.7853 RIGS − 107.4418 −1.0537 0.3359 1.7327 0.2709 −4.7713 2.2056 − 458.4274 SBK −2.9444 − 185.6498 0.0957 0.7500 0.1651 2.0438 STB −3.8584 − 241.6937 0.2475 1.9601 0.2691 3.3624 USB 0.1800 1.3786 0.0000 −1.5709 0.0004 VLY − 101.2892 UPC 0.3995 − 239.7226 2.9676 0.1230 1.4411 −3.5856 − 0.000024 −0.0107 0.0024 −2.1346 0.8381 − 0.000006 Test of overidentifying restrictions J-test: x 25 2 = 11.1657 [P-value = 0.9980]. a r it = b i,INT o INT,t +l INT +b i,FX o FX,t +l FX +b i,W o W,T +l W +o it Ö i where r it represents the excess bank stock returns, o k,t ’s are the de-meaned values of risk factors, l’s are the risk premia associated with the risk factors, and b’s are the banks’ sensitivities to the risk factors. The instruments in the GMM estimations are a constant and the risk measures. The x 2 test is the minimized values of the GMM criterion function for the system. Robust t-statistics are given in parentheses. Indicate statistically significant at the 5 level. Indicate statistically significant at the 1 level. 6 . 2 . Conditional tests of three-factor model : pricing kernel approach Since only the interest rate risk premium is detected in unconditional three-factor model and previous studies have found strong evidence of time-varying risk premia, it is interesting to see if any evidence of time-varying risk premia with respect to the assumed three risk factors can be found, in particular, the world market and exchange rate risks. The conditional three-factor model Eq. 7 for each of the three bank types is estimated based on the pricing kernel approach. The empirical results are presented in Table 4. 9 As can be seen from the table, the model applied to all three different bank types can not be rejected at any conventional levels based on the J-test of overidentifying restrictions. Consequently, the hypothesis testing concerning the pricing of risk factors can be conducted. Specifically, a Wald statistic is computed to test the null hypothesis that all 8’s coefficients of instrumental variables are zero with respect to a particular risk factor. First, none of the Wald statistics is significant for Money Center bank, indicating that the selected instruments are not useful in predicting risk premia for Money Center bank. However, in contrary to the evidence found Table 4 Conditional three-factor asset pricing model: GMM estimation a d.f. Null hypothesis Wald Money center Large Regional 49.8133 [0.0002] 1. H0: 8 j = 8 W j = 8 INT j = 8 FX j = 0; 47.4026 [0.0005] 22.4233 [0.3180] 20 Ö j = CONSTANT, SPDIV, D USTP, DUSDP, F W 19.6381 [0.1863] 15 2. H0: 8 W j = 8 INT j = 8 FX j = 0; 41.0977 [0.0003] 44.2926 [0.0001] Ö j = SPDIV, DUSTP, DUSDP, F W 2.8174 [0.7281] 3. H0: 8 W j = 0; Öj = CONSTANT, 12.4625 [0.0290] 5 8.4402 [0.1336] SPDIV, DUSTP, DUSDP, F W 4 9.8988 [0.0422] 5.9154 [0.2056] 0.9424 [0.8428] 4. H0: 8 W j = 0; Öj = SPDIV, D USTP, DUSDP, F W 5. H0: 8 INT j = 0; Öj = CONSTANT, 5 3.7484 [0.9184] 5.8297 [0.3231] 11.6315 [0.0402] SPDIV, DUSTP, DUSDP, F W 3.2053 [0.5862] 6.1020 [0.1917] 4 6. H0: 8 INT j = 0; Öj = SPDIV, 2.2828 [0.6839] D USTP, DUSDP, F W 6.3888 [0.5241] 22.3676 [0.0004] 7. H0: 8 FX j = 0; Öj = CONSTANT, 5 21.4304 [0.0007] SPDIV, DUSTP, DUSDP, F W 4 2.5891 [0.2702] 20.8759 [0.0003] 21.4631 [0.0003] 8. H0: 8 FX j = 0; Öj = SPDIV, D USTP, DUSDP, F W x 45 2 = 24.9728 x 25 2 = 17.9905 x 55 2 = 37.6710 Test of Overidentifying Restrictions [0.9642]. J-test [0.9933] [0.6287] a E[r it Z t−1 ] k d k,t−1 Co6r i,t ; F k,t Z t−1 Ö k = W, INT, FX d 0,t−1 = − Z t−1 8 d k,t−1 = Z t−1 8 k , Z t−1 = {CONSTAMT,SPDIV,DUSTP,DUSDP,F W } t−1 P-values are in the brackets. 9 Since the pricing of three risk factors is the main focus in this paper, only the hypothesis testing results are reported and parameter estimates based on the pricing kernel approach are not shown, but are available upon on request. with respect to Money Center bank, significant pricing of time-varying risk factor is detected for Large bank. That is, the joint null hypothesis of constant risk prices is strongly rejected at 1 level based on the Wald test. In particular, the time-vary- ing risk price basically comes from the exchange rate risk because the null hypothesis of constant price of exchange rate risk is rejected at 1 level with a P-value of 0.0003. Finally, for Regional bank, the joint null hypothesis of constant risk prices is strongly rejected at 1 level with a P-value of 0.0001 based on the Wald test. The time-varying risk prices come from two sources: exchange rate risk and world market risk because the null hypotheses of constant price of exchange rate risk and world market risk is rejected at 1 and 5 level, respectively. In additional, significant price of interest rate risk is detected at 5 level although it is not time varying. Overall the empirical evidence based on the pricing kernel approach indicates that exchange rate risk is an important risk factor in describing the dynamics of risk premia found in the US bank stock returns, especially for the Large and Regional banks. This evidence of time-varying price of exchange risk is consistent with previous work of Choi et al. 1998 in a domestic context, and of Ferson and Harvey 1993, Dumas and Solnik 1995 and Tai 1999a in an international context. It may explain why previous researchers are not able to detect significant pricing of exchange rate risk when restricting themselves in an unconditional framework. 10 Although the pricing kernel estimation is parsimonious in the sense that re- searchers do not need to specify the dynamics of the conditional second moments, this parsimony also comes with cost. That is its inability to answer questions like ‘‘What does the fitted risk premium look like?’’ and ‘‘What do the fitted conditional covariances look like?’’ To answer these interesting questions, the conditional second moments in the asset pricing models should be explicitly modeled. 11 In addition, one would expect the interest rate risk to receive a non-zero price for bank stock returns. Therefore, in the next section a parsimonious parameterization of multivariate GARCH in mean model is employed to explicitly deal with these problems. 6 . 3 . Conditional tests of three-factor model : MGARCH-M Given the computational complexity of estimating a multivariate system under the GARCH framework, three equally weighted bank stock portfolios are studied, namely those of Money Center bank, Large Bank, and Regional bank. The 10 For example, Jorion 1991 tests unconditional multi-factor asset pricing models and fail to find significant evidence of exchange risk pricing in the US stock market. Hamao 1988 also can not find any evidence of exchange risk pricing in the Japanese stock market. 11 Turtle et al. 1994, De Santis and Gerard 1997, 1998, Tai 1998, 1999b, and among others are good examples on how to apply a multivariate GARCH process to model the dynamics of the second moments of asset returns in testing asset pricing models. estimation results of conditional three-factor asset pricing model Eq. 25 using MGARCH-M approach are presented in Table 5. Panel A reports QML estimates of the parameters for the model. Regarding the factor betas, significant interest rate betas are found for all three bank portfolios at 1 significance level. Consistent with conventional wisdom, bank stock returns are very sensitive to the changes in interest rates, and thus expose to interest rate risk. Exchange rate beta is significant for Money Center bank at 5 level, and for Large bank at 10 level. World market beta is significant for Large bank at 5 level. Finding significant factor betas does not necessarily imply that market participants care about those factor risks because they may be diversifiable, and thus are not priced. To examine whether the chosen three risk factors are priced by the market participants, we test whether the estimated pricing parameters are statistically significant. As can be seen in Panel A, strong evidence of GARCH in mean effects are found for the dynamics of interest rate and exchange rate risk factors since the parameters l 1 , c 1 are all significant at 1 level, and thus it has significant impact on the dynamics of risk premia for the bank portfolio returns. However, the GARCH in mean effect is marginally significant at 10 level for the MSCI world equity index, implying that after accounting for the time-varying interest rate and exchange rate risk premia in asset returns, the world market risk does not play a major role in explaining the dynamics of bank portfolio returns. To further examine the factor risk pricing, we conduct several hypothesis testing in Panel C. For example, the null hypotheses of constant risk premia with respect to interest rate and exchange rate risk factors are strongly rejected at 1 level, and it is rejected at 10 level for the world market risk. The significant evidence of time-varying interest rate, and exchange rate risk premia found in the US bank data points out the advantage of MGARCH-M approach over the previous two approaches. This advantage comes directly from the explicit modeling of conditional volatilities in both asset returns and risk factors, which is ignored in the other two approaches. Next, consider the estimated parameters for the conditional variance processes. With the exception of parameter a for MSCI world equity index, all the elements in the vectors a and b are statistically significant at 1 level, implying that strong GARCH effect is present for all the return series. In addition, the estimates satisfy the stationarity conditions for all the variance and covariance processes. 12 Panel B contains some diagnostic statistics on the standardized residuals o t h t − 12 and the standardized residuals squared o t 2 h t − 1 . With the exception of Money Center bank, the null hypothesis of linear independency can not be rejected for all the series, as evidenced by the insignificant Ljung Box statistics of order 24 for the standardized residuals Q24. Similarly, the null hypothesis of nonlinear indepen- dency can not be rejected based on the insignificant Ljung Box statistics of order 24 for the standardized residuals squared Q 2 24 except for the MSCI world equity index. Overall, the conditional factor asset pricing model with MGARCH-M parameterization effectively eliminates most of the linear and nonlinear dependen- 12 For the process in H t to be covariance stationary, the condition a i a j + b i b j B 1 Öi,j has to be satisfied. Bollerslev, 1986; De Santis and Gerard, 1997, 1998 C .- S . Tai J . of Multi . Fin . Manag . 10 2000 397 – 420 415 Table 5 Quasi-maximum likelihood estimation of conditional three-factor asset pricing model: multivariate GARCH1,1-M a Panel A: parameter estimates w w 1 c l c 1 l 1 0.0177 0.0009 0.0008 − 0.0002 0.0005 − 3E-06 1.1E-06 − 0.4028 0.9691 0.0879 0.2619 0.0100 F W F INT F FX Regional Large Money center − 667.4991 − 464.5774 b INT,t − 570.1476 88.8139 67.5516 62.2521 0.1012 0.0532 F FX,t 0.1658 0.0672 0.0832 0.0669 0.0510 0.0226 b W,t 0.0445 0.0340 0.0394 0.0431 0.2037 0.1842 0.1905 0.0204 0.3966 0.1724 0.0168 − 0.0001 0.0247 A 0.0219 0.0490 0.0243 0.6269 0.9711 0.8694 0.0046 0.9662 0.0063 B 0.9736 0.0056 0.9290 0.0167 0.0057 0.0667 Panel B: residual diagnostics 11.2759 1.4823 1.2474 1.8012 Kurtosis 0.1226 0.3383 2.8664 2.7552 3012.58 48.0432 B–J 94.3194 69.7296 Q24 15.7124 43.3158 26.8584 33.5961 32.1376 19.1102 260.7326 26.8329 18.9065 22.9778 12.1728 22.0359 Q 2 24 Likelihood function: 16040.46 Panel C: hypothesis testing concerning risk premia Wald Null hypothesis d.f. P-value 0.0000 6 262.3456 1. H0: w = w 1 = l = l 1 = c = c 1 = 212.0988 3 0.0000 2. H0: w 1 = l 1 = c 1 = 3. H0: w = w 1 = 3.8815 2 0.1436 C .- S . Tai J . of Multi . Fin . Manag . 10 2000 397 – 420 416 Table 5 Continued Panel C: hypothesis testing concerning risk premia Wald d.f. P-value Null hypothesis 0.0773 1 3.1214 4. H0: w 1 = 5. H0: l = l 1 = 15.0429 2 0.0005 6. H0: l 1 = 0.0002 13.6927 1 21.0820 2 0.0000 7. H0: c = c 1 = 1 8. H0: c 1 = 0.0000 20.9928 Panel D: predicted weekly time-varying risk premium and conditional volatility Money center Large Regional 0.1997 0.1733 0.1415 Avg. risk premium 0.0035 0.0045 0.0040 Avg. world Mkt risk premium − 0.0025 Avg. FX risk premium − 0.0030 − 0.0049 0.2012 0.1718 0.1400 Avg. interest rate risk premium Avg. conditional STD. 2.2588 2.2170 2.8527 a r i,t = w + w 1 h w,t +o W,t b iW + l + l 1 h INT,t +o INT,t b iINT + c + c 1 h FX,t +o FX,t b iFX +o i,t i = Money Center, Large, Regional F W,t = w + w 1 h w,t +o w,t l W,t = EF W,t V t−1 = w + w 1 h w,t F INT,t = l + l 1 h INT,t +o INT,t l INT,t = EF INT,t V t−1 = l + l 1 h INT,t F FX,t = c + c 1 h FXT,t +o FX,t l FX,t = EF FX,t V t−1 = c + c 1 h FX,t o t V t−1 N0, H t H t = H ii−aa−bb+aao t−1 o t−1 + bbH t−1 where H t is a 6×6 conditional covariance matrix of three bank portfolio returns and three risk factors. Q24 and Q 2 24 are the Ljung-Box test statistics of order 24 for serial correlation in the standardized residuals and standardized residuals squared. B–J is the Bera-Jarque test statistic for normality. Robust standard errors are given in parentheses. Denote statistical significance at the 5 level. Denote statistical significance at the 1 level. cies found in the raw data. However, the conditional normality assumption is rejected for most cases according to the index of excess Kurtosis and Bera-Jarque test statistics, and that is why QML testing procedures are used. Because the three-factor asset pricing model with MGARCH-M process is fully parameterized, some interesting statistics can be recovered in this study. Panel D contains those statistics for the estimated time-varying risk premia and conditional volatility. For example, the estimated weekly total time-varying risk premium is 0.1997 for Money Center bank, 0.1733 for Large Bank, and 0.1415 for Regional Bank. We can further decompose the estimated total time-varying risk premium into three components: world market risk premium, foreign exchange risk premium, and interest rate risk premium. For example, the estimated weekly world market risk premium is 0.0035 for Money Center bank, 0.0045 for Large Bank, and 0.004 for Regional Bank. The estimated weekly foreign exchange risk premium is − 0.0049 for Money Center Bank, − 0.0030 for Large Bank, and − 0.0025 for Regional Bank. Finally, the estimated weekly interest rate risk premium is 0.2012 for Money Center bank, 0.1718 for Large Bank, and 0.14 for Regional Bank. Clearly, the interest rate risk premium is the major component in describing the dynamics of the US bank portfolio returns. Panel D also reports the estimated conditional volatility for each bank portfolio. The estimated weekly conditional volatilities are 2.8527, 2.2587, and 2.2170 for Money Center Bank, Large Bank, and Regional Bank, respectively.

7. Conclusion