The estimation results are reported in Table 1.4. The effect of the illiquidity risk is positive and significant in our Model as well as in (M.2), and (M.4). ˆβEupBC amounts to
0.0217 (with a standard deviation of 0.004). These results are in line with the previous findings of Chen et al. (2002), Elton et al. (2001), and Kagraoka (2010): The more illiquid the bond, the higher the expected credit spread.
Figure 1.2 shows the percentages of the ordered eigenvalues related to the unobserved factor structure in its static form as well as the eigenvalues obtained after applying on the estimated factors a VAR-regression. By using the penalty term of PC1 in our algorithm, we estimate ˆdEup= 11. The test K3 of Bai and Ng (2007), however, detects
the presence of only 2 primitive shocks. This result is confirmed by the information criterion IC1,nT of Hallin and Liˇska (2007), which suggests the presence of 2 dynamic factors in all models. The ED criterion of Onatski (2010) is optimized at 7, 6, 2, and 2 for Models (M.1), (M.2), (M.3), and (M.4) respectively. The criteria of Bai (2004) indicate the presence of at least 2 unit root factors in all models except for (M.3) and (M.4), where the unit root source seems to be automatically integrated by the lags of Yit. 1.3 (a).
Our time-varying estimates ˆαkt and their corresponding 95% confidence intervals are
depicted in Figure 1.3(a). The confidence intervals of the default risk parameters indicate that the rating effects are statistically significant, except for class A during the time between January and February in 2008. The part of the variance explained by the default risk accounts for 24.06%. This result agrees with the results of most research on
Chapter 1. Panel Models with Unknown Number of Unobserved Factors 33
Figure 1.2: (a) The screeplot of the eigenvalues obtained from the matrix of ˆ
Σ( ˆβCupBC, ˆdEup); (b) Proportions of the (squared) eigenvalues obtained from the resid-
uals of the VAR-regression (with p = 1) applied on the estimated factors (after inte- grating the first 3 I(1) factors).
Figure 1.3: (a) the time series of the estimated rating effects; (b) the EupBC estimated first and second common factors.
Regressor EupBC M.1 M.2 M.3 M.4 Xt 0.0217∗∗∗ - 0.3013∗∗∗ - 0.0047∗∗∗ (0.004) - (0.006) - (0.001) Yt−1 - - - 0.8029∗∗∗ 0.8021∗∗∗ - - - (0.005) (0.005) Yt−2 - - - 0.1061∗∗∗ 0.1055∗∗∗ - - - (0.007) (0.007) Yt−3 - - - 0.0865∗∗∗ 0.0869∗∗∗ - - - (0.005) (0.005) Number of Factors Static I(1)/I(0) gN T ,Eup 11 - - - - PC1 (dmax= 11) - 11 11 11 11 ED - 7 6 2 2 Static I(1) IPC1 (dmax= 11) 3 3 3 0 0 IPC2 (dmax= 11) 3 3 3 0 0 IPC3 (dmax= 11) 2 2 2 0 0 Primitive (dynamic) K3(m = 1, δ = 1/4) 2 3 2 2 2 ICT 1,n 2 2 2 2 2
Table 1.4: Estimation results for Models (M.1)-(M.4).
The columns labeled with M.1-M.4 respectively present the estimation results of Models (M.1)- (M.4). PC1 is the panel criterion of Bai and Ng (2002). ED is the threshold criterion of Onatski (2010). IPC1-IPC3 are from Bai (2004). K3is the selection criterion
of Bai and Ng (2007). IC1,nT is the information criterion of Hallin and Liˇska (2007). High significant coefficients (p-value < 1%) are indexed by “***”. The values between parentheses are the corresponding estimated standard deviations.
the credit spread; see, e.g., Collin-Dufresne et al. (2001), Huang and Huang (2012), and Kagraoka (2010).
From Figure 1.3(a), we can see that the time patterns of ˆαkt exhibit some structural
changes after July 16, 2007, in particular, the volatility of ˆαkt for AAA, A, and BBB.
The negative effect of the rating class A registered during the periods prior to mid- July became unstable and positive in 2008. These structural changes coincide with the beginning of the subprime crisis in the U.S. market. The market perception of the credit risk assessment performed by an external rating agency seems to depend on the market situation and is not constant over time even if bonds remain in the same rating class. The estimated factors, ˆf1t and ˆf2t, are displayed in Figure 1.3(b). These factors explain
about 84.5% of the variance in the factor structure. The forms of ˆf1t and ˆf2t over time
support the non-stationarity hypothesis. But note that these factors do not necessary affect the totality of bonds.
The first and second risk component, defined respectively as ˆCit1 = ˆλi1fˆ1t and ˆCit2 =
ˆ
Chapter 1. Panel Models with Unknown Number of Unobserved Factors 35
of the credit spread. Bonds, which had positive ˆCit1 values during the period between
Figure 1.4: (a) the first risk component ˆCit1= ˆλi1fˆ1t; (b) the second risk component
ˆ
Cit2= ˆλi2fˆ2t.
September 18, 2006 and July 16, 2007, experienced an important rise in the next period, while bonds with negative ˆCit1 experienced further decreases after July 16, 2007. This
result confirms the hypothesis of Jegadeesh and Titman (1993) and Chan et al. (1996), who assert that security returns are affected by a so-called momentum effect, because investors typically buy stocks that have performed well in the past, and sell stocks that have performed poorly. Our analysis thus sheds some light on an ongoing discussion in the literature on stock market prices. The part of variance explained by the second risk component amounts to 12.94%. The individual patterns of ˆCit2 seem to reflect the
complexity of the market behavior in the subprime period.
When re-estimating our panel model for the period spanning only the time before July 16, 2007, we detect the presence of only one primitive factor. The number of detected common factors can therefore be interpreted as an index for assessing the complexity of the market and the difficulty of diversification, as mentioned in Elton et al. (2001) and Amato and Remolona (2003). The higher the number of common risk factors, the more complex the market is.