(p.80) Appendix C: Additional Tests for Mis‐Specified Intensities
(p.80) Appendix C: Additional Tests for Mis‐Specified Intensities
(p.80) Appendix C:
Additional Tests for Mis‐Specified Intensities
This appendix reports the results of additional tests of the default intensities estimated in Chapter 4.
C.1 Testing for Independent Increments
Although the tests reported in Chapter 5 depend to some extent on the independent‐ increments property of Poisson processes, we now test specifically for serial correlation of the numbers of defaults in successive time bins. That is, under the null hypothesis of correctly estimated intensities, and fixing an accumulative total default intensity of c per time bin, the numbers of defaults N _{1},N _{2}, …, N _{K} in successive bins are independent and identically distributed. We test for independence by estimating the autoregressive model
Table C.1: Excess default autocorrelation. Estimates of the autoregressive model (C.1) of excess defaults in successive bins, for a range of bin sizes (t‐statistics are shown parenthetically). Under the hypothesis of correctly measured default intensities, fixing an accumulative total default intensity of c per time bin, the numbers of defaults N _{1},N _{2}, …, N _{K} in successive bins are independent and identically distributed. The parameters A and B are the intercept and autoregression coefficient, respectively, in (C.1).
Bin size 
No. of Bins 
A(t _{A}) 
B(t _{B}) 
R ^{2} 

2 
230 
2.091 
0.019 
0.0004 
(0.506) 
(0.286) 

4 
116 
2.961 
0.304 
0.0947 
(−2.430) 
(3.438) 

6 
77 
4.705 
0.260 
0.0713 
(−1.689) 
(2.384) 

8 
58 
5.634 
0.338 
0.1195 
(−2.090) 
(2.733) 

10 
46 
7.183 
0.329 
0.1161 
(−1.810) 
(2.376) 
Source: Das, Duffie, Kapadia, and Saita (2007).
Table C.1 presents the results of this autocorrelation analysis. The estimated autoregressive coefficient B is mildly significant for bin sizes of 4 and larger (with t‐statistics ranging from 2.37 to 3.43).
C.2 Missing Macroeconomic Covariates
Prior work by Lo (1986), Lennox (1999), McDonald and Van de Gucht (1999), Duffie, Saita, and Wang (2007), and Couderc and Renault (2004) suggests that macroeconomic performance is an important explanatory variable in default prediction. We now explore the potential role of missing macroeconomic default covariates. In particular, we examine (i) whether the inclusion of U.S. gross domestic product (GDP) or industrial production (IP) growth rates helps explain default arrivals after controlling for the default covariates that are already used to estimate our default intensities, and if so, (ii) whether the absence of these covariates could potentially explain the model test rejections reported in Chapter 5. We find that industrial production offers some explanatory power, but GDP growth rates do not.
Under the null hypothesis of no mis‐specification, fixing a bin size of c, the number of defaults in a bin in excess of the mean, Y _{k} = N _{k} − c, is the increment of a martingale and therefore should be uncorrelated with any variable in the information set available prior to the formation of the k‐th bin. Consider the regression
where GDP _{k} and IP _{k} are the growth rates of U.S. gross domestic product and industrial production observed in the quarter and month, respectively, that ends immediately prior to the beginning of the k‐th bin. In theory, under the null hypothesis of correct specification of the default intensities, the coefficients α,β _{1}, and β _{2} are zero. Table C.2 reports estimated regression results for a range of bin sizes.
We report the results for the multiple regression as well as for GDP and IP separately. For all bin sizes, GDP growth is not statistically significant, and is unlikely to be a candidate for explaining the residual correlation of defaults. Industrial production enters the regression with sufficient significance to warrant its consideration as an additional explanatory variable in the default intensity model. For each of the bins, the sign of the estimated IP coefficient is negative. That is, significantly more than the number of defaults predicted by the intensity model occur when industrial production growth rates are lower than normal, as anticipated by business‐cycle reasoning, after controlling for other covariates. (p.82)
Table C.2: Macroeconomic variables and default intensities. For each bin size c, ordinary least squares coefficients are reported for the regression of the number of defaults in excess of the mean, Y _{k} = N _{k} − c, on the previous quarter's GDP growth rate (annualized), and the previous month's growth in (seasonally adjusted) industrial production (IP). The number of observations is the number of bins of size c. Standard errors are corrected for heteroskedasticity; t‐statistics are reported in parentheses. The GDP growth rates are annualized; the IP growth rates are not annualized.
Bin Size 
No. Bins 
Intercept 
GDP 
IP 
R ^{2}(%) 

2 
230 
0.28 
−7.19 
1.06 

(1.59) 
(−1.43) 

0.15 
−41.96 
1.93 

(1.21) 
(−2.21) 

0.27 
−4.57 
−35.70 
2.31 

(0.17) 
(−0.83) 
(−1.68) 

4 
116 
0.46 
−10.61 
1.14 

(1.11) 
(−0.91) 

0.40 
−109.28 
5.49 

(1.60) 
(−2.88) 

0.53 
−5.08 
−103.27 
5.73 

(1.41) 
(−0.50) 
(−2.51) 

6 
77 
1.12 
−30.72 
4.99 

(1.84) 
(−2.12) 

0.41 
−155.09 
7.55 

(−1.00) 
(−1.89) 

0.91 
−18.09 
−124.09 
8.98 

(1.58) 
(−1.18) 
(−1.42) 

8 
58 
0.80 
−19.64 
1.81 

(0.85) 
(−0.74) 

1.35 
−357.23 
18.63 

(2.40) 
(−3.65) 

1.35 
−0.08 
−357.20 
18.63 

(1.77) 
(−0.00) 
(−3.47) 

10 
46 
1.81 
−49.00 
5.89 

(1.57) 
(−1.62) 

0.45 
−231.26 
7.66 

(0.59) 
(−2.07) 

1.96 
−41.45 
−205.15 
11.78 

(1.80) 
(−1.38) 
(−2.08) 
Source: Das, Duffie, Kapadia, and Saita (2007).
It is also useful to examine the role of missing macroeconomic factors when defaults are much higher than expected. Table C.3 provides the results of a test of whether the excess upper‐quartile number of defaults (the mean of the upper quartile less the mean of the upper quartile for the Poisson distribution of parameter c, as examined previously in Table 5.3) are correlated with GDP and IP growth rates. We report two sets of regressions; the first set is based on the prior period's macroeconomic (p.83)
Table C.3: Upper‐tail regressions. For each bin size c, ordinary least squares coefficients are shown for the regression of the number of defaults observed in the upper quartile less the mean of the upper quartile of the theoretical distribution (with Poisson parameter equal to the bin size) on the previous and current GDP and industrial production (IP) growth rates. The number of observations is the number K of bins. Standard errors are corrected for heteroskedasticity; t‐statistics are reported in parentheses.
Bin size 
K 
Intercept 
Previous Qtr GDP 
Previous month IP 
R ^{2}(%) 

2 
77 
0.28 
1.40 
0.00 

(1.55) 
(0.22) 

0.36 
−57.75 
4.92 

(2.08) 
(−2.46) 

0.16 
8.99 
−76.80 
6.94 

(1.04) 
(1.04) 
(−2.11) 

4 
48 
0.41 
−6.19 
0.97 

(1.24) 
(−0.71) 

0.29 
−65.83 
3.88 

(−1.26) 
(−1.64) 

0.29 
−22.15 
−65.26 
3.88 

(0.79) 
(−0.02) 
(−1.14) 

Bin size 
K 
Intercept 
Current bin GDP 
Current bin IP 
R ^{2}(%) 
2 
77 
0.45 
−5.98 
1.03 

(1.67) 
(−0.82) 

0.38 
−47.20 
2.82 

(2.04) 
(−2.07) 

0.36 
0.98 
−50.28 
2.84 

(1.23) 
(0.10) 
(−1.56) 

4 
48 
0.83 
−23.29 
12.67 

(1.60) 
(−2.44) 

0.48 
−77.93 
17.88 

(1.90) 
(−3.07) 

0.63 
−7.85 
−62.55 
18.63 

(1.78) 
(−0.74) 
(−2.30) 
Source: Das, Duffie, Kapadia, and Saita (2007).
We report results for those bin sizes, 2 and 4, for which we have a reasonable number of observations. Once again, we find some evidence that industrial production growth rates help explain default rates, even after controlling for estimated intensities.
(p.84) In light of the possibility that U.S. industrial production growth (IP) is a missing covariate, we re‐estimated default intensities after augmenting the list of covariates with IP. Indeed, IP shows up as a mildly significant covariate, with a coefficient that is approximately 2.2 times its standard error. (The original four covariates in (4.5) have greater significance, in this sense.) Using the estimated default intensities associated with this extended specification, we repeat all of the tests reported earlier.
Our primary conclusions remain unchanged. Albeit with slightly higher p‐values, the results of all tests reported in Chapter 5 are consistent with those reported for the original intensity specification (4.5), and lead to a rejection of the estimated intensity model. For example, the goodness‐of‐fit test rejects the Poisson assumption for every bin size; the upper‐tail tests analogous to those of Table 5.3 result in a rejection of the null at the 5% level for three of the five bins, and at the 10% level for the other two. The Prahl test statistic using the extended specification is 3.25 standard deviations from its null mean (as compared with 3.48 for the original model). The calibrated residual Gaussian copula correlation parameter r is the same for each bin size as that reported in Table B.1. Overall, even with the augmented intensity specification, the tests suggest more clustering than implied by correlated changes in the modeled intensities.
In more recent work, Lando and Nielsen (2009) have shown that substantial improvement in the intensity model estimated in Chapter 4 can be obtained by adding certain accounting ratios as additional covariates.
Notes:
(1) The within‐period growth rates are computed by compounding over the daily growth rates that are consistent with the reported quarterly growth rates.