# Semiparametric Estimation of a Corporate Bond Rating Model

## Abstract

**:**

## 1. Introduction

## 2. Data and Variable Construction

#### Measuring Conflicts of Interest

## 3. Empirical Model

#### 3.1. Model and Motivation for the Estimator

#### 3.2. Estimation Strategy

## 4. Results

#### 4.1. Simulation Evidence

#### 4.2. Empirical Illustration: Estimating Moody’s Rating Bias from 2001–2016

#### 4.3. A Placebo Test for Rating Bias

#### 4.4. Bias in Issuer Ratings

## 5. Conclusions

## Supplementary Materials

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

CRA | Credit Rating Agency |

MFOI | Moody-Firm-Ownership-Index, defined in Equation (1) |

OPM | Ordered-Probit Model |

SIM-M | Semiparametric multiple-index model with kernel (this paper) |

SIM-1 | Semiparametric single-index model (Klein and Sherman 2002) |

## Appendix A. Definitions and Assumptions

**Definition**

**A1**

**(Firm and Bond Index).**Let ${F}_{1}$ denote a continuous firm characteristics and ${F}^{\prime}$ denote the vector of firm characteristics other than ${F}_{1}$ such that ${F}_{i}\equiv [{F}_{1i},{F}_{i}^{\prime}]$ for. ${B}_{1}$ and ${B}^{\prime}$ are defined similarly for the vector of bond characteristics. I define the firm and bond index as

**Definition**

**A2**

**(Kernels).**Let ${V}_{j}\equiv ({V}_{Fj},{V}_{Bj},MFO{I}_{j})$ denotes the value of the normalized index for observation j, and ${V}_{i}\equiv ({V}_{Fi},{V}_{Bi},MFO{I}_{i})$ denote a fixed point of interest. Define a multivariate kernel function

**Definition**

**A3**

**(Density Estimator).**Let $g\left({V}_{i}\right)$ denotes the joint density of the three-dimensional index at a fixed point ${V}_{i}$, a (leave-one-out) kernel-weighted density estimator $\widehat{g}\left({V}_{i}\right)$ is defined as

**Definition**

**A4**

**(Estimated Probability).**Referring to Definitions A2 and A3, an initial estimator for the probability that a bond with characteristics ${V}_{i}=({V}_{Fi},{V}_{Bi},MFO{I}_{i})$ will be rated in category k, denoted ${P}_{k}\left({V}_{i}\right)$, is defined as

**Definition**

**A5**

**(Trimming Functions).**Let ${W}_{i}^{d}$ denote the d-th column of a $D$-dimensional continuous vector ${W}_{i}$. Define ${\widehat{\tau}}_{id}\equiv \mathrm{\U0001d7d9}\{{\widehat{a}}_{d}<{W}_{i}^{d}<{\widehat{b}}_{d}\}$ and ${\widehat{\tau}}_{i}={\prod}_{d=1}^{D}{\widehat{\tau}}_{id}$, where ${\widehat{a}}_{d}$${\widehat{b}}_{d}$ are, respectively, the lower and upper sample quantiles for ${W}^{d}$. When ${W}_{i}^{d}={X}_{i}$, I refer to ${\tau}_{ix}$ as X-trimming; With ${\widehat{V}}_{i}$ as the estimated index, when ${W}_{i}^{d}={\widehat{V}}_{i}$, I refer to ${\tau}_{iv}$ as index trimming.

**Definition**

**A6**

**(First- and Second-Stage Estimator).**Based on Definitions A1–A5, I define:

**Assumption**

**A1**

**(Data).**The vector (${Y}_{i}$,${X}_{i}$) is $iid$ over i. The categorical outcome ${Y}_{i}$ has a discrete and finite support, taking values from 1 to k. The columns of ${X}_{i}=[{F}_{i},{B}_{i},MFO{I}_{i}]$ are linearly independent with a probability of one. As stated in Definition A1 above, I require that ${F}_{i}$ and ${B}_{i}$ each contain at least one continuous regressor, termed ${F}_{1}$ and ${B}_{1}$, respectively.

**Assumption**

**A2**

**(The Error Term).**The error term ${S}_{i}$ is conditionally mean-independent of ${X}_{i}$: $E\left[{S}_{i}\right|{X}_{i}]=0$ and independent across i.

**Assumption**

**A3**

**(Continuous Firm and Bond Characteristics).**Referring to the firm and bond index defined in Definition A1, I require the index coefficient of ${F}_{1}$ and ${B}_{1}$ to be nonzero: ${\beta}_{10}^{F}\ne 0,{\beta}_{10}^{B}\ne 0$.

**Assumption**

**A4**

**(Index Assumption).**Referring to the normalized index defined in Definition A1, let ${V}_{i}\left({\theta}_{0}\right)\equiv [{V}_{Fi},{V}_{Bi},MFO{I}_{i}]$, the following index assumption is assumed to hold for all i and k:

**Assumption**

**A5**

**(Parameter Space).**The vector of the true parameters values ${\theta}_{0}\equiv [{\theta}_{0}^{F},{\theta}_{0}^{B}]$ for the model lies in the interior of a compact parameter space, Θ.

**Assumption**

**A6**

**(Conditional Densities).**Let $g\left(v\right|y,x)$ denote the density of the index ${V}_{i}\left(\theta \right)$ defined in Assumption A4 conditioning on ${Y}_{i}=y$ and ${X}_{i}=x$. Denote ${\nabla}^{d}g\left(t\right|y,x)$ as the partial or cross partial derivatives up to order d. I assume $g>0$ and ${\nabla}^{d}g\left(t\right|y,x)$ to be uniformly bounded for $d=0,1,2,3$ on the interior of its support.

**Assumption**

**A7**

**(Bandwidth Parameter).**Referring to the kernel estimator defined in Definition A2, the bandwidth parameter $h\to 0$ as $N\to \infty $. Specifically, with $h=({h}_{1},{h}_{2},{h}_{3})$, I choose ${h}_{z}=0.97{\sigma}_{z}{N}^{-r}$ according to Silverman (1982)16 where ${\sigma}_{z}$ is the standard deviation of the three indices ($z=1,2,3$) and r is a parameter that affects the rate that h goes to zero. In this paper, $r=1/7.01$.

## Appendix B. Asymptotic Theorems

**Theorem**

**A1**

**(Consistency).**Under Assumptions A1–A7 and with $1/12<r<1/7$,

**Proof.**

**Theorem**

**A1**

**(Normality).**Assumptions A1–A7 and with the bandwidth parameter $1/12<r<1/7$,

**Proof.**

## Notes

1. | For theoretical studies on the issuer-paid model and rating shopping, see Bolton et al. (2012); Sangiorgi et al. (2009); Skreta and Veldkamp (2009) and some empirical evidence (He et al. 2015; Jiang et al. 2012; Mathis et al. 2009). |

2. | Extensive literature addresses semiparametric models and the estimation of semiparametric single index models, including Härdle and Stoker (1989); Horowitz and Härdle (1996); Ichimura (1993); Klein and Spady (1993); Manski (1985); Powell et al. (1989). See Stewart (2005), Lewbel (2000), and Klein and Sherman (2002) for applications of a single-index model in the context of an ordered-response model. |

3. | Alternatively, one may also use the sieves method to estimate the rating probability. Such methods are more convenient when some prior information and constraints, such as monotonicity, additivity, and nonnegativity, needs to be incorporated in the conditional probabilities (Chen 2007). For instance, Coppejans (2007) estimates an ordered model with a quadratic-spline under the restriction that the distribution functions across all categories are the same. Such a constraint, however, is not appropriate in the current application because Moody’s rating standard can vary with categories. |

4. | Macro variables are not included because the model will be estimated separately for each year. |

5. | Moody’s was founded as a private company in 1900, acquired by Dun&Bradstreet (D&B) in 1962, and remained one of its divisions until 4 October 2000, when it was spun off and listed on the NYSE. The S&P has been a fully owned division of McGraw-Hill, a publicly traded company, since 1966. Going public makes CRAs more vulnerable to conflicts of interest. For example, Kedia et al. (2017) found that Moody’s assigned favorable ratings toward issuers that Moody’s shareholders have invested in. |

6. | From 2001 to 2010, Moody’s had two shareholders, Berkshire Hathaway and Davis Selected Advisors, which collectively own about 23.5% of Moody’s. |

7. | The numerical rating matches the seven ordinal rating categories: $Aaa=7,Aa=6,A=5,Baa=4,Ba=3,B=2$, and $C=1$ (from the highest credit quality to the lowest). |

8. | The vector $\mathbf{X}$ is assumed to be exogenous throughout. Intuitively, and as one might have expected, some information contained in S, e.g., the manager’s ability, may also drive institutional investors’ investment decisions, implying that $MFOI$ is endogenous. The problem of endogeneity can be handled, for example, using the control function approach proposed by Blundell and Powell (2004) provided with a valid exclusion restriction. |

9. | Specifically, the rating probability in an ordered-probit model is
$$\begin{array}{c}\hfill {P}_{ik}\equiv Pr({Y}_{ik}=1|{\mathbf{X}}_{i})=\mathsf{\Phi}({T}_{k}-{X}_{i}{\beta}_{0})-\mathsf{\Phi}({T}_{k-1}-{X}_{i}{\beta}_{0})\end{array}$$
$$\begin{array}{ccc}\hfill Q& =& \frac{1}{N}\sum _{i=1}^{N}\sum _{k=1}^{7}{Y}_{ik}Ln\left({P}_{ik}\right)\hfill \end{array}$$
In the case when ${y}^{*}=X{\beta}_{0}+{c}_{0}+S$ in (7) and the variance of the homoskedastic error term is ${\sigma}^{2}$, identification is up to location and scale: one can, at most, identify ${\beta}^{*}\equiv {\beta}_{0}/\sigma $, ${T}_{k}^{*}=({T}_{k}-{T}_{1})/\sigma $ (the “pseudo cutoff points”), and ${c}^{*}=({c}_{0}-{T}_{1})/\sigma $ (the “pseudo intercept”). See Amemiya (1981) for a discussion. |

10. | Since the functional form of ${P}_{k}(\xb7)$ in (6) is not specified, conditioning on the original index ${W}_{Fi},{W}_{Bi},MFO{I}_{i}$ and a linear transformation of them deliver the same amount information on ratings. Therefore, without some normalization, the limiting log-likelihood function cannot be uniquely maximized at the true parameters, which is necessary for identification. |

11. | More specifically, u is generated from a ${\chi}^{2}\left(1\right)$ distribution, standardized to have a mean of zero and unit variance. ${X}_{1},{X}_{3}\sim N(0,1)$, and ${X}_{2}$ is a standardized ${\chi}^{2}\left(1\right)$. |

12. | Confidence Intervals were constructed based on the asymptotic results derived in Appendix B. |

13. | The partial effects for MFOI in the ordered-response model are,
$$\begin{array}{ccc}\hfill {\delta}_{k}\left(MFO{I}_{i}\right)& =& \frac{\partial Prob(Y=k|{X}_{i})}{\partial MFO{I}_{i}},\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}k=Aaa,Aa,\cdots ,C\hfill \\ \hfill & \approx & {P}_{k}({V}_{Fi},{V}_{Bi},MFO{I}_{i}+\delta )-{P}_{k}({V}_{Fi},{V}_{Bi},MFO{I}_{i}),\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\hfill \end{array}$$
The Average Partial Effect, or APE, is computed by evaluating the partial effect for each bond i and averaging the computed effects,
$$\begin{array}{c}\hfill AP{E}_{k}(MFO{I}_{i};\delta )=\frac{1}{n}\sum _{i=1}^{n}{P}_{k}({V}_{Fi},{V}_{Bi},MFO{I}_{i}+\delta )-{P}_{k}({V}_{Fi},{V}_{Bi},MFO{I}_{i})\end{array}$$
The above calculation can be performed for any category. That is, even for a C-rated bond, one can compute the change in the probability of this bond being rated into AAA. To make the presentation concise and practically relevant, I only report the APE for the category that is one-notch better than the current rating grade. That is, I interpret the APE as the probabilistic change of obtaining a better rating grade if the issuer’s share-ownership relationship with Moody’s strengthens by $\delta $. |

14. | This implies that the CRA will be “punished” once a highly rated investment results in default. See Bolton et al. (2012) for a discussion. |

15. | The suggested trimming rule is indeed ad-hoc because the selected threshold may be bad for other DGP. The asymptotic theorems developed later abstracts away from the trimming issue. It is possible to develop a data-dependent optimal trimming rule similar to Ma and Wang (2019), which is left for future work. |

16. |

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**Figure 6.**Placebo test using Merrill Lynch and BNY Mellon as “False Owners”. Note: The rating difference in 2008 is not identified because all 446 bonds are issued by firms related with Merrill Lynch or BNY Mellon.

**Table 1.**Summary Statistics: A number of firm and bond characteristics were selected to predict credit ratings based on Baghai et al. (2014) and Kedia et al. (2017): (1) the value of the firm’s total assets (log(asset)), (2) long- and short-term debt divided by total assets (Book_lev). (3) Convertible debt divided by total assets (ConvDe_assets), (4) rental paymetns divided by total assets (Rent_Assets), (5) cash and marketable securities divided by total assets (Cash_assets), (6) long- and short-term debt divided by EBITDA (Debt_EBITDA), (7) EBITDA to interest payments (EBITA_int), (8) profitability, measured as EBITDA divided by sales (Profit), (9) tangibility, measured as net property, plant, and equipment divided by total assets (PPE_assets), (10) capital expenditures divided by total assets (CAPX_assets), (11) the volatility of profitability (Vol_profit), (12) the log of the issuing amount (log(issuing amount)), (13) a dummy variable indicating whether the bond is senior (seniority), and (14) a dummy variable indicating whether the bond is secured (security).

Investment Grade | High-Yield | |||
---|---|---|---|---|

Mean | Std.Dev. | Mean | Std.Dev. | |

Firm Characteristics | ||||

log(asset) | 10.88 | 1.92 | 8.26 | 1.45 |

book_lev | 0.33 | 0.18 | 0.44 | 0.20 |

convDe_asset | 0.01 | 0.03 | 0.03 | 0.07 |

rent_asset | 0.01 | 0.01 | 0.02 | 0.03 |

cash_asset | 0.11 | 0.12 | 0.08 | 0.09 |

debt_ebitda | 4.95 | 11.16 | 4.45 | 20.76 |

ebitda_int | 14.45 | 31.30 | 4.82 | 5.91 |

profit | 0.31 | 0.28 | 0.03 | 8.38 |

PPE_asset | 0.23 | 0.26 | 0.37 | 0.28 |

CAPEX_asset | 0.03 | 0.04 | 0.07 | 0.10 |

profit_vol | 0.06 | 1.84 | −0.92 | 41.74 |

Bond Characteristics | ||||

log(issuing amount) | 12.69 | 1.69 | 12.66 | 0.73 |

seniority | 0.93 | 0.26 | 0.69 | 0.46 |

security | 0.01 | 0.06 | 0.09 | 0.00 |

Small Bandwidth $\mathit{r}=1/11.99$ | Large Bandwidth $\mathit{r}=1/7.01$ | |||||||
---|---|---|---|---|---|---|---|---|

Trimming | TRUE | Mean | SD | RMSE | Mean | SD | RMSE | |

0.9 | SIM-1 | 2 | 2.569 | 0.452 | 0.776 | 2.608 | 0.460 | 0.830 |

SIM-M | 2 | 2.026 | 0.551 | 0.552 | 1.987 | 0.481 | 0.481 | |

OP | 2 | 2.634 | 0.593 | 0.995 | 2.669 | 0.621 | 1.070 | |

0.95 | SIM-1 | 2 | 2.551 | 0.434 | 0.738 | 2.602 | 0.445 | 0.808 |

SIM-M | 2 | 2.003 | 0.535 | 0.535 | 1.985 | 0.453 | 0.453 | |

OP | 2 | 2.620 | 0.599 | 0.983 | 2.622 | 0.565 | 0.953 | |

0.99 | SIM-1 | 2 | 2.539 | 0.423 | 0.714 | 2.591 | 0.450 | 0.800 |

SIM-M | 2 | 1.964 | 0.476 | 0.477 | 1.963 | 0.431 | 0.433 | |

OP | 2 | 2.607 | 0.615 | 0.983 | 2.607 | 0.547 | 0.916 |

Investment Grade (IG) | High Yield (HY) | ||||||||
---|---|---|---|---|---|---|---|---|---|

Year | Aaa | Aa | A | Baa | Ba | B | C | Total | % of IG |

2001 | 10 | 45 | 162 | 214 | 111 | 94 | 11 | 647 | 66.62% |

2002 | 1 | 78 | 142 | 212 | 71 | 105 | 7 | 616 | 70.29% |

2003 | 9 | 112 | 149 | 210 | 123 | 168 | 30 | 801 | 59.93% |

2004 | 3 | 81 | 91 | 174 | 89 | 155 | 18 | 611 | 57.12% |

2005 | 6 | 118 | 106 | 150 | 86 | 88 | 15 | 569 | 66.78% |

2006 | 3 | 164 | 161 | 189 | 58 | 65 | 22 | 662 | 78.10% |

2007 | 8 | 238 | 326 | 151 | 48 | 69 | 13 | 853 | 84.76% |

2008 | 2 | 110 | 151 | 139 | 29 | 11 | 4 | 446 | 90.13% |

2009 | 3 | 35 | 124 | 211 | 88 | 91 | 11 | 563 | 66.25% |

2010 | 7 | 51 | 101 | 172 | 90 | 110 | 26 | 557 | 59.43% |

2011 | 10 | 35 | 140 | 201 | 41 | 82 | 14 | 523 | 73.80% |

2012 | 3 | 41 | 153 | 261 | 83 | 116 | 25 | 682 | 67.16% |

2013 | 12 | 49 | 173 | 311 | 95 | 105 | 31 | 776 | 70.23% |

2014 | 8 | 32 | 139 | 303 | 92 | 92 | 20 | 686 | 70.26% |

2015 | 20 | 28 | 198 | 370 | 78 | 55 | 7 | 756 | 81.48% |

2016 | 26 | 59 | 219 | 357 | 80 | 65 | 3 | 809 | 81.71% |

Total | 131 | 1276 | 2535 | 3625 | 1262 | 1471 | 257 | 10,557 |

Shareholder | T | Mean | Max | Min |
---|---|---|---|---|

HARRIS ASSOCIATES L.P. | 21 | 2.42% | 5.02% | 0.00% |

CHILDREN’S INV MGMT (UK) LLP | 20 | 2.29% | 5.31% | 0.01% |

SANDS CAPITAL MANAGEMENT, INC. | 28 | 3.01% | 5.59% | 0.40% |

T. ROWE PRICE ASSOCIATES, INC. | 64 | 1.47% | 5.94% | 0.18% |

BARCLAYS BANK PLC | 55 | 2.52% | 6.32% | 0.03% |

GOLDMAN SACHS & COMPANY | 63 | 1.94% | 7.24% | 0.01% |

VALUEACT CAPITAL MGMT, L.P. | 13 | 5.19% | 7.77% | 0.93% |

VANGUARD GROUP, INC. | 64 | 3.79% | 7.98% | 1.64% |

MSDW & COMPANY | 57 | 2.20% | 8.14% | 0.22% |

DAVIS SELECTED ADVISERS, L.P. | 51 | 5.56% | 8.14% | 0.10% |

FIDELITY MANAGEMENT & RESEARCH | 64 | 1.99% | 9.08% | 0.00% |

CAPITAL RESEARCH GBL INVESTORS | 13 | 4.80% | 11.31% | 0.07% |

CAPITAL WORLD INVESTORS | 35 | 6.07% | 12.60% | 0.66% |

BERKSHIRE HATHAWAY INC. | 64 | 14.87% | 20.43% | 11.33% |

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Jiang, Y.
Semiparametric Estimation of a Corporate Bond Rating Model. *Econometrics* **2021**, *9*, 23.
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**AMA Style**

Jiang Y.
Semiparametric Estimation of a Corporate Bond Rating Model. *Econometrics*. 2021; 9(2):23.
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Jiang, Yixiao.
2021. "Semiparametric Estimation of a Corporate Bond Rating Model" *Econometrics* 9, no. 2: 23.
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