Exploring Common Factors in the Term Structure of Credit Spreads: The Use of Canonical Correlations
Seung C. Ahn, Stephan Dieckmann, M. Fabricio Perez *
July 7, 2011
Abstract This paper provides a new approach to model the common variation in the term structure of credit spreads. The novelty is that common factors are extracted using canonical relations between credit spreads and observable economic variables. We show how these factors can be used to test if a given set of macroeconomic and financial variables is sufficient to capture all the systematic variation in response variables, such as credit spreads. We find that credit spread innovations are subject to three common factors, two strong factors and one weak factor, and they account for 49% of the total variation. The first strong factor is related to the contemporaneous state of the economy, the second represents expectations about future economic conditions, and the weak factor is mainly related to the error correction processes in short-term spreads.
*
Ahn is affiliated with the Department of Economics at Arizona State University and Sogang University, email:
[email protected], Dieckmann with the Wharton Finance Department at the University of Pennsylvania, email:
[email protected], and Perez with the School of Business and Economics at Wilfrid Laurier University, email:
[email protected]. We would like to thank Alexander David, Federico Nardari, Spencer Martin, Iulian Obreja, Francisco Palomino, Eric Renault (Editor), Yaxuan Qi, and two anonymous referees for very useful suggestions, as well as seminar participants at Arizona State University, University of Houston, Wilfrid Laurier University, the meeting of the European Finance Association in Athens, and the meeting of the Northern Finance Association in Kananaskis Village for helpful comments.
Exploring Common Factors in the Term Structure of Credit Spreads: The Use of Canonical Correlations
Abstract This paper provides a new approach to model the common variation in the term structure of credit spreads. The novelty is that common factors are extracted using canonical relations between credit spreads and observable economic variables. We show how these factors can be used to test if a given set of macroeconomic and financial variables is sufficient to capture all the systematic variation in response variables, such as credit spreads. We find that credit spread innovations are subject to three common factors, two strong factors and one weak factor, and they account for 49% of the total variation. The first strong factor is related to the contemporaneous state of the economy, the second represents expectations about future economic conditions, and the weak factor is mainly related to the error correction processes in short-term spreads. Keywords: Factor Analysis, Credit Risk, Common Factors, Canonical Correlations. JEL Classification: C33, G12.
1
Introduction
The price of defaultable debt has received considerable attention in financial economics through the development of structural and reduced-from models. In an influential article, Collin-Dufresne et al. (2001) investigate the determinants of credit spread changes as motivated by the formulation of structural models of default risk. They consider a large set of macroeconomics and financial variables that should, in theory, explain movements in credit spreads, but find an unexplained large common factor in residuals from a multiple regression analysis. Such studies have been replicated on several data sets, see for example Ericsson et al. (2009), and the research that typically emerges is about finding a refined set of variables that better explains the variation in credit spreads, as in Cremers et al. (2008). The conventional approach to analyze common factors in this context is principal components of residuals, and “better” refers to having less correlated regression residuals. In this paper we show there is an alternative explanation for highly correlated residuals from a regression analysis. If macroeconomic and financial variables are only proxies for unobserved common factors, then individual regression residuals should contain the errors caused by the use of proxy variables instead of the true factors. We refer to such errors as “systematic measurement errors.” Highly cross-sectionally correlated residuals may not only indicate a missing common factor, but also the use of imperfect proxies. The latter raises the research question: How can we test if a set of macroeconomics and financial variables is sufficient to capture all the systematic variation of a given response variable such as credit spreads? Our test is based on the construction of factors from a canonical correlation analysis (CCA) of credit spreads (the response variable) and observed explanatory variables that in theory should explain credit spreads. We refer to these factors as CCA factors. We show the residuals obtained from regressions of the response variables on the CCA factors do not contain systematic measurement errors. Our CCA approach can be seen as the opposite of previous studies that try to find a refined set of observed variables that better explains the variation in credit spreads. Instead of testing additional structural variables, we hypothesize that a standard set of explanatory variables is sufficient to construct all common factors. Those variables might not be perfect proxies for the factors, but they should contain sufficient information to be able to construct them. As in the case of principal components, 1
canonical correlations are functions of the response variables. However, conversely to principal components, canonical correlations are estimated using the additional information from explanatory financial and macroeconomic variables. We show that the common variation in credit spreads is mainly explained by three factors. Two of them are strong factors, in terms of their explanatory power for individual credit spreads. The third is a weak factor, since it only accounts for a small fraction of total common variation relative to the amount of idiosyncratic noise. The average explanatory power of the three factors is 49% of total variation, of which the two strong factors contribute 26% and 14%, respectively. Furthermore, we find that irrelevant factors explain only the idiosyncratic noise of single test assets, while relevant ones explain all common variation. The three factors are identified using the same set of variables proposed by Collin-Dufresne et al. (2001). This is interesting, as it suggests that no strong factor exists in the corporate bond market, that is not present in the equity market, the swap market, or the market for U.S. Treasury debt. Another advantage of using canonical correlations is that interpretations are immediately available. Although having a reduced-form model, we can see more precisely what the factors capture, or whether they are correlated with multiple economic channels. The strongest factor is negatively correlated with the riskless term structure, and the factor is also present in the equity market. The cyclical consistency of credit spreads with these economic variables allow us to interpret the strongest factor as a macro-default factor. The factor also has a large positive correlation with the interest rate swap spread, which can be evidence of cyclical variation in illiquidity, see Feldhuetter and Lando (2008), and it is an affirmation of the predictions in Ericsson and Renault (2006). Our new finding here is that the common illiquidity component is related to the same factor that captures more general macroeconomic risk. The second strong factor is mainly correlated with the slope of the riskless term structure and the implied equity volatility given by VIX – both measures that have a forward-looking character. The slope of the riskless term structure, a traditional variable to capture changes in investors’ expectations about real economic activity, has the predicted impact on credit spreads. VIX can be seen to capture investors’ expectations through its correspondence with ex-ante risk premia. Therefore, we interpret the second strong factor as one representing market participants’ expectations, above and beyond those already captured by the strongest factor. The second strong factor is also confirmed to have predictive power out of sample: The strongest factor is most correlated with the contemporaneous realization of U.S. income data. The second factor, which we name “expectations factor”, however, is most significant in explaining future realizations of U.S. income data over a six-month horizon. 2
Last but not least, a part of our factor analysis is to examine the cointegration relations among credit spreads of different credit quality and their error correction effects. Some previous studies using time-series data have found that levels of credit spreads are nonstationary, see Pedrosa and Roll (1998) and Barnhill et al. (2000). It is beyond the scope of this paper to answer the question whether spread series should be non-stationary or not, since not much is known about the steady-state distribution of corporate debt from an equilibrium perspective. However, our data shows evidence of cointegrated relations and we find the inclusion of error corrections leads to some significant adjustment coefficients. The third (and weak) factor, having explanatory power of 8% on average, is strongly correlated with the error correction series of credit spreads corresponding to short-term bonds. Our results indicate that error correction effects are not a major determinant of credit spreads, but a valid representation of the data. We are not the first to investigate the relation between latent factors and observed explanatory variables. Gouriroux et al. (1995) propose to estimate factors by CCA. Our approach is different from theirs in that we estimate factors by linear functions of response variables while they propose functions of explanatory variables. The use of their factors is not immune to the above-mentioned systematic measurement error problem. Our CCA factors are not consistent estimators of true factors. Nonetheless, they preserve the factor loadings of true factors in that the factor loadings are the same as the loadings of true factors up to a linear nonsingular transformation. In addition, the residuals from the regressions of response variables on the CCA factors contain the idiosyncratic components of response variables only, unless some factors exist that are not correlated with explanatory variables. To test if some factors exist in individual credit spreads that are not correlated with the set of explanatory variables, we use the methods of Ahn and Perez (2010) to estimate two numbers: the total number of factors explaining common variation in credit spreads and the number of factors that are correlated with the explanatory variables. If the two numbers are equal, then we could conclude that the set of explanatory variables is sufficient to capture all latent factors. Gouriroux et al. (1995) provide several test methods for the number of factors correlated with explanatory variables. We also use one of their tests to check the robustness of our test results. Bai and Ng (2006) have developed a general approach to evaluate how observable variables and unobserved true factors are correlated. Their methods enable researchers to estimate canonical correlations between observed variables and true factors, and even to test if some observable variables are true factors. Their methods however require to utilize data with a large number of response variables as well as a large time series dimension. For such data, principal components of the response variables are consistent estimators of the true factors. 3
However, principal components are not consistent for data displaying a small number of response variables. Our approach is designed for data with a small number of response variables, and is therefore suited for an application to the term structures of credit spreads. Our paper proceeds as follows: We first outline our methodology in Section 2. We describe the data set in Section 3. Section 4 presents the analysis of credit spreads using multiple regressions, and Section 5 shows our results using canonical correlation factors and their economic interpretation. We conclude in Section 6, the Appendix contains Tables and Figures.
2 2.1
Empirical Methods Motivation
Consider the linear regression model commonly used in the credit spread literature: yt = α + Ξzt + et ,
(1)
where t indexes time (t = 1, 2, ..., T ), yt is an n × 1 vector of response variables, in our case credit spreads innovations with different maturities and/or different credit ratings. The vector zt contains k observed explanatory variables, et is an n × 1 vector of error terms orthogonal to (1, zt0 )0 (E [et (1, zt0 )] = 0n×(k+1) ), and α and Ξ are an n × 1 vector and n × k matrix of parameters, respectively. We focus on the cases in which n is small and T is large, so that necessary asymptotics apply as T → ∞ with fixed n. The variables in zt can be thought of as proxy variables for factors that price corporate bonds according to theoretical models of default risk, further explained in Section 3.2. Common practice after estimating the model above is to analyze if the regression residuals are highly correlated and may therefore contain common factors. Specifically, a high explanatory power of the first principal component of the residuals is taken as evidence that a factor was not captured by zt (missing factor). We, however, argue that highly correlated regression residuals may be the result of a second but equally likely scenario. We hypothesize the variables in zt may not be perfect proxies for the true factors that explain credit spreads. If this was the case, then the regression residuals capture the measurement errors in zt , which we refer to as “systematic measurement errors.” Residuals may be highly cross-sectionally correlated as result of the systematic measurement errors and not necessarily as a consequence of missing factors. But how can we distinguish if the high correlation in residuals is caused by a missing factor or by measurement error in the proxy variables? The remainder of this section is devoted to showing how CCA can be used to answer this question. In what follows, we present our main theoretical results under the assumption that population 4
parameters are known, because they hold asymptotically when the parameters are replaced by their consistent estimators.
2.2
Basic Model and Assumptions
The first step is to construct a model that involves the two possible scenarios. We assume the the n-vector of dependent variables yt follows a linear m-factor model given by yt = ξ + Bft + ut = ξ + B1 f1t + B2 f2t + ut ,
(2)
0 0 0 ) is a m × 1 vector of common factors, ut is the vector of idiosyncratic , f2t where ft = (f1t components, ξ is a n × 1 vector of intercepts, and B, B1 , and B2 are n × m, n × r, and n × (m − r) factor loading matrices corresponding to ft , f1t and f2t , respectively. The factors f1t can be predicted by a theoretical model of interest. The variables in f2t , however, are the factors that are not predicted by the theoretical model. Both, the factors in f1t and f2t are unobserved. Instead, we observe k instrumental variables correlated with the factors in f1t and we refer to them as zt , that is, the instruments zt are our proxy variables. Since the instrumental variables are chosen for the factors predicted by the model only, they would not contain information that can generate meaningful inferences about f2t . To be consistent, we assume the factors in f2t are uncorrelated with the instrumental variables zt . We do not need to determine the causal relation between f1t and zt , thus the only required assumption is non-zero correlation among them. Formal assumptions are as follows:
Assumption 1. (i) The time series of ft , zt , and ut are covariance-stationary and ergodic with finite moments of sufficiently high order such that the usual Law of Large Numbers (LLN) and Central Limit Theorem (CLT) hold. (ii) E [ut (1, ft0 , zt0 )] = 0n×(m+1+k) . Assumption 2. (i) rank(B) = m. (ii) rank(B1 ) = r. Assumption 3. (i) The linear projections of f1t and f2t on (1, zt0 )0 are given by L(f1t |1, zt ) = θ + Θ0 zt and L(f2t |1, zt ) = 0(m−r)×1 ,
(3)
where θ and Θ are parameter vector and matrix, respectively. (ii) rank(Θ) = r. Assumption 4. V ar(ut ) ≡ Ωuu is a diagonal matrix. The covariance-stationarity assumption, Assumption 1(i), could be relaxed without altering our results. Assumption 1(ii) implies that not only the factors but also the instrumental variables are uncorrelated with the idiosyncratic error terms in ut . The variables need not
5
be strictly exogenous to the errors. Assumption 1(ii) can hold even if the factors ft and the instruments zt are only weakly exogenous. Assumption 2(i) implies that m is the minimum number of linearly independent factors. Suppose that rank(B) = m−1, then there exist some matrices D and G such that B = DG0 ; see Gouriroux et al. (1995). Let ft∗ = G0 ft . Then, equation (2) reduces to yt = ξ + Dft∗ + ut , which is a (m − 1)-factor model. Assumption 2(i) rules out this possibility – Assumption 2(ii) can be motivated similarly. Assumption 3(i) indicates that only the factors f1t are correlated with the instrumental variables in zt ; it also implies that E [f2t ] = 0(m−r)×1 . We can allow the factors in f2t to have non-zero means by assuming that the projection of f2t is a constant, and this does not change our main results. Assumption 3(ii) is related to the well-known rotational indeterminacy of factor models, see Gouriroux et al. (1995). The factors f1t and the corresponding loadings B1 are defined only up to a (r × r) linear transformation – any linear transformation of the factors can be viewed as a true factor. Assumption 3(ii) assures that the instrumental variables zt have explanatory power for any rotated factor of f1t . Assumption 4 indicates that the model in equation (2) is exact, i.e. credit spreads in yt are cross-sectionally correlated only through the factors. This assumption is necessary for the consistent estimation of the total number of true factors (m), if the data contain a small number of cross-sectional units as in our analysis of credit spreads. Assumption 4 is not necessary for the CCA analysis we discuss below. However, it is needed for the consistent estimation of m for data with small n; see Ahn and Perez (2010). Substituting equation (3) into (2) yields yt = (ξ + B1 θ) + B1 Θ0 zt + (ut + B1 vt + B2 f2t ) = α + Ξzt + et ,
(4)
where vt = f1t − L(f1t |1, zt ) is the projection error vector. The errors in vt are what we have referred to as systematic measurement errors. Note the model in equation (4) is the same as model (1) if we set α = (ξ + B1 θ), Ξ = B1 Θ0 , and et = ut + B1 vt + B2 f2t . Also, under Assumptions 1 and 3 we have E [et (1, zt0 )] = 0n×(k+1) . The regression model (4) is able to capture the two possible scenarios. The error vector et has a two possible sources of a factor structure, B1 vt and B2 f2t . If there are some factors (f2t ) that are not correlated with the explanatory variables zt , then the error vector et captures this through B2 f2t . On the other hand, if the variables zt are not perfectly correlated with the factors f1t then the projection error vector vt 6= 0, and the error vector et will have a factor structure through B1 vt . When equation (4) is estimated by regressing yt on (1, zt0 )0 , 6
the principal components of the residuals may capture either case. The residuals may be highly correlated and their principal components may have significant explanatory power for the dependent variables, even if there is no missing factor, i.e. our second scenario. Hence, a principal component analysis of the regression residuals of model (4) is not sufficient to determine the existence of missing factors (f2t ). We consider a CCA-based method by which the two scenarios can be distinguished in the following subsection.
2.3
Canonical Correlation Analysis
CCA is a generalization of multiple correlation analysis, see Hasbrouck and Seppi (2001) for an application in market microstructure. In a multiple regression we use the coefficient of determination to find the linear combination that maximizes the correlation between one dependent variable and a set of predictor variables. In CCA this concept is generalized such that we find the linear combinations that maximize the correlation between a set of dependent variables and a set of independent variables. We start by a brief discussion of CCA that is relevant for the presentation of our methodology, more detailed discussions can be found in Gouriroux et al. (1995). In what follows, we use the notation Ωxw to denote the covariance matrix between two random vectors x and w; and Ωxx for the variance matrix of x. Let a and c be nonzero (and nonrandom) vectors such that a0 yt and c0 zt become scalar random variables. Let a1 and c1 be the values of a and c which maximize the absolute correlation between a0 yt and c0 zt , |corr(a0 yt , c0 zt )|. Then, g1t = a01 yt and h1t = c01 zt are referred to as the first pair of canonical covariates. Similarly, let a2 and c2 be the vectors maximizing |corr(a0 yt , c0 zt )| under the constraint that g2t = a02 yt and h2t = c02 zt are uncorrelated with g1t and h1t , respectively. The variables g2t and h2t , are referred to as the second pair of canonical covariates. The third or higher pairs of covariates are obtained following the same procedure. Letting A = (a1 , ..., ar ), we refer to gt = (g1t , ..., grt )0 = A0 yt as CCA factors. The columns of A are simply the eigenvectors corresponding to the first r largest eigenvalues of −1 0 the matrix Ω−1 yy Ωyz Ωzz Ωzy , which are normalized such that A Ωyy A = Ir . For equation (4), Gouriroux et al. (1995) have shown that the column spaces of B1 and Θ can be estimated by CCA (see Proposition 2.1). They propose to use estimates of Θ0 zt as estimated true factors. However, the use of Θ0 zt leaves the systematic measurement errors vt in the errors et . As an alternative, we propose to use A0 yt as estimated factors. The factor loading matrix B1 and the coefficient matrix A are related. For equation (4), Proposition 2.1 in Gouriroux et al. (1995) shows that the columns of Ωyy A form a basis spanning the
7
columns of B1 in Ξ. That is, there exists a r × r non-singular matrix M such that −1 B1 = Ωyy AM ; A = Ω−1 . yy B1 M
(5)
Notice that B1 is spanned by the columns of Ωyy A , not by the columns of A. A regression analysis using the CCA factors A0 yt instead of the explanatory variables zt has two advantages summarized by two propositions below. The first shows that factor loadings based on the CCA factors are the same as the factor loadings of true factors up to a linear transformation. The second shows that the projection errors from the regressions with the CCA factors do not contain systematic measurement errors. Proposition 1. Let α∗ and B1∗ be the parameters of the linear population projection of yt on (1, gt0 )0 , and let u∗t be the corresponding projection error such that yt = L(yt |1, gt ) + u∗t = α∗ + B1∗ gt + u∗t .
(6)
Then, the columns of B1∗ are spanned by the columns of B1 ; i.e. B1∗ = B1 M −1 . Proof of Proposition 1. By equation (5) and definition of projection, we have 0 −1 B1 = Ωyg Ω−1 = B1 M −1 . gg = Ωyy A(A Ωyy A)
According to Proposition 1, the CCA factors gt = A0 yt preserve the structure of the covariances between yt and the true factors f1t , up to a r × r linear transformation. This result is somewhat surprising because the CCA factors are not consistent estimators of the true factors. To see why, notice that using equations (2) and (5) we can obtain gt = A0 yt = A0 (ξ + Bft + ut ) = A0 ξ + M f1t + A0 (B2 f2t + ut ), because A0 B1 = A0 Ωyy AM = M. Due to the presence of A0 ut , the CCA factors cannot be consistent estimators of the rotated factors M ft , even if there is no missing factor (f2t = 0). For the cases in which f2t = 0 and Ωuu = σu2 IT (i.e. the idiosyncratic errors in ut are i.i.d.), Theorem 6 of Connor and Korajczyk (1986) applies, and we can show that the n × r matrix of the eigenvectors corresponding to the r largest eigenvalues of Ωyy is the same as the factor loading matrix B1 up to a r × r linear transformation. A novelty of Proposition 1 is that the factor loadings B1∗ of the CCA factors have the same properties even if the errors in ut are heteroskedastic and cross-sectionally correlated. 8
Although the CCA factors gt are not consistent, they are unbiased estimators of the rotated factors M f1t , if there is no missing factor f2t = 0, in the sense that E[gt − E(gt )|f1t ] = M [f1t − E(f1t )]. Being unbiased estimators of the rotated factors, the CCA factors can be used to analyze the correlations between true factors and instruments, although use of them may underestimate true correlations. Due to the presence of A0 ut , the correlations between the CCA factors gt and the instruments zt should be smaller than those between the rotated factors M f1t and the instruments. Proposition 2. The projection errors u∗t in equation (6) are linear functions of the idiosyncratic errors ut and the missing factors f2t only. Proof of Proposition 2. Equation 5 and Proposition 1 imply that B1∗ A0 = Ωyy AA0 and (In − Ωyy AA0 )B1 = 0. Using these results, and by definition of projection, we have α∗ = E(yt ) − B1∗ E(gt ) = (In − B1∗ A0 )E(yt ) = (In − Ωyy AA0 )α. Therefore, u∗ = yt − α∗ − B1∗ gt = (In − Ωyy AA0 )(yt − α) = (In − Ωyy AA0 )(ut + B2 f2t ).
(7)
Proposition 2 shows that the projection errors in u∗t are free from the systematic measurement error problem. If there is no missing factor (f2t = 0), then the projection errors are linear functions of the idiosyncratic errors ut only. Proposition 2, however, does not mean that the projection errors u∗t are cross-sectionally uncorrelated. Even if there is no missing factor, individual projection errors are linear functions of all, if not some, idiosyncratic errors ut in as shown in equation (7). Thus, even if the idiosyncratic errors are cross-sectionally uncorrelated as in Assumption 4, the projection errors are mutually correlated. As a consequence, the first principal component of u∗t can have some explanatory power for individual response variables in yt . This implies that statistical significance of the first principal component in the regressions with the CCA factors is not necessarily evidence for missing factors. Inferences regarding existence of missing factors should be made by testing m = r, and Ahn and Perez (2010) provide GMM estimation methods for both m and r. Even though the projection errors u∗t are correlated, it may be still worth checking the explanatory power of the first principal component of them for the response variables. If some important factors are missing, then the first principal component is expected to have strong explanatory power for all response variables. If there is no missing factor, the principal 9
component would have moderate effects for all response variables or have strong explanatory power for only a few response variables. So far we have assumed that the population variance and covariance matrices, Ωyy , Ωyz , and Ωzz are known. In reality, however, we need to estimate them. For the data with small n and large T , the matrices can be consistently estimated by their sample analogs. The matrix A can be also consistently estimated from the sample variance and covariance ˆ and the estimated CCA factors by matrices. Let us denote this estimated matrix by A, gˆt = Aˆ0 yt . Using gˆt , we still can consistently estimate all of the coefficients in B1∗ , although ˆ We the asymptotic distributions of the estimates may depend on the sampling errors in A. leave the derivation of the asymptotic distribution for future research.
3
Data
3.1
Credit Spreads
Our laboratory is a panel data set of constant maturity yield curves, following Feldhuetter and Lando (2008) among others, obtained from the Merrill Lynch bond index system. This has the advantage that idiosyncratic movements of individual bonds are averaged out, and we minimize the risk that a common factor extracts average idiosyncratic risk. In the cross-section, we utilize yield curves sorted by rating category. The rating categorization is based on Merrill Lynch’s composite rating of the Moody’s, S&P and Fitch nomenclature.1 We limit our attention to credit spreads from issues in the industrial sector, excluding issuers from the regulated financial and the utility sectors. Merrill Lynch reports the set of individual corporate bonds that are used to construct the respective yield curves for each rating category. For the time period of our study, the average number of bonds with time to maturity of less than 10 years in the joint set of AAA and AA bonds is 84 (minimum 66), reflecting the low number of highly rated issuers in recent times. In comparison, the average number of bonds with time to maturity of less than 10 years in the joint set of A and BBB bonds is 919 (minimum 556). As a result, we limit our data set to the nine rating categories with a high degree of bond coverage, ranging from A1 to BB3, thereby excluding AAA and AA bonds. The second cross-sectional dimension of our data set is the term structure. For each point in time, we select 9 equally spaced points on the term structure, i.e. 2 to 10 years to maturity. 1
In case of a split rating, the composite rating is based on the lower of the assigned ratings; for example, a Baa3/BB+ bond rating equals a BB1 composite rating. In case a bond is rated by one agency only, the composite rating will equal that individual rating.
10
We then define the credit spread as the difference between the corporate bond spot rate and the yield to maturity on a zero coupon government bond of the same maturity, see Elton et al. (2001) or Liu et al. (2006). The constant maturity yield curves are based on bond values stripped off any embedded call or put option. In each time series, we have 508 observations in levels generating 507 first differences, observed weekly each Friday and covering the time period between January 1997 and December 2006, see Tables 1 and 2. The average level of credit spreads is slightly larger than those reported by Elton et al. (2001) for a time period between 1987 and 1996. We cover a period with dramatic widening of spreads after the Russia/LTCM crisis in 1998, as well as a period of narrowing spreads to pre-1998 levels between 2002 and 2006. As an independent check on the data, we also compare the Merrill Lynch credit spreads to spreads obtained through the Bloomberg system, and find the two sources are generally consistent.2 We also study the explanatory power of our factor model out of sample. Our laboratory is a second panel data set representing the term structure of credit spreads derived from of G.B.P. denominated corporate bonds. This data set has the same time series dimension as the U.S. dollar denominated yield curves, but has a smaller cross-sectional dimension. To limit ourselves to a high degree of coverage we use the 3 rating classes ranging from AA to BBB, thereby excluding AAA and non-investment grade bonds. While our U.S. dollar denominated data set represents U.S. domestic issuers, the G.B.P. yield curves correspond to more than one country in terms of issuer origin. For example, as reported by Merrill Lynch as of January 1 2000, the portfolio of BBB-rated issues contains 86 individual bonds with maturities of less than 10 years to maturity, 71 are issued by U.K. domestic corporate issuers, the remaining 15 by non-domestic issuers. This ratio is representative for the entire sample period. Panel A of Table 3 presents summary statistics for U.K. credit spread data in levels, and Panel B for first differences, respectively.
3.2
Macroeconomic and Financial Variables
We use a set of macroeconomic and financial variables that previous literature has argued to be of importance in explaining common movements in credit spreads. 1. The Treasury debt market. A strong negative relation between the level of riskless interest rates and the credit spread has been documented for several data sets, see for example Duffee (1998). We use the 10-year yield to maturity obtained from U.S. Treasury zero coupon bonds 2
After performing a multiple regression analysis of changes in credit spreads on a set of explanatory macroeconomic and financial variables, our findings are quantitatively and qualitatively similar to previous studies based on averaged individual bond data, another confirmation for the quality of our data.
11
as the riskless spot rate. Not only the level, but also slope of the term structure carries information about economic conditions - see Harvey (1988) and Estrella and Hardouvelis (1991). We use the difference between the 10 year spot rate and the one year spot rate as the diagnostic for the slope. While it has been documented that the shape of the riskless term structure itself can be explained by multiple factors, see Litterman and Scheinkman (1991), it is unclear whether changes in the slope of the riskless term structure have explanatory power for changes in credit spreads above and beyond changes in the general level of interest rates. 2. The equity market. The volatility of firms’ assets and degree of leverage are usual ingredients for structural models of credit risk. Since both are difficult to measure on aggregate on a weekly basis, we usually rely on measures derived from observable liabilities like equity. Collin-Dufresne et al. (2001) find mixed evidence for the explanatory power of changes in leverage for changes in credit spreads; changes in equity volatility represented by VIX, however, show more consistent results. Changes in the value of equity obviously impact the likelihood of default, but it has also been shown that macroeconomic conditions affect the amount of bondholder recovery given default, as in Acharya et al. (2007) or Altman et al. (2005). Moreover, changes in the business climate might directly affect target capital structures as argued in Demchuk and Gibson (2006). We complement our data set with observations of the S&P 500 index. 3. The swap market. We should expect that credit spreads contain a reward for bearing illiquidity risk. One way to capture this empirically is to exploit an illiquidity factor already shown to exist in other fixed income market segments, see for example Liu et al. (2006) or Feldhuetter and Lando (2008) for liquidity factors in interest rate swap spreads. We add the term structure of swap spreads to our data set, such that credit spreads and swap spreads have congruent maturities. For robustness we also explore the TED spread, as well as the yield spread between Fannie Mae and Treasury bonds as suggested by Feldhuetter and Lando (2008).
4 4.1
Credit Spread Analysis Cointegration and Credit Spreads
Other studies have documented non-stationary properties in levels of credit spreads, for example Pedrosa and Roll (1998) and Barnhill et al. (2000). Motivated by this finding, it is common to use a differentiated series instead, as the first differences usually appear to be stationary. However, using a differentiated series without a careful analysis of possible 12
cointegration relations may lead to an incorrect specification of model. If a cointegration relation is not rejected, then a regression using the differentiated series as dependent variables should include error correction variables as regressors in order to be correctly specified. We first test for stationarity in our data set using an Augmented Dickey Fuller (ADF) test. As expected, we can not reject the null hypothesis of a unit root for any of the series of levels of credit spreads. All ADF p-values are larger than 0.15, and most of them larger than 0.30. Hence, the time series of levels of credit spreads appear to be non-stationary, if not near non-stationary.3 Given these non-stationarity test results, we investigate whether there is a cointegration relation in levels of credit spreads among different credit quality. Though not much is known about the steady-state distribution of corporate debt from an equilibrium perspective, it is possible that credit spreads of different credit qualities move together; a long-run relation could for example be due to target capital structures. If there is such a relation, then a deviation from the long-run relation could be due to market frictions like adjustment costs, see Welch (2004) or Leary and Roberts (2005) for empirical evidence. While market frictions could affect the level of firm’s borrowing rates as well as riskless interest rates, the use of a differentiated credit spread series alone would not allow us to detect importance of such a long-term relation. Using the Johansen Rank statistic, we test how many cointegration relations are significant, see Table 4. We find at most two cointegration relations for short-term credit spreads (MAT3 and MAT5) and at most one cointegration relation for longer-term credit spreads (MAT7 and MAT10.) We then estimate the error correction series following Johansen (1988) and Johansen (1991) for each maturity bracket, given by the specification A1 A2 A3 BB3 ecmt = δ0 − δ1 CSt−1 − δ2 CSt−1 − δ3 CSt−1 ... + δ9 CSt−1 ,
where the δ loadings represent the cointegration coefficients. The deviation from a long-run relation is corrected gradually through a series of partial short-run adjustments corresponding to the ecm term.
4.2
Multiple Regressions
Our motivation to undertake a multiple regression analysis prior to performing the factor estimation is twofold. We first establish a benchmark case in order to have an in-sample 3
It seems implausible that any credit spreads series can actually explode, as a unit root process could. The power of the unit root test is low compared to near to unit root alternatives, nonetheless ignoring the data characteristics may lead to incorrect inferences. In contrast, the null hypothesis of non-stationarity is strongly rejected for differences in credit spreads – all p-values are smaller than .0001.
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comparison of CCA and principal components. Second, we test whether the inclusion of error correction terms leads to an improvement in explaining credit spreads. 4.2.1
Standard Explanatory Variables
The base case specification for which changes in credit spreads are linear functions of the explanatory variables is given by ¡ ¢2 ∆CSit = β0,i + β1,i ∆rt10 + β2,i ∆rt10 + β3,i ∆slot + β4,i ∆vixt + β5,i s&pt + eit . (8) We estimate the previous equation for each one of 36 portfolios formed on maturity ( 3, 5, 7 and 10) and rating (A1, A2, A3, BBB1, BBB2, BBB3, BB1, BB2, BB3). Estimation results are summarized in Table 5. As expected, changes in the risk free rate are statistically and economically significant in explaining changes in credit spreads, and the estimated sign is as predicted. Larger equity volatility, for example due to a larger degree of leverage or a larger degree of asset volatility or both, leads to an increase in credit spreads. The change in the slope of the riskless term structure adds explanatory power for some credit qualities. Panel B reveals that the slope impact is the strongest for short maturities. The estimated sign is surprising as we expected the cyclical behavior of slope to be negatively associated with changes in credit spreads. The analysis of estimated residuals is crucial for our objective. First, does a multiple regression analysis applied to our data set also lead to relatively low explanatory power measured via R-square values? Second, does an inspection of estimated residuals reveal a strong common component using principal components? Our results support both observations. Average adjusted R-square values vary between .06 and .40 – the explanatory power is the largest in lower rating categories.4 Sorted by maturity, the average adjusted R-square values range from .16 to .18. Instead of interpreting the fraction of variation in the residuals explained by the principal components as in Collin-Dufresne et al. (2001) or Ericsson et al. (2009), we decide to extract the first principal component from residuals and add the extracted series to the set of explanatory variables. The explanatory power of the regression increases sharply and R-square values double in all cases, see Table 6. Hence, our data set has very similar properties as studies based on averaged individual bond data – average R-square values are low, and a strong common component appears to exists in estimated residuals. 4
In comparison, Collin-Dufresne et al. (2001) find R-square values for comparable maturities between .20 and .34. Differences in estimates compared to other studies could be due to multiple reasons. For example, our data set and the Warga Database used in other studies overlap only in the year 1997, and a dramatic widening of spreads with increased volatility started to occur in 1998. Second, we do not exclude observations with less than 4 years to maturity. Third, our time series dimension has weekly instead of monthly observations, and we verified by restricting our data set to monthly changes that the explanatory power for high rating categories increases.
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4.2.2
Explanatory Variables and Error Corrections
Changes in the dependent variable are a now function of the same set of explanatory variables as above plus the error correction term given by ¢2 ¡ ∆CSit = β0,i + β1,i ∆rt10 + β2,i ∆rt10 + β3,i ∆slot + β4,i ∆vixt + β5,i s&pt + β6,i ecmt + eit . (9) The ecmt in the above equation is maturity-specific, i.e. while running individual regressions for each of the rating-maturity portfolios we include the error component of the corresponding maturity bracket. Estimation results are summarized in Table 7. According to Panel A, the inclusion of ecmt increases explanatory power for 7 rating categories, and average coefficient estimates are significant for A1 to BBB1 rated bonds. According to Panel B, average coefficient estimates are significant for maturities 3, 5, and 7. We conclude that deviations from a cointegration relation contain some information over and above what is already contained in a standard set of explanatory variables. The reported increases in R-square values, however, are by no means substantial. The absolute of the coefficient for ecmt can be interpreted as to how quickly credit spreads revert to a long-run relation, and a larger coefficient implies a faster adjustment.5 The explanatory power of the ecm component may have to do with the informational content of the cross-sectional divergence of credit spreads. Since the main explanatory power arises in portfolios of higher rated bonds, this would be consistent with a view that higher rated bonds are relatively more liquid and can therefore incorporate information about the economy faster. Lower rated bonds, on the other hand, might trade less frequently and new information might not lead to significant adjustment coefficients. The results are also consistent with the view that short term spreads are more sensitive to economic information, and thereby adjust to a long-run relation faster. In the previous section we also acknowledged that adjustment coefficients may arise due to target capital structures. If there are deviations from target capital structures, then it appears that higher rated firms adjust to the target faster. This could be because those firms have easier access to the capital market whereas lower rated firm face more funding restrictions. In addition, the fastest adjustment activity would correspond to issuance in shorter term bonds. 5
The sign of the adjustment coefficient, however, does not lead to further insights. This is because we construct cointegration equations based on more than two variables, and even a normalization would not help us to derive a definite prediction regarding direction of adjustment.
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4.2.3
Robustness
The constant maturity yield curves in our data set are based on bond portfolios sorted by rating category. As reported by Merrill Lynch, these portfolios are rebalanced on the last business day of each month based on information about rating transitions up to the third business day prior to the last business day of the month. Since our data frequency is shorter than the rebalancing frequency, it might be possible that part of our empirical findings are driven by the mechanics of rebalancing. To address this issue, we test indicator variables corresponding to week-of-the-month effects interacted with the lagged performance of credit spread changes in several specifications. We find that our results about ecmt are robust to all specifications and are not driven by the mechanics of rebalancing. The average maturity (or duration) of individual bonds used to construct the constant maturity yield curves is not constant over time, as reported by Merrill Lynch. Hence, it might be the case that a yield curve estimation generates some term structure dynamics simply due to time-varying weights. We control for average maturity (duration) and find that our results are not affected by time-varying average maturities. The January effect, see Chang and Pinegar (1986), could also be present in our data set. Indeed, we find marginal evidence for the existence of the January effect in our time series. However, all our results are robust while controlling for the effect with indicator variables.
5
CCA Factor Analysis
5.1
Estimation of Number of Factors
In order to extract the factors using canonical correlations, two questions need to be answered. How many factors should be estimated? And, are all common factors captured by our set of observable variables? Several procedures are available to estimate the total number of factors m. For cases with a small number of cross-sectional observations, J¨oreskog (1967) has developed a maximum likelihood method under strong distributional assumptions, which are not satisfied based on the dynamics of the term structure observable in our data set.6 More recently, Bai and Ng (2002) have developed a general method for data with both large numbers of cross-sectional and time-series observations, but their method could produce inconsistent estimators if one 6
Cragg and Donald (1997) propose a methodology that requires weaker distributional assumptions and is based on the estimators of the ranks of matrices. However, as shown by Donald et al. (2005), it is computationally difficult to implement and might fail to locate a solution as it requires nonlinear optimization procedures.
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dimension is small. The estimation method we used in this paper is designed to consistently estimate the number of factors in panels with a large time series dimension and a small number of cross-sectional observations, see Ahn and Perez (2010). We follow this GMM procedure for two reasons: First, it fits the dimensionality and statistical properties of term structure data. Second, it allows for cross-sectional and time-series heteroskedasticity, autocorrelation, and requires no strict distributional assumptions. A feature of the method is that it partitions the cross-sectional observations of the response variable into two non-overlapping groups, and estimation should be repeated for several different group specifications. Ahn and Perez (2010) show that if the response variables are functions of “strong” factors only – factors with a signal-to-noise ratio greater than 0.20 – then the number of factors estimated with highest frequency is the correct result. However, if data is generated by some “strong” and some “weak” factors, then the second highest frequency should also be analyzed. Utilizing this property, we also distinguish between weak and strong factors in the term structure of credit spreads. We perform our estimation for each maturity bracket separately. This allows us to draw more precise inference about the number of factors along the second cross-sectional dimension. Though one can argue that common factors should be relevant for all maturities, specific market conditions could affect bonds with different maturities to a different extent. We present the estimation results using two criterions used by Ahn and Perez (2010), the model selection criterion (upper graph) and the sequential hypothesis testing (lower graph) in Figure 1. The model selection criterion estimates two factors more than 55% and three factors around 30% of the time for credit spreads with short maturities. For maturities larger than three years this relation is reversed, finding two factors 30-40% and three factors 45-60% of the time. Very similar results are obtained by applying the sequential testing method. For credit spreads corresponding to two and three years to maturity we can not reject the null hypothesis of two factors more that 55%, and three factors around 25% of the time. Again, for maturities larger than three this relation is reversed. Our results suggest that three common factors explain changes in the term structure of credit spreads. One of these three factors, however, appears to be a “weak” factor – it might only account for a small degree of the total variance of the response variables, and we find it to be estimated only for some maturities. Two of the three factors appear to be “strong” as their appearance is robust among all maturities, and we rarely estimate only one factor. In addition, we perform a robustness check: We expand our data set and include credit spreads from AA-rated bonds using the same data source despite lower bond coverage. If there exist three common factors that explain changes in credit spreads, then we should also expect to 17
find three common factors in the data set based on an expanded cross-section. Estimation results are quantitatively very similar, and we can confirm the finding of three factors. We are aware of two other studies that have used the ML method and the likelihood ratio (LR) test to identify the number of factors in returns or credit spreads of corporate debt contracts. Knez et al. (1994) develop three-factor and four-factor models for money market returns, although a likelihood ratio test would indicate a much larger number of common factors for their data set. More recently, Driessen (2005) proposes two common factors in credit spreads also based on a likelihood ratio test plus a common liquidity component, in addition to two common factors that explain the variation in riskless bonds and also enter credit spreads, leading to five common factors in total. In order to disentangle our factor estimation results from those studies, we apply a LR test to our data set. The null hypothesis of three factors is always rejected. If a conclusion about the number of factors was based solely on the likelihood ratio test – a test with strong distributional assumptions – we would be tempted to suggest that credit spread innovations have four or more common factors. However, the null hypothesis of normality is strongly rejected for all credit spread series based on several test statistics. We also try to estimate the number of factors following Bai and Ng (2002), but we do not find reliable results since the method always estimates the maximal number of factors tested for; this is expected given the relatively small number of cross-sectional observations. Our next objective is to test whether all three factors are captured by a set of explanatory variables. Using the notation introduced in Section 2, we are interested in estimating r, the number of the factors correlated with a set of explanatory variables zt . Again, Ahn and Perez (2010) can be used to accomplish this. Instead of partitioning the variables in two non-overlapping groups, one group is the entire vector of response variables and the other 2 group is the set of instruments zt . We first use the set given by zt (1) = {∆r10 , (∆r10 ) , ∆slo, ∆vix, s&p}. Results are presented in Figure 2. Under the model selection criterion and for credit spreads corresponding to shorter maturities (< M AT 5) the estimation procedure finds one factor more than 50% of the time, two factors around 17%-34% of the time, and three factors 3%-15% of the time. For the case of longer maturities (≥ M AT 5) the method estimates one factor around 30% of the time, two factors around 50% of the time, and three factors 17-27% of the time. The zt (1) variables are correlated with all three factors as the estimation procedure finds three factors with some positive frequency. The level of correlations with one or two of the three, however, appears to be weak. We can infer this by comparing the estimation results
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using zt (1) to the results of the previous subsection, where three factors were obtained 4560% of the time for longer maturities. In case of shorter maturities, we estimate three factors around 15% of the time using instrumental variables, compared to 25-30% obtained in the pervious subsection.7 Using a simple diagnostic, we compute the average number of factors estimated with zt (1) across all maturities, and compare it to the average number of factors estimated without instruments. Based on the model selection criterion, the average number of factors estimated with zt (1) equals 1.75, compared to 2.37 factors without instruments. We test some additional explanatory variables and check whether results improve. For example, Figure 3 shows estimation results adding the Fama and French factors smb and hml to the set of instruments, i.e. zt (2) = {zt (1), smb, hml}. The inclusion of smb and hml gives similar results compared to using zt (1) only. The average number of factors estimated with zt (2) decreases to 1.54, which is no improvement compared to zt (1). Figure 4 shows factor estimation results adding the variable ∆swap to the set of instruments, i.e. zt (3) = {zt (1), ∆swap}. The inclusion of ∆swap does not change the results consistently across all maturities. However, an improvement occurs for credit spreads corresponding to maturities larger than seven years, as three factors are estimated with a larger frequency compared to zt (1) alone. The set zt (1) alone estimates three factors only 17% to 22% of the time for maturities between 7 and 10 years; zt (1) plus ∆swap, however, estimates three factors 25% to 35% of the time. Hence, if there exists a common factor linked to illiquid market conditions and it is not captured by zt (1), then the addition of ∆swap seems to capture it for credit spreads corresponding to longer maturities.8 The average number of factors estimated with zt (3) increases to 1.78, a slight improvement over 1.75 factors estimated with zt (1) only. Finally, we test the series ecm as an instrumental variable, i.e. zt (4) = {zt (3), ecm}, see Figure 4. The results notably improve for spreads corresponding to shorter maturity bonds, as two factors are estimated more often compared to zt (3) only. For spreads with maturities shorter than five years the model selection method now estimates one factor only 9% to 26% of the time, but two factors 43-54% of the time.9 The average number of factors estimated with zt (4) increases to 2.12, a strong improvement over using zt (3) only. 7
Such weak correlations could be the cause of the strong explanatory power of the first principal component in the multiple regression, since the unexplained part of the true factors will remain part of the idiosyncratic component of the model. 8 This is consistent with Edwards et al. (2007), who find that percentage transaction costs increase with time to maturity of the underlying corporate bond. If transaction costs are a fundamental friction that correspond to bond market illiquidty, then we should expect a larger degree of bond market illiquidty for longer maturities compared to shorter maturities. 9 This is also consistent with the multiple regression results, where including the error correction term as a regressor improves adjusted R-square values for shorter maturity bonds.
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Gouriroux et al. (1995) also propose a method to estimate r. For robustness, we apply the Generalized Wald test statistic using the set zt (4) and the estimation results are similar. We find 3 factors for all maturities when the test is performed with a 1% significance level. With a 5% significance level we estimate 3 factors for maturities smaller or equal than 5, and 4 factors for longer maturities.
5.2
CCA Factor Model
We now extract the factors by estimating three canonical correlations. We utilize 36 crosssectional observations included in A0 yt = A0 ∆CS, including the 9 rating categories for maturities 3, 5, 7 and 10. We also consider “maturity-specific” CCA factors, for which only the 9 rating categories for each maturity are included. We will use the “maturity-specific” CCA factors to gain additional insight into the economic meaning of the those factors, and to check robustness. Given the results in the previous subsection, the set zt (4) will serve as explanatory variables. We include the error correction terms corresponding to three-year maturities because estimation results do not notably improve with error corrections of longer term credit spreads. Our CCA factor model is given by the specification ∆CSit = β0,i + β1,i f actor1t + β2,i f actor2t + β3,i f actor3t + u∗t ,
(10)
where the factors are ordered by their canonical correlations with the instruments zt (4), and not by their explanatory power. The estimation results in Table 8 show a large increase in explanatory power compared to the regressions in Section 4. The adjusted R-square value is now larger than 0.45 sorted by maturity, and between .20 and .76 sorted by rating class, with an average of .48 across all ratings. The explanatory power of our model is comparable to the 5-factor reduced-form approach proposed by Driessen (2005), who reports an average R-square value of .387 based on a data set of investment grade bonds only – the average R-square value of our 6 investment grade bond portfolios equals .40. As before, we extract the first principal component from residuals and add the extracted series as an explanatory variable, see Table 9. The increase in adjusted R-square values is minimal for the eight rating classes AA1 to BB2, i.e. less than .02 on average. Interestingly, the first principal component increases the explanatory power for the BB3 rating class to almost 100%, indicating that the first principal component represents the projection errors of the lowest rated credit spread series. The second principal component contains the projection errors of another rating class. Essentially, all three CCA factors can account for the common
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variation in credit spreads, and the unexplained noise is of idiosyncratic nature.10 We can compare the three CCA factors in their relative importance by studying the percentage of total variation explained by each factor, see Table 10. On average, factor 1 has explanatory power of 26.4%, followed by factor 3 with 14.1%, and by factor 2 with 7.5%.11 We can identify factor 2 as a weak factor, and factors 1 and 3 are strong factors based on a signal-to-noise ratio larger than .20. Sorted by rating, a trend appears not only in R-square values, but also in the percentage variation explained by factor 1 only. For credit spreads corresponding to BB2 and BB3 rated bonds, 90% of the total explanatory power is due to factor 1. While it is expected that lower rated corporate bonds carry a larger degree of systematic risk compared to higher credit quality bonds, see for example Elton et al. (2001) and Ericsson et al. (2009), we find it noteworthy that such a large fraction is explained by one of the three factors only. One might ask, how do our results compared to PCA factor regressions? We first obtain three PCA factors from the 36 cross-sectional observations. We then estimate the regression model in equation 10 using the three PCA factors instead, results are shown in Table 11. All three factors are significant, and adjusted R-square values are higher than in the case of CCA factors. It is possible that PCA factors can have higher explanatory power compared to CCA factors. Intuitively, this is because PCA factors are linear combinations of just the dependent variables without utilizing additional information. The adjusted R-square value is larger than 0.53 sorted by maturity, and between 0.32 and 0.93 sorted by rating class, with an average of 0.58 across all ratings. However, residuals from regressions using PCA factors may still have a factor structure that is caused by measurement error. To check if this is the case for our credit spread data we also extract the first principal component from residuals, and add the extracted series as an explanatory variable, see Table 12. The explanatory power increases sharply and R-square values increase for all maturities and almost all rating classes. The average across all ratings increases to 0.75. Clearly, the first principal component of 10
As a robustness check, we have also estimated the model in equation (10) with the fourth CCA factor as an additional regressor. The use of this factor increases the explanatory power (adjusted R-square) for A1 by 0.105. However, the increases in power are less than 0.05 for other rating classes. Adjusted R-square values decreases for BBB2. It appears that factor 4 only captures idiosyncratic error components of response variables. We do not report the results to save space, but they are available from the authors upon request. 11 The explanatory power of CCA factors may not decrease in order, in that a preceding factor has higher explanatory power than the succeeding one. The order of CCA factors are determined by the canonical correlations between instruments and credit spread changes. It is possible that a CCA factor highly correlated with instruments can have weak correlations with individual credit spread changes. For an extreme example, suppose that a set of instrumental variables is highly and weakly correlated with the weakest and strongest factors, respectively. Then, the first CCA factor should be the weakest factor.
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residuals would suggest there is indeed a missing factor that is not captured by the PCA factor model.
5.3
Interpretation
We compute the pairwise correlations of the CCA factors with the set of variables that were used to construct them, see Table 13, Panel A.12 Factor 1 shows large correlations with all economic variables, except ecm. Factor 1 is consistently negatively correlated with ∆r10 , (∆r10 )2 , and ∆slo. While ∆slo has a positive coefficient estimate in the multiple regression analysis, the predicted economic channel becomes visible in our factor construction. In improved economic times, default will be less likely and an increase in slope is associated with lower credit spreads. At the same time, the slope represents investors’ expectations about future (riskless) interest rates, such that a decreasing credit spread can also be the result of expectations about increasing riskless interest rates while holding default activity constant. Factor 1 also has a positive 31% correlation coefficient with equity volatility, and a negative 22% correlation coefficient with the equity market return. These consistent regularities with several economic variables, plus the observation that the explanatory power of factor 1 increases with lower credit quality, allow us to interpret factor 1 as a macro-default factor. In addition, factor 1 has a positive 52% correlation coefficient with the interest rate swap spread. This can be evidence of cyclical variation in the liquidity component of credit spreads with the predicted sign. While it is an affirmation of the positive correlation between illiquidity and default components made by Ericsson and Renault (2006), the new finding is that the common illiquidity component corresponds to the same factor that captures the default component. To check robustness we have computed correlations with two alternate illiquidity measures. The correlation of factor 1 with the TED spread is quantitatively similar to the swap spread correlation, and its correlation with the Fannie Mae/Treasury bond yield spread is 0.32 for levels, and 0.22 for first differences. For factor 2 and factor 3, the largest correlations occur with the error correction terms, changes in slope, and with equity volatility. To disentangle these factors further, we utilize 12
Regression results capture partial effects of individual variables, holding other variables constant. Since economic models frequently do not make predictions on partial effects of economic variables, but rather on unrestricted correlations, we believe that interpretation of the pairwise correlations is more insightful. Please note that such an interpretation is tentative. As explained in Assumption 3(ii) our factors are identified only up to a linear transformation, just like the principal component factors. However, CCA factors have an economic foundation as the explanatory variables we use are all suggested by economic theory.
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the factor construction with maturity-specific factors. Factor 2 shows the strongest correlation coefficient with ecm, ranging from .36 to .42, and therefore identifies as the common factor that relates to the adjustment of credit spreads to deviations from a cointegration relation. This interpretation is consistent with the monotone decreasing percentage of total variation explained by factor 2, see Table 10, with 11.4% for MAT3 to 4.1% for MAT10. The maturity-specific factors also reveal that factor 3 is most related to ∆slo and vix, with correlation coefficients between -.24 and -.08 for ∆slo, and between .13 and .19 for vix. Except for three rating categories, factor 3 also has a positive factor loading and therefore the predicted effect of ∆slo and vix on credit spread innovations. Please note the correlation between factor 3 and the slope is less negative for longer maturities, and the trend is monotone. As we posit that this correlation reflects investors’ expectations about future economic activity, this result might seem counterintuitive at first glance. One might think that factor 3 being constructed from longer term bonds should capture such expectations to a larger extent similar to the slope of the term structure being defined through longer term Treasury bonds. However, the sum of correlations of factor 1 and 3 with the slope variable is in fact decreasing in maturity, i.e. from -.32 to -.41. Hence, we believe that factor 3 does capture expectations about economic activity, but the severity on the time dimension is less steep as suggested by factor 1. Such effects can occur in a reduced-from analysis as ours as we simply interpret correlations without economic restrictions. We also find the correlation between factor 3 and vix is more positive for longer maturities, and the trend is monotone. We posit that this correlation reflects ex-ante risk premia, higher expected excess returns and thereby increased credit spreads, see Mele (2007) for economic channels supporting this correspondence. A plausible interpretation here would be that the risk premium component is increasing in importance with the maturity of the bonds used for constructing factor 3, for example because the holding period return of longer term bonds is more volatile and/or because of time-varying risk premia. We believe this does not contradict that the correlation between factor 1 and vix is less positive for larger maturities, as the sum of correlations of factor 1 and 3 with vix is relatively flat across maturities, i.e. between .48 and .52. Finally, to compare the CCA factor interpretation to the alternative principal component methodology, we check the pairwise correlations of the first three PCA factors with the set of explanatory variables that were not used to construct them. The interpretation of the first factor appears to be similar, due to a high correlation between the strongest CCA factor and the first PCA factor of 88%. However, the interpretation of the second and third PCA factor is diluted. This is not surprising as the correlations between the second or third CCA 23
factors and PCA factors are only .35% and .50%, respectively.13
5.4
Out of sample tests
We first split the data set into the estimation period covering 1997 to 2004 with 405 time series observations, and the test period covering 2005 to 2006 with 102 time series observations. We use the estimation period to find the canonical coefficients, use these coefficients to construct factors for the test period, and repeat the analysis only based on the test period. Estimation results are qualitatively and quantitatively similar to those in Tables 8.14 Second, since the factors are derived from the term structure of U.S. dollar denominated bonds of domestic issuers, we also test their explanatory power on changes in spreads obtained from G.B.P denominated bonds primarily from U.K. issuers. We expect domestic-wide movements to be related to movements in the U.K. market due to market integration among developed economies. Our factor model can explain close to 20% of the variation in the U.K. data set, see Table 14. This explanatory power is driven by factor 1; the loading and the significance of factor 1 is at least 3 times as large as those of factors 2 and 3. After the inclusion of the first principal component, adjusted R-square values increase to values larger than .65. Therefore, the CCA factors have satisfactory performance in explaining the term structure of spreads corresponding to G.B.P denominated corporate bonds, but at least one factor appears to be left unexplained. As a final test, we explore the predictive power of the factor model. Does the factor interpretation remain valid based on a macroeconomic variable that was not used to construct them? We integrate the CCA factor first, and regress the time series of monthly real disposable U.S. personal income data on the two strong CCA factors. We employ contemporaneous as well as lagged values. If the second strong factor indeed represents mainly investors’ expectations, then we should expect that it has predictive power for future realizations of the state of the economy. Test results are shown in Table 15 Panel A. The strongest factor is most correlated with the current realization of disposable income, lagged observations matter only marginally. This observation flips as we analyze the second strong factor. The 13
As an aside, we find that it would require more than 10 principal component factors to identify the same economic channels as the three CCA factors. After extracting 14 PCA factors, and then generating the first three canonical relations between the 14 PCA factors and the set zt (1) plus the variables ∆swap and ecm, the correlations between the original and newly obtained CCA factors are between 78% and 94%. 14 We then perform a cross-section out of sample test based on the credit spread series corresponding to AA-rated bonds. Compared to single A-rated bonds (i.e. the highest in our original data set), we expect this series to contain even less exposure to factor 1, but the second strong factor 3 to be relatively more important. The average explanatory power of the factor model for AA credit spreads is .20 R-square, and we can confirm that only .05 corresponds to factor 1, but .13 corresponds to factor 3.
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6-month lagged observation is the most significant lagged variable predicting income. Hence, we confirm the second strong factor indeed contains a component distinct from factor 1 while predicting the future state of the economy.
6
Conclusion
Several techniques are available to extract latent factors. Within a model-free environment, we find that using canonical relations to extract factors has several advantages, which should also be useful for future research in other areas of empirical finance. Two features stand out: First, residuals from regressions using the canonical correlation factors can be used to test if a given set of explanatory variables is able to capture all systematic variation in the response variables under analysis. Second, the factors are easier to interpret as they are obtained directly from canonical correlations between response variables and explanatory variables. The common variation in the term structure of credit spreads is mainly explained by three factors, and there does not exist a strong factor that is not present in the equity market, the swap market, or the market for Treasury debt. The weakest of the three factors might be specific to the corporate bond market, but its’ explanatory power appears to be small. We identify the weak factor as one that is strongly related to deviations from a cointegration relation. The economic source for this relation, however, is unknown at this point. Whether credit spreads are stationary or not is a controversial issue, and studies using a longer time series might find the existence of a weak factor to be a small sample artifact. One might ask, how do our results speak to the future development of structural models? The existence of two strong factors lend to an environment in which two macro state variables drive the term structure of credit spreads. The interpretation of one of the two as an expectation factor could also be consistent with a one-factor model under stochastic volatility.
7
Appendix
25
series A1-MAT3 A1-MAT5 A1-MAT7 A1-MAT10 A2-MAT3 A2-MAT5 A2-MAT7 A2-MAT10 A3-MAT3 A3-MAT5 A3-MAT7 A3-MAT10 BBB1-MAT3 BBB1-MAT5 BBB1-MAT7 BBB1-MAT10 BBB2-MAT3 BBB2-MAT5 BBB2-MAT7 BBB2-MAT10 BBB3-MAT3 BBB3-MAT5 BBB3-MAT7 BBB3-MAT10 BB1-MAT3 BB1-MAT5 BB1-MAT7 BB1-MAT10 BB2-MAT3 BB2-MAT5 BB2-MAT7 BB2-MAT10 BB3-MAT3 BB3-MAT5 BB3-MAT7 BB3-MAT10
min 30.41 39.11 44.15 49.91 37.98 46.50 52.30 58.01 38.62 48.95 55.56 60.22 42.15 55.00 62.72 69.78 51.73 62.68 69.64 75.62 53.25 70.78 79.99 89.02 82.08 96.53 105.84 113.48 118.38 137.24 146.32 154.15 138.24 164.09 180.66 195.62
max 153.52 166.91 170.32 183.11 165.53 178.36 190.17 201.76 191.63 206.40 218.41 228.03 273.12 284.79 286.19 281.19 281.63 275.54 265.55 250.27 325.38 327.77 325.96 318.56 576.26 554.67 532.93 505.12 612.74 599.30 589.18 597.08 745.44 735.39 725.24 712.62
mean 68.47 79.84 87.08 93.85 81.73 92.81 99.71 105.99 93.58 104.81 111.66 117.73 116.02 127.62 134.51 140.34 131.77 142.45 148.49 153.29 154.11 165.83 172.27 177.11 239.48 253.25 260.88 266.58 290.11 299.20 303.67 306.24 335.60 345.34 350.01 352.53
stdev 25.59 29.21 30.98 31.99 34.85 36.05 36.67 37.06 41.77 42.55 42.90 43.02 50.34 49.21 48.14 46.72 56.75 53.12 50.12 46.76 67.51 65.22 62.83 59.62 105.98 97.42 91.75 86.24 107.59 98.73 93.31 88.47 141.04 125.27 115.50 106.97
Table 1: Data Set - Credit Spreads - Levels. The table reports summary statistics for the indicated data series, 508 weekly observation between Jan 1997 and Dec 2006. Credit spreads are computed under semiannual compounding, expressed on an annual basis. All spreads are in basis points.
26
series A1-MAT3 A1-MAT5 A1-MAT7 A1-MAT10 A2-MAT3 A2-MAT5 A2-MAT7 A2-MAT10 A3-MAT3 A3-MAT5 A3-MAT7 A3-MAT10 BBB1-MAT3 BBB1-MAT5 BBB1-MAT7 BBB1-MAT10 BBB2-MAT3 BBB2-MAT5 BBB2-MAT7 BBB2-MAT10 BBB3-MAT3 BBB3-MAT5 BBB3-MAT7 BBB3-MAT10 BB1-MAT3 BB1-MAT5 BB1-MAT7 BB1-MAT10 BB2-MAT3 BB2-MAT5 BB2-MAT7 BB2-MAT10 BB3-MAT3 BB3-MAT5 BB3-MAT7 BB3-MAT10
min -18.78 -17.85 -16.17 -16.36 -24.98 -25.14 -24.80 -24.09 -12.99 -13.02 -12.90 -13.78 -18.72 -23.78 -31.36 -35.69 -16.18 -16.73 -16.35 -15.20 -23.99 -22.69 -24.38 -25.09 -82.46 -61.64 -47.83 -42.68 -68.32 -53.01 -56.66 -69.63 -95.16 -79.91 -70.98 -72.22
max 19.38 27.86 32.63 36.07 22.18 27.48 30.61 32.40 29.22 31.71 33.46 35.09 35.10 32.78 33.05 33.88 40.16 43.63 45.65 47.12 42.69 42.00 40.89 38.99 102.68 86.10 74.25 61.98 134.06 98.42 75.57 54.51 152.54 117.81 103.76 85.65
mean 0.025 0.035 0.043 0.051 0.011 0.034 0.050 0.065 0.028 0.042 0.052 0.062 0.032 0.054 0.071 0.088 0.028 0.059 0.080 0.101 0.049 0.074 0.091 0.106 0.054 0.098 0.128 0.156 0.026 0.068 0.095 0.121 -0.034 -0.074 -0.105 -0.139
stdev 3.57 3.68 3.88 4.02 3.74 3.82 3.92 3.96 3.86 3.85 3.94 4.02 5.21 5.10 5.15 5.15 4.73 4.60 4.66 4.72 6.21 6.13 6.24 6.33 13.52 12.04 11.53 11.26 16.75 14.88 14.37 14.33 22.89 20.01 19.10 19.00
Table 2: Data Set - Credit Spreads - First Differences. The table reports summary statistics for the indicated data series, 507 weekly observation between Jan 1997 and Dec 2006. First differences are derived from credit spreads expressed on an annualized basis. All changes in spreads are in basis points.
27
Panel A series AA1-3-MAT3 AA1-3-MAT5 AA1-3-MAT7 AA1-3-MAT10 A1-3-MAT3 A1-3-MAT5 A1-3-MAT7 A1-3-MAT10 BBB1-3-MAT3 BBB1-3-MAT5 BBB1-3-MAT7 BBB1-3-MAT10
min 17.40 31.50 41.80 53.90 29.40 49.80 60.40 66.20 54.00 65.90 77.70 89.60
max 119.00 138.80 151.70 178.10 144.90 167.60 187.50 215.50 217.50 226.10 228.90 247.40
mean 55.50 71.40 81.90 92.30 77.60 95.10 106.10 116.30 120.30 133.20 142.20 150.80
stdev 18.80 24.40 28.60 32.70 27.20 31.30 34.20 37.00 40.60 40.60 40.20 39.90
Panel B series AA1-3-MAT3 AA1-3-MAT5 AA1-3-MAT7 AA1-3-MAT10 A1-3-MAT3 A1-3-MAT5 A1-3-MAT7 A1-3-MAT10 BBB1-3-MAT3 BBB1-3-MAT5 BBB1-3-MAT7 BBB1-3-MAT10
min -10.50 -14.70 -18.00 -20.30 -13.40 -12.60 -16.00 -18.70 -39.90 -19.60 -18.30 -16.30
max 26.80 29.30 30.30 30.40 27.10 32.10 34.40 35.40 46.00 40.20 37.70 34.80
mean 0.100 0.000 0.000 0.000 0.100 0.100 0.000 0.000 0.000 0.100 0.000 0.000
stdev 2.60 2.90 3.10 3.40 3.20 3.30 3.50 3.80 7.30 5.60 5.00 4.70
Table 3: Data Set - U.K. Credit Spreads - Levels and First Differences. Panels A (508 levels) and B (507 first differences) report summary statistics for the indicated U.K. data series between Jan 1997 and Dec 2006, used for the out-of-sample test of general CCA factors. First differences are derived from credit spreads expressed on an annualized basis. All changes in spreads are in basis points.
28
Panel A - Mat3 no of ce(s) None * At most 1 * At most 2 * At most 3 At most 4
eigenvalue 0.213 0.136 0.104 0.063 0.050
statistic 122.860 75.090 56.376 33.117 26.379
crit. value 58.434 52.363 46.231 40.078 33.877
prob. 0.000 0.000 0.003 0.246 0.298
Panel B - Mat5 no of ce(s) None * At most 1 * At most 2 * At most 3 At most 4
eigenvalue 0.207 0.128 0.103 0.064 0.049
statistic 118.945 70.079 55.644 33.706 25.943
crit. value 58.434 52.363 46.231 40.078 33.877
prob. 0.000 0.000 0.004 0.219 0.324
Panel C - Mat7 no of ce(s) None * At most 1 * At most 2 At most 3 At most 4
eigenvalue 0.173 0.106 0.085 0.063 0.050
statistic 97.313 57.400 45.441 33.180 26.241
crit. value 58.434 52.363 46.231 40.078 33.877
prob. 0.000 0.014 0.061 0.243 0.306
Panel D - Mat10 no of ce(s) None * At most 1 * At most 2 At most 3 At most 4
eigenvalue 0.160 0.097 0.066 0.059 0.044
statistic 89.127 52.409 35.205 31.334 23.137
crit. value 58.434 52.363 46.231 40.078 33.877
prob. 0.000 0.050 0.447 0.341 0.520
Table 4: Unrestricted Cointegration Rank Test (Maximum Eigenvalue). Panels A and B indicate 3 cointegration equations, Panels C and D indicate 2 cointegration equations, (*) denotes rejection of the hypothesis at the 5% level. The last column shows MacKinnon-Haug-Michelis p-values.
29
Panel A sorting A1 A2 A3 BBB1 BBB2 BBB3 BB1 BB2 BB3
Panel B sorting MAT3 MAT5 MAT7 MAT10
int. 0.178 0.929 0.150 0.767 0.222 1.150 0.196 0.783 0.027 0.184 0.189 0.611 0.121 0.224 -0.575 0.944 0.046 0.108
∆r10 -2.292 1.502 -3.223 2.035 -5.451 3.486 -5.508 2.707 -7.416 3.942 -12.941 5.154 -43.260 9.606 -74.047 14.920 -81.525 11.833
(∆r10 )2 -8.498 1.567 -5.973 1.075 -10.216 1.865 -8.188 1.156 0.830 0.135 -6.675 0.763 -2.992 0.196 27.454 1.594 -19.371 0.801
∆slo -1.601 0.781 0.807 0.553 3.891 1.838 0.367 0.584 0.381 0.363 12.462 3.648 26.142 4.207 23.451 3.379 18.783 1.884
∆vix 0.528 5.254 0.349 3.357 0.510 4.969 0.840 6.301 0.551 4.475 0.511 3.105 0.935 3.158 1.129 3.467 2.262 4.942
s&p 0.009 1.239 -0.005 0.637 0.004 0.618 0.007 0.721 0.007 0.759 0.001 0.485 -0.031 1.490 -0.030 1.340 0.000 0.177
int. 0.010 0.406 0.056 0.603 0.079 0.706 0.102 0.819
∆r10 -29.038 6.256 -26.850 6.409 -25.415 6.226 -23.437 5.637
(∆r10 )2 -1.937 0.706 -3.596 1.013 -4.359 1.135 -5.055 1.214
∆slo 16.881 3.075 10.903 2.170 6.839 1.451 3.015 0.965
∆vix 0.954 4.338 0.878 4.552 0.815 4.396 0.738 4.060
s&p -0.001 0.986 -0.004 0.843 -0.005 0.763 -0.006 0.726
adj. R2 0.085 0.058 0.105 0.128 0.094 0.090 0.227 0.396 0.343
adj. R2 0.157 0.177 0.178 0.166
Table 5: Multiple Regressions - Changes in Credit Spreads on Standard Set of Explanatory Variables. Panel A shows average ols regression results sorted by rating category, Panel B sorted by maturity bracket. The averages of (absolute) t-statistics are shown below the parameter estimates. The (standard) set of explanatory variables includes the change in the 10 year riskfree spot rate, ∆r10 , the squared change, (∆r10 )2 , the change in the slope of the term structure, ∆slo, the change in VIX, ∆vix, and the return of the S&P 500 index, s&p.
30
Panel A series A1 A2 A3 BBB1 BBB2 BBB3 BB1 BB2 BB3
Panel B series MAT3 MAT5 MAT7 MAT10
int. 0.178 1.026 0.150 0.876 0.222 1.290 0.196 0.869 0.027 0.213 0.189 0.693 0.121 0.279 -0.575 1.407 0.046 0.302
∆r10 -2.292 1.644 -3.223 2.312 -5.451 3.892 -5.508 2.989 -7.416 4.541 -12.941 5.831 -43.260 12.075 -74.047 22.248 -81.525 30.993
(∆r10 )2 -8.498 1.731 -5.973 1.229 -10.216 2.091 -8.188 1.279 0.830 0.153 -6.675 0.862 -2.992 0.244 27.454 2.371 -19.371 2.120
∆slo -1.601 0.848 0.807 0.621 3.891 2.042 0.367 0.639 0.381 0.412 12.462 4.111 26.142 5.310 23.451 5.098 18.783 5.377
∆vix 0.528 5.767 0.349 3.819 0.510 5.555 0.840 6.963 0.551 5.161 0.511 3.503 0.935 3.975 1.129 5.176 2.262 13.198
s&p 0.009 1.350 -0.005 0.727 0.004 0.688 0.007 0.789 0.007 0.871 0.001 0.541 -0.031 1.874 -0.030 1.985 0.000 0.461
1st pc 0.079 10.044 0.094 12.084 0.087 11.191 0.107 10.502 0.116 12.792 0.145 11.754 0.333 16.954 0.449 24.567 0.785 54.749
adj. R2 0.237
int. 0.010 0.528 0.056 0.750 0.079 0.847 0.102 0.967
∆r10 -29.038 10.086 -26.850 10.407 -25.415 9.650 -23.437 8.313
(∆r10 )2 -1.937 1.008 -3.596 1.377 -4.359 1.479 -5.055 1.505
∆slo 16.881 4.507 10.903 3.177 6.839 2.006 3.015 1.180
∆vix 0.954 6.176 0.878 6.326 0.815 5.892 0.738 5.213
s&p -0.001 1.174 -0.004 1.060 -0.005 0.972 -0.006 0.921
1st pc 0.225 18.425 0.245 19.226 0.254 18.517 0.253 17.004
adj. R2 0.387
0.269 0.283 0.284 0.315 0.286 0.507 0.723 0.902
0.440 0.447 0.418
Table 6: Multiple Regressions - Changes in Credit Spreads on Standard Set of Explanatory Variables and 1st Principal Component. Panel A shows average ols regression results sorted by rating category, Panel B sorted by maturity bracket. The averages of (absolute) t-statistics are shown below the parameter estimates. The set of explanatory variables includes all the variables as in Table 5 plus the first principal component extracted from regression residuals.
31
Panel A series A1 A2 A3 BBB1 BBB2 BBB3 BB1 BB2 BB3
Panel B series MAT3 MAT5 MAT7 MAT10
int. 0.176 0.929 0.151 0.772 0.223 1.199 0.200 0.819 0.030 0.129 0.191 0.634 0.127 0.234 -0.583 0.956 0.048 0.115
∆r10 -1.855 1.222 -3.002 1.892 -4.562 3.030 -4.541 2.281 -6.804 3.650 -12.039 4.894 -42.025 9.364 -74.108 14.858 -81.540 11.784
(∆r10 )2 -8.344 1.554 -5.980 1.080 -10.148 1.923 -8.301 1.200 0.749 0.137 -6.679 0.786 -3.197 0.205 27.850 1.616 -19.496 0.803
∆slo -2.177 1.069 0.442 0.351 2.732 1.346 -0.896 0.494 -0.463 0.209 11.069 3.315 24.557 3.969 23.435 3.354 18.617 1.862
∆vix 0.506 5.073 0.336 3.240 0.468 4.739 0.797 6.124 0.523 4.289 0.462 2.883 0.882 2.994 1.124 3.440 2.259 4.924
s&p 0.008 1.128 -0.005 0.707 0.003 0.418 0.005 0.572 0.006 0.647 -0.001 0.469 -0.033 1.595 -0.030 1.353 0.000 0.192
ecm 0.003 3.252 0.002 2.228 0.008 6.442 0.010 5.160 0.006 3.531 0.008 4.855 0.014 2.885 -0.005 0.774 0.000 0.661
adj. R2 0.104
int. 0.040 0.498 0.041 0.565 0.066 0.671 0.103 0.837
∆r10 -28.015 5.861 -26.155 6.133 -24.926 6.022 -23.337 5.528
(∆r10 )2 -3.555 0.874 -2.688 0.949 -3.553 1.074 -5.114 1.239
∆slo 15.337 2.689 9.788 1.982 6.296 1.442 2.941 0.984
∆vix 0.920 4.210 0.832 4.296 0.781 4.203 0.736 4.049
s&p -0.002 0.953 -0.006 0.758 -0.007 0.719 -0.006 0.718
ecm 0.025 4.285 -0.007 3.869 0.001 3.207 0.002 1.878
adj. R2 0.197
0.067 0.175 0.173 0.118 0.137 0.238 0.395 0.343
0.209 0.201 0.172
Table 7: Multiple Regressions - Changes in Credit Spreads: Test of ecm. Panel A shows average ols regression results sorted by rating category, Panel B sorted by maturity bracket. The averages of (absolute) t-statistics are shown below the parameter estimates. In addition to the standard set of explanatory variables used in Table 5, we test the (maturity-specific) error correction terms stemming from the long-term relation, ecm.
32
Panel A series A1 A2 A3 BBB1 BBB2 BBB3 BB1 BB2 BB3
Panel B series MAT3 MAT5 MAT7 MAT10
int. 0.038 0.263 0.040 0.261 0.046 0.376 0.061 0.392 0.067 0.415 0.080 0.403 0.109 0.301 0.078 0.235 -0.088 0.175
factor 1 1.140 7.839 1.128 7.349 1.423 11.716 1.948 12.493 1.825 11.200 2.067 10.449 7.115 19.434 12.661 39.347 15.776 29.769
factor 2 0.450 3.074 0.693 4.527 1.344 11.190 1.416 9.092 0.944 5.780 2.672 13.569 4.394 11.946 2.501 8.034 2.355 4.214
factor 3 1.476 10.142 1.127 7.346 1.998 16.463 2.901 18.615 2.043 12.560 2.747 13.922 2.599 7.155 -1.180 4.679 1.289 2.463
int. 0.024 0.152 0.043 0.292 0.056 0.373 0.068 0.437
factor 1 5.123 15.570 5.068 17.714 5.015 17.433 4.830 15.770
factor 2 2.858 10.477 2.088 9.144 1.527 7.046 0.981 5.078
factor 3 1.176 9.800 1.625 10.850 1.854 10.545 2.012 10.292
adj. R2 0.254 0.202 0.513 0.536 0.383 0.489 0.532 0.757 0.640
adj. R2 0.463 0.503 0.492 0.454
Table 8: Multiple Regressions - Changes in Credit Spreads on General CCA Factors. Panel A shows average ols regression results sorted by rating category, Panel B sorted by maturity bracket. Explanatory variables are three factors constructed from a canonical correlation analysis using 36 cross-sectional observations, i.e. 9 rating categories and 3, 5, 7, and 10 years to maturity. The averages of (absolute) t-statistics are shown below the parameter estimates.
33
Panel A series A1 A2 A3 BBB1 BBB2 BBB3 BB1 BB2 BB3
Panel B series MAT3 MAT5 MAT7 MAT10
int. 0.038 0.267 0.040 0.271 0.046 0.380 0.061 0.392 0.067 0.423 0.080 0.406 0.109 0.302 0.078 0.239 -0.088 1.471
factor 1 1.140 7.967 1.128 7.685 1.423 11.821 1.948 12.507 1.825 11.443 2.067 10.522 7.115 19.546 12.661 39.982 15.776 272.603
factor 2 0.450 3.120 0.693 4.753 1.344 11.296 1.416 9.101 0.944 5.910 2.672 13.658 4.394 12.025 2.501 8.151 2.355 44.952
factor 3 1.476 10.305 1.127 7.684 1.998 16.611 2.901 18.636 2.043 12.830 2.747 14.017 2.599 7.187 -1.180 4.765 1.289 22.211
1st pc 0.047 4.066 0.083 6.873 0.030 3.056 0.018 1.330 0.061 4.669 0.044 2.551 0.042 2.359 -0.023 3.938 0.981 209.011
int. 0.024 0.217 0.043 0.417 0.056 0.606 0.068 0.605
factor 1 5.123 45.424 5.068 43.574 5.015 51.862 4.830 34.284
factor 2 2.858 21.086 2.088 13.959 1.527 9.714 0.981 5.448
factor 3 1.176 11.980 1.625 13.166 1.854 13.680 2.012 11.950
1st pc 0.152 34.599 0.166 25.002 0.150 28.387 0.103 17.725
adj. R2 0.278 0.271 0.522 0.537 0.408 0.495 0.537 0.766 0.995
adj. R2 0.532 0.564 0.544 0.498
Table 9: Multiple Regressions - Changes in Credit Spreads on General CCA Factors and 1st Principal Component. Panel A shows average ols regression results sorted by rating category, Panel B sorted by maturity bracket. Explanatory variables are three factors as in Table 8, plus the first principal component extracted from regression residuals. The averages of (absolute) t-statistics are shown below the parameter estimates.
34
Panel A series MAT3 MAT5 MAT7 MAT10
factor 1 0.232 0.275 0.285 0.266
factor 2 0.114 0.085 0.060 0.041
factor 3 0.121 0.146 0.150 0.149
R2 0.466 0.505 0.494 0.456
Panel B series A1 A2 A3 BBB1 BBB2 BBB3 BB1 BB2 BB3
factor 1 0.090 0.086 0.132 0.143 0.152 0.110 0.346 0.705 0.615
factor 2 0.016 0.034 0.124 0.078 0.041 0.186 0.135 0.038 0.021
factor 3 0.152 0.086 0.260 0.317 0.192 0.195 0.053 0.013 0.004
R2 0.258 0.206 0.515 0.538 0.386 0.491 0.534 0.757 0.641
Table 10: Explanatory Power of CCA Factors. The table shows the explanatory power of the three CCA factors as a fraction of the total variation of the response variable, sorted by maturity bracket in Panel A, and sorted by rating class in Panel B.
35
Panel A series A1 A2 A3 BBB1 BBB2 BBB3 BB1 BB2 BB3
Panel B series MAT3 MAT5 MAT7 MAT10
int. 0.038 0.276 0.040 0.297 0.046 0.350 0.061 0.372 0.067 0.466 0.080 0.401 0.109 0.493 0.078 0.661 -0.088 0.617
factor 1 0.034 12.088 0.038 13.809 0.040 14.914 0.051 15.371 0.051 17.458 0.064 15.943 0.173 37.258 0.257 108.872 0.372 135.031
factor 2 0.053 7.381 0.046 6.584 0.057 8.300 0.097 11.228 0.082 10.843 0.133 12.710 0.314 26.394 0.133 21.768 -0.302 42.468
factor 3 0.054 6.297 0.076 9.128 0.077 9.445 0.104 10.153 0.091 10.098 0.084 6.722 0.172 12.114 -0.402 55.221 0.133 15.935
int. 0.024 0.160 0.043 0.533 0.056 0.536 0.068 0.519
factor 1 0.128 24.866 0.123 73.455 0.118 41.545 0.111 24.908
factor 2 0.066 10.588 0.068 25.135 0.069 17.257 0.069 12.653
factor 3 0.043 9.480 0.046 23.592 0.045 16.013 0.040 10.965
adj. R2 0.320 0.383 0.427 0.477 0.506 0.474 0.808 0.928 0.925
adj. R2 0.529 0.613 0.616 0.574
Table 11: Multiple Regressions - Changes in Credit Spreads on PCA Factors. Panel A shows average ols regression results sorted by rating category, Panel B sorted by maturity bracket. Explanatory variables are three principal components constructed from 36 cross-sectional observations, i.e. 9 rating categories and 3, 5, 7, and 10 years to maturity. The averages of (absolute) t-statistics are shown below the parameter estimates.
36
Panel A series A1 A2 A3 BBB1 BBB2 BBB3 BB1 BB2 BB3
Panel B series MAT3 MAT5 MAT7 MAT10
int. 0.038 0.364 0.040 0.395 0.046 0.515 0.061 0.472 0.067 0.683 0.080 0.538 0.109 1.649 0.078 0.775 -0.088 0.640
factor 1 0.034 15.784 0.038 17.664 0.040 21.371 0.051 19.243 0.051 24.988 0.064 21.161 0.173 130.166 0.257 128.425 0.372 143.174
factor 2 0.053 9.708 0.046 8.463 0.057 11.956 0.097 14.005 0.082 15.596 0.133 16.859 0.314 91.394 0.133 25.664 -0.302 44.816
factor 3 0.054 8.309 0.076 11.678 0.077 13.557 0.104 12.702 0.091 14.485 0.084 8.919 0.172 42.282 -0.402 65.054 0.133 16.926
1st pc 0.169 18.419 0.172 17.115 0.188 22.109 0.187 16.417 0.189 22.115 0.240 19.020 -0.282 67.442 0.122 9.371 0.069 8.765
int. 0.024 0.187 0.043 0.968 0.056 0.828 0.068 0.698
factor 1 0.128 29.880 0.123 113.013 0.118 57.531 0.111 31.566
factor 2 0.066 12.895 0.068 48.248 0.069 27.693 0.069 17.147
factor 3 0.043 10.804 0.046 38.297 0.045 22.901 0.040 14.181
1st pc 0.018 12.241 0.148 33.845 0.167 24.400 0.136 18.746
adj. R2 0.585 0.602 0.693 0.655 0.734 0.686 0.948 0.933 0.941
adj. R2 0.627 0.807 0.817 0.761
Table 12: Multiple Regressions - Changes in Credit Spreads on PCA Factors and 1st Principal Component. Panel A shows average ols regression results sorted by rating category, Panel B sorted by maturity bracket. Explanatory variables are three PCA factors as in Table 11, plus the first principal component extracted from regression residuals. The averages of (absolute) t-statistics are shown below the parameter estimates.
37
Panel A - Gen series factor1 factor2 factor3
∆r10 -0.71 0.10 0.01
(∆r10 )2 -0.24 0.10 -0.19
∆slo -0.36 0.49 -0.15
∆vix 0.31 0.23 0.13
s&p -0.22 -0.14 -0.05
∆swap 0.52 -0.10 -0.16
ecm3 0.05 0.25 0.38
Panel B - Mat3 series factor1 factor2 factor3
∆r10 -0.55 0.15 -0.07
(∆r10 )2 -0.14 -0.03 -0.10
∆slo -0.08 0.16 -0.24
∆vix 0.35 0.15 0.14
s&p -0.24 -0.05 0.01
∆swap 0.39 -0.23 0.05
ecm3 0.14 0.42 -0.02
Panel C - Mat5 series factor1 factor2 factor3
∆r10 -0.62 0.12 0.00
(∆r10 )2 -0.17 -0.05 -0.10
∆slo -0.20 0.18 -0.19
∆vix 0.34 0.15 0.17
s&p -0.24 -0.08 0.00
∆swap 0.44 -0.22 0.01
ecm3 0.12 0.42 -0.03
Panel D - Mat7 series factor1 factor2 factor3
∆r10 -0.64 0.10 0.05
(∆r10 )2 -0.20 -0.04 -0.10
∆slo -0.28 0.18 -0.13
∆vix 0.33 0.13 0.18
s&p -0.22 -0.08 -0.02
∆swap 0.45 -0.20 0.00
ecm3 0.12 0.40 -0.03
Panel E - Mat10 series factor1 factor2 factor3
∆r10 -0.65 0.08 0.07
(∆r10 )2 -0.21 -0.02 -0.11
∆slo -0.33 0.17 -0.08
∆vix 0.30 0.12 0.19
s&p -0.20 -0.08 -0.06
∆swap 0.44 -0.17 0.01
ecm3 0.12 0.36 -0.04
Table 13: Pairwise Correlations - CCA Factors and Explanatory Variables. Panels show pairwise correlations between CCA factors and the set of observable explanatory variables. Panel A shows the results using the general CCA factors used in Table 8, Panels B-E are based on the maturity-specific CCA factors.
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Panel A series AA A BBB AA A BBB
Panel B series MAT3 MAT5 MAT7 MAT10 MAT3 MAT5 MAT7 MAT10
int. 0.046 0.337 0.039 0.338 0.033 0.151 0.046 0.537 0.039 0.553 0.033 0.400
factor 1 1.428 10.387 1.318 10.994 2.043 9.213 1.428 16.794 1.318 18.208 2.043 21.910
factor 2 0.377 2.743 0.273 2.267 0.856 3.888 0.378 4.426 0.273 3.751 0.855 9.261
factor 3 0.419 2.992 0.276 2.276 0.579 2.686 0.419 4.944 0.276 3.785 0.579 6.380
1st pc
int. 0.041 0.327 0.049 0.337 0.042 0.280 0.024 0.158 0.041 0.482 0.049 0.689 0.042 0.558 0.024 0.258
factor 1 1.440 8.599 1.590 10.345 1.673 11.010 1.680 10.838 1.440 13.477 1.590 22.329 1.673 22.339 1.680 17.737
factor 2 0.447 2.367 0.496 2.922 0.523 3.217 0.542 3.357 0.450 3.816 0.497 6.935 0.520 6.988 0.540 5.512
factor 3 0.240 1.302 0.413 2.613 0.498 3.213 0.546 3.478 0.240 2.104 0.413 5.761 0.498 6.590 0.546 5.690
1st pc
adj. R2 0.195 0.202 0.173
0.212 28.243 0.188 29.370 0.402 47.651
0.683 0.702 0.820
adj. R2 0.136 0.192 0.217 0.216
0.287 27.817 0.275 43.778 0.262 39.684 0.245 29.072
0.649 0.793 0.790 0.707
Table 14: Multiple Regression - Changes in U.K. Credit Spreads. Panel A shows average ols regression results sorted by rating category, Panel B sorted by maturity bracket, based on the data set of U.K. credit spreads. Explanatory variables are the three factors constructed from canonical correlation coefficients using all U.S. cross-sectional observations, plus the first principal component from estimated residuals.
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Factor1 lag 0 -7.74 -3.92 -6.36 -3.83 Factor3 lag 0 -2.46 -1.33
lag 3 0.18 0.09
lag 6 -3.35 -1.62
lag 9 0.87 0.41
lag 12 -3.12 -1.58
adj. R2 0.126 0.103
lag 3 -1.15 -0.55
lag 6 -3.96 -1.86 -5.74 -3.72
lag 9 -2.44 -1.17
lag 12 0.37 0.20
adj. R2 0.096 0.101
Table 15: Out of Sample Predictions of Income using the two strong CCA Factors between 1997 and 2006. Income is the time series of monthly U.S. real disposable income. Factor 1 and Factor 3 are the extracted CCA factors, integrated to generate monthly observations. All series are filtered with a Hodrick/Prescott filter with lambda 14400. The upper panel shows the results of a multiple regression of income on contemporaneous (lag 0) and lagged observations (3, 6, 9, and 12 months) of Factor 1, the lower panel of Factor 3, respectively. T-statistic values are shown below the point estimate, and the estimated value of the constant is not shown in the table.
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model selection criterion 1 0.9 0.8 0.7 0.6
1 2 3 >3
0.5 0.4 0.3 0.2 0.1 0
2
3
4
5
6 7 maturity
8
9
10
sequential hypothesis test 1 0.9 0.8 0.7 0.6
1 2 3 >3
0.5 0.4 0.3 0.2 0.1 0
2
3
4
5
6 7 maturity
8
9
10
Figure 1: Test of Number of Common Factors: Changes in Credit Spreads. The upper graph shows results of the model selection criterion with BIC(1) penalty function, the lower graph shows results of the sequential hypothesis testing with confidence level of 95% and bandwidth 3. Percentages are stacked per maturity, displaying the proportion the respective number of factors is estimated.
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model selection criterion with z(1) 1 0.9 0.8 0.7 0.6
1 2 3
0.5 0.4 0.3 0.2 0.1 0
2
3
4
5
6 7 maturity
8
9
10
sequential hypothesis test with z(1) 1 0.9 0.8 0.7 0.6
1 2 3 >3
0.5 0.4 0.3 0.2 0.1 0
2
3
4
5
6 7 maturity
8
9
10
Figure 2: Test of Number of Common Factors: Set of Instrumental Variables z(1). The upper graph shows results of the model selection criterion with BIC (1) penalty function, the lower graph shows results of the sequential hypothesis testing with confidence level of 95% and bandwidth 3. Instruments are ∆r10 , (∆r10 )2 , ∆slo, ∆vix, and s&p. Percentages are stacked per maturity, displaying the proportion the respective number of factors is estimated.
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model selection criterion with z(2) 1 0.9 0.8 0.7 0.6
1 2 3
0.5 0.4 0.3 0.2 0.1 0
2
3
4
5
6 7 maturity
8
9
10
sequential hypothesis test with z(2) 1 0.9 0.8 0.7 0.6
1 2 3 >3
0.5 0.4 0.3 0.2 0.1 0
2
3
4
5
6 7 maturity
8
9
10
Figure 3: Test of Number of Common Factors: Set of Instrumental Variables z(2). The upper graph shows results of the model selection criterion with BIC(1) penalty function, the lower graph shows results of the sequential hypothesis testing with confidence level of 95% and bandwidth 3. Instruments are ∆r10 , (∆r10 )2 , ∆slo, ∆vix, s&p, smb and hml. Percentages are stacked per maturity, displaying the proportion the respective number of factors is estimated.
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model selection criterion with z(3) 1 0.9 0.8 0.7 0.6
1 2 3
0.5 0.4 0.3 0.2 0.1 0
2
3
4
5
6 7 maturity
8
9
10
model selection criterion with z(4) 1 0.9 0.8 0.7 0.6
1 2 3
0.5 0.4 0.3 0.2 0.1 0
2
3
4
5
6 7 maturity
8
9
10
Figure 4: Test of Number of Common Factors: Set of Instrumental Variables z(3) and z(4). The upper graph shows results of the model selection criterion with BIC(1) penalty function, instruments are ∆r10 , (∆r10 )2 , ∆slo, ∆vix, s&p, and ∆swap. The lower graph shows results of the model selection criterion with BIC(1) penalty function, instruments are ∆r10 , (∆r10 )2 , ∆slo, ∆vix, s&p, ∆swap, and ecm. Percentages are stacked per maturity, displaying the proportion the respective number of factors is estimated.
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