یک ساختار سببی از بازدهی اوراق قرضه

This paper implements an emerging data-driven method of directed acyclic graphs to study the contemporaneous causal structure among the federal funds rate and U.S. Treasury bond yields of various maturities. Using high frequency daily data from 1994 to 2009, we find that innovations in the two-year Treasury bond yield play a central role. They contemporaneously cause most other bond yields. Therefore, monetary policy makers would benefit from closely monitoring the two-year yield in setting the interest rate target, a result echoing the policy rule suggested by Piazzesi (Journal of Political Economy, 2005). Both Fed and investors should also watch the seven-year bond yield because it explains significant portions of variability in many other yields.

Both macroeconomists and financial economists have intensively studied bond yields (prices), because the term structure of interest rates is a rich source of economic and financial information. The majority of bond-yield curve models employ a structure that consists of a small number of factors and the associated factor loadings that relate yields of different maturities to those factors (Diebold et al., 2005 and Rudebusch and Wu, 2008). In latent factor models which are popular in finance literature, the factors (or unobservable state variables) are typically backed out from yield data. These estimated factors are oftentimes given further economic interpretations. However, as evident in numerous empirical studies, economic and financial information is difficult to extract from the term structure of interest rates because of its dependence on monetary policy and the existence of time-varying term premia (Palomino, 2010). In this paper, we complement existing literature by studying bond yields from a different perspective. Specifically, we are interested in the contemporaneous casual structure of shocks (unanticipated changes or innovations) to bond yields of various maturities. Rather than estimating a few factors each as a function of all yields, here we assess the relative importance of shocks to each yield by identifying the contemporaneous causal relationship between each pair of shocks that drive the yields.1 As our later empirical results show, not all shocks are “created” equal in that the market reaction to a shock to short-term bonds is unlikely to be the same as that to long-term bonds. Some shocks cause changes in other yields immediately. Others are local in the sense that they are less likely to transmit to other yields in the contemporaneous time, although they probably also affect other yields over time. Clearly, this type of causal information can provide guidance on which bond yields the Fed Reserve and other market participants should watch more carefully.2 It is well known that inferring contemporaneous causality from observational data is by no means a simple task. In this paper we apply a data-driven method to search for the causal structure of the innovations in bond yields. The suggested method of directed graphs is the graph-theoretic analysis of causality (Pearl, 2000 and Spirtes et al., 2000). The study and application of the directed graph method is in line with the growing interest from applied researchers in automated model discovery (more on this in the Methodology section). Innovations or shocks are unexpected components of bond yields and, by definition, are unobservable. The standard strategy, also applied in this paper, is to estimate them as the residuals of a vector autoregression-type (VAR) model (e.g., a vector error correction (VEC) model with cointegration). In determining which variables to include in the VAR model, we follow a growing literature and model the term structure in combination with monetary policy.3 Namely, we include in the VAR both U.S. bond yields of maturities running from one to 20 years and a monetary policy variable. We study the U.S. data because, compared to yields on other sovereign debt, the U.S. Treasury yields are probably most investigated in the international markets, arguably due to the U.S. market size and liquidity. We use the federal funds rate as the indicator of the Fed's monetary policy stance. This is because over much of the past decades the Fed has implemented policy changes through changes in the federal funds rate (Bernanke & Blinder, 1992). Our most interesting finding is that unexpected changes in the two-year Treasury constant maturity rate contemporaneously cause unexpected changes in most other Treasury bond interest rates. Therefore, monetary policy makers should closely monitor the two-year yield in setting the interest rate target, a result echoing the policy rule suggested by Piazzesi (2005) based on a much more complicated parametric model.4 To capture the full dynamics of bond yields, we also conduct standard forecast error variance decomposition within the vector autoregressions (VARs) framework. Rather than using a restrictive recursive structure, we orthogonalize the reduced-form VAR residuals according to the identified contemporaneous causal structure (DAG). Results from this part of the analysis also suggest the importance of both two- and seven-year bonds in driving the dynamics of the bond market. The main message from our empirical results is that both the short-term and the long-term states of the economy are important determinants of interest rates. And they appear well proxied by the two-year and the seven-year rates, respectively. By pinpointing the prominent influence of the two interest rates behind the bond market, our results complement existing latent factor models which typically explain the term structure by one or two linear combinations of interest rates of all maturities. For example, Cochrane and Piazzesi (2005) show that a single linear combination of forward rates predicts excess returns on 1–5-year maturity bonds, where the largest effect comes from the three-year bond. The remainder of the paper is structured as follows. Section 2 presents econometric methodology. Section 3 describes the data. Section 4 presents major empirical results, and Section 5 discusses the results and concludes the paper.

4.1. Contemporary causal analysis We first test the integration order of the interest rates using the augmented Dickey–Fuller test. The null hypothesis is that the interest rate contains a unit root. The results are summarized in Table 2. If a trend term is not included in the test, the null hypothesis cannot be rejected for all eight interest rates at any conventional significance level. The evidence is a little mixed when a trend term is included in the test. The unit root hypothesis cannot be rejected for FF and the five short- and medium-term interest rates. However, we reject the null hypothesis for GS10 and GS20 at the 5% level. We also test for the nonstationarity of the first differences of the interest rates. The null hypothesis can be rejected for all eight interest rates.Assuming the interest rates are all nonstationary variables (more specifically integrated of order one), we proceed to estimate the vector error correction model (3) with the eight interest rates as endogenous variables. To this end, we first determine the autoregression order in Model (2). Based on Schwarz's (1978) Bayesian information criterion (BIC), we select the autoregressive order p to be 2. Given p, we test for the cointegrating rank using the popular Johansen, 1988 and Johansen, 1991 trace test and the maximum eigen-value test. Table 3 shows that both tests suggest a cointegration rank of 5. We therefore estimate the error correction model (3) imposing a cointegrating rank of 5.The maximum likelihood estimates of the parameter Π and Γ1 are not reported here. Instead, Table 4 presents their transformed parameter estimates of A1 and A2, which are used for later forecast error variance decomposition exercises. 9 Briefly, the sum of the two autoregressive coefficients is larger than 0.93 for each bond rate and is equal to 0.87 for the federal funds rate, which is consistent with the well-known (near) unit-root behavior of interest rates (e.g., Bali, Heidari, & Wu, 2009). Coefficients of the other lagged interest rates are generally small in magnitudes with a few exceptions. For example, one-day lagged interest rates help predict current FF. GS2 predicts GS1, GS3, GS5, and GS7. Based on the estimated residuals (innovations) covariance matrix Σ, we compute the correlations among the innovations to the interest rates, which are reported in the middle panel of Table 1. Note that conditional on past interest rates, seven bond interest rates remain highly correlated. The average of the correlations (absolute values) between the innovations in the bond interest rates is 0.86. By contrast, innovations in the federal funds rate FF are correlated with those in the bond interest rates with a maximum correlation of 0.042 with GS1. Nevertheless, this relatively low correlation is still statistically significant from zero (with a p-value less than 0.01) given the large sample size.We now study contemporaneous causality among innovations to the interest rates using the directed acyclic graph method as described earlier. The input is the 8 × 8 correlation matrix of the residuals (innovations) estimated from the above cointegration model in which the dependence of an interest rate on its own history and on the history of other interest rates is controlled. With 8 variables in the model, there are potentially 0.5 × 8 × (8 − 1) = 28 lines (edges) between all pairs of variables. The search based on the GES algorithm concludes with 13 edges (causal links), of which 11 are directed and the remaining 2 edges cannot be directed given the sample information (Panel A of Fig. 1). Three points stand out. First, FF is a direct cause of the one-year bond rate GS1 only, and is not caused by any other bond yields (in this sense innovations in FF are exogenous). This result is as expected given the above residuals correlation pattern between FF and the seven bond yields. Second, the two-year bond rate GS2 appears to be an important information center. It is causally linked to all other six bond yields except GS7. Third, GS7 is another important information center which contemporaneously causes four other bond yieldsTo evaluate the reliability of the above causal graph identified by the one-shot GES search, we apply the bootstrap procedure outlined in the methodology section. For each bootstrap replication, we record the existence (or lack thereof) and the direction of each edge between all 28 pair of interest rates. Table 5 summarizes the results, where the edge identification based on the original realized sample is also tabulated for the purpose of comparisons.10 While the bootstrap results generally agree with those of the one-shot search in terms of edge inclusion and exclusion, in many cases they differ from each other in edge direction.11 In particular, there are three noticeable differences, one in the edge inclusion/exclusion and the other in the edge direction. First, the one-shot search finds no edge between the two longest-term bonds (GS10 GS20), while the bootstrap selects an edge between the two 89% of the times and further directs it as GS10 → GS20 in 70% of the realizations. Similarly, second, the initial GES search concludes an edge between GS1 and GS7 while the bootstrap results suggest that there is no edge between them (the corresponding to percentage is 66%). Third, one-shot GES directs GS1 → GS2 in the realized sample. In Table 5, the bootstrap provides relatively strong evidence (49% vs. 14%) in favor of the opposite edge direction, GS1 ← GS2. Interestingly, our later analysis also supports these new findings after controlling for GARCH effects in the residuals.Panel B of Fig. 1 is the final contemporaneous causal graph modified from Panel A after incorporating the above three changes suggested by the bootstrap replications. Recall that two edges (GS5 — GS10 and GS5 — GS20) are undirected in the initial search. The bootstrap results do not provide a clear-cut answer. Nevertheless, Table 5 shows that the GES algorithm selects GS5 → GS10 and GS5 → GS20 more often than the opposite directions. In light of this evidence we direct the edges as GS5 → GS10 and GS5 → GS20 for the purpose of error variance decomposition (the main decomposition results reported later are robust to the alternative directions because the coefficients in A1 and A2 associated with these variables are small in the estimated model). To highlight the above changes to the initial DAG in Panel A, we use broken rather than the standard solid lines in Panel B. The central informational role of GS2 is clearer from Panel B. GS2 is the causal factor of bond yields of all maturities except GS7. In the meanwhile, innovations in the seven-year bond contemporaneously cause unexpected changes in GS5, GS10, and GS20. Finally, innovations in the longest-maturity bond yield (GS20) do not appear to simultaneously transmit to the bonds of shorter terms. On the contrary, they are caused by innovations in the 2-, 5-, 7-, and 10-year bonds. As Panel A, Panel B of Fig. 1 also shows that monetary policy shocks (FF) only cause GS1. The additional insight from the bootstrap is that the probability of a causal flow from FF to the two short-term bond yields GS2 and GS3 is higher (22% and 26%) than from FF to two long-term bonds yields GS10 and GS20 (8% and 8% only). This result is in line with earlier findings that monetary policy shocks affect short rates more than long ones (e.g., Piazzesi, 2005). 4.2. Error variance decompositions To study dynamic effects among the federal funds rate and the seven bond yields, we conduct an innovation accounting, forecast error variance decomposition, which measures the percentage of interest rate changes in each series at current or some future time that is due to innovations in all eight series at the current time. The decomposition is based on parameter estimates from the above VAR(2) model, which are derived from the estimates of the error correction model with the reduced rank of 5. We orthogonalize the reduced form innovations, ɛt, in Model (2), according to the contemporaneous causal pattern derived from the directed acyclic graph as summarized in Panel B of Fig. 1. We conduct the decomposition at the 1- up to the 15-day horizons. Table 6 presents the decompositions for three horizons: the 1-, 5- and 15-day horizons.At horizon 1, the forecast error variances of FF, GS2, and GS7 are explained by their own innovations, which are consistent with the qualitative results in the earlier sub-section that these three interest rates contemporaneously cause, but are not caused by, other interest rates. At this horizon, nearly all of the variance in GS20 is also explained by its own innovations even though GS20 is caused by four other rates in the DAG. GS2 is most influential to variations in the two short-term bonds GS1 and GS3, while GS7 is most influential to the longer-term GS5, GS10 and GS20. At the longest 15-day horizon we consider, the patterns are similar. More than 64% of the forecast error variances of GS1 and GS3 are attributable to the innovations related to GS2, while 80% of the variances of GS5 and GS10 are due to the innovations in GS7. Notably, at this 15-day horizon, the monetary policy instrument FF explains a significant part of variation in the long-term interest GS20 (20%), although there is no contemporaneous causal effect from FF to GS20. At horizon 15, GS2 and GS7 are the two most important factors for FF, accounting for 9.3% and 2.5% of the variation in FF, respectively. GS1, GS3 and GS5 each contributes about 1%. The second column of Table 6 suggests that, with the exception of GS1 and GS20, FF explains negligible portion of forecast error variances of the other bond yields. 4.3. Robustness check12 So far we have maintained that shocks in bond yields (ɛ t) have constant variance and covariance. However, empirical evidence from some earlier studies also suggests that bond yields, like stock returns, may contain significant time-series variation in volatility and volatility clustering. Therefore, they have fat-tailed distributions. Although simulation evidence suggests that DAG performs well as long as the underlying distribution of data is symmetric. However, it is not clear how well the GES algorithm performs in the presence of conditional heteroscedasticity. To investigate whether our basic findings are sensitive to the constant variance assumption of the interest rates, we now examine the effect of possible heteroskedasticity in the data. 13 Specifically, we study the residuals estimated from the above cointegration model (rather than the raw interest rates themselves). We assume that the residuals follow a multivariate student-t distribution with time-varying variance and covariance which are modeled as GARCH(1, 1) processes. Given the dimension of the model, we further assume constant conditional correlations (R ) between the eight interest rates and write conditional variance and covariance matrix as equation(4) Ht=StRSt,Ht=StRSt, Turn MathJax on where View the MathML sourceSt=diag(hit1/2),hit=ωi+αiεt−12+βihi,t−1,R=(ρij), with ρii=1, Turn MathJax on where the conditional volatility of shocks associated with interest rate i , hithit, is assumed to follow a univariate GARCH(1, 1) process. The GARCH model is estimated using the maximum likelihood estimation method (the detailed parameter estimates are not reported here to save space. The parameter ν , the analogous to the degrees of freedom parameter of a univariate Student's t distribution, is estimated to be 9.2). Panel C of Table 1 presents the correlation matrix of the standardized residuals View the MathML source(εt/hit1/2). The final directed acyclic graph based on this correlation matrix of filtered shocks is plotted in Fig. 2. In comparison to Panel A of Fig. 1, it can be seen that our benchmark results on causal relationships among interest rates shocks still hold after controlling for GARCH effects in the data. Innovations associated with GS2 and GS7 play a central role in the contemporaneous sense. There are some interesting changes. In Fig. 2, the causal flow is from GS2 to GS1 (GS2 → GS1), opposite to the direction in Fig. 1. A directed edge (GS10 → GS20) is present in Fig. 2, but not Fig. 1. Unlike in Fig. 1, there is no edge between GS1 and GS7. As pointed out earlier, these three findings are consistent with the bootstrap replications in Table 5 and have been incorporated in the error variance decompositions. Other changes in Fig. 2 include two new edges GS7 → GS3 and GS1 → GS3.