Chapter 5
5. Modeling Episodic Nonlinearity in Daily Bond Market Returns
5.0 Introduction*
Forming accurate expectations of inflation is an urgent task for holders of conventional, fixed-rate bonds. How do bondholders behave in the face of uncertainty about future rates of inflation? To understand bondholder behavior, we begin with Irving Fisher's (1930) observation that the nominal rate of interest can be considered the sum of an expected real return and an expected inflation rate. If conditions in the real economy are stable, then the expected real return is likely to change very little over short (e.g., daily) intervals. Changes in the nominal rate of interest, or, more precisely, in the bond returns investors actually earn, must then be due almost entirely to changes in the expected inflation rate.
Although the relationship between asset returns and inflation has been examined by Fama and Schwert (1977), Roll (1972), and others, the micro dynamics of inflation expectations are not well understood. Roll cites several studies that report evidence of the effects of various measures of price expectations on interest rates at annual, quarterly, and monthly intervals.1 However, except for anecdotal evidence from financial journalists and market participants, little is known about how bond investors form expectations of inflation over daily intervals.2 One widely held hypothesis is that bond investors look to commodities markets for early indications of broader price level changes. An obvious difficulty posed by this hypothesis is that commodity market participants are presumably as well informed about prevailing prices in the bond market as are bond market participants about prevailing prices in the commodities markets. Who is influencing whom?
In order to describe the relationship between commodity prices and bond returns, and to uncover evidence about the micro dynamics of inflation expectations, we have conducted a variety of tests on the percent change in a daily commodity price index (COMINF) and the percent change in the daily Treasury bond return index (BNDRET) from 1983 through 1993.3 Following methods suggested by Granger (1969), we have identified and estimated a 6th-order VAR model and performed Granger causality tests.4 Because the residuals of the VAR model are characterized by nonlinearity, ARCH-M models have been estimated to determine whether the findings are robust to more general plausible models. The persistence of nonlinearity has led us to perform tests for episodic nonlinearity and to graph and evaluate test results. Finally, we offer some conclusions drawn from our analysis.
1
1
Modeling Episodic Nonlinearity
5.1 Granger Causality Tests
The VAR modeling technique (Tiao and Box, 1981) has been used to perform a Granger causality test of hypothesized dynamic interactions between bond returns and commodity price inflation. A 6th order VAR model of the form of
Q(B)[COMINFt,BNDRETt]' = et(5.1-1)
has been estimated in which Q(B) is a 2 by 2 polynomial matrix in the lag operator B, defined such that BkXtXt-k and et is a 2 element vector containing the residuals for the two equations, e1t and e2t. An element of Q(B) qi,j(B) measures the effect of the jth series on the ith series. If a kth order VAR model is estimated, qi,j(B) contains k terms. In the estimated model, significant terms in q1,2(B) would indicate lags of bond returns, BNDRET, affect the percent change in commodity prices, COMINF, while significant terms in q2,1(B) would indicate that lags in the percent change in commodity prices, COMINF, affect bond returns, BNDRET.
The coefficients obtained by estimating (5.1-1) are listed in Table 5.1. The insignificance of coefficients for any lag of q1,2(B) indicates that the percent change in commodity prices is not influenced by bond returns. Because the 3rd and 5th lags of q2,1(B) are significant, with coefficient values of -0.0436 and -0.0422, respectively, our evidence suggests that the percent change in commodity prices does affect bond returns in a model that controls for lags in the bond returns series. The effect is negative, indicating that, everything else equal, if the percent change in the commodity prices rises, bond returns decline after three to five days. The lag structure of the VAR model has been extended until the estimated autocorrelations and cross correlations of the residuals are not significant. Although VAR models from 1 to 6 lags have been examined, Table 5.1 reports only the coefficients for the 6th order model. The maximum lag for k of 6 was selected so as to remove all autocorrelation from the residuals or E(eiteit-j) = 0 for all j > 0 for i=1,2. In the section 5.2, we test the residuals of this VAR model for third- order effects or nonlinearity, that is, to determine whether E(eiteit-jeit-s) = 0 for j > 0, s > 0 and j s. We first discuss the Granger (1969) causality tests, which have been performed for all estimated VAR models up to lag 6.
1
Modeling Episodic Nonlinearity
In Table 5.2, Fi,j for lag k tests whether in using a VAR(k) model, the jth series Granger causes the ith series. Details for the test are given in a note accompanying the table. The insignificance of F1,2 for VAR models of order 1 - 6, indicates that bond returns (series 2) does not Granger cause commodity price inflation (series 1). F2,1 is significant for VAR models 3 - 6 indicating that commodity price inflation does Granger cause bond returns for lags 3. This finding is expected in view of the significant q2,1(B) coefficients found for lags 3 and 5 reported in Table 5.1. Note the pattern of the t-test values in F2,1. For VAR(3) models, the significance of F2,1 is 0.9799 and presumably results from the presence of the first significant term in q2,1(B) at lag 3. For VAR(4) models, the significance of F2,1 falls slightly to 0.9624, most likely because the 4th lag term in q2,1(B) is not significant. The significance of F2,1 for VAR(5) models rises to 0.9923, most likely because there is a significant term at lag 5 in q2,1(B). Although the lag length of k was selected to remove all autocorrelation in the residuals of both equations, further testing is needed to determine whether nonlinearity remains in the residuals and, if so, whether it varies by period. These questions are addressed in the next three sections.
Table 5.1 VAR Model of Order 6 on COMINF and BNDRET
Lagq1,1(B)q1,2(B)q2,1(B)q2,2(B)
──────────────────────────────────────────────────────────────────────────
1-.0196-.0177.00100.0601*(.0191)(.0208) (.0176) (.0191)
2.0413*-.00852-.0255.00743
(.0191(.0208)(.0175)(.0191)
3.0619*-.0116-.0436*-.0146
(.0190)(.0208)(.0175)(.0191)
4.0611*-.0104-.00861-.0155
(.0191)(.0208)(.0175)(.0191)
5.0476*-.0202-.0422*.00202
(.0191)(.0208)(.0176)(.0191)
6.00200-.0163-.0218-.00992
(.0191)(.0207)(.0176)(.0191)
RESVAR(1)=.105841 RESVAR(2)=.089583
G(1)=24.26G(2)=13.04
L(1)=8.63L(2)=8.56
────────────────────────────────────────────────────────────────
Note: For variables sources, see text. The period of estimation is 1/4/1983 through 12/31/1993. Six observations are lost due to lags. COMINF = percent change in the index of commodity futures prices and BNDRET = percent change in the Lehman Composite Treasury bond index. The term in qi,j(B) for lag k measures the effect of the ith variable on the jth variable at lag k. Standard errors are listed under each coefficient. Coefficients significant at 0.95 are marked with a *. RESVAR(i) is the residual variance for the ith series. G(i) and L(i) are the Hinich Gaussianity and linearity tests for the ith series and are distributed as normal deviates. Basic documentation for these tests is contained in Hinich (1982) and Hinich and Patterson (1985). Stokes (1991) documents a modification of the testing procedure that involves searching the admissible bandwidth and averaging the statistics. In the statistics reported here, bandwidths from 30 to 53 were used. For all calculations, COMINF = series 1 and bond returns = series 2. A total of 2,742 observations were used in the calculations.
1
Modeling Episodic Nonlinearity
Table 5.2 Granger Tests of COMINF and BNDRET
LagF1,1F1,2F2,1F2,2
─────────────────────────────────────────────────────────────────────────
1.219771.3738.0698911.334*
(.3607)(.7587)(.2084)(.9992)
23.2439*.87661.40375.7420*
(.9608)(.5837)(.7541)(.9968)
35.906*.65393.2800*3.8056*
(.9995)(.4196)(.9799)(.9902)
46.9474*.468542.5465*2.9543*
(.9999)(.2411)(.9624)(.9811)
56.7567*.53333.1526*2.3392*
(.9999)(.2488)(.9923)(.9605)
65.5850*.541752.8969*1.9577
(.9999)(.2232)(.9919)(.9317)
─────────────────────────────────────────────────────────────────────────────
Note: VAR models of order 1,...,6 have been estimated for the period 1/4/1983 through 12/31/1993. For variable descriptions, see Table 5.1. Six observations are are lost for lags. All VAR models have been estimated over the same observations. Fi,j tests whether the jth series Granger causes the ith series. Significance values are listed under Fi,j and are marked with a * for all levels of significance above 95%. In all models, series 1 is COMINF and series 2 is bond returns. For further discussion of the data, see text. The Granger test statistic is distributed as F(n1,n2) where n1=6 or the number of lags and n2 is the number of observations minus the number of the right hand side variables. The value of the statistic is ((RSS - USS)/USS)*(n2/n1) where RSS is the restricted sum of squares, USS is the unrestricted sum of squares and n1 and n2 are 6 and 2729, respectively.
5.2 Hinich Tests
To test for Gaussianity and linearity, Hinich (1982) tests have been performed and are also reported in Table 5.1. Using a bispectrum procedure, the Hinich test shows whether a series is characterized by three-way systematic relationships. As we have noted previously (Neuburger and Stokes,1991), while autocorrelation measures whether the residuals eit and eit-k are related, the Hinich test measures whether eit, eit-k and eit-j are jointly related for kj. The Hinich test statistics G(i) and L(i) are normal deviates and test for Gaussianity and linearity, respectively. For example, a value of L(i) 2 indicates nonlinearity in the residuals of the ith equation.
1
Modeling Episodic Nonlinearity
In Table 5.1 both sets of Hinich test statistics are substantially above 2 and indicate violations of both Gaussianity and linearity assumptions implicit in the VAR model. ARCH techniques, as described in Bollerslev, Chou and Kroner (1992), are an appropriate next step when such results are obtained. In the next section, the VAR model is generalized by including ARCH-M terms to test whether our findings are robust to including a function of the error variance in the regression model.
5.3 ARCH-M Tests
The generalized ARCH-M with MA(1) formulation of the second row of the VAR model of order k for bond returns on COMINF maximizes
-0.5(log(vt) + (et)²/vt), (5.3-1)
where
et= BNDRETt - α - Σki=1βiBiBNDRETt - Σkj=1δjBjCOMINFt + μet-1
- Σml=1φlvt-l+1.5 (5.3-2)
vt= a0 + Σpn=1an(et-n)². (5.3-3)
To be as compatible with the VAR(k) model as possible, equations (5.3-1), (5.3-2) and (5.3-3) have been estimated assuming k=6, m=1 and p=1. ARCH-M results are reported in Table 5.3. Note that δ3 and δ5, which measure the effect of commodity price inflation lagged 3 periods (-0.05218) and lagged 5 periods (-0.03974), respectively, on bond returns, are highly negatively significant and close to the values found with the VAR(6) model reported in Table 5.1. The moving average term μ has been found to be insignificant. Although there is some evidence of variance persistence, as indicated by the significance of a1 in equation (5.3-3), the insignificant φ term shows that the residual variance does not enter into the basic equation predicting bond returns. The significant values for G and L of 13.343 and 8.591, respectively, and G* and L* of 10.201 and 8.003, respectively, indicate that the ARCH-M model has not corrected the Gaussianity and nonlinearity problems in the residual or normalized residual.
In additional unreported results, a more complex model has been estimated in which in terms of equations (5.3-1), (5.3-2) and (5.3-3) set k=6, m=4 and p=5. In this model, G and L test values are 14.60 and 8.972, respectively, indicating higher-order ARCH-M MA(1) models are not promising alternatives for resolving the Gaussianity and nonlinearity problems found in the VAR(6) model. Having found nonlinearity, which ARCH methods have not helped us to remove, we sought to determine whether this nonlinearity characterizes the entire estimation period or is centered in turbulent subperiods like 1987. This question is investigated in the next section in which tests for episodic nonlinearity are reported and discussed.
1
Modeling Episodic Nonlinearity
Table 5.3 ARCH-M(1) Model of BNDRET on COMRET
─────────────────────────────────────────────────────────────────────
α =.09926 β1 =-.25587 β2 =.032556 β3 =-.00651
(.1184)(.4740)(.0371)(.0186)
β4 =-.0085 β5 =.00726 β6 =-.00939 δ1 =.00437
(.0187)(.0194)(.01729 (.0157)
δ2 = -.02288 δ3 =-.05218*δ4 =-.01753δ5 =-.03974*
(.0142)(.0189)(.0264)(.0160)
δ6 =-.03076μ =-.32258φ =-.15377a0 =.083664*
(.0241)(.4736)(.3985)(.0026)
a1 =.064695*RESVAR = .089798G = 13.343 L = 8.591
(.0164)G*= 10.201L* = 8.003
─────────────────────────────────────────────────────────────────────
Model maximizes (5.3-1) in which et is defined in (5.3-2), with k=6, p=1 and m=1, and vt is defined in (5.3-3). Estimation used RATS386 version 4.10c, which is documented in Doan (1992). The SIMPLEX method has been used for the first 50 iterations, followed by the BFGS method. Standard errors are listed under the coefficients. Coefficients significant at 0.95 are marked with an *. Residuals were retrieved into B34S which was used to calculate Hinich residual tests. 19 observations were deleted at the start of the sample leaving 2,729 observations. G, L, G* and L* are the Hinich Gaussianity and nonlinearity test statistics for the ARCH-M residual and normalized residual respectively, where the normalized residual = et /(vt).5. For data sources and sample-period information, see Table 5.1.
5.4 Tests for Episodic Nonlinearity
Testing for episodic nonlinearity requires that the residual series be divided into subsamples and that tests for nonlinearity be calculated to detect any changes in the pattern of nonlinearity. Hinich (1993) offers a new test statistic for nonlinearity. This new test is a third-order cumulant extension of the portmanteau Q and requires substantially less computer time than the Hinich (1982) test. In contrast with the Q test, the Hinich (1993) test lets the sample size determine the number of lags used to construct the test. The new test can be performed on windows of data within the sample to detect episodic nonlinearity. The test statistic estimated for each window can be graphed and the level of nonlinearity measured. While it is beyond the scope of this chapter to discuss the test derivation in detail, we provide a brief outline of Hinich's procedure next.
1
Modeling Episodic Nonlinearity
Assume N residuals of a model are normalized to have zero mean and standard deviation equal to 1 and denote them ui. Define
D(r,s)= (N-s).5ΣN-sk=1 ukuk+ruk+s (5.4-1)
for 1<r<s=M, where M=N.4. Hinich (1993) shows that under the null hypothesis of linearity, E[D(r,s)]=0 and E(D²(r,s)) = 1. The Hinich (1993) H statistic becomes the normalized sum of the M²/2 values of D²(r,s) or
H= M-1ΣMs=2Σs-1r=1[D²(r,s)-1]. (5.4-2)
1
Modeling Episodic Nonlinearity
Using the H statistic as defined in (5.4-2), three tests have been performed. In the first test, the H statistic has been calculated for the residuals of the VAR(6) model for the complete sample of 2,742 observations reported in Table 5.1. Plots of the H statistic are given in Figure 5.1. 59 windows, each containing N/30 or 91 observations with 50% overlap, have been calculated. The first window extends from observation 1 to observation 91, the second window from observation 46 to observation 136, the third window from observation 91 to observation 181, etc. Figure 5.1 is a plot of the H statistic and gives the date associated with the center of the window. Inspection of the plot indicates twin spikes of 13.76, centered at observation 1,036, and 14.83, centered at observation 1,171. These points correspond to 1987/3/9 and 1987/9/18, respectively. Since the first date is more than 45 days prior to the stock market crash of October 1987, this spike reveals greater than usual nonlinearity in bond return residuals prior to that event. In the period prior to 1987/3/9, there is much less evidence of nonlinearity. With the possible exception of the period just prior to 1990, when the H statistic approaches 7.0, there is less evidence of nonlinearity after 1987.
1
Modeling Episodic Nonlinearity
In the second test, we have estimated the VAR(6) model in the period 1983 to 1986, a sample that contained 991 observations after lags. The Granger causality tests for this model indicate F1,1=4.5655(0.99986), F1,2=.19772(0.02527), F2,1=2.2876(0.966302) and F2,2=1.95569(0.9307) in which test statistics indicating significance are reported parenthetically ( ). The significant F2,1 value shows that commodity price inflation continues to Granger cause bond returns in the subsample,. Hinich (1982) G(2) and L(2) values are 4.7645 and 3.159, indicating failure of the Gaussianity and linearity tests for the residuals of the second equation in the VAR model. The Hinich H test has been estimated for 60 windows of 33 observations (991/30) each and the values have been plotted in Figure 5.2. Compared with the peaks in Figure 5.1, which are about 15, those in Table 5.2 are substantially smaller.
Finally, a VAR(6) model has been estimated for 1988 through 1993, which contains 1,501 observations. In this period, F1,1 = 1.45197 (0.8087), F1,2= 0.86068 (0.4768), F2,1=2.21972 (0.96153) and F2,2=1.2626 (0.7283). The significant F2,1 indicates that in this subsample commodity price inflation also Granger causes bond returns. In this subsample, the Hinich (1982) G(2) and L(2) values are 0.8858 and 1.1177, respectively, indicating no rejection of Gaussianity and linearity assumptions in the second equation which predicts bond returns.5 This finding has been validated by calculating the Hinich (1993) H test for 61 windows, each containing 49 observations. H test values are plotted in Figure 5.3. Inspection of this plot indicates maximum peaks of 5.4, substantially under the values found for the complete period in Figure 5.1.
1
Modeling Episodic Nonlinearity
5.5 Conclusion
Evidence from VAR models and Granger causality tests enables us to provide at least a partial answer to the question, "who influences whom?" This evidence indicates that bond investors do look to commodity markets for early indications of broader price level changes. We find no evidence of a reciprocal influence of bond investors on commodity markets. Our finding pertains to both the complete period (1983-1993) and the subperiods (1983-1986) and (1988-1993). The most straightforward interpretation of our VAR models and Granger causality tests is that bond investors use commodity prices of the previous six trading days in forming inflation expectations. Such microdynamics are consistent with the accounts of market participants and journalists.
1
Modeling Episodic Nonlinearity
Other evidence we have reported indicates that the relationship between daily commodity prices and daily bond returns is richer and more complex than what one can observe in the linear domain. The Hinich (1982) test results for our VAR model from 1983 through 1993 indicate that this relationship is not linear. Applying ARCH methods has not enabled us to account for the nonlinearity detected in our VAR model. Using a new Hinich test (1993) for episodic nonlinearity, we have found that the most severe episodes of nonlinearity occurred in 1987. The timing of these episodes suggests that the residuals in our VAR model reflect mounting tensions in financial markets prior to and during the stock market crash of 1987. Although it would be unwise to put too much weight on an interpretation of residuals, our results show unambiguously that this nonlinearity varies quite sharply even within a relatively short period of 11 years. An inquiry into the origins of this nonlinearity will be attempted in future work.