Nonlinearities and chaos in bond yield movements.

Pandey, Vivek K.

ABSTRACT

There are many reasons to expect the possible influence of chaotic dynamics in bond yield movements, whether it is Fed's policies, institutional trading programs, or evolutionary dynamics in the economy at play. This study attempts to detect the presence of low dimensional deterministic chaos in bond yield movements by examining four bond market aggregates. Evidence indicates an earlier presence of deterministic nonlinearity in the corporate bond funds that appears to have disappeared in recent years. No such evidence is evident for Treasury bonds. To the individual investor, this paper ends with a heartening note that, at least during recent times, the examined bond markets seem to have provided fairly priced bonds.

INTRODUCTION

Of late, chaotic and stochastic nonlinearities have been heavily investigated topics in the examination of capital markets. These studies are either fashioned as an investigation into the informational efficiency of capital markets or simply as an exercise in examining the predictability of capital markets. Very often the above two objectives complement each other in the conduct of an empirical investigation, but some recent theorists have posited that chaotic nonlinearities can exist even in informationally efficient markets (Grabbe, 1996). Many studies have examined equity markets (e.g., Kohers et al, 1997, Pandey et al, 1998, 1999), foreign exchange markets (Hseih, 1989, Pandey et al, 2000) and various economic aggregates (Brock et al, 1991) for latent chaotic and stochastic nonlinearities. However, very little evidence exists on bond market aggregates. This paper attempts to fill this void in the literature by investigating four bond market aggregates, two corporate, and two Treasury bond aggregates for latent deterministic nonlinearities.

The reasons to expect a nonlinear driving influence in bond yield movements can come from many sources. In as much as the Fed's actions in influencing interest rates have been responsive to economic conditions and not entirely discretionary, these responses may be explained by some deterministic patterns, perhaps nonlinear in nature. Hence, it is possible that the

Fed's policy decisions, and institutional traders' trading programs, induce some form of chaotic determinism in bond yield movements. Such processes have already been documented in the stock markets (Sias and Starks, 1997). Moreover, recent deliberations about viewing economies as evolutionary dynamical processes lend credence to the hypothesis that aggregate capital market behavior may be driven by a "vision of the future" (Grabbe, 1996) and hence may embody an underlying deterministic mechanism. In light of these recent developments, investigations of underlying chaotic deterministic mechanisms in capital market aggregates has taken on an increased significance.

Grabbe (1996) presents the possibility of self-organization of human societies, and thus by implication, of the economy, with a shared image or a vision of the future. At the singular level, this vision might be subconscious or nonexistent, but at the aggregate level such a vision might be discernible. In large capital markets, a large volume of the trading occurs while traders are speculating. They may not afford the luxury of acting late on any relevant news. Very often, the trader must anticipate other traders' moves and try to preempt such moves. As such, each trader must not just act on his or her expectations, but rather act on the anticipation of other traders' moves who themselves are trying to anticipate the first's and everyone else's moves and so on. Evolutionary dynamics provide a solution in the form of a spontaneous order involving dynamic feedback at a higher, or aggregate, level. Hence in the capital markets context, what appears to be competition amongst traders and institutions at the lower level, where expectations are generated, functions as co-ordination at the higher (global) level. Therefore, it is likely that even in face of rational expectations, some form of complex deterministic mechanism may generate capital market aggregates, such as the aggregate bond portfolios used in this study.

The subject of nonlinear dynamics as a driving influence in capital markets continues to receive much attention (e.g., Brock et al, 1991, Scheinkman and LeBaron, 1989, Hsieh, 1991, 1993, 1995, Kohers et al, 1997, Pandey et al, 1998, 1999). However, very little investigative work involving nonlinear dynamics and yield movements in bond aggregates appears in any of the published studies. This study intends to fill this void by examining bond portfolios of Moody's Aaa rated bonds, Baa rated bonds and Treasury constant maturity yields on ten year as well as thirty year bond portfolios.

The remainder of this paper is organized as follows. The next section details the data sources and outlines the methodology employed in this study. The third section details the results. The final section of this paper draws conclusions from the results and discusses the implications of the results derived from this exercise.

DATA AND METHODOLOGY

This study examines the bond yield movements of two corporate and two Treasury long-term bond portfolios. The corporate portfolios are comprised of Moody's seasoned Aaa issues and Baa issues respectively; whereas the Treasury portfolios are comprised of yields on actively traded bonds adjusted to constant maturities of ten years and thirty years respectively. The data is obtained from the Federal Reserve Statistical Releases and are of daily frequency.

The period examined in this study spans twenty-two years and extends from January 3, 1978 through December 31, 1999. To screen for biases arising from possible structural shifts from regime changes and other shifts in market dynamics, the overall time frame is also subdivided into five subperiods of approximately equal lengths; the subsample periods are January 1978--April 1982; May 1982--September 1986; October 1986--February 1991; March 1991--July 1995; and August 1995--December 1999.

Testing for Nonlinear Dynamics:

Each bond portfolio yield series is first-differenced in order to obtain a bond yield movement series for the examined portfolio. These differenced yield series are then subjected to a series of tests which involves: filtering for linear autocorrelation, examination of structural integrity (stationarity) of data, filtering for stochastic nonlinearity and subsequent testing for low-dimensional deterministic nonlinearities (chaos) in the multi-step procedure employed in this study.

Tests for nonlinearities employed in this study are, by their construction, highly sensitive to linear dependencies as well. Hence, prior to proceeding with their examination for nonlinear determinism, each bond yield movement series is filtered for linear correlations using autoregressive models of order p denoted AR (p) of the form:

[Y.sub.t] = [[theta].sub.0] + [P.summation over (i=1)] [[phi].sub.i][Y.sub.t-1] + [[omega].sub.t]

where [[omega].sub.t] is a random error term uncorrelated over time, while [phi] = ([[phi].sub.1], [[phi].sub.2], ... [[phi].sub.p]) is the vector of autoregressive parameters. The lags (or order 'p') used in the autoregressions for the appropriate model are determined via the Akaike Information Criterion (AIC) (see Akaike, 1974).

Examinations of chaotic dynamics have revealed that deterministic processes of a nonlinear nature can generate variates that appear random and remain undetected by linear statistics. Hence, the next step involves examining the filtered (pre-whitened) yield movement series for randomness, or more specifically the null of IID. One of the more popular statistical procedures that has evolved from recent progress in nonlinear dynamics is the BDS statistic, developed by Brock et al (1991), which tests whether a data series is independently and identically distributed (IID). The BDS statistic, which can be denoted as [W.sub.m, T]([epsilon]), is given by:

[W.sub.m, T]([epsilon]) = [square root of T][[C.sub.m, T](epsilon) - [C.sub.1, T][(epsilon).sup.m]]/[[sigma].sub.m, T](epsilon)

where T = the number of observations,

[epsilon] = a distance measure,

m = the number of embedding dimensions,

C = the Grassberger and Procaccia correlation integral, and

[[sigma].sup.2] = a variance estimate of C.

For more detail about the development of the BDS statistic, see Brock et al (1991). Simulations in Brock et al (1991) demonstrate that the BDS statistic has a limiting normal distribution under the null hypothesis of independent and identical distribution (IID) when the data series is sufficiently large, such as a series with a thousand observations. The use of the BDS statistic to test for independent and identical distribution of pre-whitened data has become a widely used and recognized process (e.g., Brorsen and Yang, 1994, Hsieh, 1991, 1993, Kohers et al, 1997, Pandey et al, 1998, 1999, 2000, Sewell et al, 1993). After data has been pre-whitened and nonstationarity is ruled out, the rejection of the null of IID by the BDS statistic points towards the existence of some form of nonlinear dynamics.

Rejection of the null hypothesis of IID by the BDS statistic is not considered evidence of the presence of chaotic dynamics. Other forms of nonlinearity, such as nonlinear stochastic processes, could also drive such results. In addition, structural shifts in the data series can be a significant contributor to the rejection of the null. To avoid biases arising from structural shifts from regime changes and other shifts in market dynamics, the sample period of January 1978--December 1999 is subdivided into five subperiods of equal lengths, which are examined individually for the violation of the IID assumption. Other subperiod divisions achieved within the constraint of maintaining at least a thousand observations per subperiod in order to preserve the robustness of tests used in this study, yielded results similar to the equal-length division of the sample period described above.

In order to minimize the possibility of stochastic (variance-driven) nonlinearity affecting the results of tests for chaotic dynamics, a series of stochastic filters were employed. As there is a wide range of identified stochastic processes in existence, no exhaustive filter exists for the general class of stochastic nonlinear processes. The alternative is to fit stochastic models to the data and capture the residuals. If these are IID, we know that stochastic nonlinearity explains all the nonlinearity identified by the BDS statistics of the pre-whitened data series. However, since one can construct an infinite number of stochastic models, fitting each model to the pre-whitened data is a near impossible task. Fortunately, prior research indicates that the Generalized Autoregressive Conditional Heteroskedasticity (Bollerslev, 1986) model of the first order, i.e., GARCH (1,1), and the first order GARCH--in-Mean process, i.e., GARCH-M (1,1), are able to explain the latent stochastic nonlinearity in a wide range of financial time series (e.g., Brock et al, 1991, Hsieh, 1993, 1995, and Sewell et al, 1996). Hence it is imperative that any pre-whitened financial series exhibiting non-IID behavior be subjected to filters for these GARCH processes first. The GARCH process may be described as:

[y.sub.t] = [[beta].sub.0] + [m.summation over (i=1)][[beta].sub.i][x.sub.t-i] + [[epsilon].sub.t]

where [[epsilon].sub.t] is conditional on past data and is normally distributed with mean zero and variance [h.sub.t] such that:

[h.sub.t] = [omega] + [q.summation over (i=1)][[alpha].sub.i][[epsilon].sup.2.sub.t-i] + [p.summation over (j=1)] [[gamma].sub.j][h.sub.t-j]

Hence, the GARCH series becomes an iterative series where past conditional variances feed into future values of the series xt and the solution is obtained when the computing algorithm achieves convergence. The GARCH (1,1) series is a GARCH model estimated with values of p = q =1 in the above scheme.

The GARCH-M model introduces an added regressor that is a function of the conditional t variance h into the GARCH function and the simple GARCH-M(1,1) function used to filter bond yield movements in this study may be described as:

[y.sub.t] = [[beta].sub.0] + [m.summation over (i=1) [[beta].sub.i][x.sub.t=1] + [delta]f ([h.sub.t]) + [[epsilon].sub.t]

[[epsilon].sub.t] = [square root of [h.sub.t][e.sub.t]]

[h.sub.t] ~ GARCH (1, 1)

Upon some experimentation, the functional form f([h.sub.t]) found to be most suited to the data series examined in this study was found to be the logged value of [h.sub.t], i.e., ln([h.sub.t]).

Both the GARCH(1,1) and the GARCH-M(1,1) models are fitted to each data series and the residuals captured in the filtering process. If these conditional heteroskedasticity models explain away any observed non IID behavior of the data series, one can be certain that stochastic nonlinearity is the contributing factor.

If the data sets examined pass the stochastic filters mentioned previously and still display non-IID behavior as per recomputed BDS statistics, then one can employ tests specifically aimed at detecting chaotic nonlinearities that are latent in the datasets. Two tests of chaos are used in this study; the GP correlation dimension analysis (Grassberger and Procaccia, 1983), and the third moment test (Brock et al, 1991, Hsieh, 1989, 1991).

Hsieh (1989, 1991) and Brock et al (1991) developed the third moment test to specifically capture mean nonlinearity in a given series. Briefly stated, this test uses the concept that mean nonlinearity implies additive autoregressive dependence; whereas variance nonlinearity implies multiplicative autoregressive dependence. Using this notion, and exploiting its implications, they constructed a test that examines the third-order moments of a given series. Additive dependencies will lead to some of these third-order moments being correlated. By its construction, this test will not detect variance nonlinearities.

The third order sample correlation coefficients are computed as:

[r.sub.(xxx)] (i, j) = [1/T [summation][X.sub.t][X.sub.t-i][X.sub.t-j]] / [[1/T [summation] [X.sup.2.sub.t]].sup.15]

[r.sub.(xxx)](i,j) = the third order sample correlation coefficient of [x.sub.t] with [x.sub.t-i] and [x.sub.t-j], and T = the length of the data series being examined.

Hsieh (1991) developed the estimates of the asymptotic variance and covariance for the combined effect of these third-order sample correlation coefficients, which can be used to construct a ?2 statistic to test for the significance of the joint influence of the [r.sub.(xxx)] (i,j)'s for specific values of j. If the [chi square] statistics for relatively low values of j are significant, then this outcome would be a strong 2 indicator of the presence of mean nonlinearity in the examined series. As chaotic determinism is a form of mean-nonlinearity, the third moment test provides strong evidence of the presence of chaos. The Grassberger and Procaccia (1983) correlation dimension test is a graphical measure of identifying a chaotic attractor in a chaotic series. The GP correlation dimension test utilizes the correlation integral, [C.sub.m, T]([epsilon]), (also used in the development of the BDS statistic) to compute probabilities of data points in clustering sequences (for a more detailed description, see Brock et al, 1991). In addition, to obtain evidence of chaotic dynamics, a graphical examination of the slope of log [C.sub.m,T]([epsilon]) over log (g) for small values of e is obtained. This value, [C.sub.m,T]([epsilon])/log ([epsilon]), is the point estimate of the Grassberger and Procaccia correlation dimension v. When plotted against m, the number of embedding dimensions, the point estimate of v (i.e., [v.sub.m]) will converge to a constant beyond a certain m if the data series is indeed chaotic.

The intuition that lies behind the search for convergence of the GP correlation dimension ([v.sub.m]) into a single value v as m (the dimensionality) increases is as follows. A two-dimensional relationship can be characterized as a line, which retains its two-dimensional form whether viewed in a map of two, three, or more dimensions. Similarly, an m dimensional relationship will retain its m dimensional form when examined in m or more dimensions. A truly random series, however, retains no form in any dimensionality and fills up the entire space. The GP correlation dimension is a measure of fractional dimensionality, and will remain constant when examined in a space-map of dimensions higher than the identified GP dimension for a deterministic series. It is important to note here that the GP correlation dimension is a relative measure of the fractional dimensionality of a series and does not actually identify the absolute dimensionality of the examined series.

Hence, first the examined bond yield movements are filtered for linear influences using autoregressive filters. The lag lengths for these AR processes are determined by using the Akaike Information Criteria (AIC) (see Akaike, 1974). Next, each of these series is tested for the null of IID using the BDS statistic. The above process is then repeated for five subperiods of equal length to examine the possibility that any observed rejection of the IID assumption in the previous step was due to nonstationarity of the data series. Next, GARCH and GARCH-M filters are employed to determine if any observed nonlinearity has its origins in one of these popular models of stochastic origin. Finally, the third moment test and the GP correlation dimension analysis are used to determine if the observed nonlinearity is indeed low-dimensional and deterministic.

RESULTS

Aside from its ability to detect nonlinear relationships, the BDS statistic by its very design is also sensitive to linear processes. Since this study concerns itself with the detection of nonlinear dynamics in bond yield movements series, autocorrelations were filtered out using autoregressive models. The appropriate lag length for each series was determined using the Akaike Information Criterion (AIC). Table 1 shows the appropriate lag lengths used for each of the bond yield series examined. Consistent with earlier observations about many types of financial series, these bond yield movements also exhibit detectable linear autocorrelation which are filtered away. The resulting data series are referred to as pre-whitened series for the remainder of this study.

Table 2 lists the BDS statistics for each data series for dimensions m = 2, ..., 10 and the distance measure g = 0.5 [sigma], 0.75 [sigma], and 1.00 [sigma]. The BDS statistic has an intuitive explanation. For example, a positive BDS statistic indicates that the probability of any two m histories, ([x.sub.t], [x.sub.t-1], ..., [x.sub.t-m + 1]) and ([s.sub.t], [s.sub.t-1], ..., [s.sub.t-m + 1]), being close together is higher than what would be expected in truly random data. In other words, some clustering is occurring too frequently in the m-dimensional space. Thus, some patterns of bond yield movements are taking place more frequently than is possible with truly random data.

An examination of the results in Table 2 reveals that all BDS statistics are significantly positive. With sufficiently large data sets (greater than 1,000 observations) simulations by Brock et al (1991) show that the BDS statistics for IID data follows a limiting standard normal distribution. For each bond yield movement series, the BDS statistics are computed for [epsilon] = 1 [sigma], 0.75 [sigma], and 0.50 [sigma]. A lower e value represents a more stringent criteria since points in the m-dimensional space must be clustered closer together to qualify as being "close" in terms of the BDS statistic. Hence, [epsilon] = 0.5 [sigma] reflects the most stringent test, while [epsilon] = 1.00 [sigma] is the most relaxed criterion used in this analysis. In this study, the values of m examined go only as high as 10. Two reasons dictate the choice of 10 as the highest dimension analyzed. First, with m = 10, only 531 non-overlapping 10 history points exist in each return series. Examining a higher dimensionality would severely restrict the confidence in the computed BDS statistic. Second, the intent of this study is to only detect low dimensional deterministic chaos. High dimensional chaos is, for all practical purposes, as good as randomness (see Brock et al, 1991).

One must remember that the BDS statistic only reveals whether or not the data series examined is different from a random identically and independently distributed (IID) series. The results in Table 2 represent a summary rejection of the null hypothesis of IID for each series examined. However, it is possible that the examined data series reject the IID hypothesis because of nonstationarity in the data. Exogenous influences such as regime changes, policy shifts or regulatory reforms, among others, could impact the bond yields in such a way that they give the appearance of not being random (although they are truly random in stable times) over the 22-year period under investigation in this study.

Tables 3, 4, 5, 6 and 7 provide the BDS statistics for the same 4-bond portfolio yield movements for the five subperiods: January 1978 through April 1982; May 1982 through September 1986; October1986 through February 1991; March 1991 through July 1995; and August 1995 through December 1999, respectively. Many other subperiod compositions were tried within the constraints of maintaining at least 1,000 observations per subperiod in order to preserve the robustness of the tests employed in this study. The results obtained from other subperiod divisions were similar to the ones obtained from the equal five-part decomposition of the sample period reported in this study. An examination of the BDS statistics for the subperiods reveals a different picture from that which emerged for the overall period.

An examination of Tables 3, 4 and 5 mostly confirms that the examined BDS statistics reject the null of IID. Hence the indication of non-IID behavior obtained from the examination of the overall sample period in Table 2 is confirmed in Tables 3, 4 and 5. However, Table 6 reveals that during subperiod 4 the pre-whitened Treasury bond portfolio yield movements do not reject the null of IID. For subperiod 5, Table 7 reveals that the pre-whitened corporate bond portfolios do not reject the null of IID either.

In conclusion, the two Treasury bond portfolios have stationary results only for the first three subperiods examined; while the two corporate bond portfolios have stationary data for the first four subperiods examined. Hence, further analysis of these data series will be restricted to the periods spanning consistent results from the BDS tests. In other words, the ten-year Treasury bond fund series and the thirty-year Treasury bond fund series will be tested for stochastic and deterministic nonlinearities using data spanning the first three subperiods only, i.e., January 1978 through March 1991. Similarly, the Moody's Aaa bond fund series and the Baa bond fund series will be subjected to further analysis using only data spanning the first four subperiods, i.e., January 1978 through July 1995.

Table 8 displays the recomputed BDS statistics for the pre-whitened data series after they are filtered for a GARCH(1,1) process. As all BDS statistics examined are significantly positive, the data series are still non-IID. Hence, for the appropriate periods under examination, none of the bond yield movement series appear to be explained by a GARCH(1,1) process.

The BDS statistics for the GARCH-M(1,1) filtered pre-whitened yield movement series shown in Table 9 are also all significant for each series examined. This indicates that the GARCH-M(1,1) process is also unable to explain the observed non-IID behavior of the examined series during stationary periods.

The results of the third moment test are presented in Table 10. This table shows the [chi square] statistics for a combined test of the significance of all examined three moment correlations [r.sub.(xxx)](i,j) up to a certain lag length. Where 1 < i < j < 5, the [chi square] statistic has 15 degrees of freedom. On the other hand, when all three moment correlations are examined up to a lag length of 10 (i.e., 1 i < j < 10), the [chi] statistic has 55 degrees of freedom. As one may observe from Table 9, both the [[chi square].sub.15] and the [[chi square].sub.55] statistics for both corporate bond portfolios examined are highly significant, indicating, that for the period under examination, i.e., January 1978 till June 1994, these two portfolio yield movements may be influenced by low-dimensional chaos. The [chi square] statistics for the two Treasury 2 bond portfolios do not display such tendency.

Whereas the third moment test presents strong evidence of a possible chaotic driving process, it does not constitute indisputable evidence of the existence of chaotic determinism in the data sets examined. Hence the whitened yield movement series of the Moody's Aaa and Baa bond portfolios are subjected to the graphical procedure of the determination of the Grassberger and Procaccia (1983) correlation dimension. Convergence of the point estimates of the GP correlation dimensions to a single number would provide evidence of the existence of a chaotic attractor. This, in turn, would give conclusive evidence of the existence of a deterministic process. As observed from Figure 1, the point estimates of the GP correlation dimension for Moody's Aaa bond portfolios remain well below 1, even when the number of embedded dimensions rises up to thirty. Similarly, the point estimates of the GP correlation dimensions for the Baa bond portfolio series stays well below 3, even as the number of embedding dimensions rises to thirty. Lacking convergence of the GP correlation dimension estimates to a single value, one is unable to provide indisputable evidence of the existence of a chaotic attractor, and hence chaotic behavior in the examined series.

[FIGURE 1 OMITTED]

To sum up the findings of this study, the filtered yield movement series of all examined bond portfolios exhibit non-IID behavior for the overall period. However, this non-IID behavior persists only for three contiguous subperiods (January 1978 until March 1991) for the Treasury portfolios examined and for four subperiods (January 1978 until July 1995) for the two corporate bond portfolios. For these relevant subperiods, the GARCH(1,1) and GARCH-M(1,1) stochastic filters are unable to remove the non-IID behavior of the examined datasets. The third moment test provides strong indications of the existence of chaotic behavior in the Aaa and Baa bond portfolios. However, in the absence of identification of a chaotic attractor by the GP correlation dimension analysis, very possibly an artifact of data limitations, one is unable to obtain conclusive evidence of the existence of deterministic chaos. And finally, it is important to observe that all observed nonlinearities appear to have disappeared in the last few years (subperiods 4 and 5) of the sample period, where linear autocorrelations appear to account for all non-IID behavior in the examined series.

CONCLUSIONS AND IMPLICATIONS

This study has been unique in its treatment of bond fund yield movements and subjecting them to an investigation for nonlinear driving influences. The results indicate some evidence of nonlinear behavior in the two Treasury bond portfolios (10 year and 30 year constant maturities) for the first three subperiods (January 1978 to March 1991) examined. However, in the case of these Treasury bonds, this study was unable to identify the source of the observed nonlinearity, which is not chaotic nor did it appear to fit the mold of the popular GARCH(1,1) or the GARCH-M(1,1) variety of stochastic models. In case of the corporate Moody's Aaa rated and Baa rated bond portfolios, the yield movements do appear to be driven by chaotic influences for the period January 1978 to July 1995. However, this study could only provide strong indication, but not indisputable evidence, of the existence of a chaotic phenomenon.

Although many arguments for the expectation of a chaotic driving influence in these series exist, some of which were outlined earlier in this manuscript, these results are also consistent with some observations made by earlier studies in stock markets (e.g., Kohers et al, 1997) which associate institutional program trading phenomenon to introduction of chaotic influence in these capital market series. Such influence is certainly deterministic, even when it is not nonlinear, as clearly demonstrated by Sias and Starks (1997). Yield movements of high quality corporate bond portfolios, such as the ones examined here, are very likely to be heavily influenced by institutional trading rules. Institutional traders are also major players in the Treasury bond markets.

Implications of the Findings in Treasury Bond Markets:

Although Institutional traders are major players in Treasury bond markets, the U.S. Treasury enters the primary markets with high frequency. The competitive bidding process used by the Treasury may well mitigate the effect of program (rule-based) trading used by Institutional investors in secondary markets. Hence any expectations of the driving influence of a chaotic phenomenon in Treasury bond markets are based mostly on the conjecture that Treasury actions may be responsive to current market conditions in a deterministic fashion. The results of this study do not support such hypothesis.

Implications of the Findings in High-quality Corporate Bond Markets:

It is important to note that the strong evidence of chaotic phenomenon found in the high quality corporate bond yield movements is only limited to the time period 1978-95. The analysis of post July 1995 data reveals that the nonlinear deterministic influence detected during the 1978-95 period has disappeared in recent times. The expectation of a chaotic driving influence in Corporate bond markets is mainly driven by the expectation of institutional program or rule-based trading. The results of this study could be indicative of the fact that institutional traders do not follow predictable paths. In addition, this recent absence of deterministic nonlinearity may also suggest that institutional investors have had less impact on bond pricing during recent times. The advent of the Internet, and ease of access to information provided by new information technology, may easily have impacted such an outcome. When individual investors have inexpensive and fast access to breaking information, they do not find the need to view actions of institutional investors as signals to buy or sell. As a result, institutional trading will have lost price-driving influence in corporate bond markets, which is one premise behind a hypothesized deterministic influence. Clearly, more research designed specifically to measure this phenomenon must be conducted to arrive at any definitive conclusion.

Implications of the Findings to Individual Investors:

In the final analysis, the results of this study advance a heartening message to the individual investor. On the aggregate, at least during recent years, both the Treasury and the high-quality Corporate bond markets have provided the individual investor with bonds that are fairly priced. In addition, these results may also be indicative of the fact that recent advances in information technology have been successful in eliminating the information asymmetry between institutional and individual investors.

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Vivek K. Pandey, The University of Texas at Tyler

Table 1: Autoregression Lags Used to Filter Returns on the
Bond Indexes Analyzed

Bond Yield Index: Autoregressive Model Used to Filter:

Aaa Bonds AR(7)
Baa Bonds AR(8)
10 Year Treasury AR(5)
30 Year Treasury AR(4)

NOTE: AR = Autoregressive model with (x) lags. Lags are
determined via the Akaike Information Criterion (AIC).

Table 2: BDS Statistics for Linear Filtered Bond Yield Movements

Sample Period: Jan 1978-Dec 1999

 [epsilon]/ 10 Year 30 Year
m [sigma] Aaa Baa Treasury Treasury

2 0.50 18.72 ** 9.08 ** 14.06 ** 13.33 **
3 0.50 25.21 ** 14.04 ** 19.07 ** 17.71 **
4 0.50 31.01 ** 20.09 ** 23.08 ** 21.81 **
5 0.50 37.28 ** 27.14 ** 27.34 ** 25.73 **
6 0.50 44.71 ** 37.00 ** 32.90 ** 31.60 **
7 0.50 53.87 ** 50.56 ** 39.60 ** 38.89 **
8 0.50 66.08 ** 72.20 ** 48.41 ** 49.45 **
9 0.50 81.71 ** 103.35 ** 59.85 ** 63.84 **
10 0.50 103.08 ** 149.42 ** 76.13 ** 85.59 **

2 0.75 18.21 ** 9.19 ** 15.00 ** 14.86 **
3 0.75 22.97 ** 13.80 ** 19.72 ** 20.33 **
4 0.75 26.11 ** 19.00 ** 23.16 ** 24.72 **
5 0.75 28.95 ** 24.50 ** 26.30 ** 28.73 **
6 0.75 32.17 ** 31.94 ** 29.90 ** 33.51 **
7 0.75 35.47 ** 42.17 ** 33.50 ** 38.72 **
8 0.75 39.23 ** 57.61 ** 37.53 ** 44.78 **
9 0.75 43.42 ** 81.00 ** 42.05 ** 51.88 **
10 0.75 48.37 ** 117.52 ** 47.40 ** 60.75 **

2 1.00 17.46 ** 9.41 ** 15.90 ** 15.38 **
3 1.00 21.09 ** 13.48 ** 20.40 ** 20.61 **
4 1.00 23.15 ** 17.62 ** 23.57 ** 24.60 **
5 1.00 24.79 ** 21.69 ** 26.23 ** 27.92 **
6 1.00 26.53 ** 26.97 ** 29.08 ** 31.61 **
7 1.00 28.06 ** 33.48 ** 31.64 ** 35.24 **
8 1.00 29.66 ** 42.69 ** 34.18 ** 39.01 **
9 1.00 31.22 ** 55.27 ** 36.85 ** 43.04 **
10 1.00 32.91 ** 72.94 ** 39.74 ** 47.65 **

NOTE: m = embedding dimension. n = Corporate: 5311, Treasury: 5256

** denotes statistics are significant at the .01 level,

Table 3: BDS Statistics for Linear Filtered Bond Yield Movements

Sample Subperiod 1: Jan 1978-Apr 1982

 [epsilon]/ 10 Year 30 Year
m [sigma] Aaa Baa Treasury Treasury

2 0.50 20.59 ** 12.49 ** 15.03 ** 14.46 **
3 0.50 32.89 ** 18.96 ** 22.59 ** 23.68 **
4 0.50 51.67 ** 29.53 ** 32.68 ** 36.67 **
5 0.50 83.13 ** 47.78 ** 48.25 ** 55.99 **
6 0.50 138.91 ** 79.06 ** 72.45 ** 87.42 **
7 0.50 239.84 ** 131.84 ** 111.53 ** 139.84 **
8 0.50 429.56 ** 225.99 ** 177.42 ** 228.43 **
9 0.50 784.76 ** 397.15 ** 289.65 ** 380.85 **
10 0.50 1463.60 ** 686.70 ** 482.61 ** 645.27 **

2 0.75 18.54 ** 12.99 ** 11.41 ** 10.96 **
3 0.75 27.44 ** 19.31 ** 15.35 ** 16.92 **
4 0.75 38.68 ** 29.55 ** 20.03 ** 24.03 **
5 0.75 56.00 ** 45.70 ** 25.90 ** 32.32 **
6 0.75 83.99 ** 72.43 ** 33.91 ** 43.96 **
7 0.75 129.34 ** 117.56 ** 45.03 ** 60.11 **
8 0.75 204.80 ** 197.23 ** 60.33 ** 82.93 **
9 0.75 330.59 ** 337.74 ** 82.15 ** 116.17 **
10 0.75 543.75 ** 588.96 ** 112.62 ** 165.04 **

2 1.00 16.21 ** 12.64 ** 9.56 ** 7.25 **
3 1.00 22.44 ** 17.69 ** 12.04 ** 10.86 **
4 1.00 29.04 ** 25.04 ** 14.74 ** 14.73 **
5 1.00 37.85 ** 35.32 ** 17.36 ** 18.39 **
6 1.00 50.51 ** 51.12 ** 20.73 ** 23.10 **
7 1.00 69.11 ** 75.12 ** 24.80 ** 28.58 **
8 1.00 96.93 ** 112.84 ** 29.58 ** 35.20 **
9 1.00 138.34 ** 172.37 ** 36.04 ** 43.82 **
10 1.00 200.28 ** 266.90 ** 44.03 ** 54.63 **

NOTE: m = embedding dimension. n = Corporate: 1062, Treasury: 1051

** denotes statistics are significant at the .01 level,

Table 4: BDS Statistics for Linear Filtered Bond Yield Movements

Sample Subperiod 2: May 1982-Sep 1986

 [epsilon]/ 10 Year 30 Year
m [sigma] Aaa Baa Treasury Treasury

2 0.50 3.32 ** -0.24 0.29 0.97
3 0.50 4.14 ** 1.16 1.35 1.04
4 0.50 4.79 ** 2.34 ** 2.61 ** 1.82 *
5 0.50 5.14 ** 3.15 ** 3.74 ** 0.10
6 0.50 5.51 ** 3.74 ** 4.91 ** -2.25
7 0.50 5.46 ** 3.29 ** 6.56 ** -2.19
8 0.50 5.45 ** 2.57 ** 8.10 ** 0.80
9 0.50 5.50 ** 5.12 ** 9.79 ** 8.17 **
10 0.50 5.46 ** 4.91 ** 10.42 ** -3.77

2 0.75 4.18 ** 0.06 0.24 0.72
3 0.75 5.03 ** 1.40 1.38 1.63
4 0.75 5.52 ** 2.51 ** 2.68 ** 3.32 **
5 0.75 5.74 ** 3.15 ** 3.85 ** 5.31 **
6 0.75 5.98 ** 3.91 ** 5.47 ** 7.54 **
7 0.75 6.04 ** 3.95 ** 6.90 ** 9.27 **
8 0.75 6.25 ** 4.09 ** 8.44 ** 10.08 **
9 0.75 6.48 ** 4.38 ** 9.50 ** 9.43 **
10 0.75 6.74 ** 3.49 ** 11.08 ** 14.38 **

2 1.00 5.19 ** 0.60 0.15 0.77
3 1.00 5.97 ** 2.14 * 1.30 1.13
4 1.00 6.27 ** 3.28 ** 2.51 ** 2.09 *
5 1.00 6.36 ** 3.81 ** 3.53 ** 2.82 **
6 1.00 6.54 ** 4.54 ** 4.77 ** 4.30 **
7 1.00 6.60 ** 4.71 ** 5.67 ** 5.53 **
8 1.00 6.73 ** 5.06 ** 6.48 ** 6.39 **
9 1.00 6.88 ** 5.71 ** 6.95 ** 6.77 **
10 1.00 7.02 ** 6.49 ** 7.29 ** 8.73 **

NOTE: m = embedding dimension. n = Corporate: 1062, Treasury: 1051

** denotes statistics are significant at the .01 level,

* denotes p-value of .05 or less

Table 5: BDS Statistics for Linear Filtered Bond Yield Movements

Sample Subperiod 3: Oct 1986-Feb 1991

 [epsilon]/ 10 Year 30 Year
m [sigma] Aaa Baa Treasury Treasury

2 0.50 8.22 ** 7.00 ** 1.84 * 2.08 *
3 0.50 9.50 ** 7.93 ** 2.45 ** 2.33 **
4 0.50 10.16 ** 8.83 ** 2.78 ** 2.88 **
5 0.50 10.51 ** 8.01 ** 2.56 ** 3.06 **
6 0.50 10.83 ** 4.09 ** 3.12 ** 3.64 **
7 0.50 11.01 ** 1.88 * 3.20 ** 4.60 **
8 0.50 11.11 ** -4.32 2.74 ** 5.42 **
9 0.50 11.33 ** -3.43 1.67 * 5.71 **
10 0.50 11.58 ** -2.78 2.05 ** 5.44 **

2 0.75 8.82 ** 4.94 ** 1.54 2.24 *
3 0.75 10.29 ** 5.18 ** 2.28 ** 3.30 **
4 0.75 10.86 ** 5.70 ** 2.89 ** 4.18 **
5 0.75 10.90 ** 6.44 ** 3.30 ** 5.04 **
6 0.75 11.03 ** 7.39 ** 4.11 ** 5.79 **
7 0.75 11.14 ** 9.56 ** 4.68 ** 6.47 **
8 0.75 11.13 ** 9.34 ** 4.94 ** 7.29 **
9 0.75 11.17 ** 9.21 ** 5.08 ** 8.27 **
10 0.75 11.28 ** -3.02 5.52 ** 9.53 **

2 1.00 8.77 ** 4.79 ** 1.63 1.83 *
3 1.00 10.35 ** 5.39 ** 2.48 ** 2.75 **
4 1.00 10.77 ** 5.75 ** 3.14 ** 3.52 **
5 1.00 10.58 ** 6.54 ** 3.38 ** 4.08 **
6 1.00 10.48 ** 7.22 ** 3.85 ** 4.61 **
7 1.00 10.56 ** 8.38 ** 4.29 ** 5.19 **
8 1.00 10.50 ** 9.03 ** 4.62 ** 5.68 **
9 1.00 10.46 ** 8.38 ** 4.80 ** 6.22 **
10 1.00 10.42 ** 8.59 ** 5.00 ** 6.72 **

NOTE: m = embedding dimension. n = Corporate: 1062, Treasury: 1051

** denotes statistics are significant at the .01 level,

* denotes p-value of .05 or less

Table 6: BDS Statistics for Linear Filtered Bond Yield Movements

Sample Subperiod 4: Mar 1991-Jul 1995

 [epsilon]/ 10 Year 30 Year
m [sigma] Aaa Baa Treasury Treasury

2 0.50 4.61 ** 1.06 -0.92 -1.81
3 0.50 8.08 ** 3.77 ** 0.06 -2.02
4 0.50 10.01 ** 6.57 ** 0.71 -3.04
5 0.50 8.70 ** 7.79 ** 1.26 -7.04
6 0.50 3.84 ** 8.35 ** 2.26 * -7.33
7 0.50 -4.84 10.51 ** 3.19 ** -5.47
8 0.50 -373% 6.76 ** 4.38 ** -4.22
9 0.50 -2.94 20.96 ** 5.46 ** -3.35
10 0.50 -2.36 -3.17 7.08 ** -2.70

2 0.75 4.96 ** 0.62 -1.15 -0.53
3 0.75 8.53 ** 2.74 ** -0.21 1.25
4 0.75 11.68 ** 4.86 ** 0.27 1.52
5 0.75 14.83 ** 6.38 ** 0.63 1.62
6 0.75 17.53 ** 8.57 ** 1.25 3.44 **
7 0.75 19.50 ** 10.62 ** 1.82 * 1.39
8 0.75 19.22 ** 15.72 ** 2.59 ** -4.57
9 0.75 22.06 ** 19.62 ** 3.08 ** -3.64
10 0.75 -2.60 29.54 ** 3.74 ** -2.96

2 1.00 5.19 ** 1.35 -0.98 -1.74
3 1.00 9.26 ** 3.44 ** -0.11 -2.84
4 1.00 13.32 ** 5.65 ** 0.28 -2.65
5 1.00 17.02 ** 7.45 ** 0.66 -3.78
6 1.00 23.20 ** 9.54 ** 1.26 -3.04
7 1.00 30.39 ** 12.49 ** 1.72 * -3.49
8 1.00 47.10 ** 17.71 ** 2.20 * -3.90
9 1.00 85.50 ** 23.36 ** 2.55 ** -3.98
10 1.00 142.78 ** 33.72 ** 2.98 ** -3.25

NOTE: m = embedding dimension. n = Corporate: 1062, Treasury: 1051

** denotes statistics are significant at the .01 level,

* denotes p-value of .05 or less

Table 7: BDS Statistics for Linear Filtered Bond Yield Movements

Sample Subperiod 5: Aug 1995-Dec 1999

 [epsilon]/ 10 Year 30 Year
m [sigma] Aaa Baa Treasury Treasury

2 0.50 0.31 -0.45 0.86 -0.11
3 0.50 1.08 -0.80 2.76 ** 1.27
4 0.50 2.49 ** -0.13 2.12 * 0.74
5 0.50 0.47 0.74 0.82 1.15
6 0.50 -4.92 2.61 ** 2.28 * 2.13 *
7 0.50 -7.85 -1.42 5.31 ** 6.59 **
8 0.50 -6.13 -6.00 -5.45 14.92 **
9 0.50 -4.91 -4.81 -4.36 -4.99
10 0.50 -4.02 -3.94 -3.56 -4.09

2 0.75 -0.89 -0.96 1.46 0.78
3 0.75 -0.41 -0.56 2.66 ** 2.18 *
4 0.75 0.14 -0.28 2.93 ** 1.92 *
5 0.75 -0.22 -0.30 3.69 ** 4.00 **
6 0.75 -4.64 0.14 6.07 ** 5.23 **
7 0.75 -3.92 1.06 7.75 ** 6.81 **
8 0.75 2.79 ** 1.56 7.84 ** 8.04 **
9 0.75 -5.18 -3.04 10.47 ** 6.10 **
10 0.75 -4.25 -4.23 16.74 ** 8.85 **

2 1.00 -0.37 -0.59 0.87 -0.83
3 1.00 0.22 -0.06 1.61 -1.75
4 1.00 0.27 0.43 1.20 -2.45
5 1.00 -0.43 0.84 1.07 -3.02
6 1.00 -0.68 0.48 1.19 -2.78
7 1.00 0.23 0.73 1.25 -2.37
8 1.00 -1.63 1.28 1.11 -2.42
9 1.00 1.39 0.96 2.99 ** -2.61
10 1.00 -4.48 -0.33 4.50 ** -2.62

NOTE: m = embedding dimension. n = Corporate: 1063, Treasury: 1052

** denotes statistics are significant at the .01 level,

* denotes p-value of .05 or less

Table 8: BDS Statistics for Garch (1,1) Filtered Pre-whitened
Bond Yield Movements

Sample Period: Corporate: Jan 1978-Jul 1995; Treasury:
Jan 1978-Mar 1991

 [epsilon]/ 10 Year 30 Year
m [sigma] Aaa Baa Treasury Treasury

2 0.50 20.55 ** 10.18 ** 13.19 ** 12.79 **
3 0.50 27.50 ** 15.42 ** 18.56 ** 18.79 **
4 0.50 33.67 ** 21.49 ** 24.50 ** 26.20 **
5 0.50 40.19 ** 28.46 ** 31.73 ** 35.36 **
6 0.50 47.93 ** 38.35 ** 42.13 ** 49.02 **
7 0.50 57.22 ** 51.09 ** 56.37 ** 69.03 **
8 0.50 69.35 ** 71.24 ** 77.11 ** 100.08 **
9 0.50 84.96 ** 100.55 ** 107.55 ** 150.30 **
10 0.50 105.85 ** 143.47 ** 155.16 ** 232.65 **

2 0.75 19.08 ** 10.41 ** 12.70 ** 12.00 **
3 0.75 24.14 ** 15.30 ** 17.10 ** 17.10 **
4 0.75 27.60 ** 20.58 ** 21.24 ** 22.21 **
5 0.75 30.60 ** 26.19 ** 25.39 ** 27.45 **
6 0.75 33.95 ** 33.92 ** 30.46 ** 34.22 **
7 0.75 37.42 ** 44.01 ** 36.19 ** 42.56 **
8 0.75 41.43 ** 59.20 ** 43.25 ** 52.92 **
9 0.75 45.96 ** 82.32 ** 51.81 ** 66.37 **
10 0.75 51.27 ** 117.88 ** 62.58 ** 84.26 **

2 1.00 17.35 ** 10.67 ** 12.23 ** 10.71 **
3 1.00 21.06 ** 14.97 ** 16.02 ** 14.91 **
4 1.00 23.26 ** 19.30 ** 19.17 ** 18.68 **
5 1.00 24.97 ** 23.59 ** 22.06 ** 22.02 **
6 1.00 26.66 ** 29.22 ** 25.30 ** 26.00 **
7 1.00 28.18 ** 35.83 ** 28.48 ** 30.15 **
8 1.00 29.82 ** 45.19 ** 31.88 ** 34.60 **
9 1.00 31.48 ** 58.18 ** 35.72 ** 39.71 **
10 1.00 33.25 ** 76.46 ** 40.05 ** 45.71 **

NOTE: m = embedding dimension. n = Corporate: 4248, Treasury: 3153

** denotes statistics are significant at the .01 level

Table 9: BDS Statistics for Garch-M (1,1,log) Filtered Pre-whitened
Bond Yield Movements

Sample Period: Corporate: Jan 1978-Jul 1995; Treasury:
Jan 1978-Mar 1991

 [epsilon]/ 10 Year 30 Year
m [sigma] Aaa Baa Treasury Treasury

2 0.50 20.56 ** 10.21 ** 13.30 ** 12.93 **
3 0.50 27.51 ** 15.64 ** 18.69 ** 19.04 **
4 0.50 33.66 ** 22.01 ** 24.64 ** 26.48 **
5 0.50 40.13 ** 29.63 ** 31.90 ** 35.79 **
6 0.50 47.84 ** 40.29 ** 42.27 ** 49.55 **
7 0.50 57.09 ** 54.79 ** 56.51 ** 69.69 **
8 0.50 69.15 ** 78.63 ** 77.29 ** 101.00 **
9 0.50 84.67 ** 115.08 ** 107.98 ** 151.79 **
10 0.50 105.42 ** 170.65 ** 155.89 ** 234.10 **

2 0.75 19.21 ** 10.42 ** 12.76 ** 11.77 **
3 0.75 24.28 ** 15.37 ** 17.18 ** 16.71 **
4 0.75 27.71 ** 20.64 ** 21.34 ** 21.67 **
5 0.75 30.68 ** 26.44 ** 25.55 ** 26.66 **
6 0.75 34.02 ** 34.32 ** 30.64 ** 33.11 **
7 0.75 37.47 ** 44.49 ** 36.40 ** 41.08 **
8 0.75 41.47 ** 59.69 ** 43.47 ** 50.98 **
9 0.75 45.98 ** 82.85 ** 52.05 ** 63.79 **
10 0.75 51.26 ** 117.99 ** 62.85 ** 80.76 **

2 1.00 17.42 10.62 ** 12.24 ** 10.57 **
3 1.00 21.11 14.97 ** 16.04 ** 14.72 **
4 1.00 23.28 19.31 ** 19.20 ** 18.52 **
5 1.00 24.97 23.66 ** 22.12 ** 21.84 **
6 1.00 26.65 29.28 ** 25.37 ** 25.75 **
7 1.00 28.17 35.88 ** 28.56 ** 29.83 **
8 1.00 29.80 45.22 ** 31.96 ** 34.22 **
9 1.00 31.44 58.12 ** 35.81 ** 39.26 **
10 1.00 33.19 76.24 ** 40.14 ** 45.14 **

NOTE: m = embedding dimension. n = Corporate: 4248, Treasury: 3153

** denotes statistics are significant at the .01 level

Table 10: Chi-Square Statistics for the Joint Influence of Three
Moment Correlations for the Filtered Bond Yield Movements

Sample Period: Corporate: Jan 1978-Jul 1995; Treasury:
Jan 1978-Mar 1991

Lags Statistic Aaa Baa

1 < i < j < 5 [chi square] (15) 53.90 * 96.30 *
1 < i < j < 10 [chi square] (55) 374.53 * 256.90 *

Lags 10 Year Treasury 30 Year Treasury

1 < i < j < 5 17.25 14.39
1 < i < j < 10 23.60 27.99

* Significant at the 1% level for a right-tailed test.