# Testing for cointegration between T- Bonds Yields adjusted to constant maturities. Essay

**Introduction**

Several theories have been arising to explain why bonds of different maturities but similar risk, tax liabilities, and redemption possibilities, have clearly diverse yields. (Elton, Gruber, et al, pp 501). Regarding this divergence one of the most relevant academic studies is The Pure Expectations Theory which is an attempt to explain the relation between expected returns of long term maturity bonds with short term interest rates.

The objective of this paper is to validate The Pure Expectations Theory using historical data; previously, the most interesting effort to prove this assumption can be found in a cointegration analysis of treasury bill yields by Anthony Hall, Heather Anderson and Clive Granger. Unfortunately, this research is applicable exclusively to money market instruments like US Treasury Bills excluding long term maturity bonds like UK Gilts; in this paper, the authors analyzed the long term stochastic trend from 5 different types of Treasury Bills returns.

At the end, the data will support the evidence of a cointegration relation between the short term yields, but at the same time will reject the existence of a stochastic common trend between short term money market instruments and long term maturity bond yields. This premise is particularly true for the bonds with more than 3 years to maturity. In addition, the results will show how a structural change in the interest rate policy can affect the stability of a cointegration model.

**Term structure theory and expectations.**

The basic equation found in Hall paper indicates the technique in which bonds with a variety of maturities are cointegrated:

Where R(k,t) is the continuously compounded yield to maturity for a k period pure discount bond. Is the expected change in the spot rate which determines the future contracts. L(k,t) is the risk premium. In this case, is very possible to find a positive correlation between yields from a bond with long term maturity and the expected returns of a short term money market investment plus a constant risk premium. (Campbell and Shiller 1991). On the other hand, the expectations and the risk premium are stationary; therefore, the difference between a short term yield like the 3 moths treasury bill representing L(1,t) and the yield of a longer maturity bond like 5 or 7 years treasury bond should be stationary as well. Consequently, the relation between the two series should be described by a cointegrated vector.

**Analyzing the Data**

The data used in this paper is similar to the one used by Boothe (1991) in: Interest parity, Cointegration, and the term structure in Canada and the United States. The historical data is composed by eight series of yields from very active bonds series adjusted to constant maturities. . The use of Interpolation techniques may cause a loss of important information regarding the general behaviour of the series. However, the use of constant maturity yields is essential in this case Unfortunately, simple average yields calculated over wide maturity classes may suffer from large swings in within-class average term to maturity. The use of constant maturity yields assures that fluctuations in term do not introduce this type of noise into the data (Boothe, 1991).

The sample consists of 277 monthly observations for each series from January 1982 until January 2005. (See figure 1) The yields are expressed in nominal terms. A detailed description of each data series is included in the appendix 2.

Figure 1. Yields on Treasury nominal securities at “constant maturity”. 3 moths, 6 moths, 1 year, 2 years, 3 years, 5 years, 7 years and 10 years. |

**Descriptive Statistics and Estimation**

The method used on this paper to test cointegration is the Johansen and Juselius approach. One of the main benefits is that Johansen maximum likelihood (ML) technique is easily applicable in multivariate frameworks.

Test for cointegration are designed to identify common stochastic trends between nonstationary variables. For that reason, is very important to test for stationarity in each one of the variables If both variables are stationary is not necessary to proceed since standard time series methods apply to stationary variables. (Enders, pp 374) The Augmented Dickey Fuller test and the Phillips Perron are the tests used to search for any evidence of stationarity.

The results of the augmented Dickey Fuller test show weakly evidence against the null hypothesis. A similar result can be found for first differences in each one of the series. The Phillips-Perron unit root test illustrates a completely different scenario because there is no evidence against the null hypothesis in any of the series. These results are summarized in table 1.

Is not totally clear why all the variables are not stationary, however the results of the Phillips Perron test confirms the existence of evidence against the null hypothesis. In addition, a visual inspection of the data is not showing a clear stochastic trend or any evidence of the existence of an intercept. At this point is possible to assume that all the variables are integrated with the same order I(1).

Table 1. Unit Root Tests (1) (2) | ||||

ADF | P-P | |||

Variable – Time to Maturity | Level | 1st difference | Level | 1st difference |

3 moths (No trend and no intercept) | -3.005 (1) | -11.437 (0)* | -2.306 | -12.252 * |

3 moths (Intercept) | -3.565 (1) | -11.583 (0)* | -2.529 | -12.318 * |

3 moths (Trend and intercept) | -4.101 (1)* | -11.687 (0)* | -3.106 | -12.396 * |

6 moths (No trend and no intercept) | -2.941 (1) | -11.432 (0)* | -2.426 | -12.306 * |

6 moths (Intercept) | -3.435 (1) | -11.578 (0)* | -2.716 | -12.321 * |

6 moths (Trend and intercept) | -3.883 (1)* | -11.693 (0)* | -3.247 | -12.407 * |

1 year (No trend and no intercept) | -2.313 (3) | -7.263 (2)* | -2.468 | -11.068 * |

1 year (Intercept) | -2.877 (3) | -7.371 (2)* | -2.739 | -11.146 * |

1 year (Trend and intercept) | -3.716 (3)* | -7.445 (2)* | -3.300 | -11.214 * |

2 years (No trend and no intercept) | -2.185 (2) | -9.544 (1)* | -2.437 | -10.680 * |

2 years (Intercept) | -2.483 (2) | -9.643 (1)* | -2.621 | -10.807 * |

2 years (Trend and intercept) | -3.335 (2) | -9.687 (1)* | -3.301 | -10.869 * |

3 years (No trend and no intercept) | -2.172 (2) | -10.046 (1)* | -2.427 | -10.608 * |

3 years (Intercept) | -2.423 (2) | -10.150 (1)* | -2.608 | -10.651 * |

3 years (Trend and intercept) | -3.897 (3)* | -10.188 (2)* | -3.418 * | -10.656 * |

5 years (No trend and no intercept) | -2.136 (2) | -10.355 (1)* | -2.461 | -10.361 * |

5 years (Intercept) | -2.338 (2) | -10.467 (1)* | -2.592 | -10.391 * |

5 years (Trend and intercept) | -4.004 (3)* | -10.498 (1)* | -3.542 * | -10.407 * |

7 years (No trend and no intercept) | -2.144 (2) | -10.507 (1)* | -2.473 | -10.771 * |

7 years (Intercept) | -2.283 (2) | -10.631 (1)* | -2.575 | -10.755 * |

7 years (Trend and intercept) | -3.958 (3)* | -10.659 (1)* | -3.650 * | -10.768 * |

10 years (No trend and no intercept) | -2.157 (2) | -10.752 (1)* | -2.437 | -11.188 * |

10 years (Intercept) | -2.259 (2) | -10.882 (1)* | -2.568 | -11.253 * |

10 years (Trend and intercept) | -3.557 (2)* | -10.913 (1)* | -3.693 * | -11.251 * |

(1) Critical values correspond to those calculated by Mckinon (1991) | ||||

(2) Number of lags used are in parentheses | ||||

*Rejection of the null hypothesis of a unit root at significance level of 5% |

As a result, the number of lags can be reduced to eleven in order to accept the null hypothesis of LM test. This number is confirmed by the LR estimation included in the Lag order selection criteria.

The results of the test shows 3 cointegrated equations at 5% level and two cointegrated equations at 1% (table 2).

H0:rank=r | Eigenvalue | trace value | 5% critical value | 1% critical value | max value | 5% critical value | 1% critical value |

r =0** | 0.21 | 215.65 | 156.00 | 168.36 | 61.64 | 51.42 | 57.69 |

r<=1** | 0.19 | 154.01 | 124.24 | 133.57 | 54.84 | 45.28 | 51.57 |

r<=2 | 0.14 | 99.17 | 94.15 | 103.18 | 39.28 | 39.37 | 45.1 |

r<=3 | 0.10 | 59.89 | 68.52 | 76.07 | 27.91 | 33.46 | 38.77 |

r<=4 | 0.05 | 31.98 | 47.21 | 54.46 | 14.34 | 27.07 | 32.24 |

r<=5 | 0.04 | 17.64 | 29.68 | 35.65 | 9.96 | 20.97 | 25.52 |

r<=6 | 0.02 | 7.68 | 15.41 | 20.04 | 6.17 | 14.07 | 18.63 |

r<=7 | 0.01 | 1.51 | 3.76 | 6.65 | 1.51 | 3.76 | 6.65 |

**Denotes rejection at the 1% significance level. | |||||||

Table 2. Results of cointegration tests |

In order to explain these results is possible to use two different approaches: first, the vectors are not cointegrated contradicting the theory and second, is possible that the series might have a structural change. To confirm the first possibility is necessary to test all the series separately using first the lag selection, LM test and checking for Heteroskedasticity and the lag order selection criteria.

The results are summarized in the following table:

Spreads between | Test statistics | 5% critical value |

m(3),m(6) | 8.68 | 3.76 |

m(3),y(1) | 7.59 | 3.76 |

m(3),y(2) | 7.06 | 3.76 |

m(3),y(3) | 6.04 | 3.76 |

m(3),y(5) | 5.75 | 3.76 |

m(3),y(7) | 5.39 | 3.76 |

m(3),y(10) | 4.99 | 3.76 |

m(6),y(1) | 6.39 | 3.76 |

m(6),y(2) | 6.07 | 3.76 |

m(6),y(3) | 5.94 | 3.76 |

m(6),y(5) | 5.59 | 3.76 |

m(6),y(7) | 5.09 | 3.76 |

m(6),y(10) | 4.48 | 3.76 |

y(1),y(2) | 5.98 | 3.76 |

y(1),y(3) | 5.93 | 3.76 |

y(1),y(5) | 5.17 | 3.76 |

y(1),y(7) | 4.60 | 3.76 |

y(1),y(10) | 3.94 | 3.76 |

y(2),y(3) | 5.37 | 3.76 |

y(2),y(5) | 3.91 | 3.76 |

y(2),y(7) | 3.80 | 3.76 |

y(2),y(10) | 3.39 | 3.76 |

y(3),y(5) | 4.95 | 3.76 |

y(3),y(7) | 4.49 | 3.76 |

y(3),y(10) | 4.28 | 3.76 |

y(5),y(7) | 4.41 | 3.76 |

y(5),y(10) | 4.18 | 3.76 |

y(7),y(10) | 4.36 | 3.76 |

Table 3. Test for cointegration between spread vectors |

All the possible combinations by pairs seem to be not cointegrated. A quick test of the stationarity of the residuals confirms that the series are not cointegrated. The graphs of all the residuals after applied OLS like in the Engle and Granger methodology are showed in the figure 2.

Figure 2. Graph of the residuals between spread vectors. |

**New estimations under structural changes.**

The data from January 1982 until September 1982 is part of a former monetary regime called New Operating procedures when the Federal Reserve did not target interest rates. Hall, Anderson and Granger, took into account both periods emphasizing: the yields were considerably more volatile during the new operating procedures regime.

The Phillips Perron tests show no evidence against the null hypothesis that there is a unit root in yield levels for all the series. At the same time, the test rejects the null hypothesis of unit root in the differences.

Table 1. Unit Root Tests (1) (2) | ||||

ADF | P-P | |||

Variable – Time to Maturity | Level | 1st difference | Level | 1st difference |

3 moths (No trend and no intercept) | -1.458 (1) | -9.723 (0)* | -1.311 | -9.850 * |

3 moths (Intercept) | -1.481 (1) | -9.765 (0)* | -1.290 | -9.739 * |

3 moths (Trend and intercept) | -2.334 (1) | -9.744 (0)* | -2.339 | -9.717 * |

6 moths (No trend and no intercept) | -1.420 (1) | -9.948 (0)* | -1.379 | -9.948 * |

6 moths (Intercept) | -1.541 (1) | -9.981 (0)* | -1.366 | -10.097 * |

6 moths (Trend and intercept) | -2.434 (1) | -9.963 (0)* | -2.305 | -10.079 * |

1 year (No trend and no intercept) | -1.239 (3) | -7.284 (2)* | -1.469 | -10.067 * |

1 year (Intercept) | -1.471 (3) | -7.310 (2)* | -1.452 | -9.856 * |

1 year (Trend and intercept) | -2.688 (3) | -7.296 (2)* | -2.382 | -9.839 * |

2 years (No trend and no intercept) | -1.431 (2) | -10.332 (1)* | -1.541 | -10.234 * |

2 years (Intercept) | -1.391 (2) | -10.377 (1)* | -1.489 | -10.254 * |

2 years (Trend and intercept) | -2.497 (2) | -10.358 (1)* | -2.508 | -10.235 * |

3 years (No trend and no intercept) | -1.460 (2) | -10.711 (1)* | -1.460 | -10.711 * |

3 years (Intercept) | -1.400 (2) | -10.742 (1)* | -1.400 | -10.761 * |

3 years (Trend and intercept) | -2.962 (3) | -10.761 (2)* | -2.962 | -10.742 * |

5 years (No trend and no intercept) | -1.473 (2) | -10.932 (1)* | -1.591 | -10.626 * |

5 years (Intercept) | -1.359 (2) | -10.989 (1)* | -1.504 | -10.615 * |

5 years (Trend and intercept) | -3.210 (3) | -10.969 (1)* | -2.887 | -10.594 * |

7 years (No trend and no intercept) | -1.522 (2) | -10.975 (1)* | -1.581 | -11.160 * |

7 years (Intercept) | -1.324 (2) | -11.045 (1)* | -1.428 | -11.333 * |

7 years (Trend and intercept) | -3.202 (3) | -11.025 (1)* | -2.913 | -11.313 * |

10 years (No trend and no intercept) | -1.550 (2) | -11.246 (1)* | -1.597 | -11.685 * |

10 years (Intercept) | -1.301 (2) | -11.322 (1)* | -1.424 | -11.603 * |

10 years (Trend and intercept) | -3.187 (2) | -11.302 (1)* | -2.973 | -11.5799 * |

(1) Critical values correspond to those calculated by Mckinon (1991) | ||||

(2) Number of lags used are in parentheses | ||||

*Rejection of the null hypothesis of a unit root at significance level of 5% |

The results showed less evidence of Heteroskedastic behaviour. (See table 5)

H0:rank=r | Eigenvalue | trace value | 5% critical value | 1% critical value | max value | 5% critical value | 1% critical value |

r =0 | 0.23 | 247.09 | 156.00 | 168.36 | 67.51 | 51.42 | 57.69 |

r<=1** | 0.17 | 179.58 | 124.24 | 133.57 | 50.66 | 45.28 | 51.57 |

r<=2** | 0.17 | 128.91 | 94.15 | 103.18 | 47.80 | 39.37 | 45.1 |

r<=3** | 0.13 | 81.11 | 68.52 | 76.07 | 37.96 | 33.46 | 38.77 |

r<=4 | 0.06 | 43.15 | 47.21 | 54.46 | 17.47 | 27.07 | 32.24 |

r<=5 | 0.06 | 25.68 | 29.68 | 35.65 | 14.94 | 20.97 | 25.52 |

r<=6 | 0.03 | 10.75 | 15.41 | 20.04 | 9.01 | 14.07 | 18.63 |

r<=7 | 0.01 | 1.73 | 3.76 | 6.65 | 1.73 | 3.76 | 6.65 |

**Denotes rejection at the 1% significance level. | |||||||

Table 5. Results of cointegration tests modified series |

The results shows 4 cointegrated vectors for 5% critical value the max value shows that is just one at 1% level. Again, there is no support of seven cointegrated vectors which implies that we cannot conclude that all the variables are cointegrated. The results are summarized in the following table:

Spreads between | Test statistics | 5% critical value |

M(3),m(6) | 2.53 | 3.76 |

M(3),y(1) | 2.42 | 3.76 |

M(3),y(2) | 1.97 | 3.76 |

M(3),y(3) | 1.70 | 3.76 |

M(3),y(5) | 6.28 | 3.76 |

M(3),y(7) | 8.74 | 3.76 |

M(3),y(10) | 7.64 | 3.76 |

M(6),y(1) | 2.15 | 3.76 |

M(6),y(2) | 1.92 | 3.76 |

M(6),y(3) | 7.08 | 3.76 |

M(6),y(5) | 9.33 | 3.76 |

M(6),y(7) | 8.08 | 3.76 |

M(6),y(10) | 7.16 | 3.76 |

Y(1),y(2) | 5.98 | 3.76 |

Y(1),y(3) | 11.41 | 3.76 |

Y(1),y(5) | 7.67 | 3.76 |

Y(1),y(7) | 7.01 | 3.76 |

Y(1),y(10) | 6.29 | 3.76 |

Y(2),y(3) | 9.89 | 3.76 |

Y(2),y(5) | 5.77 | 3.76 |

Y(2),y(7) | 6.02 | 3.76 |

Y(2),y(10) | 6.29 | 3.76 |

Y(3),y(5) | 5.75 | 3.76 |

Y(3),y(7) | 7.34 | 3.76 |

Y(3),y(10) | 6.35 | 3.76 |

Y(5),y(7) | 6.02 | 3.76 |

Y(5),y(10) | 10.80 | 3.76 |

Y(7),y(10) | 8.06 | 3.76 |

This new test illustrates the importance structural changes. Short term maturity series are most likely to be cointegrated confirming Halls paper.

**Final model from short term of maturity.**

Lag | LogL | FPE | AIC | SC | HQ |

0 | -195.3168 | 5.75E-05 | 1.588412 | 1.699199 | 1.632971 |

1 | 945.1038 | 8.81E-09 | -7.196123 | -6.863763 | -7.062449 |

2 | 1021.39 | 5.50E-09* | -7.667110* | -7.113176* | -7.444319* |

3 | 1033.196 | 5.69E-09 | -7.634346 | -6.858839 | -7.32244 |

4 | 1047.477 | 5.76E-09 | -7.620912 | -6.623831 | -7.21989 |

5 | 1055.516 | 6.14E-09 | -7.558716 | -6.340061 | -7.068578 |

6 | 1067.648 | 6.33E-09 | -7.528501 | -6.088272 | -6.949246 |

7 | 1084.057 | 6.32E-09 | -7.531697 | -5.869895 | -6.863326 |

8 | 1092.915 | 6.70E-09 | -7.475896 | -5.592521 | -6.71841 |

9 | 1107.674 | 6.78E-09 | -7.466206 | -5.361256 | -6.619603 |

10 | 1112.894 | 7.40E-09 | -7.381985 | -5.055462 | -6.446266 |

11 | 1128.131 | 7.47E-09 | -7.376021 | -4.827925 | -6.351186 |

Table 7.Var lag order selection criteria. |

All the criteria and the LM tests suggest the use two lags.

H0:rank=r | Eigenvalue | trace value | 5% critical value | 1% critical value | max value | 5% critical value | 1% critical value |

r =0 | 0.17 | 90.18 | 47.21 | 54.46 | 49.55 | 27.07 | 32.24 |

r<=1 | 0.08 | 40.63 | 29.68 | 35.65 | 23.16 | 20.97 | 25.52 |

r<=2 | 0.06 | 17.47 | 15.41 | 20.04 | 15.59 | 14.07 | 18.63 |

r<=3** | 0.01 | 1.87 | 3.76 | 6.65 | 1.87 | 3.76 | 6.65 |

**Denotes rejection at the 1% significance level. | |||||||

Table 8. Test for cointegration between spread vectors, short term series. |

The results show 3 cointegrated equations for four series, in both trace value and max value at 5% level.

Finally, the papers suggest that the cointegration should be in the form (1,-1,-1,-1). The results are shown in the Appendix 3 and Appendix 4. This is because the Value of is zero.

**Conclusions**

In general long term maturity series are not cointegrated. The evidence is not supporting the original theory about Rational Expectations, because apparently the long run series are not following a determined path. On the other hand, short term maturity series are cointegrated.

**References**

Alexander (1981) cited in Wallace, C (1987) Issues in Teaching English as a Second Language in Aburdarham, S (1987) Bilingualism and the Bilingual an interdisciplinary approach to Pedagogical and Remedial issues, Windsor: NFER Nelson.

Hall, D Anthony; Anderson, Heather and Granger, Clive. A cointegration Analysis of treasury bill yields. The review of Economics and Statistics. 74:1 pp. 116-126.

Boothe, Paul. Interest Parity, Cointegration, and the term structure in Canada and the United States The Canadian Journal of Economics. 24:3 pp. 595-603.

Enders, Walter. 1995. Applied econometric series. United States of America: John Wiley & Sons, Inc.

Elton, Edwin; Gruber, Martin; Brown, Stephen and Goetzmann William. 2003. Modern Portfolio Theory and Investment Analysis. United States of America: John Wiley & Sons, Inc.