^{1}

^{*}

^{2}

Empirical study on the factors that induce jumps in interest rates in the euro area is still missing. In this paper, maximum likelihood estimates of I-distribution parameters are extracted using as a first step, an original linear model. According to the contribution of ([1] [2]) in the case of developing a class of Poisson-Gaussian model, we try to enhance the predictive power of this model by distinguishing between a pure Gaussian and Poisson-Gaussian distributions. Such an empirical tool permits to optimizing results through a comparative analysis dealing with the fluctuation of the Euro-interbank offered rate and its statistical descriptive behaviour. The analytical and empirical methods try to evaluate the behavioural success of the ECB intervention in setting interest rates for different maturities. Jumps in euribor interest rate can mainly be linked to surprise decisions of the European Central Bank, and the too frequent meetings of the ECB before November 2001. Despite this special event that leads to a certain lack of predictability, other few day-of-week effects are modelled to prove eventual evidence of bond market overreaction. Empirical results prove that Mondays and Wednesdays are the preponderant days. Regarding monetary policy, negative surprises induce larger jumps than positive ones.

“…I think that the Maastricht Treaty and the launching of the ECB were a magnificent success and I think that when you go back to the Delors Report in 1989, it was quite remarkable when that came out, because it was a proposal for a single currency monetary union. It would have been much easier to have an 11 or 15-currency monetary union, but a single currency monetary union was quite a big step, and for a long time I thought that that was too big a step, that European governments would not be willing to accept it [the loss of sovereignty]. But the Delors gamble, and I think it was a big gamble, turned out to be successful and in retrospect, Europe is lucky that it ended up in that direction, rather than with an alternative” [

In accordance with the previous speech, it seems that searching for a convincing explanation to the ECB’s de- cisions announcement concerning the interest rate is an important subject especially when we find that the lit- erature on the ECB interest rate market has not yet covered many specific aspects studied in the euro area.

Few works are presented in literature dealing with jumps in the Federal Fund rate in general and the EONIA in the eurozone. According to the advancement of [

[

Our empirical analysis is implemented through a linear model that incorporates a fluctuation’s component. The resolution method for a linear interest rate differential equation will be obtained through a Poisson-Gaussian analysis. In this study, we aim at strengthening conclusion taken through a distinguished comparative analysis between a pure Gaussian distribution and Gaussian-Poisson process. Thus, we treat the information surprises result in discontinuous interest rate to quantify the effectiveness of the European Central Bank announcement channel. We choose as a reference the interest rate for interbank deposits in the euro zone determined as a 15% trimmed average of the interest rates contributed by the “Panel banks—banks with the highest volume of busi- ness in the euro zone money market”. It is also the rate at which a prime bank is willing to lend funds in euro to another prime bank. The EURIBOR is computed daily for interbank deposits with a maturity of one week and one to 12 months as the average of the daily offer rates of a representative panel of prime banks, rounded to three decimal places^{1}.

This research examines the role of jump-enhanced stochastic processes in modelling the Euro interbank of- fered rate for different maturities. The paper offers four distinct sets of contributions. 1) We develop an analyti- cal modelling framework for jumps in fixed income country. 2) We establish a Poisson-Gaussian model, and then deduce a pure Gaussian model. 3) We implement a comparative analysis for these models to detect limits and benefits for each one. 4) We determine which day can optimize the effectiveness of the ECB announcement channel.

The paper proceeds as follows. Section 2 deals with methodological aspects. Section 3 discusses the optimal period for estimation. Section 4 deals with the empirical results, we present those obtained through Poisson- Gaussian model and pure Gauss model (Section 4.1), then we present a method to extract day of the week effect recognition (Section 4.2). Section 5 summarizes and provides concluding remarks.

Stochastic processes governing interest rates analysis is harder than that usually encountered for resolving equi- ties and exchange rates. This complexity is due to mean reversion in models saving a surprising element with higher fluctuations. There are also very limited solutions for the stochastic differential equations. In this section, we present methodological aspects for our econometric specifications.

The mean reverting process for the interest rates can be written as:

where

Being at time t = 0, and looking ahead to time t = T, we are interested in the distribution of r(T) given the cur- rent value of the interest rate

where

where

Illustrating the solution for the previous equation, we find that:

Next, we can deduce the moments and the probability density functions for any choice in distribution where the rises or fall do not depend on the state variables.

Let

Then,

We can also compute the first, second and third derivatives of K evaluated at

Using the fact that

And the derivatives of L with respect to

Then, we can write the intermediate value as:

We can now write the analytical expressions for the moments:

In discrete time, we express the process in Equation (1) as follows:

where

Let

For x occurrences,

Under this expression, we have to classify the movements of possible fluctuations. The assumption made here is that we are searching for a jump, in each time interval either only one jump occurs or no jump occurs. Searching for a fall, in each time interval either only one fall occurs or no fall occurs. But the question that we will try to answer after the estimation is that: Does no jump mean necessary a fall or something else? According to [

Allowing the variance

In this equation noted (Equation (11)),

Designing by:

(Equation (11)) can be rewritten as:

Searching for a pure Gaussian process, we attribute a null value to

Consider a Poisson probability density function

Two values

where Q denotes the confidence level. Graphically, upon a measurement,

From Equation (12), a further Equation (15) is deduced as:

According to [

Our estimation involves maximizing the function L, where

This may be written as:

Consider a probability function of

From (19), the Equation (17) becomes:

The same for (18):

According to [

If the large-N limit has not been reached, the standard deviation can be estimated by finding the value of

The standard deviation will be symmetric with respect to

For each process, we obtain estimates that are consistent, unbiased and especially efficient attaining the Cramer-Rao lower bond due to the satisfaction of the technical regularity conditions stated in Cramer [

Given the analogy of this distribution to that of mixture distributions presented in Equation (11), ML is di- rectly achieved as a solution to a system of first order conditions

Before To avoid any unfavourable event that would bias the result in favour of finding jumps, we eliminate some critical dates (negative surprises) collected from the financial times [

First, before Wednesday 21 March 2001, the ECB had insisted that the US slowdown was unlikely to have much impact on Europe and therefore the interest rates would not change. Second, on Wednesday 21 March 2001, the ECB president (Duisenburg) said during a business school meeting in Germany that the ECB might need to consider cutting interest rates because the slowdown in growth in the USA may be stronger than earlier expected. Third, on Friday 23 March 2001, Statements of the Governor of the France ([

This section analyses the EURIBOR interest rate sample of 1945 daily observations over the period from January 1999—the starting date of Stage Three of the EMU—to February 2007 except some dates cited previ- ously. The data is daily on frequency.

As

When we observe ^{2}.

Evolution of the euro interbank rate for different maturities

Kernel density spreading

Table 1. Descriptive Statistics Euribor.

Maturity | 1 Week | 1 Month | 3 Months | 6 Months | 9 Months | 1 Year |
---|---|---|---|---|---|---|

Mean | 3.31625 | 3.34030 | 3.40076 | 3.44907 | 3.49829 | 3.55453 |

Medium | 3.34030254 | 3.638 | 3.8715 | 3.75475 | 3.49672176 | 3.48250118 |

Max | 4.844 | 5.046 | 5.14 | 5.202 | 5.251 | 5.341 |

Min | 2.06 | 2.053 | 2.073 | 2.001 | 2.103 | 2.183 |

Std | 9.63434 ´ 10^{‒1} | 9.64726 ´ 10^{‒1} | 9.79675 ´ 10^{‒1} | 9.80504 ´ 10^{‒1} | 9.84592 ´ 10^{‒1} | 9.88055 ´ 10^{‒1} |

Skew | 0.1476 | 0.1257 | 0.0941 | 0.0605 | 0.0376 | 0.0198 |

Kurt | 1.7948 | 1.7780 | 1.7761 | 1.8051 | 1.8224 | 1.8409 |

ARCH-LM (Fstatistic) | 1797.2945 | 1817.5001 | 1757.0654 | 1679.4306 | 1586.1724 | 1491.8006 |

_{2}.

Table 2. Descriptive Statistics (first difference of Euribor) R_{2}.

1 Week | 1 Month | 3 Months | 6 Months | 9 Months | 1 Year | |
---|---|---|---|---|---|---|

Mean | 9.27495 ´ 10^{‒4} | 1.7893 ´ 10^{‒3} | 2.9458 ´ 10^{‒3} | 2.6364 ´ 10^{‒3} | 2.19024 ´ 10^{‒3} | 1.75162 ´ 10^{‒3} |

Medium | −0.00035732 | 0.00039228 | 0.00044496 | 0.00038869 | 0.00026853 | 0.00029425 |

Max | 0.033 | 0.023 | 0.067 | 0.022 | 0.053 | 0.031 |

Min | −0.036 | −0.0061 | −0.038 | −0.028 | −0.0024 | −0.022 |

Std | 2.56276 ´ 10^{‒1} | 2.40427 ´ 10^{‒1} | 2.38518 ´ 10^{‒1} | 2.38053 ´ 10^{‒1} | 2.39968 ´ 10^{‒1} | 2.42285 ´ 10^{‒1} |

Skew | −0.0782 | −0.3529 | −0.2459 | 0.1366 | 0.3973 | 0.4895 |

Kurt | 167.2425 | 205.9451 | 208.3241 | 199.264 | 184.1957 | 173.5747 |

ARCH-LM (Fstatistic) | 71.5116 | 71.1644 | 71.1146 | 71.5695 | 72.4818 | 73.5728 |

To investigate the movement between two variables (in our case, it is the same variable but for different ma- turities). To detect the comovements between two maturities i/j, we resort to compare the plot with a simple OLS regression line, as well as with a non parametric estimate.

Assuming that all relations between two variables have the following form:

A possibly non linear regression function is assumed,

where K is a kernel function, h is the bandwidth defined by [

where

Use Maximum likelihood estimation (MLE) has many optimal properties in estimation. It provides a wide level of sufficiency because it encompasses a set of complete information about the parameter of interest contained in its estimator. It is consistent due to its true parameter value that generated the data recovered asymptotically, i.e. for data of sufficiently large samples. It is also efficient due to the lowest-possible variance of parameter esti- mates achieved asymptotically and parameterization invariance. MLE is useful for obtaining a good descriptive measure for the purpose of summarizing observed data, it is a standard approach to parameter estimation and inference in statistics.

We needed an optimization algorithm that could efficiently handle the complicated log-likelihood function in Equation (14). After some exploration, we have chosen to use unconstrained function minimization routine of

. Estimation of Euribor for different maturities versus euribor one week (in level)

t-statistic | t-statistic | |||||
---|---|---|---|---|---|---|

1 Month | 0.0322 | 5.6554 | 0.9976 | 605,9804 | 0.9925 | 0.0838 |

3 Months | 0.0853 | 7.0284 | 0.9998 | 284.3872 | 0.9666 | 0.1790 |

6 Months | 0.2001 | 11.1104 | 0.9797 | 187.8398 | 0.9267 | 0.2656 |

9 Months | 0.3112 | 10.1025 | 0.9543 | 106.6543 | 0.8123 | 0.3103 |

12 Months | 0.4153 | 16.1025 | 0.9466 | 126.7516 | 0.8520 | 0.3803 |

. Estimation of Euribor for different maturities versus euribor one week (in first difference)

t-statistic | t-statistic | |||||
---|---|---|---|---|---|---|

1 Month | −0.0001 | −0.6174 | 0.8803 | 143.1836 | 0.8804 | 0.0832 |

3 Months | −0.0022 | 1.1746 | 0.8504 | 118.6820 | 0.8350 | 0.0969 |

6 Months | −0.0019 | 0.9136 | 0.8280 | 103.7491 | 0.7945 | 0.108 |

9 Months | −0.0008 | −0.6098 | 0.8942 | 100.6960 | 0.0745 | 0.1267 |

12 Months | −0.0001 | −0.0525 | 0.9165 | 169.3001 | 0.9113 | 0.0527 |

Euribor in level Nadaraya-Watson regression (one week versus one month)

the Optimization Toolbox of the MATLAB Software to locate the minimum of the negative log-likelihood func- tion. The routine implements a subspace trust region method which is based on the interior-reflective Newton method described in [

Euribor in first difference Nadaraya-Watson regression (one week versus twelve months)

. Basic Poisson-Gaussian estimation

Parameter | 1 Week | 1 Month | 3 Months | 6 Months | 9 Months | 1 Year |
---|---|---|---|---|---|---|

0.0075 (−3.21) | 0.0055 (4.25) | 0.0075 (2.03) | 0.0097 (3.62) | 0.0027 (2.94) | 0.0111 (2.54) | |

0.001 (2.74) | 0.001 (1.98) | 0.001 (2.95) | 0.001 (2.98) | 0.001 (2.59) | 0.001 (2.98) | |

0.0182 (2.15) | 0.0185 (2.55) | 0.0182 (3.25) | 0.0177 (4.25) | 0.0185 (2.54) | 0.0175 (2.94) | |

0 | 0^{−} | 0^{−} | 0^{−} | 0 | 0^{−} | |

0.0149 (4.21) | 0.0199 (5.25) | 0.018 (3.87) | 0.0178 (2.95) | 0.0162 (3.56) | 0.0188 (2.90) | |

0.0317 (4.98) | 0.0221 (4.23) | 0.0257 (3.65) | 0.0316 (2.59) | 0.0288 (3.09) | 0.0255 (3.19) | |

1.2083 × 10^{19} | 3.0413 × 10^{29} | 5.1599 × 10^{27} | 4.3081 × 10^{29} | 5.2747 × 10^{28} |

Bollerslev and Wooldridge (1992) robust t-statistics are in parentheses [

. Basic Pure Gaussian estimation

Parameter | 1 Week | 1 Month | 3 Months | 6 Months | 9 Months | 1 Year |
---|---|---|---|---|---|---|

0 | 0 | 0 | 0 | 0 | 0 | |

3.05 (4.25) | 3.05 (4.25) | 3.05 (4.25) | 3.05 (4.25) | 3.05 (4.25) | 3.05 (4.25) | |

0.08 (11.78) | 0.08 (11.75) | 0.08 (11.75) | 0.08 (11.75) | 0.08 (11.75) | 0.08 (11.75) | |

0.0098 (4.78) | 0.0098 (4.78) | 0.0098 (4.78) | 0.0098 (4.78) | 0.0098 (4.78) | 0.0098 (4.78) | |

0 | 0 | 0 | 0 | 0 | 0 |

Bollerslev and Wooldridge (1992) robust t-statistics are in parentheses [

Intuitive results emanate from this analysis. There is no evidence of skewness

The jump intensity or the ex-ante probability of a jump occurring is better seen in a Poisson-Gaussian model than in a pure Gauss model because it encompasses maximum number of parameter estimated.

Gaussian-Poisson model estimation for different maturities

Probability Plots (first derivative of kernel density)

Note that the likelihood function takes the form of a curve if there is only one parameter beside h; which is assumed to be known. For example, if the model has two parameters, the likelihood function will be a surface sitting above the parameter space. In general, for a model with k Parameters, the likelihood function takes the shape of a k-dim geometrical “surface” sitting above a k-dim hyperplane spanned by the parameter vector.

The unconditional probability density function from the raw data and the plots from the best fitted models of each maturity is presented^{3}. The upper panel plots the full distribution over the range 3 × 10^{−}^{2}. The middle panel presents the same distribution but for a bigger value of h which is equal to 0.5, the closer plot clearly brings out the good fit from the ARCH-jump model compared to the other models. The lower panel deals with a represen- tation from first derivative of kernel density which deviates negatively from the origin axe.

In this section, we shall employ the model to examine various phenomena in the bond markets via the lens of the model. Our jump model is facile in permitting many different analyses. We explore whether fluctuations are more likely to happen in predetermined days only for five operative days of the week (Monday, Tuesday, Wednesday, Thursday, and Friday). Our purpose is to determine which day is the favourable for announcing a supervising decision that can affect the market.

It is well known that fluctuations would be more likely on Monday since the release of non observable infor- mation over the weekend may lead to a larger volatility of the interest rate. Moreover, option expiry may inject fluctuations into the behaviour of interest rate on Wednesday and Thursday.

We focus our analysis on the last day of the operative week for the ECB and we determine the contribution of the other days in amplifying the arrival intensity of jumps in that day.

To be more precise, let illustrate what we had said before in a simple linear model that enable us to take into consideration the arrival intensity of fluctuation (rise or full) in the interest rate for different maturities:

where

the possible day

The following

We treat in this paper the evolution of the daily euro interbank offered rate to describe the announcement chan- nel of the European Central Bank. The latter is computed daily for interbank deposits with a maturity of one week, one month, three months, six months, nine months and twelve months.

To provide a tutorial exposition of the maximum likelihood estimation, we evaluate results from basic Gaus- sian and Poisson-Gaussian models and try to compare the eventual illustrative results adopting these processes. Moreover, we conclude that jumps are an essential component for modeling EURIBOR. The illustrative Poisson and Gauss processes implemented in a linear model contribute to a much better in-sample fit once jumps are considered under either one or two models. We conclude that models do not lead to the same conclusions. It is the Poisson-Gaussian model that gives better performance. Searching for the contribution of day of the week in amplifying operative actions in the announcement channel of the ECB that consolidates the link with the market, we have resorted to a third linear model deeply linked to the previous parameter estimated through a Poisson-

. Jump estimation parameter with day of the week effects

Maturity | 1 Week | 1 Month | 3 Months | 6 Months | 9 Months | 1 Year |
---|---|---|---|---|---|---|

0.2318 (2.15) | 0.1687 (13.25) | 0.1587 (7.45) | 0.1698 (15.16) | 0.1458 (4.25) | 0.1236 (3.65) | |

0.1418 (4.52) | 0.1587 (11.25) | 0.1625 (2.62) | 0.1478 (1.21) | 0.1345 (−1.95) | 0.1298 (−1.92) | |

0.1353 (−2.35) | 0.1354 (−0.25) | 0.1024 (0.55) | 0.2524 (1.25) | 0.0214 (1.36) | 0.0254 (0.65) | |

0.2258 (4.98) | 0.2135 (10.25) | 0.2153 (6.64) | 0.2054 (8.36) | 0.2456 (6.65) | 0.2354 (7.25) | |

0.1442 (1.25) | 0.1145 (1.36) | 0.0214 (−0.98) | 0.0058 (−2.65) | 0.1025 (0.25) | 0.0254 (1.25) | |

0.0001 (1.98) | 0.0001 (2.96) | 0.004 (−2.39) | 0.0003 (3.63) | 0.0007 (2.01) | 0.0007 (2.06) | |

0.032 (1.67) | 0.1859 (9.78) | 0.1566 (8.63) | 0.1673 (9.62) | 0.0937 (8.09) | 0.0937 (6.21) | |

0 | 0 | 0 | 0 | 0 | 0 | |

0.0001 (11.92) | 0.0098 (8.26) | 0.0098 (13.53) | 0.0098 (19.21) | 0.0098 (7.25) | 0.0098 (6.25) | |

0.4769 (1.25) | 0.1999 (2.65) | 0.0697 (3.25) | 0.0568 (4.25) | 0.1881 (3.56) | 0.1881 (11.25) |

Bollerslev and Wooldridge (1992) robust t-statistics are in parentheses [

Mondays effect (estimation with, maturity = one month)

Wednesdays pertinent effect (estimation with, maturity = one month)

Gaussian model. Therefore, we have added dummy variables to conclude after estimation that only Mondays and Wednesdays for each maturity taken and especially for one month can represent the preponderant days that contribute to amplifying the jumps that may occur on Fridays (