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This paper tests the popular continuous-time interest rate models for Chinese repo market to address what and how the interest rates change with the marketlization in China. Using Bandi [1]’s method, we get the functional nonparametric estimation of drift and diffusion terms and the local time of the process. We find that the interest rates of China during the period from 1993 to 2003 are bimodal distributed and propose a two-regime model which can fit the data better. We also study the probabilities that the process will stay the two regimes respectively and its transition probability that the process transfers from one regime to another regime.

The short rate is fundamental to the pricing of fixed-income securities. Large literature devotes itself to the esti- mation of the short term interest rate process using different models and methods. In continuous time finance, the dynamic evolution of the spot interest rate process is usually driven by a Markov stochastic differential equ- ation. Diffusion processes have become the standard tool for modelling prices in financial markets for derivative pricing and risk management purposes. Although such continuous time processes offer analytic tractability, the parameters of the process are often difficult to estimate from the data because sample data are available only at discrete time points.

Literature has documented different parametric models for short rate dynamics, each attempting to capture particular features of observed interest rate movements. However, empirical tests of these models have yielded mixed results. Therefore, nonparametric techniques are well-used to remove some distributional restrictions im- posed by parametric models.

Ait-Sahalia [

An assumption commonly made in nonparametric methods is the stationarity of the process. Notwithstanding the advantages of assuming stationarity, it would be helpful to allow for martingale and other possible forms of non-stationary behavior in the process. Motta and Hafner [

Bandi and Philips [

There is no large literature investigating Chinese short interest rate market. Interest rates can be regarded as a benchmark to distribute rare capital by interest rate mechanism in the financial market. It is meaningful to study whether the interest rate is decided by the mechanism of market competition or not. Hong and Lin [

In this paper, we study the interest rate behavior of China based on the observed 7 days repo rate for Shanghai market. The repo rate provides the benchmark for the interest rate of marketability and pricing of national debt futures. With the interest rate marketlization of China, the movement of interest rate reflects the principle of the supply and demand tightly. We follow Bandi and Philips [

The paper is organized as follows. Section 2 introduces the data and method. Section 3 gives the empirical results. Section 4 discusses the two-regime model and its properties implied by the empirical results. Conclusions are given in Section 5.

We use 7-day repo rate of Shanghai market of China as the proxy of Chinese repo market. The data are retrieved from database of China Center for Economic Research (CCER) of Peking University. On the pre-holiday days such as the one-week holiday on the labor day, National day and Chinese new year, the interest rates are abnor- mally high since they are not real interest rates for 7 days, so I removed these from my observations. The final data set is composed of 2052 daily observations from January 4, 1995 to December 31, 2003. The short rate is continuously compounded yield to maturity.

From

We study the behavior of short interest rates by two sub-samples 1995.01-1998.12 and 1999.01-2003.12. The results of preliminary analysis of the whole sample and sub-samples are shown in

A. Summary statistics of daily repo rate | |||||
---|---|---|---|---|---|

Sample period | Mean | Std | Skewness | Kurtosis | First Autocorr |

1995.01-2003.12 | 0.0606 | 0.0416 | 0.7804 | −0.4660 | 0.9626 |

1995.01-1998.12 | 0.0989 | 0.0310 | 0.1340 | 0.8521 | 0.8789 |

1999.01-2003.12 | 0.0283 | 0.0097 | 2.6744 | 13.0090 | 0.7757 |

B. Summary statistics of daily change of repo rate | |||||

Sample period | Mean | Std | Skewness | Kurtosis | First Autocorr |

1995.01-2003.12 | −4.38E−05 | 0.0114 | −0.4823 | 18.1432 | −0.1928 |

1995.01-1998.12 | −7.46E−05 | 0.0152 | −0.3962 | 9.8552 | −0.1620 |

1999.01-2003.12 | −1.79E−05 | 0.0065 | −0.4006 | 35.0510 | −0.3350 |

C. Hypothesis test for daily rate of two subperiods | |||||

result: | |||||

result: | |||||

D. Hypothesis test for daily change of two subperiods | |||||

result: | |||||

result: | |||||

E. Wilcoxon Rank Sum Test for daily repo rate | |||||

result: | |||||

F. Wilcoxon Rank Sum Test for daily change rate of two subperiods | |||||

result: |

This table presents the mean, standard deviation, skewness, kurtosis, and the first autocorrelation of the daily data and daily change of entire sample period and two subperiods. It also gives the hypothesis test about mean and variance of daily rate and daily change rate of two subperiods respectively. Panel E and F give the Wilcoxon Rank Sum test to test whether two subperiods have the same distribution.

the statistics of continuously compounded annualized daily repo rate

Panel C shows the result of hypothesis test for daily rate. It shows that the two subperiods have significant different means and variances. But this may induce that the stationarity of the whole data cannot be guaranteed. Using the same hypothesis test with the Panel C, I tested for daily change of the two subperiods in Panel D. The null hypothesis that the two subperiod samples have the volatility was rejected at 1% level.

Furthermore, from Panel A and Panel B of

Based on the above analysis using the repo rate data sample, we add a state variable into our model for our empirical study.

We assume that the short rate follows a stochastic differential equation as follows:

where

But parametric interest rate models may not fit historical data well. Ait-Sahalia [

A. Estimates of parameters | ||||||
---|---|---|---|---|---|---|

Parameters | ||||||

Estimation | 0.0012 | −0.0212 | −0.2394 | −0.1666 | −0.1597 | −0.1324 |

Standard error | 0.0004 | 0.0059 | 0.0222 | 0.0224 | −0.0219 | −0.02062 |

T value | 2.8044 | −3.5900 | −10.7800 | −7.4100 | −7.1300 | −6.0390 |

B. Augmented Dickey-Fuller T Test | ||||||

T statistics: | ||||||

Result: the null hypothesis of a unit root is rejected at 5%. |

This table presents the statistics of Augmented Dickey Fuller T test for the daily annualized yield on repo rate for Shanghai market. The model used in the test is:

parametric drift estimator performed very poorly. Therefore, we follow the nonparametric estimation techniques which is popular in recent literature related.

The basis for our Monte Carlo simulation is a time-discretization of (1) over a daily interval (

where

After the drift and diffusion estimates are obtained, the next short rate will be simulated according to this data-generating process. After repeating this process a large number, G, sample paths from the true continuous- time model are produced, then the Mento Carlo confidence bands can be determined.

As Johannes [

Based on the nonparametric model of Stanton [

where the parameters in Equations (3) and (4) are the same as in Equation (2).

The estimators of drift and diffusion terms are:

where

where

This nonparametric method has been developed but they either rely on the existence of a time-invariant mar- ginal density for the underlying process (Jiang [

Bandi [

Definition 1 If

This formula gives the amount of time in real time units that the process

Recurrence requires the continuous trajectory of the process to visit any set in its range an infinite number of times over time almost surely. It makes economic sense because interest rates are expected to return to the values in their range over and over again. It is meaningful to estimate the drift and diffusion functions at each point in the range of the sample interest rate process. The density of the observations plays a role in the operation of the asymptotic. This information is contained in the estimated local time of the spot interest rate process.

In order to show precise inference on the drift of process of a point (i.e., to achieve statistically consistent esti- mates), we require the estimated local time of the process at that point to be large. Its properties and estimation are shown in the following section.

According to the previous analysis, we derive the estimation of drift and diffusion from the above estimators in Equations (5) and (6) and obtain the 1000 simulated interest rate paths using the Monte Carlo simulation method. Then we estimate the drift and diffusion for every path.

Drift and diffusion estimates for the single-factor model in Equation (1) and their Monte Carlo confidence bands are given in the

The simulation results indicate that the estimates are unbiased. Because there are few observations are high rates, the confidence intervals are relatively wide. Especially the diffusion estimation fits well based on

Local time gives the amount of time that the process spends in the vicinity of one point. Bandi [

By virtue of recurrence, interest rates may visit every level over time which opens up the possibility of re- covering the true function by using a single trajectory of the process over a long time, through a combination of

infill and long span asymptotic. Bandi [

where the parameters in Equations (8) and (9) are consistent in the whole paper. These asymptotic confidence bands resemble conventional intervals for probability densities.

graph of the estimated local time, we anticipate that problems would arise in the 17% - 21% range, as the time spent by the sample process in this range is quite small. The density in the figure is bimodal, the spatial density of the process appears to be bimodal. Compared with the frequency histogram of the repo rate in

From the feature of the data, the interest rates had a higher level before 1999,, but after 1999, interest rates went down and kept a lower level until 2003. Therefore, two different time horizon can be considered: 1995.01- 1998.12 and 1999.01-2003.12.

We find that the two peaks in

This pattern appears again for their diffusion estimation in

From the previous analysis, we consider the effect from the state variable. The model is the following:

where

When process

When

We then obtain the estimation of the conditional probability

babilities

Then from the Equation (10), we consider the model relying on the state variable. The probability

Based on the previous analysis for drift and diffusion terms, we assume that

where j = 1 or 2 and

Given the discrete data, the data generating process is:

where

In our model, we only have two states and

are:

So for a two-state Markov chain, the transition matrix is

No loss of generality, we assume that this two-state Markov chain is ergodic provided that

The unconditional probability that the process will be in regime 1 at any given date should be the follows:

It is obvious that

The matrix of m-period-ahead transition probabilities for an ergodic two-state Markov chain is given by:

Thus, if the process is currently in state 1, the probability in state 2 after m periods later is given by

where

Given a short rate, whether it stay in regime 1 or 2 is unknown, but we can estimate the probability for any states.

From Hamilton [

where let

bability

We suppose virtually certainty from observations from regime j, so that

Following the method of Hamilton [

where

Because we know

and

Using the whole data sample (2052 observations), we calculate the smoothed probabilities

It is known that

This means that once the process enters a regime, it will remain in that state with a high transition probability. Furthermore, in regime 1, mean-reversion parameter is larger, but it is different for regime 2 in which the drift coefficient is very close to zero. These are very reasonable, because the interest rates are lower and not so volatile as regime 1. Both average change rates of two regimes are very close to 0, but their variances differ.

The inference about the value of

It is obvious that after 1999 probability was very high and close to 1 most of the time. In reality, it is known that when Chinese interest rates remain at a lower level, high economic growth rate gives pressure towards lower rates. Interest-rate liberalization in China is necessary.

In this paper, I study the interest rate behavior of China based on the observed 7 days repo rate of Shanghai market. The repo rate provides the benchmark for the interest rate of marketability and pricing of national debt futures.

Following Bandi and Philips [

We find that the density of the process is bimodal. Two regime model could be better to capture the interest

Parameters | ||||
---|---|---|---|---|

Estimation | 0.0089 | 0.000034 | 0.8925 | 1.0009 |

Standard error | 0.008 | 0.00018 | 0.1 | 0.002 |

Parameters | ||||

Estimation | 0.00033 | 0.0000048 | 0.8783 | 0.9224 |

Standard error | 0.00001 | 0.000000003 | 0.0254 | 0.029 |

This table presents the result of two regime model and the data generating process is:_{t}, follows a two-state Markov chain model with

rates of China. Based on the evidence of local time of sub-sample data, we estimate the parameters and examine the properties of two-regime model. Using functional nonparametric method, we test the Vasicek model at different states. The short rates behave like a martingale in regime 2. We also calculate the probabilities that the process will stay in regime 1 and regime 2, and the probability that process will transfer from one state to another and the inference probability for a single date.

From our results, China’s recent interest rate stays in regime 2 in which the interest rate keeps at a low level with a high probability. Interest rate marketization of China will enable market forces to play a greater role in determining the allocation of credit, and economy will be more responsive to changes in rates. The liberalization of rates is a landmark change, and it represents another major milestone in China’s transformation to a market economy.

We acknowledge helpful comments from Chu Zhang, Jiang Luo, Jin Zhang and seminar participants at the Hong Kong University of Science and Technology. The research of this paper has been partially supported by grants from the Fundamental Research Funds for the Central Universities (Project No. 1209022), National Natural Science Foundation of China (Project No. 71303265, No. 71272201 and No. 71231008).