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. Figure 1 gives the changing of time series of the sample data.

From Figure 1, it is clear that the data has a different feature before and after 1999. Before 1999, interest rates stayed at a higher level, but they dropped dramatically after 1999. This is consistent with the change of term structure in Chinese money market. Figure 1 also shows the daily change (difference between the two successive days) of the spot rates. It also shows similar pattern with the daily data: interest rates become less volatile since 1999.

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 Table 1. Panel A presents

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. The first autocorrelation of whole sample is close to 1, and two subperiods have significant different means, standard deviations and skewness. After 1998, interest rates have a higher mean, higher positive skewness and lower volatility. With the market- lization of interest rates, the distribution of them may become more asymmetry because of the stochastic market. Panel B gives the summary statistics of daily change of repo rate. The first autocorrelation of the daily change is lower and negative with a negative and positive kurtosis.

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 Table 1, the skewness and kurtosis of repo rate and daily change rate are not consistent based on the Wilcoxon Rank Sum test in Panel E and F. This test is a nonparametric alternative to test whether the two samples have the same distribution when their distribution are not known. We find that the two subperiod data follow the different distributions with the different mean and variance. This means that the stationarity of the whole data process may not be guaranteed. Table 2 shows the result for the linear stationary test. The null hypothesis of a unit root was rejected at 5% level based on the augmented Dickey-Fuller test (ADF, see Harvey [16] ).

Figure 2 gives the frequency histogram of the whole data. The height presents the times that the repo rate appears in a small vicinity of a point. It is clear that there are two peaks in the figure at about 3% and 11%.

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 is a standard Brownian motion, and are the drift and diffusion of interest rate process respectively which depend on the values of the short rate and a state variable which has two states 1 and 2. Models such as interest rate models of Cox, Ingersoll and Ross (CIR) [17] , Vasicek [18] , Hull and White [19] are special cases of this model.

But parametric interest rate models may not fit historical data well. Ait-Sahalia [2] reject “every para- metric model of the spot rate [previously] proposed in the literature”. Jiang and Knight [4] also think that the

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:. Panel A reports the estimates of parameters with standard error and t-statistics. Panel B reports the test of unit root. The sample period is 1995.01-2003.12.

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 (day)

where is a standard normal process with zero mean and variance.

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 [11] mentioned, nonparametric estimation method firstly requires little prior information relating to the functional form of the conditional expectations, so it doesn’t need to estimate the type of the function as parametric estimation. Second, nonparametric estimators focus on local effects. This implies that the abnormal or very volatile sub-sample will not change any of the conclusion. The final advantage of nonparametric esti- mation method is that the estimators are feasible and easy to evaluate.

Based on the nonparametric model of Stanton [3] and econometric estimation, which is wildly used by Jiang [20] , Bandi [1] and Johannes [11] , we suppose that the short rate process follows one factor model, not consider- ing the state variable with observations of interest rates at, i.e.,. The model and data-generating process are the following:

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

The estimators of drift and diffusion terms are:

where is a Gaussian Kernel,

is the window width depending on the size and disperse of observations. Scott [21] suggest the window width

where is standard deviation of observations, is the number of observations and m is the dimension. The approximations converge to the true functions at a rate, where is the time between successive obser- vations and is an arbitrary positive integer.

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 [20] , Jiang and Knight [4] ), or stationarity which is assumed despite robustness to deviation from it (Stanton [3] ). So Bandi [1] proposes local time to describe the data. Based on our previous analysis, stationarity of the short rate process cannot be guaranteed, so we also use local time to grasp more information of data.

Bandi [1] uses new fully functional methods to exploit the spatial properties, embodied in the local time (classical references are Chung and Williams [22] ; karatzas and Shreve [23] ; Revuz and Yor [24] ) of interest rate which is robust against deviations from stationarity. Spatial densities and their functionals can be regarded as new descriptive tools for the series that are non-stationary or stationarity cannot be guaranteed, as in Bandi [1] which assume recurrence, a weaker assumption than the stationary condition.

Definition 1 If is a continuous semi-martingale, then exists a nondecreasing stochastic process (non- decreasing in, that is), called the chronological local time of at a. This process is defined, almost surely, as

This formula gives the amount of time in real time units that the process spends in the spatial neigh- borhood of a point. This spatial density assumes importance particularly when the underlying process is non- stationary, as they furnish the possibility of characterizing some of the features of the data, i.e., the location of the process. In fact, in the presence of non-stationarity, conventional descriptive statistics fail to provide reliable information given the tendency of the data to drift away from a particular point. So spatial densities can be regarded as new descriptive tools for series that are non-stationary or stationary cannot be guaranteed.

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 Figure 3. We report estimates from Equations (3) and (4) for, which cover the 99.6% of the data.

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 Figure 3. At lower interest rate levels, it has a lower variance. As interest rates go up, variance increases accordingly.

Local time gives the amount of time that the process spends in the vicinity of one point. Bandi [1] also derive the estimator of local time:

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 [1] suggest that the asymptotic 95% confidence interval for is given by

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

Figure 4 gives the plot of local time of the short rate of the entire data sample (2167 daily observations). The modes show up at around 3% and 11%. Given the features of the estimation procedure in Bandi [1] , we expect to be able to identify the functions of interest rate at points that are visited frequently. After a quick look at the

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 Figure 3, we can find that they are very similar. Therefore, the local time can be the approximation of density of the one path for the underlying process.

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. Figure 5 presents their local time estimation respectively.

We find that the two peaks in Figure 4 appear in Figure 5 separately. For the time horizon 1995.01-1998.12, the interest rates below 5% have a very low frequency. For the time horizon 1999.01-2003.12, because 98% of data is below 5%, local times for interest rates above 5% are close to zeros. These features provide the evidence to consider the effect of a state variable.

Figure 6 shows that the drift estimation using the non-parametric estimation method for the two subperiods 1995.01-1998.12 and 1999.01-2003.12 respectively. It can be seen that the drifts are very close to zeros for two subperiods, but other parts below 4% and above 14% for subperiod 1995.01-1998.12 are mean-reversion. It is surprising that for subperiod 1999.01-2003.12, mean-reversion speed is very low for 1% - 5%, especially from 2.5% - 5%, the drifts behave like a martingale. At the same time, from the corresponding local time figures, they have higher local time and cover more than 98% of subperiod data respectively.

This pattern appears again for their diffusion estimation in Figure 7. The corresponding variances over two subperiods are low and relative stable. The Monte Carlo 95% confidence bands are very close. This means that the data have a big change after 1999. Considering their different states, we use two-regime model to fit the data in the follows.

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

where is a stochastic state variable which satisfies:

When process equals to 1 at time, the interest rates stay at the state 1 with probability and the process follows the following model with probability:

When equals to 2, the interest rates stay at the state 2 with probability and the process follows the following model with probability:

We then obtain the estimation of the conditional probability based on our sample data. For example,

is estimated as from our discrete data in the below section. The transition pro-

babilities, , and are the the transition probabilities and respectively in the below.

Then from the Equation (10), we consider the model relying on the state variable. The probability will be estimated and plotted in the following sections based on our sample data and parameters estimated by. Then we assume the short rates follow one of interest rate models (Vasicek model here) with a probability relying on the short rate at time and and parameters.

Based on the previous analysis for drift and diffusion terms, we assume that and have the same form as the Vasicek model:

where j = 1 or 2 and, and mean that the process is in regime 1, which is also and, and mean regime 2 which is (, ,). The change in regimes is itself a random variable and unobservable. A complete time series model would therefore include a description of the probability law governing the change from, and and, and.

Given the discrete data, the data generating process is:

where. So there is a relationship:

In our model, we only have two states and equals to 1 or 2. With the daily data, we then test the model in Equation (13). We assume that follows a normal distribution with mean and variance for

. Such a process is described as a two-state Markov Chain with transition probabilities which

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, and.

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 [25] , there is the maximum likelihood estimation from the observed data as the following:

where let be a vector containing all observations obtained through date T. Our first pro-

bability and begin from and.

We suppose virtually certainty from observations from regime j, so that equals to unity for those observations that came from regime j and equals to zero for those observations that came from other regimes.

Following the method of Hamilton [26] , the EM algorithm is:

where.

Because we know

and

Using the whole data sample (2052 observations), we calculate the smoothed probabilities . These smoothed probabilities are used in equations of EM algorithm to calculate the parameters. The estimated results are reported in Table 3.

It is known that and are 87.83% and 92.24%. From Equation (13), we find that our two-regime model is the following Vasicek model:

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 for a single date is obtained. A probabilistic inference in the form of

can be calculated for each date in the sample. The resulting series is plotted as a function of

in Figure 8.

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.