12.9 Simulation Study and Illustration

We investigate our testing procedure in two simulation studies. In our first simulation we consider the time series model

$\displaystyle Y_i=2 Y_{i-1}/(1+Y_{i-1}^2) + c_n \sin(Y_{i-1}) + \sigma(Y_{i-1}) \eta_i$

where $ \{\eta_i\}$ are independent and identically distributed uniform random variables in $ [-1,1]$, $ \eta_i$ is independent of $ X_i=Y_{i-1}$ for each $ i$, and $ \sigma(x)=exp(-x^2/4)$. Note that the mean and the variance functions are both bounded which ensures the series is asymptotically stationary. To realize the stationarity, we pre-run the series 100 times with an initial value $ Y_{-100}=0$. The empirical likelihood test statistic is calculated via the 25427 elmtest quantlet.


{el,p,kn,h2} = 25437 elmtest (x,y,model{,kernel{,h{,theta}}})
calculates the empirical likelihood test statistic

The first and the second parameter are the vectors of observations of $ X$ and $ Y$. The third parameter model is the name of a quantlet that implements the parametric model for the null hypothesis. The optimal parameter kernel is the name of the kernel $ K$ that is used to calculate the test statistic and h is the bandwidth used to calculate $ \bar{U}_1$ and $ \bar{U}_2$ in (12.18). theta is directly forwarded to the parametric model.

25441 XFGelsim1.xpl

For the simulation study the sample sizes considered for each trajectory are $ n=500$ and $ 1000$ and $ c_n$, the degree of difference between $ H_0$ and $ H_1$, takes value of 0, $ 0.03$ and $ 0.06$. As the simulation shows that the two empirical likelihood tests have very similar power performance, we will report the results for the test based on the $ \chi^2$ distribution only. To gauge the effect of the smoothing bandwidth $ h$ on the power, ten levels of $ h$ are used for each simulated sample to formulate the test statistic.

Figure: Power of the empirical likelihood test. The dotted lines indicate the 5% level
\includegraphics[width=0.7\defpicwidth]{XFGELfigure3a.ps} \includegraphics[width=0.7\defpicwidth]{XFGELfigure3b.ps}

Figure 12.3 presents the power of the empirical likelihood test based on 5000 simulation with a nominal 5% level of significance. We notice that when $ c_n=0$ the simulated significance level of the test is very close to the nominal level for large range of $ h$ values which is especially the case for the larger sample size $ n=1000$. When $ c_n$ increases, for each fixed $ h$ the power increases as the distance between the null and the alternative hypotheses becomes larger. For each fixed $ c_n$, there is a general trend of decreasing power when $ h$ increases. This is due to the discretization of $ H_0$ by binning as discussed at the end of the previous section. We also notice that the power curves for $ c_n=0.06$ are a little erratic although they maintain the same trend as in the case of $ c_n=0.03$. This may be due to the fact that when the difference between $ H_0$ and $ H_1$ is large, the difference between the nonparametric and the parametric fits becomes larger and the test procedure becomes more sensitive to the bandwidths.

In our second simulation study we consider an Ornstein-Uhlenbeck process $ Z$ fluctuating about 0 that satisfies the stochastic differential equation

$\displaystyle dZ(t) = a Z(t) dt + \sigma d W(t) $

where $ W$ is a standard Brownian Motion. The speed of adjustment parameter $ a$ has to be negative to ensure stationarity. To apply the empirical likelihood test we construct the time series $ X$ and $ Y$ as in Section 12.2, i.e.
$\displaystyle X_i$ $\displaystyle =$ $\displaystyle Z^{\Delta}(t_{i}) \; , \qquad X = (X_1, \ldots, X_n)$  
$\displaystyle \varepsilon_i$ $\displaystyle =$ $\displaystyle W(t_{i+1}) - W(t_i) \; ,
\qquad \varepsilon = (\varepsilon_1 , \ldots , \varepsilon_n)$  
$\displaystyle Y_i$ $\displaystyle =$ $\displaystyle X_{i+1} - X_i
= a X_i \Delta
\ + \sigma \varepsilon_i \; , \qquad
Y = (Y_1, \ldots, Y_n)$ (12.33)

It is well known that the transition probability of an Ornstein-Uhlenbeck process is normal with conditional mean

$\displaystyle \textrm{E}[Z_{t+\Delta} \vert Z_{t} = x] = \textrm{E}[X_{i+1} \vert X_i = x] = x e^{a \Delta}$    

and conditional variance

$\displaystyle \textrm{Var}(Z_{t+\Delta} \vert Z_{t} = x) = \textrm{Var}(X_{i+1}...
...rt X_i = x) = \frac{\gamma^2}{-2 \beta} \left( e^{-2 \beta \Delta} -1 \right) .$    

To simulate the process we use the 25452 simou quantlet.


x = 25462 simou (n,a,s,delta)
simulates a discretely observed path of an Ornstein-Uhlenbeck process via its transition probability law.

The number of observations is given by n+1/, a is the speed of adjustment parameter $ a$, s is the diffusion coefficient $ \sigma$ and delta is the time difference $ \Delta$ between two observations.

The proposed simulation procedure and the Goodness-of-Fit test are illustrated in 25465 XFGelsim2.xpl .

25469 XFGelsim2.xpl

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