17.3 Nonparametric GARCH Estimates

In the following, we consider nonparametric GARCH(1,1) models which depend symmetrically on the last observation:

$\displaystyle \varepsilon _t$ $\displaystyle =$ $\displaystyle \sigma _t Z_t,$ (17.5)
$\displaystyle \sigma_t^ 2$ $\displaystyle =$ $\displaystyle g(\varepsilon _{t-1}^ 2 , \sigma_{t-1}^ 2 )\,.$  

Here, $ g$ denotes a smooth unknown function and the innovations $ Z_t$ are chosen as in as in Section 17.2. This model covers the usual parametric GARCH(1,1) process (17.1) but does not allow for representing a leverage effect like the TGARCH(1,1) process. We show now how to transform (17.5) into an ARMA model. First, we define

$\displaystyle X_t = \log (\varepsilon _t^ 2),\ e_t = \log (Z_t^ 2). $

By (17.5), we have now
$\displaystyle X_{t+1}$ $\displaystyle =$ $\displaystyle \log (\varepsilon _{t+1}^ 2) = \log \sigma _{t+1}^ 2 +
e_{t+1}$  
  $\displaystyle =$ $\displaystyle \log g (\varepsilon _t^ 2 , \sigma_t^ 2) + e_{t+1}$  
  $\displaystyle =$ $\displaystyle \log g_1 (\log (\varepsilon _t^ 2), \log (\sigma _t^ 2)) + e_{t+1}$  
  $\displaystyle =$ $\displaystyle \log g_1 (X_t, X_t - e_t) + e_{t+1}$  
  $\displaystyle =$ $\displaystyle f(X_t, e_t) + e_{t+1}$  

with

$\displaystyle g_1 (x,u) = g(e^ x, e^ u),\quad f(x,v) = \log g_1 (x, x-v).$

Now, we can estimate the ARMA function $ f(x,v)$ from the logarithmic squared data $ X_t = \log (\varepsilon _t^ 2)$ as in Section 17.3 using the nonparametric ARMA estimate $ \widehat{f}_{b,h,h^*} (x,v)$ of (17.5). Reverting the transformations, we get

$\displaystyle \widehat{g}_1 (x, u) = \exp \{ \widehat{f}_{b,h,h^*} (x,x - u)\}, \, \, \widehat{g}_{b,h,h^
*} (y,z) = \widehat{g}_1 (\log y, \log z) $

or, combining both equations,

$\displaystyle \widehat{g}_{b,h,h^
*} (y,z) = \exp \left\{ \widehat{f}_{b,h,h^*} \left(\log y, \log (y/z)
\right) \right\}, \quad y,
z > 0 , $

as an estimate of the symmetric GARCH function $ g(y,z).$

We have to be aware, of course, that the density $ p_e$ used in the deconvolution part of estimating $ f(x,v)$ is the probability density of the $ e_t = \log Z_t^ 2 $ , i.e. if $ p_z (z) $ denotes the density of $ Z_t,$

$\displaystyle p_e (u) = \frac{1}{2} \, \left\{ e^ {u/2} p_z (e^ {u/2}) +
e^ {- u/2} p_z ( e^{- u/2} ) \right\} . $

If $ \varepsilon _t$ is a common parametric GARCH(1,1) process of form (17.1), then $ g(y,z) = \omega + \alpha y + \beta z,$ and the corresponding ARMA function is $ f(x,v) = \log (\omega + \alpha e^ x + \beta e^ {x-v}).$ This is a decreasing function in $ v$ which seems to be a reasonable assumption in the general case too corresponding to the assumption that the present volatility is an increasing function of past volatilities.

As an example, we simulate a GARCH process from

  proc(f)=gf(x,e,c)
    f=c[1]+c[2]*x+c[3]*e 
  endp

  proc(e,s2)=mygarch(n,c)
    e=zeros(n+1)
    f=e
    s2=e
    z=normal(n+1)
    t=1
    while (t<n+1)
      t=t+1
      s2[t]=gf(e[t-1]^2,s2[t-1]^2,c)
      e[t]=sqrt(s2[t]).*z[t]
    endo
    e=e[2:(n+1)]
    s2=s2[2:(n+1)]
  endp



f = 33731 npgarchest (x {,h {,g {,N {,R } } } } )
estimates a nonparametric GARCH process


The function 33734 npgarchest computes the functions $ f(x,v)$ and $ g(y,z)$ for a GARCH process using the techniques described above. Consider a GARCH(1,1) with

$\displaystyle g(y,z) = 0.01 + 0.6\,y + 0.2\,z.$

Hence, we use
  n=1000
  c=0.01|0.6|0.2
  {e,s2}=mygarch(n,c)
and call the estimation routine by
  g=npgarchest(e)
33738 XFGnpg05.xpl

Figure 17.5 shows the resulting graph for the estimator of $ f(x,v)$ together with the true function (decreasing in $ v$) and the data ($ X_{t+1}$ versus $ X_t$). As in the ARMA case, the estimated function shows the underlying structure only for a part of the range of the true function.

Figure 17.5: Nonparametric estimation of $ f(x,v)$ for a (linear) GARCH process. True vs. estimated function, data $ X_t = \log (\varepsilon _t^ 2)$.
\includegraphics[width=1.25\defpicwidth]{XFGnpg05.ps}

Finally, we remark how the the general case of nonparametric GARCH models could be estimated. Consider

$\displaystyle \varepsilon _t$ $\displaystyle =$ $\displaystyle \sigma_t Z_t$ (17.6)
$\displaystyle \sigma_t^ 2$ $\displaystyle =$ $\displaystyle g( \varepsilon _{t-1}, \sigma_{t-1}^ 2 )$  

where $ \sigma_t^ 2$ may depend asymmetrically on $ \varepsilon_{t-1}.$ We write

$\displaystyle g(x,z) = g^ + (x^ 2, z) \, \boldsymbol{1}(x\geq 0) + g^ - (x^ 2, z)\,\boldsymbol{1}(x<0). $

As $ g^ +, g^ -$ depend only on the squared arguments we can estimate them as before. Again, consider $ X_t = \log (\varepsilon _t^ 2) , e_t = \log
(Z_t^ 2).$ Let $ N_+$ be the number of all $ t \le N$ with $ \varepsilon _t \ge 0,$ and $ N_- = N - N_+.$ Then, we set

$\displaystyle \widehat{p}^+_b (x)$ $\displaystyle =$ $\displaystyle \frac{1}{N_+b} \, \sum^ N_{t=1} K^ x (\frac{x-X_t}{b}) \boldsymbol{1}(\varepsilon_t \geq 0)$  
$\displaystyle \widehat{q}^+_{b,h} (u\vert x)$ $\displaystyle =$ $\displaystyle \frac{1}{N_+hb} \, \sum^ N_{t=1} K^h \left(
\frac{u-X_{t+1}}{h}\r...
..._t}{b} \right) \boldsymbol{1}(\varepsilon_t \geq 0)\ \big/\ \widehat{p}^+_b
(x)$  
$\displaystyle \widehat{q}^+_{h^*} (u)$ $\displaystyle =$ $\displaystyle \frac{1}{N_+h^*} \, \sum^ N_{t=1} K_{h^*} \left( \frac{u-X_t}{h^*}
\right) \boldsymbol{1}(\varepsilon_t \geq 0).$  

$ \widehat{Q}_{b,h} ^ + (v\vert x),\ \widehat{P}_{e,h^ *}^ + (v\vert x)$ are defined as in Section 17.2 with $ \widehat{q}^ +_{b,h}, \widehat{p}^+_b$ replacing $ \widehat{q}_{b,h}$ and $ \widehat{p}_b$, and, using both estimates of conditional distribution functions we get an ARMA function estimate $ \widehat{f}^ +_{b,h,h^ *} (x,v).$ Reversing the transformation from GARCH to ARMA, we get as the estimate of $ g^ + (x^ 2, z) $

$\displaystyle \widehat{g}^ +_{b,h,h^ *} (x^ 2, z) = \exp \left\{ \widehat{f}^ +_{b,h,h^ *}
\left(\log x^ 2,\log (x^2/z)\right) \right\} . $

The estimate for $ g^ - (x^ 2, z)$ is analogously defined

$\displaystyle \widehat{g}^ -_{b,h,h^ *} (x^ 2, z) = \exp \left\{ \widehat{f}^ -_{b,h,h^ *}
\left(\log x^ 2,\log (x^2/z)\right) \right\} . $

where, in the derivation of $ \widehat{f}^ -_{b,h,h^ *}, \, N_+ $ and $ \boldsymbol{1}(\varepsilon _t \geq 0)$ are replaced by $ N_- $ and $ \boldsymbol{1}(\varepsilon _t<0). $