Editing Multiplicative function (section)

== Multivariate ==
[[Multivariate function]]s can be constructed using multiplicative model estimators. Where a matrix function of {{math|1=''A''}} is defined as <math display="block">D_N = N^2 \times N(N + 1) / 2</math>

a sum can be [[logarithmic distribution|distributed]] across the product<math display="block">y_t = \sum(t/T)^{1/2}u_t = \sum(t/T)^{1/2}G_t^{1/2}\epsilon_t</math>

For the efficient [[estimation]] of {{math|1=Σ(.)}}, the following two [[nonparametric regression]]s can be considered: <math display="block">\tilde{y}^2_t = \frac{y^2_t}{g_t} = \sigma^2(t/T) + \sigma^2(t/T)(\epsilon^2_t - 1),</math>

and <math display="block">y^2_t = \sigma^2(t/T) + \sigma^2(t/T)(g_t\epsilon^2_t - 1).</math>

Thus it gives an estimate value of <math display="block">L_t(\tau;u) = \sum_{t=1}^T K_h(u - t/T)\begin{bmatrix} ln\tau + \frac{y^2_t}{g_t\tau} \end{bmatrix}</math>

with a local likelihood function for <math>y^2_t</math> with known <math>g_t</math> and unknown <math>\sigma^2(t/T)</math>.