Why is ordering values necessary for nonparametric regression

Question

Suppose we are want to consider the asset price $P_t$ (business daily) of some stock. The log return is defined as

$$X_t:=\log{\frac{P_t}{P_{t-1}}}$$

suppose we are considering the prices between some interval, eg. June 1986 and March 1990). Log returns can be modelled by $X_t=\sigma_t\epsilon_t$, with $E[\epsilon_t]=0,Var(\epsilon_t)=1$. We assume Markov-property, thus $\sigma^2_t=v(X_{t-1})$. The model can be fitted by nonparametric regression of the function $v$ in

$$Z_t:=X^2_t=v(X_{t-1})+\phi_t$$ where $\phi_t=\sigma^2_t(\epsilon^2_t-1)$ can be seen as error. We want to use ksmooth, smooth.spline and loess. In my notes they say: "First we sort the values, in order not to get problems with ksmooth, which orders the values internally and gives back results corresponding to the ordered values."

I do not understand why we have to do that. Basically we have a function $v$, with values on the $y$-axis and the $x$-axis is our time interval between June 1986 and March 1990. So why do we have to order the data?

Answer 1

It is an implementation detail. ksmooth is documented to return the points at which the smooth was evaluated in increasing order of x, the input data.

Value:

     A list with components

       x: values at which the smoothed fit is evaluated. Guaranteed to
          be in increasing order.

       y: fitted values corresponding to ‘x’.

Note also that what ksmooth() returns is a vector of n.points points, evenly spread across the range of x (unless argument x.points is supplied during fitting), not the evaluation of the smoother at the input location x.

Read ?ksmooth for more details.