# How to randomly generate a data with an error which is not normally distributed but follows an empirical distribution $(0, \sigma^2)$

I have this model AR model for multiple time series

$Y_{it} = \phi y_{it-1} + \delta_1y_{i-1 ,t-1} + \delta_2 y_{i+1 , t-1} + \lambda_i + \epsilon_{it}$

where

$\epsilon_{it}$ is a function of nonparametric realized volatility.

$\phi$ is the autoregressive coefficient common to all

$\lambda$ is the individual random part or the unique characteristic of the $ith$ assets/stocks

$\delta_1$ and $\delta_2$ are the some coefficient of previous and future asset/stocks the investor wanted to invest.

The $\phi$ , $\lambda$ and $\delta$ is assumed normal

The motivation for this model model is the stock market.

My problem is how to generate a random data where the error ($\epsilon$) is not normally distributed but under empirical distribution of $(0, \sigma^2)$. How would I implement this in R to meet all the constrains and to really characterized my desired generated data which is needed for estimation part.

I really need some help.

• What do you mean by "empirical distribution"? If you want to sample from empirical distribution use bootstrap. – Tim Nov 16 '17 at 22:42
• before I go for bootstrapping, I need data first and I have to generate a data that would characterize my model. Empirical cumulative distribution function that is approaching to normal but not normal. Most of the codes I know, the error are all assumed normal or have some specific distribution that is common to us, but what if the error is not distributed normally but approaching Normal but not really normal. I don't know how to implement this in R. – Remly Nov 17 '17 at 20:23
• It is not clear what do you mean by "empirical distribution". Empirical distribution is the distribution of the data, so if you don't have any data, then there is no empirical distribution of it. So it seems that you need to make some distributional assumptions (e.g. normal). There is no "empirical distribution" without data. – Tim Nov 17 '17 at 21:11
• I want to generate a data that don't have a distribution. the error term is non-parametric. ithas a $(0, \sigma^2)$ but no distribution. not normal not any. – Remly Nov 17 '17 at 23:15
• Would that be possible? – Remly Nov 17 '17 at 23:15