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# System prepended metadata

title: Introduction

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# Introduction

We intend to give a review of [Starica2003] in this note. 

The main theme of the paper, in my point of view, is to argue that 

1. GARCH(1,1) model has no statistically significant difference than a nonparametric nonstationary model proposed in the paper. 

2. A simple moving average prediction method is superior to GARCH(1,1) model in terms of long-term volatility prediction, and better produces some stylized facts observed in real-world financial market. 

3. With a high statistical confidence, the volatility of real-world financial products, such as S&P 500 or Dow Jones index, cannot be considered statinary in the sense of GARCH(1,1) model. 

# A nonparametric nonstationary model

The paper proposed a model that is later shown to be statistically equivalent to GARCH(1,1) in terms of the ACF of the fitted residuals. 

## Model

## Estimation Method

## Result


# Methodology of Prediction

## Criterion

The aim is to, at each time $t$, predict the realized volatility of future $p$ steps, defined as: 

$$\bar{r}_{t,p}^2 := \sum_{i=1}^{p} r_{t+i}^2$$

(note that there is a typo in the original paper).

The criterion is mean-squared error (MSE) of the prediction and the realized volatility defined as: 

$$MSE^*(p) := \sum_{t=1}^n(\bar{r}_{t,p}^2 - \bar{\sigma}_{t,p}^{2, *})^2$$

where $n$ is the total time horizon and  $\bar{\sigma}_{t,p}^{2, *}$ the prediction based on either of the methods detailed below: 

## GARCH(1,1)

$$r_t = z_t h_t^{1/2}$$

$$h_t = \alpha_0 + \alpha_1 r^2_{t-1} + \beta_1 h_{t-1}$$

....

## Simple Moving Average

$$\bar{\sigma}_{t,p}^{2, BM} := \frac{p}{250}{\sum_{i=1}^{250}{r^2_{t-i+1}}}$$

## Results

![](https://i.imgur.com/Y7sV4pT.png)


# References

[Starica2003]: Starica C. Is GARCH (1, 1) as good a model as the accolades of the Nobel prize would imply?. Available at SSRN 637322, 2003.