eviews Tutorial

How to Estimate ARCH models in EViews

ARCH model EVIews
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arch models

Overview

ARCH stands for: Autoregressive Conditional Heteroskedasticity. The model was introduced by R. Engle (1982), in his paper entitled “Autoregressive Conditional Heteroskedasticity with estimates of the variance of United Kingdom Inflation”, and,  it was was later expanded in 1986 by Bollerslev when he introduced GARCH models.

So far, we have focused on estimating the mean equation (i.e., trying to predict next quarters CPI using an ARIMA model). However, in this tutorial, we will focus on estimating the variance.

why are we interested in estimating the variance?

Most econometric models assume that the variance is constant over time (homoskedastic), however, the variance can change over time. For example, most equity markets suffer volatility over time. There are periods where the returns are stable, but some other periods of instability can arise. COVID-19 has recently been a factor that contributed to time series volatility. Consequently, asset holders are not only interested in understanding and estimating the mean of the returns, but also the variance.

In the image below, we can see the Toronto Stock Exchange Index and its returns. The beginning of 2020 has been a period of high volatility and instability.

ARCH Model JDEconomics
Note: Toronto Stock Exchange (TSX) and returns. Frequency: Daily data. Range: 2016-2021. Source: Yahoo Finance
volatility clustering

When we talk about arch models and volatility, the concept “volatility clustering” comes across. But, what is volatility clustering? Volatility clustering was first observed by Mandelbrot (1963), when he indicated that small changes tend to be followed by small changes, while large changes tend to be followed by large changes. Consequently, volatility clustering refers to periods of high volatility which are  followed by periods of high volatility (and viceversa).

arch models formalities

Part 1: Mean equation

ARCH models are estimated in two steps. First, you need to estimate the mean equation (1), which in this case is an AR model. Please note that you can estimate an ARIMA model as well (in such case, you can add MA components to equation (1)).

 y_t=\theta_0 + \theta_1y_{t-1}+...+\theta_ny_{t-n}+ \epsilon_t ~~~(1)

The error term, is conditional on past information. As you can see in equation (2), the errors follow a normal distribtion with mean 0, but the variance depends on time. Furthermore, the term “h”, stands for heteroskedasticity. We are asuming that the variance is not constant over time and changes depending on previous information.

  \epsilon_t|{I_{t-1}}~N(0,h_t) ~~~(2)
Part 2: Variance equation

Finally, you need to estimate the variance equation (3). What does the variance equation mean? R.Engle (1982), specified that the variance is explained by a constant and the past squared residuals. In other words, we are saying that the variance is conditional on previous information.

  h_t=\alpha_0+\alpha_1\epsilon^2_{t-1}+...+\alpha_n\epsilon^2_{t-n} ~~~(3)

arch models requirements

Mean Equation

The mean equation can be estimated using an AR, MA, ARMA or ARIMA model. As we know from ARIMA models, the variable of interest has to be stationary. If our variable in levels is not stationary, we need to apply the corresponding transformations (logs, differences or both)

Variance Equation

Once we estimated the model, the variance equation has to satisfy the following requirements:

 1)~\alpha_0~ \&~ \alpha_i>0 to guarantee a positive variance
 2)~\alpha_1> \alpha_2>...>\alpha_n implies that the recent information has more influence/weight than older information. In other words, something that happened yesterday should have more influence than something that happened some years ago.
 3)~ 0\leq\sum_{i=1}^{i=n} \alpha_i\le1 to ensure the errors return to its mean (avoids explosive results).

arch models diagnostics

We have estimated our ARCH model. Now what?

Significance of ARCH effects

We need to ensure that the ARCH components we included in the model are statistically significant (i.e., at the 5% significance level, the p-vale needs to be small than 0.05 for the arch component to be significative).

autocorrelation

The model needs to exhibit no autocorrelation. The correlogram of the model should be white noise (no significat AC or PACF terms). If you are using EViews, the last column of the correlogram shows the P-Value for the Ljung-Box Q-Statistic test.

Ljung Box (Q) Null hypothesis: Residuals are white noise.

Note: if P<0.05, residuals are not white noise. You model suffers from autocorrelation

Heteroskedasticity

After estimating the ARCH model, you should test again for heteroskedasticity. If your model still shows heteroskedasticity, that’s a solid indicator that your model still needs to incorporate more ARCH components.

Possible Solution: Review the correlogram of the squared residuals and see if you can identify more ARCH components to include in the model.

Recommended literature

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