1. Vera Tabakova, East Carolina University
An Introduction to Financial Econometrics: Time-Varying Volatility and ARCH Models
Modified JJ
Vera Tabakova, East Carolina University

2. 14.2 Time-Varying Volatility 14.3 Testing, Estimating and Forecasting
Chapter 14: An Introduction to Financial Econometrics: Time-Varying Volatility and ARCH Models
14.1 The ARCH Model
14.2 Time-Varying Volatility
14.3 Testing, Estimating and Forecasting
14.4 Extensions
Below y denotes a return series. Returns are assumed: uncorrelated in levels, with a constant expected value beta_0 , and autocorrelated in squared values:
Principles of Econometrics, 3rd Edition

3. 14.1 The Arch Model
The noise is serially uncorrelated, has marginal mean 0 and marginal variance sigma square which is constant. Its marginal distribution is left unspecified. We’ll see later that it is not Normal.
The return y_t has the same marginal distribution, only with nonzero mean beta_0. This means that expected return is constant and equal to beta_0.
(14.1a) E
(14.1b)
Principles of Econometrics, 3rd Edition

4. 14.1 The Arch Model (14.2a) (14.2b) (14.2c)
Moreover, as we see here the noise has a conditional Normal distribution given its own past.
Its conditional mean is constant and, like its marginal mean, it is 0. Its conditional variance h_t is time varying: h_t has ARCH(1) dynamics.
(14.2a)
(14.2b)
(14.2c)
Principles of Econometrics, 3rd Edition

5. The positivity conditions are needed to make sure the conditional variance h_t stays positive.
The marginal and conditional variance of (demeaned) returns are related
Principles of Econometrics, 3rd Edition

6. 14.1.1 Conditional and Unconditional Moments
Conditional expected return=marginal expected return:
E(y_t| I_t-1) =E(y_t) = beta_0
Conditional variance of returns is time varying h_t:
E[(y_t-beta_0) ^2 | I_t-1]= h_t
Principles of Econometrics, 3rd Edition

7. 14.1.1 Conditional and Unconditional Moments
Marginal expected return :
E(y_t) = beta
Marginal variance of returns:
E[(y_t-beta)^2] = sigma = alpha_0/(1-alpha_1)
Principles of Econometrics, 3rd Edition

8. 14.2 Time Varying Volatility
Figure 14.1 Examples of Returns to Various Stock Indices
Principles of Econometrics, 3rd Edition

9. 14.2 Time Varying Volatility
Figure 14.2 Histograms of Returns to Various Stock Indices
Principles of Econometrics, 3rd Edition

10. 14.2 Time Varying Volatility
Figure 14.3 Simulated Examples of Constant and Time-Varying Variances
Principles of Econometrics, 3rd Edition

11. 14.2 Time Varying Volatility
Figure 14.4 Frequency Distributions of the Simulated Models
Principles of Econometrics, 3rd Edition

12. Empirical properties of returns
When returns are plotted over time, they display volatility clustering. Volatility varies over time( if you calculate variance from subsamples, you will get different results)
The marginal distributions of returns have fat tails (kurtosis>3)-because they admit more extreme values than a Normal variable
The marginal distributions of returns are often skewed
Principles of Econometrics, 3rd Edition

13. 14.3 Testing, Estimating and Forecasting
Testing for ARCH effects
You first estimate beta by regressing returns on a constant. You get hat(beta). Next, compute the OLS residual as y_t- hat(beta). Square the residual, regress on its lag
Testing for ARCH effects
(14.3)
Principles of Econometrics, 3rd Edition

14. 14.3 Testing, Estimating and Forecasting
Figure 14.5 Time Series and Histogram of Returns
Principles of Econometrics, 3rd Edition

15. 14.3.2 Estimating ARCH Models
This is a one-step estimation . On this and the next slides, the fitted returns are denoted by r_t hat
(14.4a)
(14.4b)
Principles of Econometrics, 3rd Edition

16. 14.3.3 Forecasting Volatility
Here (r_t-hat(beta_)) is the residual or demeaned return
(14.5a)
(14.5b)
Principles of Econometrics, 3rd Edition

17. 14.3.3 Forecasting Volatility
Figure 14.6 Plot of Conditional Variance
Principles of Econometrics, 3rd Edition

18. 14.4 Extensions
In many applications, ARCH(1) with one lag is insufficient and q lags are used instead
ARCH(q) captures the volatility persistence (clustering), in the sense large squared returns are followed by large squared returns and small are followed by small.
In many applications q tends to be very large. Hence instead of a long ARCH(q), estimate the GARCh(p,q) given next.
(14.6)
Principles of Econometrics, 3rd Edition

19. 14.4.1 The GARCH Model - Generalized ARCH
(14.7)
Principles of Econometrics, 3rd Edition

20. 14.4.1 The GARCH Model - Generalized ARCH
Principles of Econometrics, 3rd Edition

21. 14.4.1 The GARCH Model - Generalized ARCH
Figure 14.7 Estimated Means and Variances of Various ARCH Models
Principles of Econometrics, 3rd Edition

22. 14.4.2 Allowing for an Asymmetric Effect
GARCH(p,q) disregards the empirical fact that large negative returns increase volatility more than large positive returns:
(14.8)
Principles of Econometrics, 3rd Edition

23. 14.4.2 Allowing for an Asymmetric Effect
Principles of Econometrics, 3rd Edition

24. 14.4.3 GARCH-in-Mean and Time-varying Risk Premium
This model is motivated by the CAPM: high- risk assets have higher expected return than low-risk assets.
Therefore conditional variance (volatility) appears in the conditional expectation of returns.
(14.9a)
(14.9b)
(14.9c)
Principles of Econometrics, 3rd Edition

25. 14.4.3 GARCH-in-Mean and Time-varying Risk Premium
Principles of Econometrics, 3rd Edition

26. Keywords ARCH Conditional and Unconditional Forecasts
Conditionally normal
GARCH
ARCH-in-mean and GARCH-in-mean
T-ARCH and T-GARCH
Time-varying variance
Principles of Econometrics, 3rd Edition