KYIV SCHOOL OF ECONOMICS

Financial Econometrics (2nd part):

Introduction to Financial Time Series

May 2011

Instructor: Maksym Obrizan

Lecture notes II

# 2. This lecture:

(i) Very brief summary of ARCH-GARCH and their shortcomings

(ii) A few more advanced models (TAR, MSA)

Sometimes series {rt} may be with no or minor serial correlation but it is still dependent…

Log of Intel stock returns from January 1973 to December 1997 on the bottom left slide # 3 and of squared Intel returns on slide # 4

# 3. # 4.

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ACF for Returns of Intel

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ACF for Squared Returns of Intel

# 5. # 6. Series are uncorrelated but dependent:

volatility models attempt to capture such dependence

Define shock or mean-corrected return

# 7. Then ARCH(m) model assumes

In practice, the error term is assumed to follow the standard normal of a standardized Student-t distribution

# 8. Practical way of building an ARCH model

(i) Build an ARMA model for the return series to remove any linear dependence in data

If the residual series indicates possible ARCH effects – proceed to (ii) and (iii)

(ii) Specify the ARCH order and perform estimation

(iii) Check the fitted ARCH model for necessary refinements

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ACF for Modulus Returns of Intel

# 9. Fitting an ARCH Model:

Model Checking:

Obtain standardized shocks

Then use Ljung-Box statistic on to check the adequacy of the mean equation and on to check the validity of the volatility equation.

Use kurtosis, skewness and QQ-plot of to check if normal distribution is applicable

# 10. Shortcomings of ARCH model

(i) Positive and negative shocks have the same effects on volatility

MOTIVATION FOR TAR, MSA MODELS!

(ii) ARCH model is restrictive – parameters are constrained by certain intervals for finite moments etc

(iii) Sometimes not parsimonious models: use GARCH

# 11. GARCH model:

Weaknesses of GARCH model are similar to those of ARCH: symmetric response to negative and positive shocks etc

# 12. NOTES

# 13. Application: daily log returns of IBM stock from July 3, 1962 to December 31, 1999

All the estimates (except the coefficient of rt-2) are highly significant

# 14. In addition, the Ljung-Box statistics of the standardized residuals is Q(10) = 11.31 (p-value of 0.33) and of the squared standardized residuals is Q(10) = 11.86 (p-value of 0.29)

# 15. The unconditional mean of rt in this model is

while in the sample it is only 0.045.

What if the model is misspecified?

Motivation for nonlinear models such as TAR and MSA

# 16. Threshold Autoregressive (TAR) model

# 17. Consider a simple two-regime TAR model

# 18.

# 19. Observe that this model has coefficient -1.5

However, despite this fact it is stationary and geomertically ergodic if

Ergodic theorem – statistical theorem showing that the sample mean of xt converges to the mean of xt

# 20. Model behavior depends on xt -1:

When it is negative then

When it is positive then

Question: Which regime will have more observations?

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# 21. In addition, TAR model has non-zero mean even though the constant terms are zero (think of an AR(m) model with zero constant for a comparison)

Re-consider slide # 15 with AR(2)-GARCH(1,1) model of IBM stock: the unconditional mean of 0.065 overpredicted the sample mean of 0.045

Estimate AR(2)-TAR-GARCH(1,1) model and refine it (remove insignificant term in volatility equation)

# 22. AR(2)-TAR-GARCH(1,1) of IBM stock

# 23. Model fit

All coefficients are significant at 5%

The unconditional mean?

The Ljung-Box statistics applied to standardized residuals does not indicate serial correlations or conditional heteroscedasticity

# 24. NOTES:

# 25. Convenient to re-write TAR-GARCH(1,1): # 26. Recall the integrated GARCH model (IGARCH is a unit-root GARCH model)

For example, IGARCH(1,1) is defined as

The unconditional variance of at, and thus of rt, is not defined

Meaning: Occasional level shifts in volatility?

IGARCH(1,1) with is used in RiskMetrics (Value at Risk calculating)

# 27. Thus, under nonpositive deviation the volatility follows an IGARCH(1,1) model without a drift

With positive deviation, the volatility has a persistent parameter 0.046+0.885=0.931 which is <1 giving rise to GARCH(1,1)

Conclusion:

# 28. NOTES:

# 29. Markov Switching Model # 30. Application to the US quarterly real GNP

# 31. Cont’d # 32. Notes

# 33. Nonlinearity tests: Parametric tests

The RESET Test for a linear AR(p) model

Basic idea: if a linear AR(p) model is adequate then a1 and a2 should be zero.

# 34. Apply F statistic

with g and T-p-g degrees of freedom

# 35. Nonlinearity tests: Nonparametric tests

Q-statistic of Squared Residuals

The null hypothesis of the statistic is

# 36. NOTES

# 37. Application to the US quarterly civilian unemployment from 1948 to 1993 based on Montgomery, Zarnowitz, Tsay and Tiao (1998)

# 38. TAR model

# 39. MSA model # 40. NOTES