6. ARCHGARCH

Question 1

How are ARCH effects tested using regression?

Answer 1

ARCH effects are tested by regressing squared residuals on lagged squared residuals: ε̂²t = a0 + Σ(ai * ε̂²t-i) + νt for i = 1 to q, and assessing the significance of coefficients ai.

Question 2

How can the variance persistence in a GARCH(1,1) model be evaluated?

Answer 2

Variance persistence can be evaluated by the sum of α1 and β1 from the GARCH(1,1) model, σ²t = ω + α1 * ε²t-1 + β1 * σ²t-1. A sum close to 1 indicates high persistence.

Question 3

How can you interpret the chi-square test statistic in the context of testing for ARCH effects?

Answer 3

The chi-square test statistic χ² = T * R² tests the null hypothesis that there are no ARCH effects (αi = 0). A significant χ² value indicates the presence of ARCH effects.

Question 4

How do GARCH models extend ARCH models?

Answer 4

GARCH models include lagged conditional variances in addition to lagged squared residuals, allowing for better modeling of volatility persistence.

Question 5

How do you interpret the R² value in the regression for testing ARCH effects?

Answer 5

R² indicates the proportion of variation in squared residuals explained by lagged squared residuals; higher R² suggests stronger ARCH effects.

Question 6

How does an ARCH(q) model extend the ARCH(1) model?

Answer 6

ARCH(q) includes multiple lagged squared residuals in the conditional variance, σ²t = ω + Σ(αi * ε²t-i) for i = 1 to q, to capture more complex dependencies.

Question 7

How does the EGARCH model differ from the standard GARCH model?

Answer 7

EGARCH models log(σ²t) and incorporate asymmetry, allowing for negative shocks to have a different effect than positive shocks.

Question 8

How does the GARCH(1,1) model formula differ from the ARCH(1) model formula?

Answer 8

GARCH(1,1) adds a β1 * σ²t-1 term to the ARCH(1) formula, allowing past variances to influence current conditional variance.

Question 9

How does the inclusion of β terms in a GARCH(p,q) model improve volatility modeling?

Answer 9

β terms incorporate past conditional variances, allowing the model to capture longer memory effects in volatility dynamics compared to ARCH models, which rely only on past residuals.

Question 10

How is the log-likelihood function for GARCH estimation expressed?

Answer 10

log(LT(θ)) = Σ(-1/2 * log(2π) - 1/2 * log(σ²t) - ε²t / (2σ²t)) from t = m+1 to T

Question 11

How is the variance of residuals in an ARCH(1) model calculated?

Answer 11

Var(εt) = ω / (1 - α)

Question 12

How is the variance of residuals in an ARCH(q) model calculated?

Answer 12

Var(εt) = ω / (1 - Σ(αi)) for i = 1 to q, provided Σ(αi) < 1.

Question 13

What are non-normal innovations, and why are they used in GARCH models?

Answer 13

Non-normal innovations, like t-distributions, capture heavy tails and skewness in financial return data, which normal distributions fail to model.

Question 14

What are the main assumptions of residuals in an ARCH(1) model?

Answer 14

Residuals are assumed to have zero mean, conditional variance that changes over time, and potential leptokurtosis, indicating heavy tails.

Question 15

What does ARCH stand for?

Answer 15

Autoregressive Conditional Heteroscedasticity.

Question 16

What does the AR term in AR(p)-ARCH(q) models represent?

Answer 16

The AR term models the mean structure of the time series, while ARCH(q) models the conditional variance.

Question 17

What does the GARCH(p,q) conditional variance formula include?

Answer 17

σ²t = ω + Σ(αi * ε²t-i) + Σ(βj * σ²t-j) for i = 1 to q and j = 1 to p

Question 18

What does the parameter ω represent in ARCH models?

Answer 18

ω represents the constant baseline level of variance in the conditional variance equation.

Question 19

What does the persistence parameter α1 + β1 indicate in a GARCH(1,1) model?

Answer 19

It indicates the degree of volatility persistence; values close to 1 suggest long-lasting volatility shocks.

Question 20

What does the term “volatility clustering” describe?

Answer 20

It describes the phenomenon where periods of high volatility tend to cluster together, followed by periods of low volatility.

Question 21

What does σ² represent in ARCH models?

Answer 21

The conditional variance of the residuals.

Question 22

What implications does leptokurtosis in ARCH residuals have for modeling financial returns?

Answer 22

Leptokurtosis indicates that ARCH models need to account for heavy tails, suggesting the potential use of non-normal innovations like t-distributions for better model fit.

Question 23

What is a key characteristic of volatility in ARCH models?

Answer 23

Volatility clustering, where high volatility periods are followed by high volatility, and low volatility by low volatility.

Question 24

What is the assumption about residuals in ARCH models?

Answer 24

Residuals are assumed to be conditionally heteroscedastic with a mean of zero and variance changing over time.

Question 25

What is the conditional variance formula for an ARCH(1) model?

Answer 25

σ²t = ω + α * ε²t-1

Question 26

What is the main difference between ARCH(1) and ARCH(q) models?

Answer 26

ARCH(1) uses only the most recent squared residual, while ARCH(q) uses a weighted sum of q lagged squared residuals to model conditional variance.

Question 27

What is the main purpose of ARCH models?

Answer 27

To model time-varying volatility in time series data.

Question 28

What is the null hypothesis in hypothesis testing for ARCH effects?

Answer 28

H0: αi = 0 for all i = 1 to q

Question 29

What is the primary goal of ARCH models in time series analysis?

Answer 29

To model and predict time-varying conditional variances in time series data, particularly for financial data with volatility clustering.

Question 30

What is the purpose of the chi-square test for ARCH effects?

Answer 30

To determine whether lagged squared residuals significantly affect current conditional variance, indicating ARCH effects.

Question 31

What is the significance of the sum Σ(αi) in an ARCH model?

Answer 31

It determines the stationarity of the process; if Σ(αi) < 1, the model is stationary.

Question 32

What type of data is ARCH commonly applied to?

Answer 32

Financial time series data, such as stock returns.

Question 33

Why are asymmetric GARCH models like EGARCH used over standard GARCH models?

Answer 33

Asymmetric GARCH models, such as EGARCH, account for leverage effects, where negative shocks have a larger impact on volatility than positive shocks of the same magnitude, which standard GARCH models cannot capture.

Question 34

Why are non-normal innovations used in GARCH models?

Answer 34

Non-normal innovations, like t-distributed or skewed-t, better capture the heavy tails and asymmetry often observed in financial time series data.

Question 35

Why do financial return series often exhibit leptokurtic distributions?

Answer 35

Due to heavy tails and extreme observations that are not well captured by normal distributions, reflecting periods of high volatility.

Question 36

Why does volatility clustering occur in ARCH models?

Answer 36

Volatility clustering occurs because large residuals (both positive and negative) tend to follow large residuals, indicating that volatility depends on past squared residuals.

Question 37

Why is maximum likelihood estimation (MLE) used in GARCH model fitting?

Answer 37

MLE provides parameter estimates by maximizing the likelihood of the observed data under the assumed GARCH process.

Question 38

Why is stationarity important in ARCH and GARCH models?

Answer 38

Stationarity ensures that the variance process is well-defined and that the model parameters are interpretable and stable over time.

Question 39

Why is the conditional variance in ARCH models time-dependent?

Answer 39

Because it is modeled as a function of past squared residuals, reflecting how past shocks influence current volatility.

Question 40

Why might an asymmetric GARCH model like EGARCH be preferred?

Answer 40

Because it captures the leverage effect, where negative shocks have a larger impact on volatility than positive shocks of the same size.