Autoregressive Conditional Heteroskedasticity Flashcards
What is the ARCH process?
The Autoregressive Conditional Heteroskedastic process that allows the conditional variance to change over time as a function of past errors.
Who introduced the ARCH process?
Engle in 1982.
What does GARCH stand for?
Generalized Autoregressive Conditional Heteroskedasticity.
What is the main advantage of the GARCH process over the ARCH process?
It allows for a more flexible lag structure in the conditional variance equation.
What is the GARCH(1, 1) process?
A specific case of the GARCH process that is often used for modeling.
What is the condition for wide-sense stationarity in the GARCH(p, q) process?
A(1) + B(1) < 1.
What does the GARCH(p, q) process allow that the ARCH(q) process does not?
It allows lagged conditional variances to enter the conditional variance equation.
What is a necessary condition for the existence of the 2mth moment in the GARCH(1, 1) process?
a_1 + b_1 < 1.
Fill in the blank: The GARCH(p, q) process can be interpreted as an ______ process.
autoregressive moving average.
True or False: The GARCH process can be justified through Wald’s decomposition.
True.
What is the role of autocorrelation and partial autocorrelation functions in GARCH models?
They are useful for identifying and checking time series behavior in the conditional variance equation.
What is the relationship between the GARCH process and the ARMA process?
The extension of the ARCH process to the GARCH process resembles the extension of the AR process to the ARMA process.
What type of economic phenomena has the ARCH process proven useful in modeling?
Uncertainty of inflation, among others.
What does the term ‘leptokurtic’ refer to in the context of GARCH(1, 1)?
It indicates that the distribution has heavy tails.
What is the significance of the parameterization in the GARCH process?
It facilitates easier practical implementation compared to other representations.
What empirical example is discussed in relation to GARCH models?
The uncertainty of the inflation rate.
What happens when the roots of 1 - B(z) = 0 lie outside the unit circle?
The GARCH process can be rewritten as a distributed lag of past innovations.
What is the impact of a long memory in the GARCH process?
It allows for more accurate modeling of volatility over time.
What is often a problem with negative variance parameter estimates in ARCH models?
It leads to the imposition of a fixed lag structure.
What does the notation E(e_t) = 0 imply in the context of GARCH?
It indicates that the expected value of the innovations is zero.
What is the conditional variance in the GARCH process dependent on?
It is a function of past errors and past conditional variances.
What is the implication of a non-negativity constraint in GARCH models?
It ensures that variance estimates remain valid.
How does the GARCH process relate to empirical work in economics?
It captures the time-varying nature of volatility in economic time series.
What is the significance of the appendix in the paper?
It contains proofs for the theorems presented in the main text.
What is the condition for the inverse of a qXq matrix in the context of GARCH models?
3/(I - ¢) < 1
What does the notation q’ represent in GARCH models?
q’ = (a, …., ,), where i = 1, …, q
In the context of GARCH processes, what is the maximum value of m?
m = max{p, q}
What does the notation a_i = 0 signify for i > q in GARCH models?
It indicates that the parameters a are zero for indices greater than q.
What do the Yule-Walker equations relate to in GARCH processes?
They relate the first p autocorrelations for e to the parameters a and B.
True or False: The autocorrelations for an ARCH(q) process depend only on the parameters a and B.
True
What is the behavior of the partial autocorrelation function for a GARCH(p,q) process?
It is generally non-zero but dies out.
What does the notation s_k represent in the context of partial autocorrelation?
It denotes the kth partial autocorrelation.
Fill in the blank: The maximum likelihood estimate for the parameters in a GARCH model is denoted as _____.
θ
What is the purpose of the Lagrange multiplier test in GARCH models?
To test for the presence of GARCH by evaluating the null hypothesis a_i = 0.
What is the asymptotic distribution of the Lagrange multiplier test statistic under the null hypothesis?
Chi-square with degrees of freedom equal to the number of elements in a.
What does the notation R^2 represent in the context of the GARCH model?
The squared multiple correlation coefficient between the residuals.
What is the significance of the block diagonality in the information matrix for GARCH models?
It allows for separate estimation of parameters without loss of asymptotic efficiency.
What is the expected behavior of the LM test statistic for GARCH alternatives when the null is an ARCH process?
It is singular if both p > 0 and q > 0.
What does the notation h_t signify in GARCH models?
It represents the conditional variance.
True or False: The maximum likelihood estimation for GARCH is independent of the recursive terms in the model.
False
What algorithm is mentioned as convenient for maximum likelihood estimation in GARCH models?
Berndt, Hall, Hall and Hausman (BHHH) algorithm.
What is the significance of the fourth-order moment of e_t in GARCH models?
It is necessary for the existence of certain statistical properties.
What is the implication of the LM test statistic for the inclusion of additional parameters in GARCH models?
It assesses the significance of those parameters in improving model fit.
In the empirical example, what economic variable is examined using the ARCH framework?
The rate of growth in the implicit GNP deflator.
What does the notation m_t represent in the empirical example?
m_t = 100·In(GD_t/GD_{t-1})
Fill in the blank: The model estimation is based on ____ observations from 1948.2 to 1983.4.
143
What is indicated by the high significance of the LM test for ARCH models in the empirical example?
The presence of autoregressive conditional heteroskedasticity.
What does the notation h_t = 0.282 signify in the context of the empirical model?
It represents the estimated conditional variance.
What is the LM test statistic for the inclusion of the eighth-order linear declining lag structure?
2.33
Corresponds to the 0.87 fractile in the distribution.
What is the LM test statistic for GARCH(1,2) or locally equivalent GARCH(2,1)?
3.80
Not significant at the 5% level.
What is the value of the LM test statistic for the inclusion of e; ,s…,e;_,?
5.58
Equal to the 0.77 fractile in the distribution.
What is the sample coefficient of kurtosis for e,h, / from model (31)?
3.81
Differs from the ‘normal’ value of 3.00 by slightly less than two asymptotic standard errors.
What are the coefficients of kurtosis for models (29) and (30)?
- Model (29): 6.90
- Model (30): 4.07
What are the sample coefficients of skewness for e,h, / from the three models?
- Model (31): -0.13
- Model (29): 0.18
- Model (30): 0.11
All are within one asymptotic standard error.
What are the estimated mean and median lag in the conditional variance equation in (31)?
- Mean lag: 5.848
- Median lag: 3.696
What are the forced mean and median lags in model (30)?
- Mean lag: 3
- Median lag: 2
What does the lag structure in the GARCH(1,1) model suggest?
It can be rationalized by some sort of adaptive learning mechanism.
What do figures 3 and 4 illustrate regarding the actual inflation rate?
They graph the actual inflation rate together with 95% asymptotic confidence intervals for the one-step-ahead forecast errors.
How was the inflation rate characterized from the late forties until the mid-fifties?
Very volatile and hard to predict.
What characterized the inflation rate during the sixties and early seventies?
Stable and predictable.
What happened to the uncertainty of the inflation rate starting with the second oil crisis in 1974?
There was a slight increase in uncertainty.
What is the basic idea of the proof of Theorem I?
It follows that of Theorem 1 in Milhaj (1984).
What is the condition for convergence in the context of the proof of Theorem I?
e; converges almost surely.
What does the binomial formula yield in the proof of Theorem 2?
A specific expression involving a, and p.
What is the significance of the eigenvalues of matrix C in the proof of Theorem 2?
All eigenvalues must lie inside the unit circle for the limit to exist.
What does the condition (a,,B,,i) < 1 imply?
(a,,B,,i-1) < 1 for a, + < 1.
What does the reference by Engle (1982) focus on?
Autoregressive conditional heteroskedasticity with estimates of the variance of U.K. inflation.
What is the significance of the GARCH model in relation to the ARCH model?
It provides a slightly better fit and exhibits a more reasonable lag structure.
