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ARMA-GARCH versus GARCH, questions about skewness and kurtosis.
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I am currently researching the following statement:
The ARMA-GARCH model better captures the skewness and leptokurtosis of financial time series than a GARCH model would.
What I know so far:
For a GARCH(p,q) process defined by:
$epsilon_t = sigma_t eta_t$
$sigma^2_t = omega + sum_i=1^qalpha_i epsilon^2_t-i + sum_j=1^p beta_j sigma^2_t-j $
The kurtosis is given by:
$K^(epsilon) = fracmathbbE(eta_t^4)mathbbE(eta_t^4)-[mathbbE(eta_t^4)-1]sum_j=0^infty Psi^2_j$, where $Psi_j$ are the constants of the infinite ARCH representation.
For an ARMA(P,Q)-GARCH(p,q) process defined by:
$y_t = mu + sum_i=1^P alpha_i (y_t-i-mu) + epsilon_t + sum_j=1^Q beta_j epsilon_t-j$
$epsilon_t = sigma_t eta_t$
$sigma^2_t = omega + sum_i=1^q phi_i epsilon_t-i^2 +sum_j=1^p theta_j sigma^2_t-j $
The kurtosis is given by:
$K^(y) = fracK^(epsilon)[sumlimits^infty_j=0Psi^4_j]=6 sumlimits_i<j^infty Psi^2_i Psi^2_j(sumlimits^infty_j=0 Psi^2_j)^2$
Where $K^(epsilon)$ is given by:
$K^(epsilon) = fracmathbbE(eta_t^4)mathbbE(eta_t^4)-[mathbbE(eta_t^4)-1]sum_j=0^infty Psi^2_j$
In both equations the $Psi$ are the constants of the infinite Moving Average representation.
My questions:
The relationship between the ARMA-GARCH kurtosis and GARCH kurtosis is clear. How can I quantify the difference though?
Why does an ARMA-GARCH process capture the kurtosis of financial time series better and how is this clear from these equations?
How do I find the equations for the skewness of a ARMA-GARCH/GARCH process?
statistics stochastic-processes time-series
asked Aug 2 at 12:15
Jeannot van den Berg
111
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up vote
0
down vote
favorite
I am currently researching the following statement:
The ARMA-GARCH model better captures the skewness and leptokurtosis of financial time series than a GARCH model would.
What I know so far:
For a GARCH(p,q) process defined by:
$epsilon_t = sigma_t eta_t$
$sigma^2_t = omega + sum_i=1^qalpha_i epsilon^2_t-i + sum_j=1^p beta_j sigma^2_t-j $
The kurtosis is given by:
$K^(epsilon) = fracmathbbE(eta_t^4)mathbbE(eta_t^4)-[mathbbE(eta_t^4)-1]sum_j=0^infty Psi^2_j$, where $Psi_j$ are the constants of the infinite ARCH representation.
For an ARMA(P,Q)-GARCH(p,q) process defined by:
$y_t = mu + sum_i=1^P alpha_i (y_t-i-mu) + epsilon_t + sum_j=1^Q beta_j epsilon_t-j$
$epsilon_t = sigma_t eta_t$
$sigma^2_t = omega + sum_i=1^q phi_i epsilon_t-i^2 +sum_j=1^p theta_j sigma^2_t-j $
The kurtosis is given by:
$K^(y) = fracK^(epsilon)[sumlimits^infty_j=0Psi^4_j]=6 sumlimits_i<j^infty Psi^2_i Psi^2_j(sumlimits^infty_j=0 Psi^2_j)^2$
Where $K^(epsilon)$ is given by:
$K^(epsilon) = fracmathbbE(eta_t^4)mathbbE(eta_t^4)-[mathbbE(eta_t^4)-1]sum_j=0^infty Psi^2_j$
In both equations the $Psi$ are the constants of the infinite Moving Average representation.
My questions:
The relationship between the ARMA-GARCH kurtosis and GARCH kurtosis is clear. How can I quantify the difference though?
Why does an ARMA-GARCH process capture the kurtosis of financial time series better and how is this clear from these equations?
How do I find the equations for the skewness of a ARMA-GARCH/GARCH process?
statistics stochastic-processes time-series
asked Aug 2 at 12:15
Jeannot van den Berg
111
I really don't understand the downvotes for this question. The OP is trying to justify a statement that they've read in a time series or econometrics textbook (I presume) and has provided their understanding of the question. I can't imagine there are too many time series experts on this website to justify the downvotes I've seen already. (+1)
– Clarinetist
Aug 2 at 12:50
If you don't get an answer here, I would recommend trying the Stats StackExchange website, but take into account that it (in my opinion, at least) is significantly less active than this website. If you end up posting to Stats SE, you should either delete the question here and repost it to Stats SE, or leave your question, wait about 3-4 days, and then post it again to Stats SE.
– Clarinetist
Aug 2 at 12:52
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
I am currently researching the following statement:
The ARMA-GARCH model better captures the skewness and leptokurtosis of financial time series than a GARCH model would.
What I know so far:
For a GARCH(p,q) process defined by:
$epsilon_t = sigma_t eta_t$
$sigma^2_t = omega + sum_i=1^qalpha_i epsilon^2_t-i + sum_j=1^p beta_j sigma^2_t-j $
The kurtosis is given by:
$K^(epsilon) = fracmathbbE(eta_t^4)mathbbE(eta_t^4)-[mathbbE(eta_t^4)-1]sum_j=0^infty Psi^2_j$, where $Psi_j$ are the constants of the infinite ARCH representation.
For an ARMA(P,Q)-GARCH(p,q) process defined by:
$y_t = mu + sum_i=1^P alpha_i (y_t-i-mu) + epsilon_t + sum_j=1^Q beta_j epsilon_t-j$
$epsilon_t = sigma_t eta_t$
$sigma^2_t = omega + sum_i=1^q phi_i epsilon_t-i^2 +sum_j=1^p theta_j sigma^2_t-j $
The kurtosis is given by:
$K^(y) = fracK^(epsilon)[sumlimits^infty_j=0Psi^4_j]=6 sumlimits_i<j^infty Psi^2_i Psi^2_j(sumlimits^infty_j=0 Psi^2_j)^2$
Where $K^(epsilon)$ is given by:
$K^(epsilon) = fracmathbbE(eta_t^4)mathbbE(eta_t^4)-[mathbbE(eta_t^4)-1]sum_j=0^infty Psi^2_j$
In both equations the $Psi$ are the constants of the infinite Moving Average representation.
My questions:
The relationship between the ARMA-GARCH kurtosis and GARCH kurtosis is clear. How can I quantify the difference though?
Why does an ARMA-GARCH process capture the kurtosis of financial time series better and how is this clear from these equations?
How do I find the equations for the skewness of a ARMA-GARCH/GARCH process?
statistics stochastic-processes time-series
asked Aug 2 at 12:15
Jeannot van den Berg
111
I am currently researching the following statement:
The ARMA-GARCH model better captures the skewness and leptokurtosis of financial time series than a GARCH model would.
What I know so far:
For a GARCH(p,q) process defined by:
$epsilon_t = sigma_t eta_t$
$sigma^2_t = omega + sum_i=1^qalpha_i epsilon^2_t-i + sum_j=1^p beta_j sigma^2_t-j $
The kurtosis is given by:
$K^(epsilon) = fracmathbbE(eta_t^4)mathbbE(eta_t^4)-[mathbbE(eta_t^4)-1]sum_j=0^infty Psi^2_j$, where $Psi_j$ are the constants of the infinite ARCH representation.
For an ARMA(P,Q)-GARCH(p,q) process defined by:
$y_t = mu + sum_i=1^P alpha_i (y_t-i-mu) + epsilon_t + sum_j=1^Q beta_j epsilon_t-j$
$epsilon_t = sigma_t eta_t$
$sigma^2_t = omega + sum_i=1^q phi_i epsilon_t-i^2 +sum_j=1^p theta_j sigma^2_t-j $
The kurtosis is given by:
$K^(y) = fracK^(epsilon)[sumlimits^infty_j=0Psi^4_j]=6 sumlimits_i<j^infty Psi^2_i Psi^2_j(sumlimits^infty_j=0 Psi^2_j)^2$
Where $K^(epsilon)$ is given by:
$K^(epsilon) = fracmathbbE(eta_t^4)mathbbE(eta_t^4)-[mathbbE(eta_t^4)-1]sum_j=0^infty Psi^2_j$
In both equations the $Psi$ are the constants of the infinite Moving Average representation.
My questions:
The relationship between the ARMA-GARCH kurtosis and GARCH kurtosis is clear. How can I quantify the difference though?
Why does an ARMA-GARCH process capture the kurtosis of financial time series better and how is this clear from these equations?
How do I find the equations for the skewness of a ARMA-GARCH/GARCH process?
statistics stochastic-processes time-series
asked Aug 2 at 12:15
Jeannot van den Berg
111
asked Aug 2 at 12:15
Jeannot van den Berg
111
asked Aug 2 at 12:15
Jeannot van den Berg
111
asked Aug 2 at 12:15
Jeannot van den Berg
111
111
I really don't understand the downvotes for this question. The OP is trying to justify a statement that they've read in a time series or econometrics textbook (I presume) and has provided their understanding of the question. I can't imagine there are too many time series experts on this website to justify the downvotes I've seen already. (+1)
– Clarinetist
Aug 2 at 12:50
If you don't get an answer here, I would recommend trying the Stats StackExchange website, but take into account that it (in my opinion, at least) is significantly less active than this website. If you end up posting to Stats SE, you should either delete the question here and repost it to Stats SE, or leave your question, wait about 3-4 days, and then post it again to Stats SE.
– Clarinetist
Aug 2 at 12:52
add a comment |Â
I really don't understand the downvotes for this question. The OP is trying to justify a statement that they've read in a time series or econometrics textbook (I presume) and has provided their understanding of the question. I can't imagine there are too many time series experts on this website to justify the downvotes I've seen already. (+1)
– Clarinetist
Aug 2 at 12:50
If you don't get an answer here, I would recommend trying the Stats StackExchange website, but take into account that it (in my opinion, at least) is significantly less active than this website. If you end up posting to Stats SE, you should either delete the question here and repost it to Stats SE, or leave your question, wait about 3-4 days, and then post it again to Stats SE.
– Clarinetist
Aug 2 at 12:52
I really don't understand the downvotes for this question. The OP is trying to justify a statement that they've read in a time series or econometrics textbook (I presume) and has provided their understanding of the question. I can't imagine there are too many time series experts on this website to justify the downvotes I've seen already. (+1)
– Clarinetist
Aug 2 at 12:50
I really don't understand the downvotes for this question. The OP is trying to justify a statement that they've read in a time series or econometrics textbook (I presume) and has provided their understanding of the question. I can't imagine there are too many time series experts on this website to justify the downvotes I've seen already. (+1)
– Clarinetist
Aug 2 at 12:50
If you don't get an answer here, I would recommend trying the Stats StackExchange website, but take into account that it (in my opinion, at least) is significantly less active than this website. If you end up posting to Stats SE, you should either delete the question here and repost it to Stats SE, or leave your question, wait about 3-4 days, and then post it again to Stats SE.
– Clarinetist
Aug 2 at 12:52
If you don't get an answer here, I would recommend trying the Stats StackExchange website, but take into account that it (in my opinion, at least) is significantly less active than this website. If you end up posting to Stats SE, you should either delete the question here and repost it to Stats SE, or leave your question, wait about 3-4 days, and then post it again to Stats SE.
– Clarinetist
Aug 2 at 12:52
add a comment |Â
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I really don't understand the downvotes for this question. The OP is trying to justify a statement that they've read in a time series or econometrics textbook (I presume) and has provided their understanding of the question. I can't imagine there are too many time series experts on this website to justify the downvotes I've seen already. (+1)
– Clarinetist
Aug 2 at 12:50
If you don't get an answer here, I would recommend trying the Stats StackExchange website, but take into account that it (in my opinion, at least) is significantly less active than this website. If you end up posting to Stats SE, you should either delete the question here and repost it to Stats SE, or leave your question, wait about 3-4 days, and then post it again to Stats SE.
– Clarinetist
Aug 2 at 12:52