Podcast Beta
Questions and Answers
Which of the following is a characteristic of financial asset returns that linear structural models cannot explain?
In the context of non-linear data generating processes, what is the definition provided by Campbell, Lo and MacKinlay (1997)?
What does volatility clustering refer to in financial markets?
What is the more compact form of the 'traditional' structural model $y = Xeta + u$?
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What is the Quasi-Maximum Likelihood (QML) method?
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What are some possible problems with GARCH(p,q) models?
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What is the advantage of modeling log($\sigma_t^2$) in the EGARCH model?
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What does the GJR model allow for?
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What does the news impact curve plot?
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What does a GARCH model help in forecasting?
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In ARCH models, what does the current error variance plausibly depend on?
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What is the key assumption that leads to the use of Autoregressive Conditionally Heteroscedastic (ARCH) models?
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How can ARCH effects be tested?
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What is a potential issue in estimating ARCH and GARCH models?
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What does GARCH stand for?
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How do GARCH models extend ARCH models?
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Non-linear data generating processes can be written as $y_t = f(u_t, u_{t-1}, u_{t-2}, ...)$ where $u_t$ is an iid error term and $f$ is a non-linear function
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Volatility clustering refers to the tendency for volatility in financial markets to appear in bunches
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The 'traditional' structural model $y_t = \beta_1 + \beta_2x_{2t} + ... + \beta_kx_{kt} + u_t$ is an example of a non-linear model
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Leverage effects refer to the tendency for volatility to rise more following a large price fall than following a price rise of the same magnitude
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Is it necessary to test for normality when using GARCH models?
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The EGARCH model always ensures that the conditional variance parameters are non-negative.
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The GJR model allows for a leverage effect by including an additional parameter.
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GARCH models can account for non-negativity constraints and leverage effects without any issues.
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The news impact curve plots the relationship between volatility and returns in the GARCH model.
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The ARCH-M specification suggested by Engle, Lilien, and Robins incorporates risk premium into the model.
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GARCH-type models are not suitable for modeling the volatility clustering effect.
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Forecasting variances using GARCH models requires iterating with the conditional expectations operator.
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The GARCH(1,1) model can be represented as $\sigma_{t+1}^2 = \alpha_0 + \alpha_1u_t^2 + \beta\sigma_t^2$.
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The two-step ahead forecast for $\sigma^2$ in the GARCH model is calculated using the conditional expectation of the second equation.
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The GARCH model assumes that $u_t$ follows a normal distribution with mean 0 and variance 1.
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The QML method in GARCH modeling is used to estimate the parameters by minimizing the likelihood function.
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ARCH models assume that the current error variance plausibly depends on previous squared error terms.
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The ARCH(1) model is a particular case of ARCH models.
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GARCH models extend ARCH models by allowing the conditional variance to depend on previous own lags.
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The GARCH(1,1) model is a particular case of GARCH models, which is sufficient to capture volatility clustering in the data.
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The unconditional variance of a time series under the GARCH specification can exhibit non-stationarity, which can lead to non-convergence of conditional variance forecasts as the horizon increases.
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Estimation of ARCH and GARCH models involves maximizing the log-likelihood function using numerical methods due to non-linear variance equations.
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Potential issues in maximizing the log-likelihood function include local optima or multimodalities in the likelihood surface.
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Models with non-linear mean functions have non-linear mean, while those with non-linear variance functions have non-linear variance.
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Autoregressive Conditionally Heteroscedastic (ARCH) models are used when the assumption of constant variance (homoscedasticity) is not met.
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To test for ARCH effects, we first estimate a linear regression model, then test for ARCH by regressing the squared residuals on their own lags.
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Problems with ARCH(q) models include deciding on the value of q and potential violation of non-negativity constraints.
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GARCH models are more parsimonious, avoid overfitting, and less likely to violate non-negativity constraints than ARCH models.
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Study Notes
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Models with non-linear mean functions have non-linear mean, while those with non-linear variance functions have non-linear variance.
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Autoregressive Conditionally Heteroscedastic (ARCH) models are used when the assumption of constant variance (homoscedasticity) is not met.
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ARCH models assume that the current error variance plausibly depends on previous squared error terms.
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ARCH models can be written as: yt = β1 + β2x2t + . + βkxkt + ut, ut ～ N(0, ht), where ht = α0 + α1 + α2 + ... + αq.
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The ARCH(1) model is a particular case of ARCH models.
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To test for ARCH effects, we first estimate a linear regression model, then test for ARCH by regressing the squared residuals on their own lags.
-
Problems with ARCH(q) models include deciding on the value of q and potential violation of non-negativity constraints.
-
GARCH models extend ARCH models by allowing the conditional variance to depend on previous own lags.
-
The GARCH(1,1) model is a particular case of GARCH models, which is sufficient to capture volatility clustering in the data.
-
GARCH models are more parsimonious, avoid overfitting, and less likely to violate non-negativity constraints than ARCH models.
-
The unconditional variance of a time series under the GARCH specification can exhibit non-stationarity, which can lead to non-convergence of conditional variance forecasts as the horizon increases.
-
Estimation of ARCH and GARCH models involves maximizing the log-likelihood function using numerical methods due to non-linear variance equations.
-
Potential issues in maximizing the log-likelihood function include local optima or multimodalities in the likelihood surface.
-
Models with non-linear mean functions have non-linear mean, while those with non-linear variance functions have non-linear variance.
-
Autoregressive Conditionally Heteroscedastic (ARCH) models are used when the assumption of constant variance (homoscedasticity) is not met.
-
ARCH models assume that the current error variance plausibly depends on previous squared error terms.
-
ARCH models can be written as: yt = β1 + β2x2t + . + βkxkt + ut, ut ～ N(0, ht), where ht = α0 + α1 + α2 + ... + αq.
-
The ARCH(1) model is a particular case of ARCH models.
-
To test for ARCH effects, we first estimate a linear regression model, then test for ARCH by regressing the squared residuals on their own lags.
-
Problems with ARCH(q) models include deciding on the value of q and potential violation of non-negativity constraints.
-
GARCH models extend ARCH models by allowing the conditional variance to depend on previous own lags.
-
The GARCH(1,1) model is a particular case of GARCH models, which is sufficient to capture volatility clustering in the data.
-
GARCH models are more parsimonious, avoid overfitting, and less likely to violate non-negativity constraints than ARCH models.
-
The unconditional variance of a time series under the GARCH specification can exhibit non-stationarity, which can lead to non-convergence of conditional variance forecasts as the horizon increases.
-
Estimation of ARCH and GARCH models involves maximizing the log-likelihood function using numerical methods due to non-linear variance equations.
-
Potential issues in maximizing the log-likelihood function include local optima or multimodalities in the likelihood surface.
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Description
Explore the motivation for modelling volatility in finance and the limitations of linear structural and time series models in explaining financial features. Understand the need for non-linear models to capture important financial characteristics.
