Flashcards in 2016 - Modelling non-normality Deck (13):

1

## GARCH: alpha and beta

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- alpha: captures the heteroskedasticity

- beta: captures the persistence in the variance

- alpha + beta < 1: process is cov stationary

- alpha + beta ~ 1: process is strictly stationary. Persistence in the variance

2

## GARCH: H0: alpha + beta ~ 1

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- Null of integrated conditional var

- Wald test

- TS distributed as a Khi^2(1): 3.85

3

## GARCH: Wald statistic

### W = (alpha + beta - 1)^2/(sig^2_alpha + 2*sig_alpha*sig_beta*rho_alphabeta + sig^2_beta) ~ Khi^2(1): 3.85

4

## GARCH: Jarque-Bera test

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- Null of normally distributed log-returns (or residuals)

- If rejected for residuals: the non-normality of the data generating process is higher than of the uncond distr

- In other words, it means that the assumption of normality for the cond distr isn't appropriate to capture the non-normality of returns

5

## GARCH: Ljung-Box test

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- Null of no autocorr in log-returns (or residuals) for the first p

- If rejected for log-returns: that's why we use an AR process to capture this effect

- If rejected for residuals: model isn't successful in capturing autocorr

6

## DGT: DGT test

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- First part: null of time dependency of u's (different test for each power)

- If rejected for some u: GARCH doesn't capture all the heteroskedasticity

- Second part: null of uniformly distributed u's

- If rejected: goes against the adequacy of normality assumption

7

## STUDENT T: parameter estimates

### - If v is significant: confirms fat-taildness of returns distr

8

## STUDENT T: test the null of normality H0: v=%

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- Null of normally distributed return process and alternative that return process has t-student distr

- LR test ~ Khi^2(1): 3.85

- LR = 2*(logL_alt - logL_null)

- If rejected: GARCH model with alternative distr outperforms the null hyp GARCH model -> new likelihood is significantly higher than the old one

9

## STUDENT T: DGT test

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- If part 1 rejected: reject iidness of the u's

- If part 2 rejected: reject uniformity of the u's. Goes against the adequacy of the t-student assumption

10

## SKEWED STUDENT T: parameter estimates

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- If lambda significant: confirms presence of neg skewness in the return distr

- which is partly or fully captured by assuming the skew-t distr

11

## SKEWED STUDENT T: null of Student t (H0: lambda = 0)

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- Null that return process has t-student distr vs alt that return has skew-t distr

- Wald test or LR test (both khi^2(1): 3.85)

- LR = 2*(logL_alt - logL_null)

- If rejected: the diff btw the 2 likelihoods is significant

- Wald = squared t-test = lambda^2/sig^2_lambda

- Wald is about the significance of parameters and LR about the model's perf compared to another

12

## SKEWED STUDENT T: DGT test

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- If first part rejected: iidness of the u's is rejected

- If second part rejected: uniformity of the u's is rejected. Goes against the adequacy of skew-t distr

13