Chapter 3 - Conditional heteroskedastic mdoells

Question 1

what is the objective here

Answer 1

Modeling volatility of asset returns

Question 2

important to understand about volatility

Answer 2

Not observable. We can only estimate it

Question 3

what is implied volaitlity

Answer 3

Volatility output from BS model

Question 4

weakness of IV

Answer 4

relies on model assumptions, like geometric brownian motion

Question 5

elaborate on the characteristics of volatility

Answer 5

1) Tendency to cluster
2) jumps are rare
3) mean reverting (generalyl speaking stationairy)
4) leverage effect (volatility react different to a price increase of X vs price decrease of X)

Question 6

elaborate on the foundation of volatility study

Answer 6

The foundation is that volatility of log returns are serially uncorrelated, or with only minor lower order correlations, but at the same time volatility is a dependent series. This means that there is a structure there, but the structure is not in regard to serial correlations (autocorrelation).

By the 4 characteristics of voaltility, we know that there is structure there. We want to capture this structure.

Empirically, we can view the dependency in the series by plotting the ACF of magnitudes (abs or square) of log returns.

Question 7

what can we say about the equation for conditional mean of log returns

Answer 7

should be simple, as there is no evidence of a pattern

Question 8

discuss the 4 steps in building a volatility model

Answer 8

1) specify a mean equation by testing for serial dependencies in the data. Build a model to remove linear dependencies from the return series.

2) use the residuals of the mean equation to test for ARCH effects

3) if ARCH effects are present, specify a volatility model.

4) check the fitted model and refine if necessary

Question 9

what is ARCH effects

Answer 9

conditional heteroskedasticity

Question 10

elaborate on conditional heteroskedasticity, and regular heteroskedasticity

Answer 10

the difference is that heterskedasticity refers to that variance is not constant. Variance has a structrure.

Conditional heteroskedasticity refers to a variance structure where variance depends on past informaiton in general.

Question 11

what is this:

“The expected value of the squared difference between a value and the mean of that value”

Answer 11

Variance

Question 12

elaborate on what we are actually trying to do in this chapter

Answer 12

WE want to build a model for the conditional variance. In the simplest form, conditional variance is given by:

sigma_t^2 = Var(r_t | F_{t-1})

We know that variance of r_t is the same as “E[(X - E[X])^2]”, which in our case is E[(r_t - mu)^2]

For the mean, we can make use of the fact that the mean equation should be very simple. Perhaps just constant, since this is asset returns we are talking about.

r_t = mu + a_t

Subbing this into the variance equation:

sigma_t^2 = E[(r_t - (r_t - a_t))^2 |F_{t-1}]

sigma_t^2 = E[(a_t)^2 |F_{t-1}]

So, in the simple scenario of mean equation being only the constant term plus error, we get that the variance equation, the conditional variance equation is equal to the conditional expectation of the squared shock/error term.

Because of how the expected value of a_t is 0, this expression is a collapsed version of variance, which gives us final result of :

sigma_t^2 = var(a_t |F_{t-1})

it should be immediately obvious what this result entails. It entails that we can change the perspective from estiamting variance of return series to estimating variance of the shock series.

Question 13

GARCH models belong to one category of models. What is this category, and what is the other category?

Answer 13

GARCH belong to the category that use an exact function to govern how sigma_t^2 change.

The other category is a stochastic process

Question 14

what do we use to test for ARCH effects

Answer 14

We use the squared residuals. The residuals are the results from using the following mean equation:

r_t = mu + a_t

and solvign for a_t

a_t = r_t - mu

Question 15

elaborate on testing for ARCH effects

Answer 15

there are 2 types of tests, but both use the squared residuals. The first test is Ljung Box on the squared residual series. If the first m lags have zero ACF, then there is no evidence of ARCH effects. However, if one of them have non zero ACF (statisticaly significant ACF) then we reject the null hypothesis and claim that there is ARCH effects present.

Recall that ARCH effects refer to there being conditional heteroskedastic patterns, meaning that the variance is dependent on previous levels. This specific test is about the square.

The other test is the ARCH-LM test.

Question 16

elaborate on conditional heteroskedasitcity vs ARCH effects

Answer 16

conditional heteroskedasticity is a broad class of all patterns that have some sort of structure where variacne depends on earlier information.

ARCH effects, GARCH effects, ARCH-X effects etc are examples of types of dependencies. For instance, ARCH, which is short for Auto Regressive Conditional Heteroscedasticity, includes a structure of variance being dependent on past squared shocks.

ARCH effects specifically refer to cases where the conditional heteroskedasticity structure is dependent on the square values of earlier shocks. An ARCH process strictly produce the structure where past shocks squared along with coefficients to give their contributions assign a new level of variance. If the process is pure ARCH, or have ARCH tendencies, some of this structure can be captured by using an ARCH model. Also, we can verify the presence of ARCH process (ARCH effects) by using either Ljung Box or ARCH-LM, as both make use of squared residuals

Question 17

do we have var(r_t | prior) == var(a_t | prior) for all kinds of mean equaitons?

Answer 17

Yes.

Question 18

what is a_t? where does it come from

Answer 18

shock. residual. it comes from the mean equation.

Question 19

what do we mean by “a_t = sigma_t epsilon_t”

Answer 19

it tells us what we assume about the residual.

We assume that the residual is equal to the standard deviation/volatility multiplied by some randomness. epsilon is iid white noise.

Question 20

elaborate on ARCH model

Answer 20

it assumes 2 things:

1) shocks comes from the play between volatility and white noise randomness: a_t = sigma_t eps_t

2) sigma_t^2 = alpha_0 + alpha_1 a_(t-1)^2 …

Question 21

why do we need the a_t = sigma_t eps_t part?

Answer 21

We need to establish how the data is generated. At the end of this is a recursive structure that use white noise.

if we do not include it, we will only have a model saying that “if this, then that happens”. However, when defining what the ARCH model assume, we are talking about the sort of process that it assumes. And specifically, it assumes that a shock is a result of volatility at the same level multiplied by white noise.

And the reason why specifically this assumption is included, is that it gives us E[a_t | prior] = 0 AND
Var[a_t] = sigma_t^2 (because epsilon has variance 1 and is uncorrelated).

Question 22

elaborate on what ARCH believe

Answer 22

we build some model for our return series, for instnace “r_ t = mu + a_t”. Under the assumption of ARCH process, a_t is a result of an interplay between the volatility at that current time t and some random component. The random component basically ensure that we can never know for sure what happens, meaning that it is not a purely deterministic process. So this basically means that the asset return, r_t, can be understood as given by some mean component mu, and some degree of volatility. ARCH believe that when the return series is generated, it is always produced as a sum of the mean level and some sort of volatility oscillation around this level.

Also, it is worth noting why ARCH believe/assume the specific relationship between shocks and sigma and epsilon.

Var(a_t) = var(sigma_t eps_t) = sigma_t var(eps_t) = sigma_t *1 = sigma_t

rmeember that sigma_t is a constant.

Question 23

why is it nice that ARCH establish that the variance of the shocks are equal to sigma_t^2?

Answer 23

because we know from before that variance of returns and variance of shocks are the same thing when conditioned upon all prior information.

Therefore, when the ARCH process assume that the shocks are generated according to sigma_t eps_t, it is equivalent to say that ARCH process assumes that the asset return series is generated with volatility sigma_t.

This is why it makes a lot of sense to include the data generation process with the ARCH process. If we dont include it, then we have no basis to use.

Question 24

important but subtle point regarding hte shcoks in ARCH model

Answer 24

the current shock is not included. This makes the model recursive inherently. because of this, it makes sense to define the additional data generation thing as well, and not making the argument be circular.

Question 25

requirements of the ARCH model?

Answer 25

alpha_0 must be strictly positive, and the other coefficeiunts must be non-zero.

Question 26

elaborate on the nested structure of expectations:

Answer 26

it is needed because we cannot take the sigma_t out from the expectation. we cannot do this because sigma_t is not constant. it varies. therefore we must etablish the condiitonal part aswell.

Question 27

elaborate on moments

Answer 27

We need a random variable. The moment of a random variable is given by the expected value of that variable increased to some power. for instanc, in the case of ARCH, we have moments of a_t, the shock variable: fourth moment = E[a_t^4] skewness is third kurtosis is fourth

Question 28

what do we need in order t ostudy tail behavior of a random variable?

Answer 28

We require that the fourth moment is finite.

Question 29

Consider the fourth moment: E[a_t^4] elaborate on the procedure here

Answer 29

It is tempting to use: E[sigma_t^4 eps_4] = sigma_t^4 E[eps_4] = sigma_t^2 0 = 0. but we cannot do this because sigma_t is not constant. it is conditional, and wil lchange. instead we do: E[a_t^4] = E[E[a_t^4 | prior]] = E[E[sigma_t^4 eps_t^4 | prior]] Here is the KEY: Because we use conditional expectation, we can take sigma_t outsied of the expectation: = E[sigma_t^4 E[eps_t^4 | prior]] but then, eps is white noise and doesnt depend on prior, so we get: = E[sigma_t^4 E[eps_t^4]] THen we need to recognize that E[eps_t^4] is the fourth moment from a normal distribution. It has value 3. this gives us: = 3 E[sigma_t^4], or alternatively: = 3 E[(sigma_2)^2] sigma_t^2 is our model, basically. = 3 E[alpha0 + alpha1 a_{t-1}^2]^2

Question 30

why do we care about moments of the a_t?

Answer 30

a_t is observable and a random variable. it also relates to our return series: r_t = mu + a_t we must understand how the random component in a_t works because this is what produce the volatility ultimately. At least according to the ARCH model/process.

Question 31

elaborate on finding the unconditional mean of ARCH model

Answer 31

reclal that ARCH model assume the following: sigma_t^2 = alpha0 + alpha1 a_{t-1}^2 ... a_t = sigma_t eps_t Assume for now order 1, so that we do not have more than 1 recursive part. We want unconditional mean, the first moment, of the random variable: E[a_t] however, a_t depend on prior, so we need: E[E[a_t | prior]] = E[E[sigma_t eps_t | prior]] move sigma_t out, recognize that eps_t doesnt depend on prior, we get: E[sigma_t E[eps_t]] = E[sigma_t *0]. =0

Question 32

elaborate on finding unconditional variance of the ARCH model

Answer 32

Recall that we are interested in a_t. It makes no sense to look for sigma_t directly becasue it is not obsdervable. We find it through a_t. var(a_t) = E[a_t^2] = E[E[a_t^2 | prior]] Recognize that E[a_t^2 | prior] is conditional variance, which is basically our model. This makes a lot of sense. the uncodnitional variance is the expected conditional variance... var(a_t) = E[alpha0 + alpha1 a_{t-1}^2] var(a_t) = alpha0 + alpha1 E[a_{t-1}^2] Since var(a_t) == var(a_{t-1}) == E[a_t^2] we have: E[a_t^2] = alpha0 + alpha1 E[a_t^2] E[a_t^2] = alpha0 / (1 - alpha1) Since variance must be positive, we must require that alpha1 is less than 1.

Question 33

elaborate on the nested expcation with priors etc

Answer 33

informally we have that "unconditional expectation is equal to the average of conditional expectations". basocally, we take the expectation of a variable given some information, and then do this over all possible realization of that information. In our context, it means that whne we have a_t, and want E[a_t], we know that it is equal to E[E[a_t | prior]], as in the outer expectation we impliclty place all possible combinations of prior.

Question 34

elaborate on weaknesses of ARCH

Answer 34

there is no leverage effect. Positive and negative shocks are treated the same way since shocks are squared. It provides no interpretation for understanding variations, it just describe conditional variance.

Question 35

elaborate on why we are actually removing the linear dependencies from the residual/shock

Answer 35

It is because we need ot make the shock aligned with what the ARCH model assume about the data generation process. ARCH assume: a_t = sigma_t eps_t Therefore, it assumes that the shock is purely determined by the volatiltiy and white noise. Therefore, when we observe the shocks, we need to account for this shit so that it is as close to it as possible.

Question 36

what is correlation?

Answer 36

a measure of the degree of **linear** association between two variables. This is a crucial definition. Only linear.

Question 37

what does "dependency" mean+

Answer 37

In this contex,t dependency is used as a category representing all kinds of relationships in the time series. For instance, an asset return series is dependent if a variable r_t depends on its earlier values or prior information in any way.

Question 38

elaborate on why the book present the mean equation as it does etc

Answer 38

We need a structure of the mean equation (it is useful later). We know from research that it should be simple, with very little patterns. Specifically, this model assume that the asset return series {r_t} is produced by "r_t = mu_t + a_t". This means, the random variable (asset return at time t) equals some mea nlevel and some shock. the shock is generally assumed to be random, or at least not within our abilities to model. the mean equation is conditionally dependent since it various through time. Therefore, it is not a constant level, as mu is sometimes used as in other models. Here, mu_t represent the conditional mean of the asset return series. In other words: mu_t = E[r_t | prior information]. Sometimes, the best we can do is to model mu_t = constant. However, the model specification open up for hte possibility of removing the effect of explanatory variables, and modeling the adjusted return series (adjusted after effect of explanatory variables) as an ARMA model. I believe the ARMA model is to remove serial correlation that might exist. Given the mean equation, it is possible to remove this mean from the asset return series, and get residuals. Our hope is that these residuals show dependencies that we can capture. One possibility, is that they follow a relationship based on their squares. This is known as ARCH effect, because we are trying to see whether a residual (squared) a_t^2 can be predicted by using the square of its earlier values up to some order. Volatility is generally difficult to understand, and we can only hope to estimate it. If we observe that it tend to follow a systemic pattern based on its squared earlier values, then we want to leverage this and build an ARCH model. However, there is no guarantees here. ARCH is a nice first candidate because it ticks some boxes based on what volatility tends to do.

Question 39

what is the idea behind volatility study

Answer 39

An asset return sereis is extrmeely little serially correlated, but the asset return series is still somehow dependent on earlier occurrings.

Question 40

explain why Var[r_t | prior] == Var[a_t | prior]

Answer 40

We know that the variance of r_t can be writte nas: var[r_t | prior] = E[(r_t - mu)^2 | prior] => E[a_t^2 | prior] This is the second moment, we therefore know that it can alternatively be written as: Var[a_t | prior] This tells us that when we use a mean equation to find the residuals, according to the MODEL, finding the conditional variance of the asset return sereis is equivalent to finding the conditional variance of the shock series.

Question 41

conditional heteroskedastic models are concerned with how sigma_a^2 evolves through time. how does it evolve?

Answer 41

Depends on the model. Different models explain it differently. They assume different things.

Question 42

elaborate on the outcome of Var[r_t |prior] = Var[a_t | prior]

Answer 42

it is built on the idea of a mean equation, mu_t. the mean equaiton does not produce the mean, but it produce the conditional mean. The conditional mean is the expected value of r_t given all prior information. When we use a model, this will typically be deterministic, as we do not have teh ifnromation on the current a_t shock, and we assume it has expected value 0. So what we do is that we are interested in the conditiona lvariance, meaning the variance of the next period. This is the expected squared difference between the actual r_t value we will later observe, and the outcome from our mean equation. By using the fact that the difference between the actual value and the model prediction is the residual, a_t, also called the shock, we have the result that finding the conditional variance of an asset return sereis is the same as finding the variance of these error terms. This assumes, however, that the mean equation is purely linear. This is because the remaining pattern is non-linear, and is what we want to capture. The fact that conditional variance of asset return is equal to conditional variance of shcok series also place emphasis on the fact that volatility, or variance, is a relationship that we have not been able to model using linearities. Meaning, our asset return series may depend on many things in a linear way, like its past values and influences from macro economic factors and other things, and the remaining "error" is what we have not been able to capture. This is a variation that the linear model is not explaining. From empirical evidence, we know that asset return series are typically not serially correlated to any significant degree. However, they usually are dependent. Meaning, the squared shock series (squared residual series) is typically depednent on its lags. In addition, we know that historical volatility tend to have certain patterns: 1) clustering 2) mean reversion 3) Leverage effect 4) rarely big jumps, usually smooth Different models for volatility have different assumptions and ways to capture these relationships. It is important to add that volatility in real life does not follow a process. However, our models assume they do. Meaning, when we use a model, liek ARCH, the model makes certain assumptions on the behavior of the asset return series. Therefore, the model will likely not be a perfect fit. However, our goal is to find a suitable model that explain as much as possible of the relationship. Final remarks, volatility is defined as standard deviation of an asset return. Since our "mean equation" is a model that ultimately predict the asset return in a deterministic way, we can consider the conditional voaltility as the deviation from this conditional mean equation model.

Question 43

what is the assumption of ARCH models

Answer 43

Assumes that: 1) The residuals, shocks, are produced as sigma_t eps_t, where sigma_t is some 'deterministic' value, while eps_t is random white noise, standard normally distirbuted. 2) the conditional heteroskedasticity can be modleled using a simple quadratic function of the residuals.

Question 44

why are we interested in the moments of a_t, and not sigma_t?

Answer 44

sigma_t is the second moment of a_t, as we have established earlier. We have established that the model try to explain linear relationships. whats left is the non linear relationships. we have stablished that these relationships are all captured in a_t. a_t is THE difference, and we believe that the volatility depend on the square of it. The ARCH model takes this one step further, and say that "this model assume that a_t contains purely the hidden variance and random component. by understanding the dynamics of the residual, we can udnerstand the volatility etc".

Question 45

difference between ARCH and GARCH

Answer 45

GARCH just adds variables of itself (previous conditional variances)

Question 46

does a GARCH model have ARCH in it?

Answer 46

Yes, we say that the parameters associated with shocks (squared) is the ARCH params. While the params associated with the squared past random variables of the conditional variance is the GARCH parameters.

Question 47

what is GARCH-M

Answer 47

Garch in mean