Financial time series - compelte Flashcards
why use returns vs prices regarding time series
Returns are complete and have a scale-free characteristic. Returns also have favorable statistical characteristicas
Define simple gross return
1+R_t = P_t / P_{t-1}
Define simple net return
R_t = P_t / P_{t-1}
Define a time series in the context of asset returns
A time series is a collection of random variables over time, where the random variables are the asset returns.
how do we denote a time sereis
{r_t}
define autocorrelation
correlation between past values of the same random variable
what is the foundation of time series analysis and why
Stationarity.
Important to be able to make predictions
elaborate on stationarity
Strict: The joint distribution of a subset of the time series is invariant under time shifts.
Weak: The mean of r_t is time invariant and the covariance between r_{t} and r_{t-l} is also time invariant and depends only on l. meaning the covariance of lag-l is constant.
Weak stationarity should show constant fluctuation arounda fixed level.
when is weak stationarity equal to strict?
If the random variables are normally distributed, and weakly distriubted, then it is also strict.
what do we call “cov(r_t, r_{t-l})”?
lag-l autocovariance
two important properteis from lag-l autocovariance
1) cov(r_t, r_t) = var(r_t) = gamma_0
2) gamma_k = gamma_(-k)
how do we check for stationarity, broadly speaking
divide tiem series into subsections. Check for consistency among each period
when is autocorrelation between two variables 0?
if they are normally distirubted, they have correlation 0 if they are indpendent.
important regarding samples of time series
we need to use the sample counterpart formulas
how do we denote lag-l autocorrelation
p_l
important result regarding variance in time sries
variance is constant, so variance of various r_k is the same:
var(r_k) = var(r_l) etc
why cant we use this on a sample time series?
Becasue it requires that we know the true values of covariances. These are not observable, so we need to estimate them. And estimation is subject to error, so with small samples there is a differnece between those formulas. However, in the limit they approach the true value.
what can we say about the properties of the sample autocorrelation function for some lag?
if {r_t} is iid sequence, then the estimate is asymptotically normal with mean zero and variance 1/T.
This is the base we use to test for autocorrelation differnet to 0, by using a regular t-ratio
what do we get from Bartletts formula?
gives the variance of sample lag-l autocorrelation, given that r_t satisfy the linear time series requirement (r_t = my + ∑constant_i a_{t-i}), where a_t is a time series of its own with zero mean.
how do we test for individual lag-l autocorrelations
Some assumptions first: We need to use an estimator of the lag-l correlation because this is not a directly observable size. The estimator we use is the sample correlation function. A great property of this estimator is that it is asymptotically normal with mean 0 and variance 1/T if the returns are IID.
This allows us to use a t-ratio test. Subtract the mean (0) and divide on standardized variance or whatever and we can use Bartletts formula for this.
We use t-ratio, with Bartlets fromula for variance to get the standard deviation. Since t-ratio, the null hyp is that the lag-l autocorreletion is 0.
what do we need to know regarding the sampel lag-l autocorrelation function+
it is biased wiht order 1/T. Substantial in smaller samples, but is negligible with larger samples.
more effective way of testing autocorrelation functions
Portmanteau test.
elaborate on Portmanteau test
Asymptotically chi squared with m degrees of freedom. Null hyp is that all m autocorrelations are 0.
better version of Portmanteau test
Ljung box
what do we actualyl get from ljung box and portmanteau testing
Indication to whether there is “some” lag with non-zero autocorrelation
what is ACF
autocorrelation function, we take AC of lags from 0 to M, and plot.
elaborate on white noise
white noise is a time series, iid, finite mean and finite variance.
We typically work with gaussian white noise, whcih simply means that the random variables are normally distriubted.
Important regarding white noise: ACF is zero entirely. This means that it is common to test whether something is white noise to see if there are meaningful patterns in the data or not.
define a linear time seires
IMPORTANT: a_t is a white noise series.
a_t
shock, innovation, new information
does the variance of a time series depend on the weights?
yes obvioisuly. if the time series react violently to shocks, then it is more volatile and the variance should be larger.
what can we say about the varaince of a linear time series
The variance is given as:
var(r_t) = sigma_a^2 ∑w_i^2
The sum goes from 0 to infinity. As a result, if the time series is weakly stationariy, this is a convergent property that makes sure that hte variance if finite and bounded. This is a result of how the squared weights approach 0 as the iteration number grow.
what can we say about the autocorrelation function (ACF) of a weakly stationariy linear time series
Since the squared weights approach 0 as the iteration number grow larger, the covariance is smaller and smaller for larger lags. As a result, the corrleation will be smaller and smaller as well. As a result, the ACF is expected to diminish. Converge to 0.
Reasonable, but very important result.
important regarding the white noise series variance
constant
give AR model
autoregressive:
r_t = ø_0 + ø_1 r_{t-1} + a_t
IMP: the shock is that of the same time as the varialbe we are predicting.
what is the conditional mean of the simpel AR(1) model
E[r_t | r_t-1, r_t-2, ….] = ø_0 + ø_1r_{t-1}
what is the conditional variance of simple AR(1) model (conditional on r_t-1, …)
var(r_t | r_t-1, r_t-1,….) =
what is the property we get from the conditiona lmean and variance of the AR model?
Markov property. Conditional on the return r_t-1, the return r_t is not correlated with r_{t-i}, for i>1.
we only care about the order of the model.
how do we find the unconditional mean of the AR(1) model
E(r_t) = ø_0 + ø_1 E(r_{t-1})
now, if the model is stationary, the mean (true mean) is constant. AS a result, the expectaiton we take on both sides are equal, which allow us to simplify:
mu = ø_0 + ø_1 mu
mu (1 - ø_1) = ø_0
mu = ø_0 / (1 - ø_1)
Thus, we have arrived at an expression for the unconditional mean of a simple AR(1) model. The crucial part is stationarity. if we do not have it, we are not able to make this nice closed form result.
how can we prove that AR(1) is a true linear (valid) time series model+
We use the result from the unconditional mean, and by repeated substitutions we can show that the model can be written as an infinite sequence of weights and shock variables from the past.
Specifically for this case, the weights are powered to the sum iteration number, so they converge rapidly to 0.
what is the stationarity condition for AR(p) models
We find the characteristic equation. Its roots are not characteristic roots, but their inverses (the inverses of the roots) are the characteristic roots.
In order to have stationarity, all characteristic roots must be less than 1.
alternatively, all the regular roots must be greater than 1.
2 ways to determine orders of AR models
1) PACF
2) Information criteria
what is PACF
Added contribution of only the direct relationship between a specific lag and the variable of interest. This means, we do not want all of the intermediary effects.
We want to isolate for instance how the returns on Friday and correlated with the returns on Monday without accounting for ripple effects.
The way we can do this for AR models, is to create consecutive models of icnrementally increasing orders. For a true AR(p) process, lags greater than p should have zero influence, and thus zero PACF. We therefore want to identify the cutoff point.
what can we say about PACF is regards to a correlation coefficeint
PACF is more than just a correlation. It is not limited to [-1, 1]. Therefore, it represent more of a direct effect relationship. If it is 2.5, we interpret this as saying that each unit increase in the specific lagged variable has a direct influence of 2.5 on the variable we are estimating. At the same time, the overall effect on the main variable as a result of increasing the lagged variable by a unit, might not be exactly 2.5, because of the regular correlation as well.
elaobrate on using informaiton criteria
Select one or several criteria. AIC, etc.
then compute the value for the criteria for each lagged variable we test for. So, we get one value per added order.
We select the order that gave the lowest information criteria value.
how do we typically estimate hte paramters of a AR(p) model
conditional least squares.
what is important regarding the fitted AR(p) model
it doesnt include the white noise error term. We need to compute this based on the difference between fitted and observed r_t values.
WE call the residuals the “residual series”
how do we do goodness of fit of some model on AR(p) models?
R^2 = 1 - RSS/TSS
Requires stationarity
elaborate on forecasting
We use a model to forecast, which means that we are not actually inserting the noise terms.
The noise is the error. We can measure the expected error, but generally we care about hte forecast itself.
Forecasting is done using the conditional expectation. We can take unconditional as well, but this is much more involved.
Using the conditional expectation, we can generalize it to multi-step forecasting, where we use the recursive structure to build the sequence of forecasts. Meaning, to forecast 3 periods into the future, we must first forecast 1 period, then use this result to forecast 2 periods, then use this new result to forecast 3 periods.
From analyzing the variance of forecasting, we know the variance of the forecasts will increase with the number of steps into the future (variance increase with the forecasting horizon). This is also common sense.
general form MA(q) model
what is required for stationarity?
MA models are always weakly stationary because of how they are finite linear combinations of white noise series.
characteristics of ACF of MA model
cuts of sharply at order
how to determine order of MA model
Because it cuts off hard at some ACF point, we use the ACF to determine order.
Likewise, for AR models, since PACF cuts off sharp at some order, we use PACF to determine order of the AR model.
Moving onto volatility
what do we mean by volatility and why?
Conditional standard deviation of the underlying asset returns.
This is important, because it highlights a crucial point we make: conditional. In this context, conditional does not mean exactly rhe same as in regular statistics. It refers to the fact that we are assuming that there is a possibility that volatility depends on its earlier, previous values.
what is special about volatility
not observable
what is the result of volatility not being observable in the context of volatility models
It is difficult to assess their performance
what is IV
implied volatility refers specifically to the volatility we get from reversing the BS equation
what can we say about IV vs volatility frm GARCH models?
IV tends to be higher
There are some characteristics that are curcial to understadning volatility. elaborateo n them
1) Volatility tends to cluster
2) Volatility moves in a continuous way, meaning that there are no jumps.
3) Volatility does not diverge to infinity. It stays within a range. Mean reverting. typically this means that volatility is close to stationariy.
4) Leverage effect: volatility react more to negative returns than to positive returns.
These 4 characteristics are empirical results mainly. The models we consider will be strong on some of them, and weak on others.
what is the “idea” of volatility study in regards to correlaton?
There is very little autocorrelation, but volatility as a time series is still a dependent series.
What we can see, is that when plotting ACF of log returns, ACF of absolute log returns, ACF of squared log returns, we see that volatility is not serially correlated. however, the squared and absolute ACF show relation. But this is not correlation in the typical sense. It is correlation of the sign of the volatility. This means that volatility is not autocorrelated in the typical way, but is clearly dependent on earlier values.
These dependencies is the key issue of volatility models.
what is the mean equation and what is the volatility equation?
mean equation is a model for the mean mu.
Volatility equation is mostl ikely built on top of the mean equation, and gives the way volatility evolve.
Both are conditional equations.
discuss model building of volatility models
We must remove linear dependence from the model. we do this by building a model for the conditional mean.
Use residuals from the mean equation to test for ARCH effects.
Specify a volatility model if ARCH effects are present (statistically significant).
Check model and refine etc.
elaborate on removing linear dependenceis from a model
“Removing dependencies” means that we are identifying the dependency, and accounting for it. The aim is to produce a model that create residuals that behave like white noise. If there is a predictable pattern in the residual series, then we know that there is a relationship we have not accounted for.
For most asset return series, the serial correlations are weak, if any.
Therefore, building a mean equation amounts to removing the mean (sample mean) from the data, if the sample mean is statistically significant differnet from 0.
The mean equaiton would look like this:
r_t = mu + a_t
If there are no other dependencies, the idea is that by assuming that the best way to predict r_t is by the simple mean constant, the a_t residual series (resulting from fitting a model and obtaining hte residuals), can be proved to be white noise.
In some cases, we might need more than this to remove linear dependencies.
In terms of the volatility equation, since this builds on the mean equation (volatility of the returns etc) we will only be able to accurately predict volatility to the highest level of precision if the mean equation accounts for all the dependencies.
Note on why we say “remove dependencies”: I’d assume that if we take r_t, and subtract the mean and whatever relationships we have accounted for, we “remove dependencies from the return series”. The resulting variable is the a_t residual. So therefore, we can look at the process as trying to reduce a_t to “nothing”.
elaborate on the thought process of volatility models
The conditional volatility of a model is given as Var(model | all prior info). Given all prior info, this quickly becomes Var(a_t | all prior info), as the other terms become constants. Therefore, we are taking the variance of the residuals. The question is, is there a pattern here? Can we predict sigma_t^2 more precisely than the unconditional variance?
And to answer this, we must first try to reduce the residuals down to as close to noise as possible. This is because if we dont, the residuals reflect certain relationships. While possibly important, if we study for volatility, we dont give a fuck about them. Therefore, we want to remove them from consideration. This actually gives us an incentive to create a very good model for the asset return series in itself. If we are able to capture more actual relationships between various variables, our volatility model will also become better. Therefore, if we build a great asset return series model, we also have a great foundation for a volatility model.
Typically, when working with volatility, we say that we want to remove linear dependencies. In the simple case where there are no dependencies, we simply remove the mean, and we have the white noise series as our residuals.
The residuals are white noise, which means that their variance is finite over time, and they have ZERO autocorrelation for all lags. It is important to understand that white noise does NOT have constant variance. We want to see if the variance follow a specific pattern over time that depends on previous values. Volatility is typically close to stationary when looking at long time frames, but the dependent patterns are also present. These patterns are called ARCH effects, and represent what we want to capture with the volatility models
very briefly, how do we test for ARCH effects?
Acquire a mean equation, and produce their residuals
Then square the residual series. The squared residual series is the basis we use to test for ARCH effects.
another word for ARCH effects
conditional heteroskedasticity effects
what tests are available for us to test for ARCH effects?
1) Ljung box
2) Lagrange multiplier test
elaborate on the lagrange multiplier test for ARCH effects
equivalent to the usual F test used in linear regression. But we use a_t squared series.
elaborate on the idea of ARCH models
ARCH is built on the idea that there is no serial autocorrelation between the shocks of an asset return, but dependent, and that this dependence can be described by a simple quadratic relationship.
what are ARCH effects?
ARCH effects refer to the fact that we can use past squared residuals to predict the next squared residual BETTER than the simple mean. IF we can do this, we say that there are ARCH effects present.
We test for ARCH effects by lagrange method that use an F-distribution to test for a number of parameters (correspondng to squared “lags” of the residulas”) and checks whether some of hte coefficients are non-zero.
IF some are non-zero, there are arch effects (statistically significant non-zero, not actually purely non-zero).
what is the general form of ARCH(q) model=
negatives with ARCH models
not accounting for leverage effects.
limited in general.
likely to overpredict.
does the GARCH model have ARCH parameters?
Yes. GARCH is basically the same as ARCH, but with another summation term that includes lagged variables of the volatility as well.
We refer to these components as ARCH and GARCH parameters.
what can we say about the fourth moment of ARCH and GARCH models?
Fatter tails than normal distribution. This is nice because this is more in line with outliers, which we know appear empirically.
what do we mean by this? What defines the residuals?
We obtain residuals from the mean equation.
However, if the process is a true garch(1,1)-m process, the residuals will follow the structure provided by the equation with the epsilon.
what is special about garch m models?
allow returns to be dependent on the volatility.
what is a multi variate time series model?
We are no longer in the world of volatility.
we consider for instance logged asset returns for a number of differnet assets. we put them together into a vector, at treat is as a multivariate model.
what is VAR
Vector AutoRegressive models.
in vector-matrix notation it looks exactly like the regular AR model. but now, we have vectors and matrices.
This is the VAR(1) model. Elaborate on what it actually entails and represent
most significant is the matrix.
The matrix holds the contribution that the other return series has on each other, conditionally on their own values.
So, if we have VAR(1), and we have 2 asset return series, then this matrix is a 2x2 matrix.
For r_{1t}, which represent the return series 1 at time t, the matrix has the row “a_11 r_{1, t-1} + a_12 r_{2, t-1}” and for r_{2,t}:
“a_21 r_{1, t-1} + a_22 r_{2, t-1}”.
the order of the terms is determiend by the matrix, so there is no “sense” here.
The important part is the coefficients that actually establish whether there is a relationship between some asset return at time t-1 and the other asset return at time t.
difference between concurrent and dynamic relationships in multivariate models?
Concurrent refers to relationships at the same tiem period.
Dynamic refers to relationships at different time periods.
how do we find concurrent relationships?
We use the covariance metrix, ∑, of the residuals a_t.
If there is a non zero eleemnt at some place other than the main diagonal, we know that there is a concurrent relationship.
