Chapter 2 - Time series Flashcards
(55 cards)
in general, what can we say that simple tiem series models attempt to do?
capture teh linear relationship between r_t and all prior information.
why is correlation/covariance linear
we take two random variables. Then subtract their mean, which gives us only the “differnece” in both cases (movement, relative, not absolute). We can write this as : E[AB], where A and B is the mean-adjusted variables. Since A and B are random variables, they are associated with a probability distribution. When multiplied together, we get a joint distribution. When taking expectation, we average over the distribution to find the most likely outcome (the weighted joint average). but since A and B hold relative movements, the expectation gives us the “expected joint movement”. We can consider this as the average movement in X relative to average movement in Y, and if they move opposite of each other, we get something negative as the covriance. This is inherently linear, because we compare movemnt in one variable as a function of another linear variable. I suppose that if we did E[AB^2] instead, so that B is squared, we’d model something non linear.
REMEMBER: The whole concept is to explore correlation. Movement in one variable is associated with movement in some other variable. Therefor,e it makes sense to consider locksteps.
recall the variance property of weakly stationary time series
constant variance.
this means that var(r_t) == var(r_{t-1}) etc
elaborate on very basic hypothesis testing of lag correlation
We use lag-l sample correlation estimator. Under some conditions, lag-l sample correlation is a consistent estimator to the real lag-l correlation (autocorrelation).
For instance, we require the secodn moment of the return series to be bounded. Meaning, bounded variance.
we also require r_t to be iid.
Under these assumptions, lag-l sample correlation is asymptotically normal with mean 0 and variance 1/T.
Then we’d use the t-ratio test to figure out if the coefficient is statistically significant or not.
However, Bartlett is better because it assumes weakly stationary and bounded second moment, wheras the previous one require iid sequence. So the different variances test for different things.
there is sometihng key to rememver that simplify the lag-l sample correlation estimator?
var(r_t) == var(r_{t-1})
This means that we can simply use var(r_t) in the denominator instead of both lags.
briefly mention null and alternaive hypothesis when using Bartlett for testing of individual lag autocorrelations
null: p=0
h1: p!=0
If the time series is weakly stationary and bounded second moment, we expect the sample lag-l autocorrelation estimator to asynmptotically follow normal distirbution with mean 0 and variance as given by bartletts formula. Based on THIS,. we test whether the value observed was extreme enough to say that the correlation is most definitely not 0 or not.
when using the t-ratio, we operate with a standard normal vairable.
what is portmantaeu
sammentrekning, flere i en
What does Q statistic and Ljung Box do?
Test multiple lags at once. Chi squared
Q-statistic is simple. Sums together “m” squares of lag-l autocorrelations for differnet lags. Since they are assumed normal given the estimator (sample, aysymptotic etc) we know that the sum of squares gives a chi squared variable with degrees of freedom equal to the number of squared entries: m.
Q is bad in small samples.
if we have estimates for p_1, p_2, … what do we have?
sample autocorrelation function
elaborate on white noise
A tiem series is defined as white noise if it is iid with finite mean and variance.
Gaussian noise has mean 0 and variance sigma^2.in
in practice, what qualifies as white noise?
if all sample ACF’s are 0, we have white noise.
Assume we use 1/T (iid + finite second moment finite/bounded) to apply a t-ratio to test for ACF. We get significant result and reject null. What can we say in regards to Bartlett+
IMPORTANT CARD
The thing is that they are used to test for different things. 1/T is used to test for whether the tiem series is iid or not (white noise, basically). Bartlett is an assumption (weak stationarity) that we can use. We need to test whether this assumption is valid or not by testing for stationarity. if stationary, we can use bartlett for variance to test for significant ACF lags.
if the series is iid, the ACF should have variance 1/T. If the estimator gives high values, it would indicate that the series is NOT iid. it would not be iid because there is a clear structure where ACF show signs of linear correlation.
If the series is weakly stationary, the variance is Bartlett. therefore, a large value from ACF relative to Bartlett would idnicate that the series is autocorrelated. We have to assume weak stationarity to use Bartlett.
IT IS WORHT ADDING that if ACF is insignificant at 1/T, it is also insig at Bartlett. Therefore, 1/T is a nice quick-test.
define a linear time series
A tiem series is linear if it can be written as:
r_t = mu + ∑w_i a_{t-i}
where a_t is a sequence of iid random variables with mean zero and a well defined distribution (white noise).
give the simple AR(1) model
r_t = ø_0 + ø_1 r_{t-1} + a_t
given the simple AR model, what is its conditional mean and variance?
E[r_t | r_{t-1}] = ø_0 + ø_1 r_{t-1}
Var[r_t |r_{t-1}] = Var(a_t) = sigma_a^2
for a simple AR(1) model, give the requirements for stationarity
We need constant mean ,constant variance, and constant lag-l autocovariance.
Start with constant mean.
E[r_t] = mu
E[r_t] = ø_0 + ø_1 E[r_{t-1}]
if the mean is constant, we have :
mu(1-ø_1) = ø_0
mu = ø_0 / (1 - ø_1)
This entails that ø_1 cannot be 0, and we know that if ø_0 is 0 then the mean mu is 0 as well.
VARIANCE:
what is the stationarity requirement for AR models
unit roots larger than 1 in absolute value
what is characteristic root?
1/x, where x
briedly itnroduce how we can determine the order of AR model
1) PACF
2) Information criteria
how do we use AIC and BIC?
Select the one with the smallest value
when forecasting, how do we do it?
We forecase from some origin and up to some point.
We need to represent forecasting as conditional expectation of r_{t+1}. This basically removes the error term / shocks term at time t+1, as it has zero mean.
The variance of the error is sigma_a^2.
Basides from this, we simply use the model in a deterministic way.
build the ACF for MA
WE start wit hthe MA model, and subtract the mean. This is equivalent to making it zero-mean. Since MA models have mean equal to the constant term, this is done by subtracting this constant from both sides.
The new time series is adjusted.
We take r_t and r_{t-l}, multiply them together and take expectation. This is the covariance computation. Normally, we’d include subtracting the mean here, for both variables, but since we have established a time series with zero mean, we dont need to.
E[r_t r_{t-l}].
then we need to change both into their “model” form.
Taking the expectation of this will yield -w_i sigma_a^2 for lag 1, 0 for all above.
stationairty condition for MA models
THey are always stationary because they are linear combinations of white noise sequence for which the first two moments are time invariant.
elaborate on estimating the parameters in the MA models
MLE is the way to go.
There are 2 ways of performing MLE:
1) Conditional maximum liklihood
2) Exact MLE
Conditional:
It is typical to assume that the very first hsocks are 0. Then, we can start at r_0 with shocks at earlier times than 0, and 0 itself, being 0. We then get shocks according to:
a_t = r_t - c_0
so, a_1 = r_1 - c_0.
a_2 = r_2 - c_0 + a_1 ø_1
EXACT:
Treats earlier shocks as parameters to be estimated as well. More expensive computationally.
Under large sample sizes, these two methods bhave similarily.
