Tsay 2

Question 1

Define a time series

Answer 1

A collection of random variables over time, e.g. asset returns over time

Question 2

What is AR

Answer 2

AutoRegressive models

Question 3

what is MA

Answer 3

Moving Average models

Question 4

What is ARMA

Answer 4

mixed autoregressive moving average models

Question 5

In general, very broadly speaking, what do we say that simple timeseries models attempt to do?=

Answer 5

Use information prior to the current point to predict the next point

Question 6

name the fundamentals correlations

Answer 6

serial correlation and autocorrelations

Question 7

What is the foundation of time series forecasting/analysis

Answer 7

Stationarity

Question 8

Define strict stationarity

Answer 8

In other words, the distribution of the values is not affected by a shift in time.

Strict stationarity entails that any subset of the random variables have the same joint distribution regardless of WHEN we observe them. This is difficult to empirically validate, and is typically considered too strict in finance.

Question 9

Define weak stationarity. Also elaborate on staitonarituy

Answer 9

A tiem series is weakly stationariy if both its mean and covariance between r_t and r_{t-l} are time invariant, where l is an arbitrary integer.

This simply means: r_t, which is a time series and consists of a number of random variables through time, and r_{t-l} have a covariance that is time invariant. This means that their covariance does not depend on time, but depend only on l.

Recall that if the lag is 0, we get covariance of itself, which is the variance of the time series. Therefore, by requering covariance being cosntant, we also get that the variance must be constant.

Stationarity therefore entails:
1) same mean across all time series of the same length (same number of random variables).
2) Same variance
3) Same covariance

The covariance has an important interpretaiton. constant Covariance entail that the impact a lagged variable has on future variable remains constant. this basically enforce no seasonality.

CRUCIAL: At first, stationarity appears abstract. Difficult to place. For instance, stock prices doesnt exactly resemble stationarity. Same goes for most other time series patterns as well.
However, the key lies in what we can do to enforce stationarity on it. For instance, if we take the daily returns instead of daily stock prices as a time series, the resulting seires is much closer to a stionary process.

Question 10

what can we say about hte first two moments of a time series when assuming weak stationarity

Answer 10

they are fixed constants

Question 11

what is lag-l autocovariance?

Answer 11

the result of taking the covariance between r_t and r_t shifted by l: r_{t-l}. It gives a value. With weak staitonarity, this value is finite.

lag-l autocovariance has 2 important properties:
1) when l is 0, the lag-l autocovariance is equal to the variance of r_t.
2) gamma_l = gamma_{-l}, where gamma_l is the lag-l autocovariance between r_t and r_{t-l}.

The second property is that we simply say the lag-l autocovariance is the same whether we shift the time series up or down.

Question 12

give definition of the correlation coefficient between two random variables

Answer 12

p = cov(x,y/sqrt(var(x) var(y))

alternaively:

p = cov(x,y)/SD(x)SD(y)

Question 13

when is correlation coefficent zero

Answer 13

If X and Y are normal variables, they are uncorrelated if and only if they are independent.

Question 14

how do compute the correlation of a sample {(x_t, y_t)}^T_(t=1)

Answer 14

we extend the formula to:

Question 15

what is ACF

Answer 15

AutoCorrelationFunction

Question 16

what is lag-l autocorrelation?

Answer 16

the correlation coefficent of the time series r_t and r_{t-l}.

Question 17

how do we compute lag-l autocorrelation?

Answer 17

here, the property that var(r_t) = var(r_{t-l}) for weak stationarity is applied.

Question 18

what is the difference between serial correlation and autocorrelatoin?

Answer 18

It is the same tihng

Question 19

what is a time series not autocorrelated at all? a weakly stationary one

Answer 19

the lag-l autocorrelation must be 0 for all values of l greater than 0.

Question 20

what is the formula for lag-1 autocorrelation for a sample of returns, {r_t}^T_(t=1)?

Answer 20

Question 21

Give the sample lag-l autocorrelation formula for a sample of returns time series

Answer 21

Question 22

what is important to understand about sample lag-l autocorrelation formula

Answer 22

It is estimator.

if {r_t} is iid and E[r_t^2]< infinity, then the estimator for p_l is asymptotically normal with mean zero and variance 1/T. We can use this to test using t-ratio.

Question 23

elaborate on testing individual ACF

Answer 23

We can use the t-ratio to test against whether the lag-l autocorrelation is 0 or not:

Question 24

elaborate on this formula

Answer 24

it say that each random variable in the time series r_t is the result of applying a funciton that consists of a mean mu, plus the sum-product of shocks and their coefficients that took place during the last q time steps. Therefore, the formula look q steps back in time, and compute today’s value based on previous events. It is a linear way of doing it.

If {a_j} is a sequence of iid random variables with mean zero, then the sample lag-l autocorrelation coefficnet is asynptotically normal with mean zero and varianec as given by Bartletts formula.

Question 25

what is Bartlett's formula

Answer 25

the variance of the sample estimator of lag-l autocorrelation as discussed earlier is equal to (1+2∑(p_i)^2)/T for l>q.

Question 26

what does it mean that a sequence is iid?

Answer 26

independent and identicially distributed. It means that each variable doesnt depend on earlier variables (since we're dealing with time series) and identically disitrbuted means that they follow the same distribution. They must also have the same parameters of the distributons (mean, variance etc).

Question 27

What is the Portmanteau statistic?

Answer 27

Useful when we want to test that several lag-l autocorrelations of r_t are 0. We'll have the null hypothesis that: H0: p_1 = p_2 = ... = p_m = 0 Under the assumption that {r_t} is iid, with certain moment conditions, this test statistic is asymptotically chi-squared with m degrees of freedom. The moment conditions is basically: finite variance, weak stationarity, finite fourth moment etc

Question 28

why is portmanteau good/bad?

Answer 28

require a lot of sample size. It is asymptotically normal, meaning that when the sample size grow large (T grow large) the test statistic receive normal behavior.

Question 29

Elaborate on Ljung and box

Answer 29

They modify the portmanteau statistic so that the power of the test is better for finite sample sizes.

Question 30

what exactly is the sample size we're discussing+

Answer 30

The number of random variables in the time series. So, if we are modeling a stock, and we have one random variable per day, and we track for 100 days, the sample size will be 100.

Question 31

what should we choose as the value for m? First, recall what m is

Answer 31

m is the number of autocorrelated lags, so that we test for lag-l autocorrelation for m different shifts or lags. Simulation studies suggests that the best performance is to set m=ln(T). Performance in terms of the power of the test. Recall that the power of the test is the probability of rejecting the null hypothesis when it is false.

Question 32

what exactly is the ACF

Answer 32

AutoCorrelationFunction refers to the correlation estimator that is based on a sample.

Question 33

define white noise

Answer 33

we say that a time series {r_t} is white noise if it has a sequence of iid variables with finite mean and finite variance. For instance, if it has mean zero and variance sigma^2, it is called gaussian white noise.

Question 34

what is special about a white noise series?

Answer 34

All the ACF's are zero. In practice, it holds for ACF's that are close to zero to be defined as white noise.

Question 35

white noise series has ACF's all equal to 0. Why?

Answer 35

An assumption of white noise is that all of the random variables in the time series are iid. As a result of independence, we have zero autocorrelation.

Question 36

define a linear time series

Answer 36

A time series is linear if it can be described as: This notation say: For each random variable r_t in the time series {r_t}^T_{t=1}, we go through an infinite sum back in time to accumulate all of the effects.

Question 37

what do we typically use the fucking trident symbol for?

Answer 37

Some kind of weight, as it resemble a w.

Question 38

elaborate on the convergence of the time series of the weights that are used in the linear model

Answer 38

Since the definition of a linear time series if infinitely long, we need the weights to be such that they converge. Otherwise, we dont get nice results.

Question 39

What is AR(1)

Answer 39

Auto regressive model of order 1. Auto regressive refer to regression models that are used on time series. With order 1, we have: r_t = ø_0 + ø_1r_(t-1) + a_t, where {a_t} is assumed to be a white noise series with mean 0 and variance sigma_a^2. In time series litterature, this is called autoregressive. But it is just the same as dependent and independent variables used in linear regresson.

Question 40

is AR same as linear regression?

Answer 40

there are some similariaties but also important differences

Question 41

for AR(1), what is E[r_t | r_{t-1}] and same for variance?

Answer 41

E = ø_0+ø_1r_{t-1} Var = sigma_a^2

Question 42

if we were to generalize the AR model, what would it look like?

Answer 42

Question 43

what is the expectation of an AR(1) model?

Answer 43

E[r_t] = ø_0+ø_1E[r_{t-1}]. because of stationarity, we know that the expected value of one random varialbe in the tiem series is equal to all the others, and we get: E[r_t] = E[r_{t-1}] = mu. Hence, we get: mu = ø_0 + ø1mu, or equivalently: E[r_t] = mu = ø_0/(1-ø_1)

Question 44

what are the implications of this?

Answer 44

1) The expected value is not defined for ø_1 = 1., also the mean is not defined for ø_1 = 1. 2) ø_0 = 0 imply that the mean of the time series if also 0.

Question 45

I am missing sometihng on properties of AR(1) model, but idgaf

Question 46

Recall how the AR(1) model looks like

Answer 46

r_t = ø_0 + ø_1r_{t-1} + a_t

Question 47

Recall the expectation of the AR(1) model

Answer 47

E[r_t] = ø_0+ø_1E[r_{t-1}]+0 mu = ø_0 / (1- ø_1)

Question 48

elaborate on the covariance of r_{t-1} and a_t

Answer 48

it is zero. This can be understood by considering how the r_t-1 variable occured earlier in time than a_t, and a_t is assumed to be white noise and not dependent on earlier informaiton.

Question 49

Find the variance of r_t in AR(1)

Answer 49

Ar(1): r_t = ø_0 + ø_1r_{t-1}+a_t Var(r_t) = (ø_1)^2Var(r_r-1) + var(a_t) Var(r_t) = (ø_1)^2var(r_t) + sigma_a^2 var(r_t) = sigma_a^2 / (1-(ø_1)^2)

Question 50

what form is taken by AR(2) models

Answer 50

Question 51

recall what ACF is

Answer 51

autocorrelation function.

Question 52

what is gamma_l

Answer 52

gamma_l represent the covariance between a time series and its lag-l corresponding time series. as a result, gamma_l is called the autocovariance of lag-l.

Question 53

give a intuitive definition of stationarity

Answer 53

the time series must sort of act the same the entire time. We need constant volatility, constant mean, no seasonality.

Question 54

is white noise stationary?

Answer 54

Yes. but obvs not everything that is statonary is white noise

Question 55

if sometihng is not stationary, what can we do?

Answer 55

We can apply some fucntion to make the time series into a new time series that is a little different. The key is that we need the mean and the volatility to be constant, and also have no seasonality.

Question 56

elaborate on why time series are treated differently than other lienar regression models

Answer 56

time series forecasting is ultiamtely extrapolation, and not interpolation. It is also very intersting to consider what happens when we extrapolate multiple time steps into the future. The error is going to grow larger and larger in a cone shaped way. This is a huge problem, and it require careful consideration. Time series is simply more difficult than their interpolation (past) counterparts.

Question 57

What is ACF and PACF

Answer 57

AutoCorrelationFunction and PartialAutoCorrelationFunction

Question 58

why do we lag multiple shifts?

Answer 58

Because prices multiple periods before can have an effect on current price. It doesnt only have to propagate from period to period.

Question 59

what is autocorrelation?

Answer 59

The correlation between time points

Question 60

what is important to understand regarding the ACF

Answer 60

it incorporates both the direct correlation effect between lagged series and real series, and also the propagated effect.

Question 61

elaborate on PACF

Answer 61

We only want to isolate the effect that go from lagged time step to current time step, without the propagated effect. In other words, if we have 2-month lag, we're looking at the effect that january alone has on march etc.

Question 62

why might we be interested in PACF?

Answer 62

If we want to see whether the lagged time series is a good indicator/predictor of the current. If we only use ACF we get the propagated effect as well, and not only the direct effect. Therefore, isolating the direct effect tells us whether the lag is good on its own or not.

Question 63

how can we find PACF?

Answer 63

set up regression model. The weights represent the PACF for the various lagged variables.

Question 64

broadly speaking, what is our goal with the whole topic of autocorrelation?

Answer 64

Our ultiamte goal is to predict future values. We need to understand what the past data can say about the future, and isolate their effects.

Question 65

define white noise

Answer 65

White noise is just a type of time series that meet a list of specific criteria: - mean = 0 - standard deviation is constant with time - correlation between lags is 0 correlation 0 mean that if we look at a time series that is basically white noise, we will find no pattern of correlation between any prior time step value vs any other time step value

Question 66

what is the biggest "thing" about white noise?

Answer 66

NOT PREDICTABLE

Question 67

why do we care about white noise?

Answer 67

Whenever we make a model, we assume that we have two things: y_t = signal + noise The signal can be predicted, but the noise (white) cannot. However, we can test to see whether the noise is white or not. If it is white noise, we basically have a very good model, because we have predicted what can be predicted.

Question 68

how to test if something is white noise

Answer 68

1) visual test 2) global vs local test 3) Check the ACP plot

Question 69

what exactly does Auto regressive model mean?

Answer 69

Regression where we predict somthing by using the same thing but at past time

Question 70

elaborate on ACF

Answer 70

we have a time series, and the goal is to be able to say something about how much correlation (or autocorrelation since it is a time series) that there is between a point and some point in the past. The ACF is simply computing the autocorrelation for a specific lag case. Generally speaking, given a time series with many random variables, all of the variables will be used to consider the effect. So, the sample correlation coefficient computed for say lag-4 will be a generalized number that represent how much correlation there is between a value ,and the value that is 4 time steps ahead. The coefficient value is important because it represent how much autocorrelation there is. We can use it in combination with hypothesis tests to understand whether a specific lag-variable contribute, or hold any predictive power in regards to the future

Question 71

elaborate on the properties of a sample PACF for a stationary gaussian AR(p) model

Answer 71

First of all, the sample PACF is the parameter that has j=p. properties are: 1) the estimator ^ø_jj converge to the true value when the sample size T goes to infinity. 2) the estimator ^Ø_ll converge to 0 for all l>p 3) the asynptotic varaince of the estimator ^Ø_ll is 1/T for l>p.

Question 72

elaborate on the derivation of the MA's

Answer 72

we start by entertaining the idea of infinite AR series. Then we make the constraint that each parameter must be ø_i=-theta^i_1. Dont forget the negative. Thus, the series has the same negative theta term, but to increasing power. with the restriction that theta must have absolute vlaue less than 1, these values decay exponentially, which is nice. Then we basically have: r_t = ø_0 - theta_1 r_(t-1) ..... + a_t We re-write to get ø_0 and a_t on RHS, and r_t - everything else on the LHS. Then we do the very same thing for the r_(t-1) series. the r_(t-1) series will have the powers of the theta values shifted by 1 relative to the r_t series. Therefore, we multiply by theta_1. Then we take it, and subtract it from the original series. The result is simply a clean LHS and a RHS that makes sense. We get: r_t = ø_0 + a_t - theta_1(ø_0+a_(t-1)) r_t = ø_0 - theta_1ø_0 + a_t - theta_1 a_(t-1) r_t = ø_0 (1 - theta_1) + a_t - theta_1 a_(t-1) We can make it more general by collapsing the constnat into "c": r_t = c + a_t - theta_1 a_(t-1) The idea is that this is a weighted average of the shocks.

Question 73

give the MA(q) extension

Answer 73

Question 74

is MA models stationary?

Answer 74

Yes. They limit the thetas to smaller than 1, so we have that the r_t is basically just a linear combinaiton of white noise variables which is always finite. The expectation is simply the constant term, while the variance is a little bit more hairy, but still time invariant. it is basiclaly just variance of some white noise terms, and white noise terms are uncorrelated, iid.

Question 75

elaborate on MA cutoff

Answer 75

cuts off at q. Sharply. This comes as a result of how the model assumes how many lags it need. This may not be beneficial. the assumption of zero c-term just change the ACF values, not the cutoff point.

Question 76

is MA finite memory?

Answer 76

Yes, because it sharply cuts off at some point. Doesnt rmember anything beyond that

Question 77

how do we estimate the parameters of MA models?

Answer 77

typically use maximum liklihoof

Question 78

what variants of maximum likelihood do we havefor MA models?

Answer 78

conditional-likelihood exact-likelihood

Question 79

elaborate on conditional likelihood

Answer 79

THe conditional part comes from an assumption we make which is that "no data point before our time series start will influence the model". This imply that we set a_t = 0 for t<=0. With this in mind, we build a recursive function that express the shock series. a_1 = r_1 - c_0 a_2 = r_2 - c_0 + theta_1 a_1 and so on. Recall that the MA Model only use the shocks and theta somehow.

Question 80

elaborate on the exact likelihood method

Answer 80

The exact method treat the earlier shocks are random variables that are estimated with the model.

Question 81

when is exact likelihood method preferred

Answer 81

especially when the MA model is close to being non-invertble.

Question 82

what can we say about the multi-step forecasting abilities of MA models?

Answer 82

for MA(1) it becomes just the mean, c_0. Single step forecasting is c_0 - theta_1 a_h. In general, the MA(q) model go to the mean after q steps in the future.

Question 83

define an ARMA(1,1) model

Answer 83

A time series r_t follow ARMA(1,1) if it has the following form: r_t - ø_1 r_(t-1) = ø_0 + a_t - theta_1 a_(t-1) the series {a_t} is white noise. The LHS is the AR model and the RHS is the MA model.

Question 84

elaborate on properties of ARMA(1,1) model

Answer 84

if we take the expectation, we get that the expected value is the same as for the AR(1) model, provided that the time series is weakly stationary.

Question 85

stationarity is one case. Elaborate on others

Answer 85

WE have non-stationairty, typically referred to as "unit root non-stationary time series". This happens when the model is basically just random. For instance, if we have the model: r_t = ø_1r_(t-1) + a_t, and ø_1 = 1, we get that r_t = r_(t-1) + a_t, which is just a random walk. If ø is greater than 1, the time series explode.

Question 86

what is a change series

Answer 86

the series we get by looking at differences between the values of the time series rather then the tiem series values themselves

Question 87

what is differencing

Answer 87

differencing is the process of making a non-stationary time series into a stationary time series by finding its change series.

Question 88

define a seasonal time series

Answer 88

A seasonal time series is just a time series that follow cyclic patterns

Question 89

how can we remove exponential growth

Answer 89

take the logarithm. This gives us linear growth instead.

Question 90

what is the difference between conventional differencing and seasonal differencing?

Answer 90

Conventional differnecing, also referred to as regular differencing, is given by: ∆y_t = y_t - y_{t-1} = (1-B)y_t. Seasonal differencing is: ∆_t = (1- B^4). If a time series y_t has periodicity "s", we have: ∆_sy_t = (1-B^s)y_t = y_t - y_{t-s}