Introduction and Basic principles

Question 1

Define a time series

Answer 1

A set of observations {xt}t element of A recorded across time and is a realisation fo a sequence of random variables {Xt}t element of A. Main characteristic iod time series data has dependence and usually here A is finite as we observe over fixed time

Question 2

Why does time series data need spearate analysis methods/ studies

Answer 2

Correlation with sampling over adjacent time points can restrict use of traditional stats methods which often assume observations are iid.

Question 3

When is a time series gaussian ?

Answer 3

Sequence {Xt} is gaussian time series if Xt1, Xt2, … , Xtk follow a multivariate normal distribution for all time indexes

Question 4

Define statistical noise

Answer 4

Unexplained variability with a data sample

Question 5

What is independent white noise?

Answer 5

When White noises are all IID

Question 6

What is the value of delta in a random walk

Answer 6

The drift. If drift = 0 Xt is simply a random walk, otherwise this drift term measures the trend of the random walk. Graph of Xt will be based loosely around line Y = Delta*x

Question 7

Why is it called a random walk?

Answer 7

Value of time series at time t is value of the time series at t-l plus a completely random movement determined by Wt

Question 8

Explain the idea of signal?

Answer 8

In general we employ simple additive model of some unknown signal + time series that may be white or correlated. Many realistic models for time series assume a signal with some consistent period variation which then gets contaminated by adding of random noise

Question 9

Why do we rely on mean functions of time series?

Answer 9

A complete description of a time series (n RVs observed at K times) is provided by a multidimensional distribution function that is not easily written out. Often we compare different realisations of the time series to the mean function

Question 10

What does autocovariance measure?

Answer 10

Linear dependence between two points on the same series observed at different times.

Question 11

Explain how we can tell if a series is smooth or choppy over time through autocovariance

Answer 11

Smooth series autocovariance functions stay large even if t and s are far apart, choppy series will have autocovariance functions close to 0 for large separations.

Question 12

If Autocovariance function is 0 what does that mean?

Answer 12

Xs and Xt are not linearly related

Question 13

If Xs and Xt are bivariate Normal what doesn Autocovariance function of 0 mean

Answer 13

Xs and Xt are also independent

Question 14

If X and Y are idnependent what is their covariance

Answer 14

0

Question 15

Covariance (X,Y) = Covariance of (Y,X)?

Answer 15

Yes

Question 16

What is the autocorrelation of a time series?

Answer 16

Correlation of Xs and Ys across time for s and t. The ACF measures the linear predictability of the series at time t using only the values of the xs

Question 17

What value can the Autocorrelation function take?

Answer 17

Between -1 and 1 only

Question 18

If cor(Xt , Xs)= 1 what does this imply?

Answer 18

We can predict xt perfectly from xs and there is a positive linear relationship xt= a+bxs

Question 19

If cor(Xt , Xs)= -1 what does this imply?

Answer 19

We can predict xt perfectly from xs and there is a negative linear relationship xt= a-bxs