1. Introduction to Time Series Analysis
Gloria González-Rivera
University of California, Riverside
and
Jesús Gonzalo U. Carlos III de Madrid
Spring 2002
Copyright(© MTS-2002GG): You are free to use and modify these slides for educational purposes, but please if you improve this material send us your new version.

2. Brief Review of Probability
Sample Space: , the set of possible outcomes of some random experiment
Outcome: , a single element of the Sample Space
Event: , a subset of the Sample Space
Field: , the collection of Events we will be considering
Random Variables: , a function from the Sample Space  to a State Space S
State Space: S, a space containing the possible values of a random variables –common choices are the integers N, reals R, k-vectors Rk, complex numbers C, positive reals R+, etc
Probability: , obeying the three rules that you must very well know
Distribution: is the Borel sets (intervals, etc)

3. Brief Review (cont)
Random Vectors: Z= (Z1, Z2 , ..., Zn) is a n-dimensional random vector if its components Z1 , ..., Zn are one-dimensional real-valued random variables
If we interpret t=1, ..., n as equidistant instants of time, Zt can stand for the outcome of an experiment at time t . Such a time series may, for example, consists of Toyota share prices Zt at n succeeding days.
The new aspect now, compared to a one-dimensional radnom variable, is that now we can talk about the dependence structure of the random vector.
Distribution function FZ of Z : It is the collection of the probabilities

4. Stochastic Processes
We suppose that the exchange rate €/$ at every fixed instant t between 5p.m and 6p.m. this afternoon is random. Therefore we can interpret it as a realization Zt(w) of the random variable Zt, and so we observe Zt(w), 5<t<6. In order to make a guess at 6 p.m. about the exchange rate Z19(w) at 7 p.m. it is reasonable to look at the whole evolution of Zt(w) between 5 and 6 p.m. A mathematical model describing this evolution is called a stochastic process.

5. Stochastic Processes (cont)
A stochastic process is a collection of time indexed random variables
defined on some space W.
Suppose that
(1) For a fixed t
This is just a random variable.
(2) For fixed
This is a realization or sample function
Changing the time index, we can generate several random variables:
Add graph to generate intuition
From which a realization is:
This collection of random variables is called a STOCHASTIC PROCESS
A realization of the stochastic process is called a TIME SERIES

6. Examples of stochastic processes
E1: Let the index set be T={1, 2, 3} and let the space of outcomes (W) be the possible outcomes associated with tossing one dice:
W={1, 2, 3, ,4 ,5, 6}
Define
Z(t, w)= t + [value on dice]2 t
Therefore for a particular w, say w3={3}, the realization or path would be (10, 20, 30).
Q1: Draw all the different realizations (six) of this stochastic process.
Q2: Think on an economic relevant variable as an stochastic process and write down an example similar to E1 with it. Specify very clear the sample space and the “rule” that generates the stochastic process.
E2: A Brownian Motion B=(Bt, t [0, infty]):
It starts at zero: Bo=0
It has stationary, independent increments
For evey t>0, Bt has a normal N(0, t) distribution
It has continuous sample paths: “no jumps”.

7. Distribution of a Stochastic Process
In analogy to random variables and random vectors we want to introduce non-random characteristics of a stochastic process such as its distribution, expectation, etc. and describe its dependence structure. This is a task much more complicated that the description of a random vector. Indeed, a non-trivial stochastic process Z=(Zt, t  T) with infinite index set T is an infinite-dimensional object; it can be inderstood as the infinite collection of the random variables Zt, t  T. Since the values of Z are functions on T, the distribution of Z should be defined on subsets of a certain “function space”, i.e.
P(X  A), A  F,
where F is a collection of suitable subsets of this space of functions. This approach is possible, but requires advanced mathematics, and so we will try something simpler.
The finite-dimensional distributions (fidis) of the stochastic process Z are the distributions of the finite-dimensional vectors
(Zt1,..., Ztn), t1, ..., tn T,
for all possible choices of times t1, ..., tn  T and every n  1.

8. Stationarity
Consider the joint probability distribution of the collection of
random variables
1st order stationary process if
2nd order stationary process if
n-order stationary process if
Definition.
A process is strongly (strictly) stationary if it is a n-order stationary process for any n.

9. Moments

10. Moments (cont)
For strictly stationary process:
because
provided that
The correlation between any two random variables depends on the
time difference

11. A process is said to be n-order weakly stationary if all its joint
Weak Stationarity
A process is said to be n-order weakly stationary if all its joint
moments up to order n exist and are time invariant.
Covariance stationary process (2nd order weakly stationary):
constant mean
constant variance
covariance function depends on time difference between R.V.
Write graphs showing a process with time-variant mean, and time-variant variance

12. Autocovariance and Autocorrelation Functions
For a covariance stationary process:
Make a picture of the autocorrelogram

13. Properties of the autocorrelation function

14. Partial Autocorrelation Function (conditional correlation)
This function gives the correlation between two random variables
that are k periods apart when the in-between linear dependence
(between t and t+k ) is removed.
Motivation
Think about a regression model
(without loss of generality, assume that E(Z)=0)

15. Dividing by the variance of the process:
Yule-Walker
equations

16. Examples of stochastic processes
Yt if t is even
E4: Zt=
Yt if t is odd
where Yt is a stationary time series. Is Zt weak stationary?
E5: Define the process
St = X Xn ,
where Xi is iid (0, s2). Show that for h>0
Cov (St+h, St) = t s2,
and therefore St is not weak stationary.

17. Examples of stochastic processes (cont)
E6: White Noise Process
A sequence of uncorrelated random variables is called a white noise
process. k

18. Dependence: Ergodicity
See Reading 1 from Leo Breiman (1969) “Probability and Stochastic Processes: With a View Toward Applications”
We want to allow as much dependence as the Law of Large Numbers (LLN) let us do it
Stationarity is not enough as the following example shows:
E7: Let {Ut} be a sequence of iid r.v uniformly distributed on [0, 1] and let Z be N(0,1) independent of {Ut}.
Define Yt=Z+Ut . Then Yt is stationary (why?), but
The problem is that there is too much dependence in the sequence {Yt}. In fact the correlation between Y1 and Yt is always positive for any value of t.

19. Ergodicity for the mean
Objective: estimate the mean of the process
Need to distinguishing between:
1. Ensemble average
2. Time average
Which estimator is the most appropriate?
Ensemble average
Problem: It is impossible to calculate
Under which circumstances we can use the time average?
Is the time average an unbiased and consistent estimator of the mean?

20. Ergodicity for the mean (cont)
Reminder. Sufficient conditions for consistency of an estimator.
1. Time average is asymptotically unbiased
2. Time average is consistent for the mean

21. Ergodicity for the mean (cont)
A covariance-stationary process is ergodic for the mean if
A sufficient condition for ergodicity for the mean is

22. Ergodicity under Gaussanity
is a stationary Gaussian process, If
is sufficient to ensure ergodicity for all moments

23. Parametric and linear models
Where are We?
The Prediction Problem as a Motivating Problem:
Predict Zt+1 given some information set It at time t.
The conditional expectation can be modeled in a parametric way or in a non-parametric way. We will choose in this course the former. Parametric models can be linear or non-linear. We will choose in this course the former way too. Summarizing the models we are going to study and use in this course will be
Parametric and linear models

24. Some Problems
P1: Let {Zt} be a sequence of uncorrelated real-valued variables with zero means and unit variances, and define the “moving average”
for constants a0, a1, ... , ar . Show that Y is weak stationary and find its autocovariance function
P2: Show that a Gaussian process is strongly stationary if and only if it is weakly stationary
P3: Let X be a stationary Gaussian process with zero mean, unit variance, and autocovariance function c. Find the autocovariance functions of the process

25. Appendix: Transformations
Goal: To lead to a more manageable process
Log transformation reduces certain type of heteroskedasticity. If we assume mt=E(Xt) and V(Xt) = k m2t, the delta method shows that the variance of the log is roughly constant:
Differencing eliminates the trend (not very informative about the nature of the trend)
Differencing + Log = Relative Change