100% satisfaction guarantee Immediately available after payment Both online and in PDF No strings attached
logo-home
Summary Time Series and its Applications $14.46
Add to cart

Summary

Summary Time Series and its Applications

 32 views  0 purchase
  • Course
  • Institution

Summary of the Time Series and its Applications course, taught at Tilburg University in the master (semester 2).

Preview 4 out of 77  pages

  • May 30, 2024
  • 77
  • 2023/2024
  • Summary
avatar-seller
Tilburg University

Master Program


Summary TSA

Author: Supervisor:
Rick Smeets Kojevnikov, D

May 30, 2024

,Table of Contents
1 Stochastic Processes 3
1.1 Time Series Data . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Second-order Properties: Weak Stationarity . . . . . . . . . . 6
1.3 Construction of Stochastic Processes . . . . . . . . . . . . . . 8

2 Asymptotic Results 11
2.1 Linear Processes . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2 Estimation of the Mean and the ACVF . . . . . . . . . . . . . 12
2.2.1 Estimation of µX . . . . . . . . . . . . . . . . . . . . . 12
2.2.2 Estimation of γX . . . . . . . . . . . . . . . . . . . . . 15
2.2.3 Estimation of v . . . . . . . . . . . . . . . . . . . . . . 18

3 ARMA Models 19
3.1 The Lag Operator, Lag Polynomials . . . . . . . . . . . . . . . 19
3.2 AR and MA Polynomials . . . . . . . . . . . . . . . . . . . . . 22
3.3 Causality and Invertibility of ARMA Processes . . . . . . . . . 24

4 Forecasts, the Wold Decomposition 29
4.1 Forecasting Stationary Time Series . . . . . . . . . . . . . . . 29
4.2 Forecasting From the Infinite Past . . . . . . . . . . . . . . . . 32
4.3 The Wold Decomposition . . . . . . . . . . . . . . . . . . . . . 36

5 Estimation of ARMA Models Part I 38
5.1 The ACF and PACF of an ARMA Process . . . . . . . . . . . 38
5.2 The Autocorrelation Function . . . . . . . . . . . . . . . . . . 38
5.3 The Partial Autocorrelation Function . . . . . . . . . . . . . . 40
5.4 Interpretation of ACF and PACF . . . . . . . . . . . . . . . . 43

6 Estimation of ARMA Models Part II 44
6.1 The Yule-Walker Estimator (Method of Moments) . . . . . . . 44
6.2 Maximum Likelihood Estimation . . . . . . . . . . . . . . . . 47
6.3 Diagnostic Checking and Order Selection . . . . . . . . . . . . 51

7 Models of Volatility 53
7.1 GARCH Models . . . . . . . . . . . . . . . . . . . . . . . . . . 53
7.2 Estimation of GARCH Models . . . . . . . . . . . . . . . . . . 59
7.3 Forecasting Volatility . . . . . . . . . . . . . . . . . . . . . . . 60

1

,8 Generalizations of the ARMA model 62
8.1 Integrated Processes . . . . . . . . . . . . . . . . . . . . . . . 62
8.1.1 Forecasting Integrated Processes . . . . . . . . . . . . . 64
8.2 Unit Root Tests . . . . . . . . . . . . . . . . . . . . . . . . . . 67
8.2.1 Unit Root in Autoregressions . . . . . . . . . . . . . . 68
8.2.2 Unit Root in Moving-Averages . . . . . . . . . . . . . . 71
8.3 Seasonal ARIMA Models . . . . . . . . . . . . . . . . . . . . . 72
8.3.1 Forecasting Seasonal ARIMA Processes . . . . . . . . . 75




2

, 1 Stochastic Processes
1.1 Time Series Data
In this course we are concerned with data whose observations are recorded
at discrete time intervals. Such a dataset is referred to as a time series.

Formally, we think about a regular time series as a particular realization
of a (discrete-time) stochastic process, i.e., the observation xt at time t is
a realization of a certain random variable Xt . Such modelling allows for
the unpredictable nature of future observations. Let T denote a set of time
points, e.g., T = Z = {. . . , −2, −1, 0, 1, 2, . . .} or T = N = {1, 2, 3, . . .}.


Definition 1.1. A stochastic process is a family of random variables {Xt :
t ∈ T } defined on a common probability space (Ω, F, P). Recall that Ω is
the sample space (possible outcomes), F is a collection of events (F ⊂ 2Ω ),
and P is a probability measure. A random variable X defined on this space
is a function X : Ω → R.


When the index set T is obvious, we write {Xt } for short. If {Xt } is a
stochastic process, Xt is a random variable for each t ∈ T . For example,
{Xt : t ∈ N} is an infinite sequence of random variables, and {Xt : t ∈ Z}
is a doubly infinite sequence of random variables. For a particular outcome
ω ∈ Ω, we obtain a realization of the whole process by varying the time index
t.


Definition 1.2. For a given outcome ω ∈ Ω, the function t 7→ Xt (ω) is called
a sample path of the stochastic process {Xt }. This means that for a specific
outcome ω, the sample path shows how the random variable Xt evolves over
time t.


Thus, {Xt } can be described as the collection of all possible sample paths or
trajectories, realized according to the underlying probability space. Hence-
forth, for a process {Xt } consisting of i.i.d. random variables with mean µ
and finite variance σ 2 , we write Xt ∼ IID(µ, σ 2 ).

3

The benefits of buying summaries with Stuvia:

Guaranteed quality through customer reviews

Guaranteed quality through customer reviews

Stuvia customers have reviewed more than 700,000 summaries. This how you know that you are buying the best documents.

Quick and easy check-out

Quick and easy check-out

You can quickly pay through credit card or Stuvia-credit for the summaries. There is no membership needed.

Focus on what matters

Focus on what matters

Your fellow students write the study notes themselves, which is why the documents are always reliable and up-to-date. This ensures you quickly get to the core!

Frequently asked questions

What do I get when I buy this document?

You get a PDF, available immediately after your purchase. The purchased document is accessible anytime, anywhere and indefinitely through your profile.

Satisfaction guarantee: how does it work?

Our satisfaction guarantee ensures that you always find a study document that suits you well. You fill out a form, and our customer service team takes care of the rest.

Who am I buying these notes from?

Stuvia is a marketplace, so you are not buying this document from us, but from seller rickprive611. Stuvia facilitates payment to the seller.

Will I be stuck with a subscription?

No, you only buy these notes for $14.46. You're not tied to anything after your purchase.

Can Stuvia be trusted?

4.6 stars on Google & Trustpilot (+1000 reviews)

52510 documents were sold in the last 30 days

Founded in 2010, the go-to place to buy study notes for 14 years now

Start selling
$14.46
  • (0)
Add to cart
Added