100% satisfaction guarantee Immediately available after payment Both online and in PDF No strings attached
logo-home
Previously searched by you
Previously searched by you
logo
Complete exam material of Introduction to Time Series and Dynamic Econometrics, Bachelor Econometrics, Vrije Universiteit Amsterda, $9.77   Add to cart
Quickly navigate to
Preview
  2.  
  3. Vrije Universiteit Amsterdam (VU)
  4.  
  5. Introduction to Time Series andDynamicEconometrics (E_MFAE_ITSDE)

Class notes

Complete exam material of Introduction to Time Series and Dynamic Econometrics, Bachelor Econometrics, Vrije Universiteit Amsterda,
 75 views  3 purchases
  • Course
  • Introduction to Time Series andDynamicEconometrics (E_MFAE_ITSDE)
  • Institution
  • Vrije Universiteit Amsterdam (VU)

Complete summary of the exam material for the course Introduction to Time Series and Dynamic Econometrics in the 3th year of the Bachelor of Econometrics at the Vrije Universiteit Amsterdam, or the minor Applied Econometrics. The summary is in English. All lectures are in the summary, with extra i...

[Show more]

Preview 3 out of 26  pages

Document information

  • October 11, 2023
  • 26
  • 2023/2024
  • Class notes
  • K. moussa
  • All classes

Subjects

  • econometrics
  • time series
  • dynamic econometrics
  • forecast
  • spurious regression
  • arma model
  • adl model
  • impulse response function irf

Written for

  • Vrije Universiteit Amsterdam (VU)
  • Econometrics and Data Science
  • Introduction to Time Series andDynamicEconometrics (E_MFAE_ITSDE)
All documents for this subject (1)

Seller

avatar-seller
charhoog

Reviews received

Content preview

Introduction to Time Series and Dynamic
Econometrics
Charlotte Hoogteijling
October2023


Contents
0 Preparatory Notes 3
0.1 Mean . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
0.2 Law of Total Expectation . . . . . . . . . . . . . . . . . . . . . . 3
0.3 Geometric series . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
0.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1 Basic Properties of Time Series 4
1.1 Basic definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2 Strict and weak stationarity . . . . . . . . . . . . . . . . . . . . . 4
1.3 Unconditional and conditional moments . . . . . . . . . . . . . . 5
1.4 Sample moments . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.5 Autocorrelation function (ACF) . . . . . . . . . . . . . . . . . . . 5
1.6 White Noise (WN) and Random Walk (RW) processes . . . . . . 6
1.7 Sources of non-stationarity . . . . . . . . . . . . . . . . . . . . . 6
1.8 Lag and difference operator . . . . . . . . . . . . . . . . . . . . . 6
1.9 Wold decomposition . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.10 Linear process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2 Properties of ARMA Models 9
2.1 Autoregressive moving-average (ARMA) model . . . . . . . . . . 9
2.1.1 Moments . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.1.2 MA(∞) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.1.3 AR(∞) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.2 Autoregressive (AR) model . . . . . . . . . . . . . . . . . . . . . 10
2.2.1 Moments of stable AR(p) process . . . . . . . . . . . . . . 11
2.3 Moving average (MA) model . . . . . . . . . . . . . . . . . . . . 12
2.3.1 Moments of MA(q) process . . . . . . . . . . . . . . . . . 12
2.4 Extra notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12




1

,3 Estimation and Specification of ARMA Models 13
3.1 ARMA coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.2 Maximum likelihood estimator (MLE) . . . . . . . . . . . . . . . 13
3.3 Multivariate normal likelihood of ARMA model . . . . . . . . . . 13
3.3.1 Variance-covariance matrix of AR(1) . . . . . . . . . . . . 14
3.4 Prediction error decomposition of ARMA likelihood . . . . . . . 14
3.4.1 Likelihood function of AR(1) . . . . . . . . . . . . . . . . 15
3.4.2 MLE of an AR(1) with NID(0, 1) innovations . . . . . . . 15
3.5 Least Squares Estimator (LSE) . . . . . . . . . . . . . . . . . . . 15
3.5.1 MLE and LSE properties . . . . . . . . . . . . . . . . . . 15
3.6 Asymptotic properties . . . . . . . . . . . . . . . . . . . . . . . . 16
3.7 Forecasting ARM A(p, q) processes . . . . . . . . . . . . . . . . . 16
3.7.1 Confidence interval for X̂T +h . . . . . . . . . . . . . . . . 17
3.7.2 Optimal forecast under quadratic loss . . . . . . . . . . . 17

4 Autoregressive distributed lag and error correction models 18
4.1 Various models of this course . . . . . . . . . . . . . . . . . . . . 18
4.1.1 Box-Jenkins approach to modeling time series . . . . . . . 18
4.1.2 Structural models . . . . . . . . . . . . . . . . . . . . . . 18
4.1.3 Statistical (reduced form) models . . . . . . . . . . . . . . 18
4.2 Autoregressive distributed lag model (ADL) . . . . . . . . . . . . 18
4.3 Long and short run multipliers . . . . . . . . . . . . . . . . . . . 19
4.4 Forecasting with ADL(1,1): triangular System . . . . . . . . . . 19
4.5 Impulse response function (IRF) . . . . . . . . . . . . . . . . . . 20
4.6 Error correction model (ECM) . . . . . . . . . . . . . . . . . . . 20

5 Spurious regression unit-roots 21
5.1 Spurious regression problem . . . . . . . . . . . . . . . . . . . . . 21
5.2 Dickey Fuller (DF) test . . . . . . . . . . . . . . . . . . . . . . . 21
5.2.1 Augmented Dickey-Fuller (ADF) - AR(p) . . . . . . . . . 22
5.2.2 ADF General-to-specific (G2S) . . . . . . . . . . . . . . . 22
5.3 Extra notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

6 Cointegration and Granger causality 24
6.1 Cointegration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
6.1.1 Cointegration tests . . . . . . . . . . . . . . . . . . . . . . 24
6.2 Modeling strategy . . . . . . . . . . . . . . . . . . . . . . . . . . 25
6.2.1 Estimation based on ECM . . . . . . . . . . . . . . . . . . 26
6.2.2 Engle and Granger 2-step procedure . . . . . . . . . . . . 26
This document contains the contents of the lecture slides and notes. The expla-
nations are summarized. To understand the properties of the definitions, proofs,
and formulas, it is recommended to derive the formulas yourself.




2

, 0 Preparatory Notes
0.1 Mean
If X and Y are independent of each other.


E(XY ) = E(X)E(Y )

0.2 Law of Total Expectation
We can find the expected value of a variable X by considering all different
scenarios A under which X can occur.


E(X) = E(E(X | Y ))
= P (A1 )E(X | A1 ) + · · · + P (An )E(X | An )



0.3 Geometric series
P∞
A geometric series is a sum of the type i=0 ri = 1 + r + r2 + . . . .
• If r < 1, the series will converge to 1
1−r .

• If r = 1, the terms in the series will oscillate (positive-negative).
• If r > 1, the series will diverges (goes to infinity).

0.4 Notes
• The joint probability density function (joint pdf) is a function used to
characterize the probability distribution of a continuous random vector.
• The covariance is a measure of joint variability of two random variables.
It measures the directional relationship.




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 charhoog. Stuvia facilitates payment to the seller.

Will I be stuck with a subscription?

No, you only buy these notes for $9.77. You're not tied to anything after your purchase.

Can Stuvia be trusted?

4.6 stars on Google & Trustpilot (+1000 reviews)

75057 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
$9.77  3x  sold
  • (0)
  Add to cart