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Complete exam material of Introduction to Time Series and Dynamic Econometrics, Bachelor Econometrics, Vrije Universiteit Amsterda, $9.68   Add to cart
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Complete exam material of Introduction to Time Series and Dynamic Econometrics, Bachelor Econometrics, Vrije Universiteit Amsterda,
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  • 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...

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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)
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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

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