Complete exam material of Introduction to Time Series and Dynamic Econometrics, Bachelor Econometrics, Vrije Universiteit Amsterda,

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




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,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.




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, 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.




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