Time Series Analysis
Complete Course Summary
ARMA Models, ARCH/GARCH, Trends & Seasonality,
Nonlinear & Structural Break Models, VAR & Cointegration,
Forecast Combination & Density Forecasts
Lecture Notes, Tutorials, Worked Exam Exercises & Figures
,Time Series Analysis Complete Course Summary 1
Contents
I Lecture Notes 3
1 Week 1 Foundations of Time Series 3
1.1 Characteristics of Time Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Stationarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.4 White Noise Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.5 Autocorrelation Function (ACF) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.6 Autoregressive AR(1) Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.7 The k -th Order Autocovariance for an AR(1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.8 Non-Stationary AR(1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2 Moving Average & ARMA Models 5
2.1 Moving Average (MA) Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 Invertibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.3 Autoregressive Moving Average (ARMA) Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.4 Partial Autocorrelation Function (PACF) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.5 ARMA(1,1) Worked Derivation Variance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3 ARCH & GARCH Models 6
3.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3.2 ARCH(1) Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
3.3 The ARCH(1) Model Can Describe Data with Volatility Clustering . . . . . . . . . . . . . . . . . . . 7
3.4 ARCH(p) and GARCH(p,q) Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
3.5 Stochastic Volatility & EGARCH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
3.6 Leverage Eect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
3.7 Combined Model: AR(1)ARCH(1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
3.8 Testing for ARCH Eects: The Ljung-Box Test on Squared Residuals . . . . . . . . . . . . . . . . . 8
3.9 Forecasting the Conditional Variance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
4 Week 3 Trends, Stochastic Trends & Forecasting 9
4.1 Deterministic vs. Stochastic Trends . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
4.2 Stochastic Trend (ST) Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
4.3 Dierences Between an AR(1) and a RW/Drift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
4.4 Trend Stationary Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
4.5 Forecasting ARIMA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
4.6 Holt-Winters Trend Forecast . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
5 Dickey-Fuller & Augmented Dickey-Fuller Tests 11
5.1 Dickey-Fuller F-test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
5.2 Dickey-Fuller t-test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
5.3 Augmented Dickey-Fuller (ADF) Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
6 Seasonality & SARIMA 12
6.1 Seasonality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
6.2 Seasonal Dierencing: Some Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
6.3 The Hold-Winters Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
7 Week 4 Outliers, Structural Breaks & Nonlinear Models 12
7.1 Outliers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
7.2 Level Shifts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
7.3 Nonlinear Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
7.4 Bilinear Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
7.5 Threshold AR (TAR) Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
7.6 Smooth Transition AR (STAR) Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
7.7 Markov Switching Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
7.8 Arti
cial Neural Networks (ANN) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
8 Dynamic Regression Models 15
8.1 AR with Distributed Lags . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
,Time Series Analysis Complete Course Summary 2
8.2 Partial Adjustment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
8.3 Adaptive Expectations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
8.4 Error Correction Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
9 Forecasting ARMA & ADL Models 16
9.1 Forecasting ADL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
II Model Averaging & Forecast Combination 16
10 Model Averaging 17
10.1 Model Averaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
10.2 Bates-Granger Averaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
10.3 Equal-Weights Averaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
10.4 Weights Based on Information Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
10.5 Convergence Properties of OLS-based Weights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
10.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
III VAR Models & Cointegration 18
11 Week 5 Cointegrated Time Series 18
11.1 Cointegration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
11.2 VAR(1) Model and Stationarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
11.3 VAR(p) Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
11.4 Properties of VAR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
11.5 Granger Causality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
11.6 Intro to Cointegration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
11.7 Cointegration Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
11.8 CI in VAR(p) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
11.9 Johansen Trace Test for Cointegration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
12 Worked Exercises VAR, VECM & Cointegration 21
IV Density Forecasts, Scoring Rules & Model Con
dence Sets 23
13 Week 6 Density Forecast Evaluation 23
13.1 Density Forecasts & Scoring Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
13.2 How to Score the Forecast? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
13.3 Comparing Density Forecasts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
13.4 Weighted Scoring Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
13.5 Density Forecasts in Model Averaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
13.6 Multivariate Density Forecasts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
13.7 Sklar's Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
13.8 Constructing Copulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
13.9 Score Dierence Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
14 Testing Density Forecasts & Model Con
dence Sets 25
14.1 Giacomini-White Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
14.2 Using Hypothesis Testing to Check if Dierent & Sign . . . . . . . . . . . . . . . . . . . . . . . . . . 25
14.3 Model Con
dence Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
14.4 Con
dence Sets (Reader) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
14.5 Scoring Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
14.6 Density Forecasts in Model Averaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
14.7 Time-Varying Weights in the GAS Setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
,Time Series Analysis Complete Course Summary 3
Part I
Lecture Notes
1 Week 1 Foundations of Time Series
1.1 Characteristics of Time Series
Trend: usually appears over T →∞ (most matters up in the long run).
Seasonality: repeated cycle within a year (e.g. Christmas sales).
Abrupt shift/interventions: could be due to cycles or extreme events.
Conditional variance: time-varying volatility (heteroskedasticity).
Non-linearity.
Irregular component: like random disturbance.
1.2 Models
Regression based models: yt = β1 x1t + · · · + βp xpt + εt disturbance with known pdf
Parametric time series models (BoxJenkins):
xt = ϕ1 xt−1 + · · · + ϕp xt−p + εt ⇒ yt = βxt−1 + · · · + ϕp xt−p + εt
this is called the p-th order autoregressive model: AR(p)
yt = εt + θ1 εt−1
dierent time series
Aim of TSA:
nding a suitable form of approximation of the unknown, possibly non-linear regression function
E[yt+1 |yt , yt−1 , . . . ] = E[yt+1 |yt , yt−1 , . . . ] ⇒ yt = F (yt−1 , yt−2 , . . . ) + εt = yt − E[yt |yt−1 , yt−2 , . . . ]
Forecasting/estimating ,→ use ŷT +h = E[yT +h |yT , yT −1 , . . . ]
,→ analysis of the eect of shocks: ∂εt
1.3 Stationarity
A stochastic process yt having a
nite mean and variance is strictly stationary (or second-order stationary)
if:
1. Mean: E[yt ] = E[yt+s ] = µ (constant)
2. Variance: Var(yt ) = E[(yt − µ)2 ] = γ0 · σ 2 (constant)
3. Lag k autocovariance: E[(yt − µ)(yt+k − µ)] = γk (constant)
Note that γ0 = σ 2 and the autocorrelation:
E[(yt − µ)(yt+k − µ)] γk
ρk = =
E[(yt − µ)2 ] γ0
(symmetry)
1.4 White Noise Process
A stationary process denoted by {εt }:
1. E[εt ] = 0, zero mean
2. Var(εt ) = E[ε2t ] = σ 2 , constant variance
3. Cov(εt , εt−k ) = E[εt εt−k ] = 0, k ̸= 0, no autocorrelation
We require the disturbances εt to be i.i.d. (without systematic eects, correlations). Should be put in the regression
function, the remaining part should be unpredictable.
, Time Series Analysis Complete Course Summary 4
1.5 Autocorrelation Function (ACF)
ACF helps to estimate correlation, which is needed for examination. For a stationary process, the sequence {γk :
k = 0, ±1, ±2, . . . } is called the autocovariance function. ACF:
γk
ρk =
γ0
−1 ≤ ρk ≤ 1, ρ0 = 1, ρk = ρ−k : correlation with self
In a class that ρk ̸= 1 (correlation with self ). Recall ρ0 = 1.
Example is the MA process εt , for which E[εt ] = 0 & ρk = 0 ∀k ̸= 0,
because Cov(εt , εt−k ) = E[εt εt−k ] = 0.
1.6 Autoregressive AR(1) Process
AR(1) process: yt = α + ϕyt−1 + εt
⇒ model implies yt = α + ϕy0 + ϕεt−1 + εt , where y0 is a starting value:
yt = α + ϕy0 + ϕεt−1 + εt + · · · + ϕt ε0
t−1
X
= · · · = α(1 + ϕ + · · · + ϕt−1 ) + ϕt y0 + ϕj εt−j
j=0
An innovation εt−j at time t−j aects yt with multiplier ϕj :
|ϕ| ≥ 1: yt displays explosive behavior (it does not forget what happened long ago)
|ϕ| < 1: impact dies out (forgets)
It does not the same and univariate group; concept stationarity unde
ned.
α
Mean of yt : E[yt ] = E[α + ϕyt−1 + εt ] = α + ϕE[yt−1 ], s.t. E[yt ] = (ϕ ̸= 1)
1−ϕ
Variance of yt : Var(yt ) = ϕ2 Var(yt−1 ) + Var(εt ) + 2ϕCov(yt−1 , εt ) = ϕ2 Var(yt−1 ) + σ 2
if |ϕ| ≥ 1: Var(yt ) > Var(yt−1 ) s.t. variance grows, contradicts stationarity violated.
2
σ
if |ϕ| < 1: Var(yt ) =
1 − ϕ2
Determine whether AR-process has unit/stationary root
1. Solve ϕ(z) = 0 for z
2. Stationary if |z| > 1
1.7 The k-th Order Autocovariance for an AR(1)
γk = Cov(yt , yt−k ) = E[(yt − µ)(yt−k − µ)]
= E[(εt + ϕεt−1 + . . . )(εt−k + ϕεt−k−1 + . . . )]
1
= ϕk σ 2 (1 + ϕ2 + ϕ4 + . . . ) = ϕk σ 2 ·
1 − ϕ2
σ2
⇒ γk = ϕk · γ0 = ϕk ·
1 − ϕ2
The autocorrelations are: γk
ρk = = ϕk , k = 0, 1, 2, . . .
γ0
Note: autocorrelation →0 for k → ∞, depending on ϕ; if ϕ close to 1, they die out slowly.
1.8 Non-Stationary AR(1)
ϕ = 1: now yt = yt−1 + εt , which is called a random walk (RW) with drift.
,→ this is an example of a process which is integrated of order 1.
∆yt = yt − yt−1 = εt
depends on a constant & WN ⇒ stationary
Complete Course Summary
ARMA Models, ARCH/GARCH, Trends & Seasonality,
Nonlinear & Structural Break Models, VAR & Cointegration,
Forecast Combination & Density Forecasts
Lecture Notes, Tutorials, Worked Exam Exercises & Figures
,Time Series Analysis Complete Course Summary 1
Contents
I Lecture Notes 3
1 Week 1 Foundations of Time Series 3
1.1 Characteristics of Time Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Stationarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.4 White Noise Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.5 Autocorrelation Function (ACF) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.6 Autoregressive AR(1) Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.7 The k -th Order Autocovariance for an AR(1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.8 Non-Stationary AR(1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2 Moving Average & ARMA Models 5
2.1 Moving Average (MA) Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 Invertibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.3 Autoregressive Moving Average (ARMA) Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.4 Partial Autocorrelation Function (PACF) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.5 ARMA(1,1) Worked Derivation Variance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3 ARCH & GARCH Models 6
3.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3.2 ARCH(1) Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
3.3 The ARCH(1) Model Can Describe Data with Volatility Clustering . . . . . . . . . . . . . . . . . . . 7
3.4 ARCH(p) and GARCH(p,q) Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
3.5 Stochastic Volatility & EGARCH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
3.6 Leverage Eect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
3.7 Combined Model: AR(1)ARCH(1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
3.8 Testing for ARCH Eects: The Ljung-Box Test on Squared Residuals . . . . . . . . . . . . . . . . . 8
3.9 Forecasting the Conditional Variance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
4 Week 3 Trends, Stochastic Trends & Forecasting 9
4.1 Deterministic vs. Stochastic Trends . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
4.2 Stochastic Trend (ST) Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
4.3 Dierences Between an AR(1) and a RW/Drift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
4.4 Trend Stationary Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
4.5 Forecasting ARIMA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
4.6 Holt-Winters Trend Forecast . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
5 Dickey-Fuller & Augmented Dickey-Fuller Tests 11
5.1 Dickey-Fuller F-test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
5.2 Dickey-Fuller t-test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
5.3 Augmented Dickey-Fuller (ADF) Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
6 Seasonality & SARIMA 12
6.1 Seasonality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
6.2 Seasonal Dierencing: Some Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
6.3 The Hold-Winters Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
7 Week 4 Outliers, Structural Breaks & Nonlinear Models 12
7.1 Outliers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
7.2 Level Shifts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
7.3 Nonlinear Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
7.4 Bilinear Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
7.5 Threshold AR (TAR) Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
7.6 Smooth Transition AR (STAR) Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
7.7 Markov Switching Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
7.8 Arti
cial Neural Networks (ANN) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
8 Dynamic Regression Models 15
8.1 AR with Distributed Lags . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
,Time Series Analysis Complete Course Summary 2
8.2 Partial Adjustment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
8.3 Adaptive Expectations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
8.4 Error Correction Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
9 Forecasting ARMA & ADL Models 16
9.1 Forecasting ADL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
II Model Averaging & Forecast Combination 16
10 Model Averaging 17
10.1 Model Averaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
10.2 Bates-Granger Averaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
10.3 Equal-Weights Averaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
10.4 Weights Based on Information Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
10.5 Convergence Properties of OLS-based Weights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
10.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
III VAR Models & Cointegration 18
11 Week 5 Cointegrated Time Series 18
11.1 Cointegration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
11.2 VAR(1) Model and Stationarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
11.3 VAR(p) Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
11.4 Properties of VAR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
11.5 Granger Causality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
11.6 Intro to Cointegration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
11.7 Cointegration Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
11.8 CI in VAR(p) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
11.9 Johansen Trace Test for Cointegration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
12 Worked Exercises VAR, VECM & Cointegration 21
IV Density Forecasts, Scoring Rules & Model Con
dence Sets 23
13 Week 6 Density Forecast Evaluation 23
13.1 Density Forecasts & Scoring Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
13.2 How to Score the Forecast? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
13.3 Comparing Density Forecasts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
13.4 Weighted Scoring Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
13.5 Density Forecasts in Model Averaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
13.6 Multivariate Density Forecasts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
13.7 Sklar's Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
13.8 Constructing Copulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
13.9 Score Dierence Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
14 Testing Density Forecasts & Model Con
dence Sets 25
14.1 Giacomini-White Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
14.2 Using Hypothesis Testing to Check if Dierent & Sign . . . . . . . . . . . . . . . . . . . . . . . . . . 25
14.3 Model Con
dence Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
14.4 Con
dence Sets (Reader) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
14.5 Scoring Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
14.6 Density Forecasts in Model Averaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
14.7 Time-Varying Weights in the GAS Setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
,Time Series Analysis Complete Course Summary 3
Part I
Lecture Notes
1 Week 1 Foundations of Time Series
1.1 Characteristics of Time Series
Trend: usually appears over T →∞ (most matters up in the long run).
Seasonality: repeated cycle within a year (e.g. Christmas sales).
Abrupt shift/interventions: could be due to cycles or extreme events.
Conditional variance: time-varying volatility (heteroskedasticity).
Non-linearity.
Irregular component: like random disturbance.
1.2 Models
Regression based models: yt = β1 x1t + · · · + βp xpt + εt disturbance with known pdf
Parametric time series models (BoxJenkins):
xt = ϕ1 xt−1 + · · · + ϕp xt−p + εt ⇒ yt = βxt−1 + · · · + ϕp xt−p + εt
this is called the p-th order autoregressive model: AR(p)
yt = εt + θ1 εt−1
dierent time series
Aim of TSA:
nding a suitable form of approximation of the unknown, possibly non-linear regression function
E[yt+1 |yt , yt−1 , . . . ] = E[yt+1 |yt , yt−1 , . . . ] ⇒ yt = F (yt−1 , yt−2 , . . . ) + εt = yt − E[yt |yt−1 , yt−2 , . . . ]
Forecasting/estimating ,→ use ŷT +h = E[yT +h |yT , yT −1 , . . . ]
,→ analysis of the eect of shocks: ∂εt
1.3 Stationarity
A stochastic process yt having a
nite mean and variance is strictly stationary (or second-order stationary)
if:
1. Mean: E[yt ] = E[yt+s ] = µ (constant)
2. Variance: Var(yt ) = E[(yt − µ)2 ] = γ0 · σ 2 (constant)
3. Lag k autocovariance: E[(yt − µ)(yt+k − µ)] = γk (constant)
Note that γ0 = σ 2 and the autocorrelation:
E[(yt − µ)(yt+k − µ)] γk
ρk = =
E[(yt − µ)2 ] γ0
(symmetry)
1.4 White Noise Process
A stationary process denoted by {εt }:
1. E[εt ] = 0, zero mean
2. Var(εt ) = E[ε2t ] = σ 2 , constant variance
3. Cov(εt , εt−k ) = E[εt εt−k ] = 0, k ̸= 0, no autocorrelation
We require the disturbances εt to be i.i.d. (without systematic eects, correlations). Should be put in the regression
function, the remaining part should be unpredictable.
, Time Series Analysis Complete Course Summary 4
1.5 Autocorrelation Function (ACF)
ACF helps to estimate correlation, which is needed for examination. For a stationary process, the sequence {γk :
k = 0, ±1, ±2, . . . } is called the autocovariance function. ACF:
γk
ρk =
γ0
−1 ≤ ρk ≤ 1, ρ0 = 1, ρk = ρ−k : correlation with self
In a class that ρk ̸= 1 (correlation with self ). Recall ρ0 = 1.
Example is the MA process εt , for which E[εt ] = 0 & ρk = 0 ∀k ̸= 0,
because Cov(εt , εt−k ) = E[εt εt−k ] = 0.
1.6 Autoregressive AR(1) Process
AR(1) process: yt = α + ϕyt−1 + εt
⇒ model implies yt = α + ϕy0 + ϕεt−1 + εt , where y0 is a starting value:
yt = α + ϕy0 + ϕεt−1 + εt + · · · + ϕt ε0
t−1
X
= · · · = α(1 + ϕ + · · · + ϕt−1 ) + ϕt y0 + ϕj εt−j
j=0
An innovation εt−j at time t−j aects yt with multiplier ϕj :
|ϕ| ≥ 1: yt displays explosive behavior (it does not forget what happened long ago)
|ϕ| < 1: impact dies out (forgets)
It does not the same and univariate group; concept stationarity unde
ned.
α
Mean of yt : E[yt ] = E[α + ϕyt−1 + εt ] = α + ϕE[yt−1 ], s.t. E[yt ] = (ϕ ̸= 1)
1−ϕ
Variance of yt : Var(yt ) = ϕ2 Var(yt−1 ) + Var(εt ) + 2ϕCov(yt−1 , εt ) = ϕ2 Var(yt−1 ) + σ 2
if |ϕ| ≥ 1: Var(yt ) > Var(yt−1 ) s.t. variance grows, contradicts stationarity violated.
2
σ
if |ϕ| < 1: Var(yt ) =
1 − ϕ2
Determine whether AR-process has unit/stationary root
1. Solve ϕ(z) = 0 for z
2. Stationary if |z| > 1
1.7 The k-th Order Autocovariance for an AR(1)
γk = Cov(yt , yt−k ) = E[(yt − µ)(yt−k − µ)]
= E[(εt + ϕεt−1 + . . . )(εt−k + ϕεt−k−1 + . . . )]
1
= ϕk σ 2 (1 + ϕ2 + ϕ4 + . . . ) = ϕk σ 2 ·
1 − ϕ2
σ2
⇒ γk = ϕk · γ0 = ϕk ·
1 − ϕ2
The autocorrelations are: γk
ρk = = ϕk , k = 0, 1, 2, . . .
γ0
Note: autocorrelation →0 for k → ∞, depending on ϕ; if ϕ close to 1, they die out slowly.
1.8 Non-Stationary AR(1)
ϕ = 1: now yt = yt−1 + εt , which is called a random walk (RW) with drift.
,→ this is an example of a process which is integrated of order 1.
∆yt = yt − yt−1 = εt
depends on a constant & WN ⇒ stationary
