Financial Econometrics
Complete Course Summary
ARMA Models & Model Selection, Volatility Modelling (ARCH/GARCH),
Value-at-Risk, Expected Shortfall & Extreme Value Theory,
Multivariate Models: VAR, Cointegration & Multivariate GARCH,
Realized Volatility & Jumps
Lecture Notes, Tutorials & Worked Exam Exercises
,Financial Econometrics Complete Course Summary 1
Contents
I ARMA Models & Model Selection 3
1 Week 1 Stationarity, ARMA & Financial Returns 3
1.1 Financial Returns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Stationarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Wold Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.4 AR(1) Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.5 AR(p) Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.6 MA(q) Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.7 ARMA(p,q) Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.8 Integrated Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2 Model Selection & Diagnostic Checking 5
2.1 Box-Jenkins Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 Sample ACF & Standard Errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.3 Step-by-Step Approach to Identify ARIMA(p,d,q) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.4 How to Estimate ARMA Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.5 Diagnostic Checking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.6 Information Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.7 Dickey-Fuller Test for Unit Roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.8 ARMAX Models & Mean-Reversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
II Volatility Modelling 7
3 Week 2 ARCH & GARCH Models 7
3.1 Volatility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
3.2 ARCH Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
3.3 Kurtosis of ARCH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
3.4 GARCH Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
3.5 Leverage Eects: What Standard GARCH Cannot Capture . . . . . . . . . . . . . . . . . . . . . . . 8
3.6 EGARCH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
3.7 GJR-GARCH / TGARCH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
3.8 GARCH-in-Mean (GARCH-M) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
3.9 ML Estimation of GARCH Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
3.10 Diagnostic Checking for GARCH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
3.11 Forecasting with GARCH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
3.12 Stochastic Volatility (SV) Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
3.13 Implied Volatility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
III Value-at-Risk, Expected Shortfall & Extreme Value Theory 10
4 Week 3 Risk Measures 10
4.1 Value-at-Risk (VaR) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
4.2 Parametric (Model-Based) VaR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
4.3 Historical Simulation VaR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
4.4 RiskMetrics Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
4.5 Multi-Period VaR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
4.6 Portfolio VaR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
4.7 Expected Shortfall . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
4.8 Coherent Risk Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
4.9 Extreme Value Theory (EVT) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
4.10 Hill Estimator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
4.11 QQ-plot for Tail Diagnostics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
4.12 Why Backtest VaR? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
4.13 Backtesting: Unconditional Coverage (Kupiec Test) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
4.14 Conditional Coverage (Christoersen Test) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
4.15 Backtesting Expected Shortfall . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
,Financial Econometrics Complete Course Summary 2
IV Multivariate Models: VAR, Cointegration & Multivariate GARCH 13
5 Week 4 VAR Models & Cointegration 13
5.1 VAR(p) Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
5.2 Estimation & Lag-Length Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
5.3 Structural VAR & Impulse Responses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
5.4 Cointegration & Vector Error-Correction Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
5.5 Johansen Trace Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
5.6 Estimation & Identi
cation of Cointegrating Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
6 Week 5 Multivariate GARCH Models 15
6.1 Multivariate GARCH: The Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
6.2 VEC and BEKK Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
6.3 Factor GARCH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
6.4 Constant Conditional Correlation (CCC) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
6.5 Dynamic Conditional Correlation (DCC) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
V Realized Volatility & Jumps 16
7 Week 6 High-Frequency Data 16
7.1 Realized Variance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
7.2 Continuous-Time Price Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
7.3 Bipower Variation & Jump Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
7.4 Choosing the Sampling Frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
7.5 Market Microstructure Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
7.6 Forecasting Realized Volatility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
VI Worked Exam & Tutorial Exercises 18
8 ARMA & Model Selection Exercises 18
9 Volatility Modelling Exercises 19
10 Risk Management Exercises 20
11 Multivariate & High-Frequency Exercises 21
,Financial Econometrics Complete Course Summary 3
Part I
ARMA Models & Model Selection
1 Week 1 Stationarity, ARMA & Financial Returns
1.1 Financial Returns
One-period simple return: R = P P− P = PP − 1
t
t t−1 t
Continuously compounded (log) return: r = log PP
t−1 t−1
t
t = log Pt − log Pt−1
t−1
Disadvantage: from continuous compounding of simple returns.
First-order Taylor approximation: log(1 + R ) ≈ R , so r ≈ R for short-horizon returns.
Advantage of log returns: additive over time. If r is the k-period log return, then
t t t t
t,k
Pt
rt,k = log = rt + rt−1 + · · · + rt−k+1
Pt−k
The sum of capital gain and dividend yield.
Stylized facts of
nancial returns:
Have (almost) mean close to zero and very little autocorrelation
Display volatility clustering: alternating periods of high and low volatility
Have non-Gaussian distribution: fat tails (excess kurtosis)
Related facts (squared/absolute returns):
Display long-range, very slow mean-reversion
Changes have similar characteristics as returns
1.2 Stationarity
A process x is weakly stationary if it has constant mean, constant variance, and Cov(x , x depends only
on k:
t t t−k )
E[x ] = µ, t Var(x ) = σ , Cov(x , x ) = γ t
2
t t−k k
ACF: ρ = γγ k
k
0
Strict stationarity: the joint distribution does not change over time this is a much stronger requirement.
Weak stationarity: only mean, variance and autocovariance structure are constant.
ACF: ρ is a correlation with sign; |ρ | ≤ 1, ρ = 1, ρ = ρ .
White noise (all-zero ACF) is an example of a stationary process: E[ε ] = 0, ρ .
k k 0 k −k
t k = 0 ∀k ̸= 0
1.3 Wold Decomposition
Wold decomposition: every zero-mean stationary process that can be written as the sum of a linear and a
deterministic process: ∞
X ∞
X
xt = µ + ψj εt−j , ψ0 = 1, ψj2 < ∞
j=0 j=0
Linear process: xt = µ + ψ(L)εt
1.4 AR(1) Process
AR(1) : xt = ϕ0 + ϕ1 xt−1 + εt
is a necessary condition for stationarity, i.e. Cov(x , x ) → 0 as k → ∞
,→ |ϕ1 | < 1
For stationarity, we need: Var(x ) = Var(x ), so:
t t−k
t t−1
Var(x ) = ϕ Var(x ) + σ ⇒ γ = 1 −σ ϕ
2
2 2
t 1 t−1 0 2
1
,Financial Econometrics Complete Course Summary 4
So, order stationarity condition: µ = E[x ] = 1 −ϕ ϕ and γ = 1 −σ ϕt
0
1
0
2
2
1
Backward substitution gives: x = 1 −ϕ ϕ + P ϕ ε (in
nite MA representation)
t
0 ∞
j=0
j
1 t−j
,→ So, AR(1) is actually a linear process, where ψ = ϕ , so the weights decay geometrically.
1
We
nd ACF: ρ = ϕ
j
j 1
k
k 1
1.5 AR(p) Process
AR(p) : xt = ϕ0 + ϕ1 xt−1 + · · · + ϕp xt−p + εt
Some intuition when inverting the polynomial: (1 − ϕ L − · · · − ϕ L )x = ϕ + ε
Lag polynomial representation: ϕ(L)x = ϕ + ε with ϕ(L) = 1 − ϕ L − · · · − ϕ L
p
1 p t 0 t
p
An AR(p) model is stationary if all roots of ϕ(z) = 0 lie outside the unit circle.
t 0 t 1 p
Stationary AR(p) model has mean: µ = 1 − ϕ −ϕ · · · − ϕ 0
Some simple expressions for ACF: for AR(p), the ACF autocorrelations can be derived recursively from:
1 p
ρk = ϕ1 ρk−1 + · · · + ϕp ρk−p
Some general decay in exponential rate: ρ ∝ |λ| , cλ , c < 1, implying exponential decay.
k
,→ For an AR(p) model, the PACF has cut-o point at lag p, i.e. ϕ = 0 for k > p.
k 1
kk
1.6 MA(q) Process
M A(q) : xt = c0 + θ1 εt−1 + · · · + θq εt−q + εt
M A(q) models are always stationary!
ACF has cut-o point at lag q : ρ = 0 for k > q
k
PACF decays exponentially
M A(q) model is invertible if all roots of θ(z) = 0 lie outside the unit circle
θ(z) = 1 + θ1 z + · · · + θq z q = 0
1.7 ARMA(p,q) Process
ARM A(p, q) : xt = ϕ0 + ϕ1 xt−1 + · · · + ϕp xt−p + θ1 εt−1 + · · · + θq εt−q + εt
or, in lag polynomial form: ϕ(L)x = ϕ + θ(L)ε
Stationary: if all roots of ϕ(z) = 0 are outside the unit circle
t 0 t
Invertible: if all roots of θ(z) = 0 are outside the unit circle
Both ACF and PACF of ARMA models decay exponentially; neither has cut-o point.
Identi
cation problem. Consider the M A(∞) representation of an ARM A(p, q) model:
θ(L)
xt = ψ(L)εt = εt
ϕ(L)
If ϕ(z) and θ(z) share a common root, the model is not identi
ed. To avoid identi
cation problems, reduce
the model to ARM A(p − 1, q − 1).
ACF PACF
AR(p) exponential decay cut-o at lag p
MA(q) cut-o at lag q exponential decay
ARMA(p, q) exponential decay exponential decay
,→ MA uses past errors (εt−1 , εt−2 , . . . ); AR uses past observations (xt−1 , xt−2 , . . . ).
Complete Course Summary
ARMA Models & Model Selection, Volatility Modelling (ARCH/GARCH),
Value-at-Risk, Expected Shortfall & Extreme Value Theory,
Multivariate Models: VAR, Cointegration & Multivariate GARCH,
Realized Volatility & Jumps
Lecture Notes, Tutorials & Worked Exam Exercises
,Financial Econometrics Complete Course Summary 1
Contents
I ARMA Models & Model Selection 3
1 Week 1 Stationarity, ARMA & Financial Returns 3
1.1 Financial Returns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Stationarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Wold Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.4 AR(1) Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.5 AR(p) Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.6 MA(q) Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.7 ARMA(p,q) Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.8 Integrated Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2 Model Selection & Diagnostic Checking 5
2.1 Box-Jenkins Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 Sample ACF & Standard Errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.3 Step-by-Step Approach to Identify ARIMA(p,d,q) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.4 How to Estimate ARMA Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.5 Diagnostic Checking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.6 Information Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.7 Dickey-Fuller Test for Unit Roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.8 ARMAX Models & Mean-Reversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
II Volatility Modelling 7
3 Week 2 ARCH & GARCH Models 7
3.1 Volatility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
3.2 ARCH Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
3.3 Kurtosis of ARCH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
3.4 GARCH Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
3.5 Leverage Eects: What Standard GARCH Cannot Capture . . . . . . . . . . . . . . . . . . . . . . . 8
3.6 EGARCH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
3.7 GJR-GARCH / TGARCH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
3.8 GARCH-in-Mean (GARCH-M) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
3.9 ML Estimation of GARCH Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
3.10 Diagnostic Checking for GARCH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
3.11 Forecasting with GARCH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
3.12 Stochastic Volatility (SV) Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
3.13 Implied Volatility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
III Value-at-Risk, Expected Shortfall & Extreme Value Theory 10
4 Week 3 Risk Measures 10
4.1 Value-at-Risk (VaR) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
4.2 Parametric (Model-Based) VaR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
4.3 Historical Simulation VaR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
4.4 RiskMetrics Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
4.5 Multi-Period VaR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
4.6 Portfolio VaR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
4.7 Expected Shortfall . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
4.8 Coherent Risk Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
4.9 Extreme Value Theory (EVT) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
4.10 Hill Estimator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
4.11 QQ-plot for Tail Diagnostics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
4.12 Why Backtest VaR? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
4.13 Backtesting: Unconditional Coverage (Kupiec Test) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
4.14 Conditional Coverage (Christoersen Test) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
4.15 Backtesting Expected Shortfall . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
,Financial Econometrics Complete Course Summary 2
IV Multivariate Models: VAR, Cointegration & Multivariate GARCH 13
5 Week 4 VAR Models & Cointegration 13
5.1 VAR(p) Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
5.2 Estimation & Lag-Length Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
5.3 Structural VAR & Impulse Responses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
5.4 Cointegration & Vector Error-Correction Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
5.5 Johansen Trace Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
5.6 Estimation & Identi
cation of Cointegrating Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
6 Week 5 Multivariate GARCH Models 15
6.1 Multivariate GARCH: The Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
6.2 VEC and BEKK Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
6.3 Factor GARCH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
6.4 Constant Conditional Correlation (CCC) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
6.5 Dynamic Conditional Correlation (DCC) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
V Realized Volatility & Jumps 16
7 Week 6 High-Frequency Data 16
7.1 Realized Variance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
7.2 Continuous-Time Price Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
7.3 Bipower Variation & Jump Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
7.4 Choosing the Sampling Frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
7.5 Market Microstructure Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
7.6 Forecasting Realized Volatility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
VI Worked Exam & Tutorial Exercises 18
8 ARMA & Model Selection Exercises 18
9 Volatility Modelling Exercises 19
10 Risk Management Exercises 20
11 Multivariate & High-Frequency Exercises 21
,Financial Econometrics Complete Course Summary 3
Part I
ARMA Models & Model Selection
1 Week 1 Stationarity, ARMA & Financial Returns
1.1 Financial Returns
One-period simple return: R = P P− P = PP − 1
t
t t−1 t
Continuously compounded (log) return: r = log PP
t−1 t−1
t
t = log Pt − log Pt−1
t−1
Disadvantage: from continuous compounding of simple returns.
First-order Taylor approximation: log(1 + R ) ≈ R , so r ≈ R for short-horizon returns.
Advantage of log returns: additive over time. If r is the k-period log return, then
t t t t
t,k
Pt
rt,k = log = rt + rt−1 + · · · + rt−k+1
Pt−k
The sum of capital gain and dividend yield.
Stylized facts of
nancial returns:
Have (almost) mean close to zero and very little autocorrelation
Display volatility clustering: alternating periods of high and low volatility
Have non-Gaussian distribution: fat tails (excess kurtosis)
Related facts (squared/absolute returns):
Display long-range, very slow mean-reversion
Changes have similar characteristics as returns
1.2 Stationarity
A process x is weakly stationary if it has constant mean, constant variance, and Cov(x , x depends only
on k:
t t t−k )
E[x ] = µ, t Var(x ) = σ , Cov(x , x ) = γ t
2
t t−k k
ACF: ρ = γγ k
k
0
Strict stationarity: the joint distribution does not change over time this is a much stronger requirement.
Weak stationarity: only mean, variance and autocovariance structure are constant.
ACF: ρ is a correlation with sign; |ρ | ≤ 1, ρ = 1, ρ = ρ .
White noise (all-zero ACF) is an example of a stationary process: E[ε ] = 0, ρ .
k k 0 k −k
t k = 0 ∀k ̸= 0
1.3 Wold Decomposition
Wold decomposition: every zero-mean stationary process that can be written as the sum of a linear and a
deterministic process: ∞
X ∞
X
xt = µ + ψj εt−j , ψ0 = 1, ψj2 < ∞
j=0 j=0
Linear process: xt = µ + ψ(L)εt
1.4 AR(1) Process
AR(1) : xt = ϕ0 + ϕ1 xt−1 + εt
is a necessary condition for stationarity, i.e. Cov(x , x ) → 0 as k → ∞
,→ |ϕ1 | < 1
For stationarity, we need: Var(x ) = Var(x ), so:
t t−k
t t−1
Var(x ) = ϕ Var(x ) + σ ⇒ γ = 1 −σ ϕ
2
2 2
t 1 t−1 0 2
1
,Financial Econometrics Complete Course Summary 4
So, order stationarity condition: µ = E[x ] = 1 −ϕ ϕ and γ = 1 −σ ϕt
0
1
0
2
2
1
Backward substitution gives: x = 1 −ϕ ϕ + P ϕ ε (in
nite MA representation)
t
0 ∞
j=0
j
1 t−j
,→ So, AR(1) is actually a linear process, where ψ = ϕ , so the weights decay geometrically.
1
We
nd ACF: ρ = ϕ
j
j 1
k
k 1
1.5 AR(p) Process
AR(p) : xt = ϕ0 + ϕ1 xt−1 + · · · + ϕp xt−p + εt
Some intuition when inverting the polynomial: (1 − ϕ L − · · · − ϕ L )x = ϕ + ε
Lag polynomial representation: ϕ(L)x = ϕ + ε with ϕ(L) = 1 − ϕ L − · · · − ϕ L
p
1 p t 0 t
p
An AR(p) model is stationary if all roots of ϕ(z) = 0 lie outside the unit circle.
t 0 t 1 p
Stationary AR(p) model has mean: µ = 1 − ϕ −ϕ · · · − ϕ 0
Some simple expressions for ACF: for AR(p), the ACF autocorrelations can be derived recursively from:
1 p
ρk = ϕ1 ρk−1 + · · · + ϕp ρk−p
Some general decay in exponential rate: ρ ∝ |λ| , cλ , c < 1, implying exponential decay.
k
,→ For an AR(p) model, the PACF has cut-o point at lag p, i.e. ϕ = 0 for k > p.
k 1
kk
1.6 MA(q) Process
M A(q) : xt = c0 + θ1 εt−1 + · · · + θq εt−q + εt
M A(q) models are always stationary!
ACF has cut-o point at lag q : ρ = 0 for k > q
k
PACF decays exponentially
M A(q) model is invertible if all roots of θ(z) = 0 lie outside the unit circle
θ(z) = 1 + θ1 z + · · · + θq z q = 0
1.7 ARMA(p,q) Process
ARM A(p, q) : xt = ϕ0 + ϕ1 xt−1 + · · · + ϕp xt−p + θ1 εt−1 + · · · + θq εt−q + εt
or, in lag polynomial form: ϕ(L)x = ϕ + θ(L)ε
Stationary: if all roots of ϕ(z) = 0 are outside the unit circle
t 0 t
Invertible: if all roots of θ(z) = 0 are outside the unit circle
Both ACF and PACF of ARMA models decay exponentially; neither has cut-o point.
Identi
cation problem. Consider the M A(∞) representation of an ARM A(p, q) model:
θ(L)
xt = ψ(L)εt = εt
ϕ(L)
If ϕ(z) and θ(z) share a common root, the model is not identi
ed. To avoid identi
cation problems, reduce
the model to ARM A(p − 1, q − 1).
ACF PACF
AR(p) exponential decay cut-o at lag p
MA(q) cut-o at lag q exponential decay
ARMA(p, q) exponential decay exponential decay
,→ MA uses past errors (εt−1 , εt−2 , . . . ); AR uses past observations (xt−1 , xt−2 , . . . ).
