Financial Econometrics
Prof J. Annaert
Academic year 2025-2026
1. Stylized return facts......................................................................................................................... 2
Introduction to R.......................................................................................................................................... 2
Descriptive statistics.....................................................................................................................................3
Stylised return facts......................................................................................................................................5
2. Refresher on regression...................................................................................................................10
Regression analysis.....................................................................................................................................10
R implementation....................................................................................................................................... 11
Specification checks................................................................................................................................... 12
3. Linear time series models................................................................................................................ 15
Motivation.................................................................................................................................................. 15
Time series concepts...................................................................................................................................15
Moving Average Processes......................................................................................................................... 16
Autoregressive Processes............................................................................................................................17
Autoregressive Moving Average Processes................................................................................................. 18
Estimation and testing (refresher).............................................................................................................. 19
Non-stationarity......................................................................................................................................... 19
4. Predicting asset returns.................................................................................................................. 20
Random Walk hypothesis........................................................................................................................... 20
Serial correlation tests................................................................................................................................20
Predictive regressions.................................................................................................................................22
Out-of-sample forecasting..........................................................................................................................23
5. Volatility modelling........................................................................................................................ 24
Introduction volatility dynamics................................................................................................................. 24
ARCH(1)..................................................................................................................................................... 24
GARCH models...........................................................................................................................................26
Asymmetric GARCH models....................................................................................................................... 27
GARCH in mean..........................................................................................................................................28
6. Multivariate GARCH models............................................................................................................ 29
Motivation and background........................................................................................................................29
Multivariate GARCH models....................................................................................................................... 30
Direct generalisations of univariate models................................................................................................ 30
Nonlinear combinations of univariate models............................................................................................. 32
1
,1. Stylized return facts
Introduction to R slide 5
1. Returns are growth rates
When talking about returns, we can divide it in three types, given that all wealth is reinvested in the investment
𝑊𝑡+1
● gross return: 1 + 𝑅𝑡+1 = 𝑊𝑡
𝑊𝑡+1 𝑟
● simple return: 𝑅𝑡+1 = 𝑊𝑡
− 1 =𝑒𝑡− 1
● log return: 𝑟𝑡+1 = 𝑙𝑛 ( )𝑊𝑡+1
𝑊𝑡 (
= 𝑙𝑛 1 + 𝑅𝑡+1 = 𝑙𝑛 𝑊𝑡+1 − 𝑙𝑛 𝑊𝑡) ( ) ( )
Returns can also be divided into their cashflow components (dividend yield and price appreciation)
𝑅𝑡+1 =
𝑃𝑡+1+𝐷𝑡+1
𝑃𝑡
− 1 or 𝑅𝑡+1 =
𝐷𝑡+1
𝑃𝑡
+ ( 𝑃𝑡+1
𝑃𝑡
− 1 )
2. Returns in multiperiod settings (time-series): log returns have an advantage
In financial econometrics, the use of log returns is widespread because for small |𝑥| we have that 𝑙𝑛(1 + 𝑥) ≈ 𝑥,
meaning that 𝑅 ≈ 𝑟. This small difference gives us the chance to use the logarithm properties when applying the chain
link rule in a multiperiod setting.
𝑊𝑡+1 = 𝑊𝑡 1 + 𝑅𝑡+1 ( )
(
𝑊𝑡+2 = 𝑊𝑡+1 1 + 𝑅𝑡+2 = 𝑊𝑡 1 + 𝑅𝑡+1 1 + 𝑅𝑡+2) ( )( )
Thereby obtaining the chain-link rule for the multiperiod return
τ τ
⎡
(
𝑊𝑡+τ = 𝑊𝑡⎢ ∏ 1 + 𝑅𝑡+𝑠
⎢𝑠=1 )⎤⎥⎥ ⇒ 𝑅𝑡+τ(τ) = ∏ 1 + 𝑅𝑡+𝑠 − 1 ( )
⎣ ⎦ 𝑠=1
When using log returns, this rule becomes much easier to estimate since we know the statistical properties of sums of
random variables but not the properties of products of random variables.
𝑟𝑡+1
𝑊𝑡+1 = 𝑊𝑡 𝑒
𝑟𝑡+2 𝑟𝑡+1 𝑟𝑡+2 𝑟𝑡+1+𝑟𝑡+2
𝑊𝑡+2 = 𝑊𝑡+1 𝑒 = 𝑊𝑡 𝑒 𝑒 = 𝑊𝑡 𝑒
Thereby changing the chain-link rule to an additive expression
τ
∑ 𝑟𝑡+𝑠 τ
𝑊𝑡+τ = 𝑊𝑡𝑒𝑠=1 ⇒ 𝑟𝑡+τ(τ) = ∑ 𝑟𝑡+𝑠
𝑠=1
3. Returns in a portfolio context: simple returns have an advantage
In a portfolio, single-asset returns are weighted by their importance in the portfolio to create a weighted average of all
assets. Simple returns are additive in a portfolio setting.
𝑁
𝑅𝑝,𝑡+1 = ∑ 𝑤𝑗,𝑡𝑅𝑗,𝑡+1
𝑗=1
Instead of using market capitalization to weight the components, stock prices can also be used as a measure.
→ eg. the Dow Jones Industrial Average uses stock prices to weight its index components
2
,Descriptives slide 25
In order to answer all questions regarding probabilities of returns, volatility, systematic risk … we need to have an
understanding about the return distribution (cdf) either of a single asset or in a multivariate setting
[ ] [ ] [
𝐹(𝑥) = 𝑃 𝑅𝑡 ≤ 𝑥 or 𝐹(𝑥) = 𝑅𝑡 ≤ 𝑥 = 𝑃 𝑅𝑗𝑡 ≤ 𝑥𝑗, 𝑗 = 1, 2,... 𝑁 ]
1. Random sample
To make an inference, a sample has to be random. This implies that the observations have to be independently and
identically distributed (iid) to satisfy the CLT requirements. If this is the case, then its distribution in large samples will
converge towards a normal distribution making inference possible.
A random sample’s empirical distribution 𝐹 can be used to inform us about the characteristics of the population
distribution 𝐹.
2. The empirical distribution function
For a random sample 𝑅𝑡{ }𝑇𝑡=1 we can define two random variables (for each sample, different results will be obtained)
𝑇
● the empirical probability distribution function (pdf): 𝑝(𝑥) =
1
𝑇 [
Σ𝑡=1 𝐼 𝑅𝑡 = 𝑥]
𝑇
Σ𝑡=1 𝐼[𝑅𝑡 ≤ 𝑥]
1
● the empirical cumulative distribution function (cdf): 𝐹(𝑥) = 𝑇
Following the (weak) Law of Large Numbers, when 𝑇 → ∞ then 𝐹(𝑥) → 𝐹(𝑥).
● 𝐹(𝑥) is a consistent estimator of 𝐹(𝑥) because lim 𝑃 ⎡⎢|||𝐹𝑇(𝑥) − 𝐹(𝑥)||| > ϵ⎤⎥ = 0
𝑇→∞ ⎣ ⎦
● “the probability to get deviations from the population cdf goes to zero”, also called the probability limit
If this is the case, we can use the empirical distribution function instead of the population cdf to infer the stochastic
properties of a random variable 𝑋.
To use this theorem, we need to test whether the assumptions of random sampling and iid observations hold. However,
the data need not be "completely" independent for the fundamental theorem to hold. Some dependencies (correlations)
do not break down the characteristics of random sampling.
An alternative way to get an idea of the return distribution’s shape is to plot a histogram, which resembles a pdf.
3. Asymptotic properties statistics
3.1 Consistency descriptive statistics
We saw that 𝐹(𝑥) is a consistent estimator for 𝐹(𝑥). This property also carries over to the descriptive statistics as long as
the population moment exists and the data are identically distributed.
𝑇 𝑋𝑡
● raw sample moments are consistent: 𝐸[𝑋] is estimated by 𝑋 = ∑ 𝑇
𝑡=1
2
2
● central sample moments are consistent: 𝑣𝑎𝑟(𝑋) = 𝐸[(𝑋 − 𝐸(𝑋)) ] is estimated by ∑
𝑇
(𝑋 −𝑋)
𝑡
𝑇
𝑡=1
● sample quantiles are consistent
3
, This leads to the analog or plug-in principle of estimation where we use the empirical pdf as a substitute for the real pdf
to estimate a statistic of interest.
Often we are interested in functions of moments (eg. volatility, Sharpe ratio, skewness and kurtosis). Luckily, the concept
of consistency carries over to functions of random variables.
3.2 Central limit theorem (CLT)
We know that the sample mean µ𝑇 is a consistent estimator for the population expected value µ. But can we also say
something useful about (probabilities of) deviations from µ?
A Central Limit Theorem (CLT) states that the pdf of “inflated” random deviations is well approximated by a normal
distribution when T is large.
(
𝑇 µ− µ )
( )
Consider 𝑇 iid random variables with 𝐸 𝑋𝑡 = µ and 𝑣𝑎𝑟 𝑋𝑡 = σ². ( )
𝑋𝑡−µ 𝑇 𝑍𝑡
𝑍𝑡 = σ
and 𝑍𝑇 = ∑ 𝑇
𝑡=1
for all 𝑧 ∈ 𝑅: lim 𝑃⎡⎢ 𝑇 𝑍𝑇 < 𝑧⎤⎥ = Φ(𝑧)
𝑇→∞ ⎣ ⎦
We say that 𝑍𝑇 converges in distribution to a standard normal random variable. Alternatively, we call its asymptotic
distribution a standard normal distribution.
𝑑
𝑇 𝑍𝑇 → 𝑍 ~ 𝑁(0, 1)
𝑇 ( )
𝑋𝑇−µ
σ
𝑑
→ 𝑍 ~ 𝑁(0, 1) or 𝑇 ( ) → 𝑍 ~ 𝑁(0, 1)
µ−µ
σ
𝑎
3.3 Asymptotic distribution
Under some regularity conditions
● raw sample moments are asymptotic normal
● central sample moments are asymptotic normal
● quantiles are asymptotic normal
This proposition holds for data of all distributions as long as the moments exist and are finite.
We now know ( ) ( )
𝑇 𝑋𝑇 − µ ~𝑁(0, σ²), but we are interested in the distribution of a function 𝑌𝑇 = 𝑔 𝑋𝑇 . For many
functions, the asymptotic distribution of 𝑇 𝑌𝑇 is also normal. The parameters for 𝑌𝑇 are related to µ and σ² and can be
found by applying the delta method. This property is asymptotic, meaning that it is only valid in large samples.
2
( )
𝑇 𝑌𝑇 − 𝑔(µ) ~ 𝑁 0, [𝑔'(µ)σ] ( )
3.4 Slutsky’s theorem
The asymptotic distribution depends on unknown population parameters (eg. σ² or 𝑘). However, Slutky’s theorem allows
us to replace these parameters by consistent estimators without affecting the asymptotic distribution.
4
Prof J. Annaert
Academic year 2025-2026
1. Stylized return facts......................................................................................................................... 2
Introduction to R.......................................................................................................................................... 2
Descriptive statistics.....................................................................................................................................3
Stylised return facts......................................................................................................................................5
2. Refresher on regression...................................................................................................................10
Regression analysis.....................................................................................................................................10
R implementation....................................................................................................................................... 11
Specification checks................................................................................................................................... 12
3. Linear time series models................................................................................................................ 15
Motivation.................................................................................................................................................. 15
Time series concepts...................................................................................................................................15
Moving Average Processes......................................................................................................................... 16
Autoregressive Processes............................................................................................................................17
Autoregressive Moving Average Processes................................................................................................. 18
Estimation and testing (refresher).............................................................................................................. 19
Non-stationarity......................................................................................................................................... 19
4. Predicting asset returns.................................................................................................................. 20
Random Walk hypothesis........................................................................................................................... 20
Serial correlation tests................................................................................................................................20
Predictive regressions.................................................................................................................................22
Out-of-sample forecasting..........................................................................................................................23
5. Volatility modelling........................................................................................................................ 24
Introduction volatility dynamics................................................................................................................. 24
ARCH(1)..................................................................................................................................................... 24
GARCH models...........................................................................................................................................26
Asymmetric GARCH models....................................................................................................................... 27
GARCH in mean..........................................................................................................................................28
6. Multivariate GARCH models............................................................................................................ 29
Motivation and background........................................................................................................................29
Multivariate GARCH models....................................................................................................................... 30
Direct generalisations of univariate models................................................................................................ 30
Nonlinear combinations of univariate models............................................................................................. 32
1
,1. Stylized return facts
Introduction to R slide 5
1. Returns are growth rates
When talking about returns, we can divide it in three types, given that all wealth is reinvested in the investment
𝑊𝑡+1
● gross return: 1 + 𝑅𝑡+1 = 𝑊𝑡
𝑊𝑡+1 𝑟
● simple return: 𝑅𝑡+1 = 𝑊𝑡
− 1 =𝑒𝑡− 1
● log return: 𝑟𝑡+1 = 𝑙𝑛 ( )𝑊𝑡+1
𝑊𝑡 (
= 𝑙𝑛 1 + 𝑅𝑡+1 = 𝑙𝑛 𝑊𝑡+1 − 𝑙𝑛 𝑊𝑡) ( ) ( )
Returns can also be divided into their cashflow components (dividend yield and price appreciation)
𝑅𝑡+1 =
𝑃𝑡+1+𝐷𝑡+1
𝑃𝑡
− 1 or 𝑅𝑡+1 =
𝐷𝑡+1
𝑃𝑡
+ ( 𝑃𝑡+1
𝑃𝑡
− 1 )
2. Returns in multiperiod settings (time-series): log returns have an advantage
In financial econometrics, the use of log returns is widespread because for small |𝑥| we have that 𝑙𝑛(1 + 𝑥) ≈ 𝑥,
meaning that 𝑅 ≈ 𝑟. This small difference gives us the chance to use the logarithm properties when applying the chain
link rule in a multiperiod setting.
𝑊𝑡+1 = 𝑊𝑡 1 + 𝑅𝑡+1 ( )
(
𝑊𝑡+2 = 𝑊𝑡+1 1 + 𝑅𝑡+2 = 𝑊𝑡 1 + 𝑅𝑡+1 1 + 𝑅𝑡+2) ( )( )
Thereby obtaining the chain-link rule for the multiperiod return
τ τ
⎡
(
𝑊𝑡+τ = 𝑊𝑡⎢ ∏ 1 + 𝑅𝑡+𝑠
⎢𝑠=1 )⎤⎥⎥ ⇒ 𝑅𝑡+τ(τ) = ∏ 1 + 𝑅𝑡+𝑠 − 1 ( )
⎣ ⎦ 𝑠=1
When using log returns, this rule becomes much easier to estimate since we know the statistical properties of sums of
random variables but not the properties of products of random variables.
𝑟𝑡+1
𝑊𝑡+1 = 𝑊𝑡 𝑒
𝑟𝑡+2 𝑟𝑡+1 𝑟𝑡+2 𝑟𝑡+1+𝑟𝑡+2
𝑊𝑡+2 = 𝑊𝑡+1 𝑒 = 𝑊𝑡 𝑒 𝑒 = 𝑊𝑡 𝑒
Thereby changing the chain-link rule to an additive expression
τ
∑ 𝑟𝑡+𝑠 τ
𝑊𝑡+τ = 𝑊𝑡𝑒𝑠=1 ⇒ 𝑟𝑡+τ(τ) = ∑ 𝑟𝑡+𝑠
𝑠=1
3. Returns in a portfolio context: simple returns have an advantage
In a portfolio, single-asset returns are weighted by their importance in the portfolio to create a weighted average of all
assets. Simple returns are additive in a portfolio setting.
𝑁
𝑅𝑝,𝑡+1 = ∑ 𝑤𝑗,𝑡𝑅𝑗,𝑡+1
𝑗=1
Instead of using market capitalization to weight the components, stock prices can also be used as a measure.
→ eg. the Dow Jones Industrial Average uses stock prices to weight its index components
2
,Descriptives slide 25
In order to answer all questions regarding probabilities of returns, volatility, systematic risk … we need to have an
understanding about the return distribution (cdf) either of a single asset or in a multivariate setting
[ ] [ ] [
𝐹(𝑥) = 𝑃 𝑅𝑡 ≤ 𝑥 or 𝐹(𝑥) = 𝑅𝑡 ≤ 𝑥 = 𝑃 𝑅𝑗𝑡 ≤ 𝑥𝑗, 𝑗 = 1, 2,... 𝑁 ]
1. Random sample
To make an inference, a sample has to be random. This implies that the observations have to be independently and
identically distributed (iid) to satisfy the CLT requirements. If this is the case, then its distribution in large samples will
converge towards a normal distribution making inference possible.
A random sample’s empirical distribution 𝐹 can be used to inform us about the characteristics of the population
distribution 𝐹.
2. The empirical distribution function
For a random sample 𝑅𝑡{ }𝑇𝑡=1 we can define two random variables (for each sample, different results will be obtained)
𝑇
● the empirical probability distribution function (pdf): 𝑝(𝑥) =
1
𝑇 [
Σ𝑡=1 𝐼 𝑅𝑡 = 𝑥]
𝑇
Σ𝑡=1 𝐼[𝑅𝑡 ≤ 𝑥]
1
● the empirical cumulative distribution function (cdf): 𝐹(𝑥) = 𝑇
Following the (weak) Law of Large Numbers, when 𝑇 → ∞ then 𝐹(𝑥) → 𝐹(𝑥).
● 𝐹(𝑥) is a consistent estimator of 𝐹(𝑥) because lim 𝑃 ⎡⎢|||𝐹𝑇(𝑥) − 𝐹(𝑥)||| > ϵ⎤⎥ = 0
𝑇→∞ ⎣ ⎦
● “the probability to get deviations from the population cdf goes to zero”, also called the probability limit
If this is the case, we can use the empirical distribution function instead of the population cdf to infer the stochastic
properties of a random variable 𝑋.
To use this theorem, we need to test whether the assumptions of random sampling and iid observations hold. However,
the data need not be "completely" independent for the fundamental theorem to hold. Some dependencies (correlations)
do not break down the characteristics of random sampling.
An alternative way to get an idea of the return distribution’s shape is to plot a histogram, which resembles a pdf.
3. Asymptotic properties statistics
3.1 Consistency descriptive statistics
We saw that 𝐹(𝑥) is a consistent estimator for 𝐹(𝑥). This property also carries over to the descriptive statistics as long as
the population moment exists and the data are identically distributed.
𝑇 𝑋𝑡
● raw sample moments are consistent: 𝐸[𝑋] is estimated by 𝑋 = ∑ 𝑇
𝑡=1
2
2
● central sample moments are consistent: 𝑣𝑎𝑟(𝑋) = 𝐸[(𝑋 − 𝐸(𝑋)) ] is estimated by ∑
𝑇
(𝑋 −𝑋)
𝑡
𝑇
𝑡=1
● sample quantiles are consistent
3
, This leads to the analog or plug-in principle of estimation where we use the empirical pdf as a substitute for the real pdf
to estimate a statistic of interest.
Often we are interested in functions of moments (eg. volatility, Sharpe ratio, skewness and kurtosis). Luckily, the concept
of consistency carries over to functions of random variables.
3.2 Central limit theorem (CLT)
We know that the sample mean µ𝑇 is a consistent estimator for the population expected value µ. But can we also say
something useful about (probabilities of) deviations from µ?
A Central Limit Theorem (CLT) states that the pdf of “inflated” random deviations is well approximated by a normal
distribution when T is large.
(
𝑇 µ− µ )
( )
Consider 𝑇 iid random variables with 𝐸 𝑋𝑡 = µ and 𝑣𝑎𝑟 𝑋𝑡 = σ². ( )
𝑋𝑡−µ 𝑇 𝑍𝑡
𝑍𝑡 = σ
and 𝑍𝑇 = ∑ 𝑇
𝑡=1
for all 𝑧 ∈ 𝑅: lim 𝑃⎡⎢ 𝑇 𝑍𝑇 < 𝑧⎤⎥ = Φ(𝑧)
𝑇→∞ ⎣ ⎦
We say that 𝑍𝑇 converges in distribution to a standard normal random variable. Alternatively, we call its asymptotic
distribution a standard normal distribution.
𝑑
𝑇 𝑍𝑇 → 𝑍 ~ 𝑁(0, 1)
𝑇 ( )
𝑋𝑇−µ
σ
𝑑
→ 𝑍 ~ 𝑁(0, 1) or 𝑇 ( ) → 𝑍 ~ 𝑁(0, 1)
µ−µ
σ
𝑎
3.3 Asymptotic distribution
Under some regularity conditions
● raw sample moments are asymptotic normal
● central sample moments are asymptotic normal
● quantiles are asymptotic normal
This proposition holds for data of all distributions as long as the moments exist and are finite.
We now know ( ) ( )
𝑇 𝑋𝑇 − µ ~𝑁(0, σ²), but we are interested in the distribution of a function 𝑌𝑇 = 𝑔 𝑋𝑇 . For many
functions, the asymptotic distribution of 𝑇 𝑌𝑇 is also normal. The parameters for 𝑌𝑇 are related to µ and σ² and can be
found by applying the delta method. This property is asymptotic, meaning that it is only valid in large samples.
2
( )
𝑇 𝑌𝑇 − 𝑔(µ) ~ 𝑁 0, [𝑔'(µ)σ] ( )
3.4 Slutsky’s theorem
The asymptotic distribution depends on unknown population parameters (eg. σ² or 𝑘). However, Slutky’s theorem allows
us to replace these parameters by consistent estimators without affecting the asymptotic distribution.
4
