Empirical Finance Summary
MSc Finance ’22 – ’23, period 1
Contents
Part I – Linear regression ....................................................................................................................4
1.1 Explanation of linear models and transformation ......................................................................4
1.2 Dummies...................................................................................................................................5
1.3 Model adequacy and outliers (𝑅2, 𝑅2, 𝐴𝐼𝐶 & 𝐵𝐼𝐶) ....................................................................6
1.4 OLS framework..........................................................................................................................7
1.5 t-test & F-test ............................................................................................................................8
Part II – Logit ....................................................................................................................................10
2.1 Limited dependent variable models ........................................................................................ 10
2.2 Logit/Probit models and their estimation through maximum likelihood .................................. 10
2.3 Parameter interpretation and marginal effects ........................................................................ 10
2.4 Model adequacy: R2 and ROC/CAP curve ................................................................................. 11
2.5 Other Logit/Probit models ...................................................................................................... 13
Part III – Panel data ..........................................................................................................................14
3.1 Motivation for panel data........................................................................................................ 14
3.2 Pooled regression.................................................................................................................... 14
3.3 Fixed effects models................................................................................................................ 15
3.4 Within and between estimator ................................................................................................ 16
3.5 Random effects models ........................................................................................................... 16
3.6 Take fixed effects or random effects? ...................................................................................... 16
3.7 Clustered standard errors ........................................................................................................ 17
Part IV – Endogeneity and Diff-in-Diff ..............................................................................................18
, 4.1 Endogeneity explanation ......................................................................................................... 18
4.2 Source 1: Measurement error ................................................................................................. 18
4.3 Source 2: Omitted variable bias ............................................................................................... 19
4.4 Solution 1: Instrumental Variables (IV) .................................................................................... 20
4.5 Solution 2: Diff-in-Diff approach .............................................................................................. 20
Part V – Misspecification ..................................................................................................................22
5.1 Assumption 0a; correction specification .................................................................................. 22
5.2 Assumption 0b; no perfect multicollinearity ............................................................................ 23
5.3 Assumption 1; zero mean errors.............................................................................................. 24
5.4 Assumption 4’; non-random regressors ................................................................................... 24
5.5 Assumption 4; no covariance between errors and regressors .................................................. 24
5.6 Assumption 5; normality ......................................................................................................... 24
5.7 Assumption 2; homoskedasticity ............................................................................................. 25
5.8 Assumption 3; No cross-correlation ......................................................................................... 26
Part VI – Event studies .....................................................................................................................27
6.1 Motivation and overview ........................................................................................................ 27
6.2 Event study methodology ........................................................................................................ 27
6.3 Cross-sectional regressions ..................................................................................................... 29
6.4 Extensions and some important aspects .................................................................................. 29
Part VII – Univariate linear time series models ................................................................................31
7.1 Specification of AR, MA & ARMA models ................................................................................. 31
7.2 (Partial) Autocorrelations ........................................................................................................ 32
7.3 Diagnostics checking ............................................................................................................... 37
7.4 Forecasting with univariate time series models ....................................................................... 39
7.5 Forecasting an AR(1) model Empty sheet to write on .......................................................... 40
7.6 Forecasting an ARMA(2,1) model Empty sheet to write on ................................................. 41
7.7 Evaluation of forecasts ............................................................................................................ 42
Part VIII – Unit roots.........................................................................................................................44
8.1 Motivation .............................................................................................................................. 44
8.2 Non-stationarity and unit roots ............................................................................................... 44
8.3 Testing for unit roots ............................................................................................................... 48
Part IX – Univariate volatility models ...............................................................................................51
9.1 Motivation and overview ........................................................................................................ 51
9.2 Different types of volatility ...................................................................................................... 51
9.3 ARCH and GARCH models, their varieties and motivation ........................................................ 52
9.4 Estimation and diagnostics ...................................................................................................... 56
2
Empirical Finance summary
, 9.5 Forecasting and evaluation...................................................................................................... 58
Part X – Extra: Overview tables ........................................................................................................61
10.1 OLS Assumptions ................................................................................................................... 61
10.2 Linear probability model overview ........................................................................................ 62
10.3 Various models overview ...................................................................................................... 62
3
Empirical Finance summary
, Part I – Linear regression
1.1 Explanation of linear models and transformation
Most simple model to explain a relationship between certain factors:
𝑌𝑖 = 𝛽0 + 𝛽1 𝑋𝑖 + 𝛽2 𝑋𝑖 + 𝜖𝑖
This is also called the ‘’Classical Linear Regression Model’’ (CLRM), and it consist of
the following main components:
Dependent variable Is observed and random
Parameters Are fixed & non-random, however they are not known,
therefore we estimate them.
Regressors Are observed and could be random or non-random
Error term Is unobserved and random
Anything that depends on data will be (a) random (variable) and will have
distributional properties.
This linear relationship can also be described through vector notation:
First standard notation: 𝑦𝑖 = 𝑥𝑖′ 𝛽 + 𝜖𝑖
where the beta’s and regressors are stored in column vectors:
𝛽0 1
𝛽 = (𝛽1 ) , 𝑥𝑖 = (𝑋…1 ) Note that the Xi vector needs to be transposed in order to
….
βi 𝑋𝑖
make matrix multiplication possible.
The second standard notation stacks all the observations i = 1, …., 3010:
𝑦 = 𝑋𝛽 + 𝜖
A linear regression can be done when the model is linear in the parameters, e.g.:
𝑊𝑎𝑔𝑒𝑖 = 𝛽0 + 𝛽1 𝐸𝑑𝑢𝑐𝑖 + 𝜖𝑖
4
Empirical Finance summary
MSc Finance ’22 – ’23, period 1
Contents
Part I – Linear regression ....................................................................................................................4
1.1 Explanation of linear models and transformation ......................................................................4
1.2 Dummies...................................................................................................................................5
1.3 Model adequacy and outliers (𝑅2, 𝑅2, 𝐴𝐼𝐶 & 𝐵𝐼𝐶) ....................................................................6
1.4 OLS framework..........................................................................................................................7
1.5 t-test & F-test ............................................................................................................................8
Part II – Logit ....................................................................................................................................10
2.1 Limited dependent variable models ........................................................................................ 10
2.2 Logit/Probit models and their estimation through maximum likelihood .................................. 10
2.3 Parameter interpretation and marginal effects ........................................................................ 10
2.4 Model adequacy: R2 and ROC/CAP curve ................................................................................. 11
2.5 Other Logit/Probit models ...................................................................................................... 13
Part III – Panel data ..........................................................................................................................14
3.1 Motivation for panel data........................................................................................................ 14
3.2 Pooled regression.................................................................................................................... 14
3.3 Fixed effects models................................................................................................................ 15
3.4 Within and between estimator ................................................................................................ 16
3.5 Random effects models ........................................................................................................... 16
3.6 Take fixed effects or random effects? ...................................................................................... 16
3.7 Clustered standard errors ........................................................................................................ 17
Part IV – Endogeneity and Diff-in-Diff ..............................................................................................18
, 4.1 Endogeneity explanation ......................................................................................................... 18
4.2 Source 1: Measurement error ................................................................................................. 18
4.3 Source 2: Omitted variable bias ............................................................................................... 19
4.4 Solution 1: Instrumental Variables (IV) .................................................................................... 20
4.5 Solution 2: Diff-in-Diff approach .............................................................................................. 20
Part V – Misspecification ..................................................................................................................22
5.1 Assumption 0a; correction specification .................................................................................. 22
5.2 Assumption 0b; no perfect multicollinearity ............................................................................ 23
5.3 Assumption 1; zero mean errors.............................................................................................. 24
5.4 Assumption 4’; non-random regressors ................................................................................... 24
5.5 Assumption 4; no covariance between errors and regressors .................................................. 24
5.6 Assumption 5; normality ......................................................................................................... 24
5.7 Assumption 2; homoskedasticity ............................................................................................. 25
5.8 Assumption 3; No cross-correlation ......................................................................................... 26
Part VI – Event studies .....................................................................................................................27
6.1 Motivation and overview ........................................................................................................ 27
6.2 Event study methodology ........................................................................................................ 27
6.3 Cross-sectional regressions ..................................................................................................... 29
6.4 Extensions and some important aspects .................................................................................. 29
Part VII – Univariate linear time series models ................................................................................31
7.1 Specification of AR, MA & ARMA models ................................................................................. 31
7.2 (Partial) Autocorrelations ........................................................................................................ 32
7.3 Diagnostics checking ............................................................................................................... 37
7.4 Forecasting with univariate time series models ....................................................................... 39
7.5 Forecasting an AR(1) model Empty sheet to write on .......................................................... 40
7.6 Forecasting an ARMA(2,1) model Empty sheet to write on ................................................. 41
7.7 Evaluation of forecasts ............................................................................................................ 42
Part VIII – Unit roots.........................................................................................................................44
8.1 Motivation .............................................................................................................................. 44
8.2 Non-stationarity and unit roots ............................................................................................... 44
8.3 Testing for unit roots ............................................................................................................... 48
Part IX – Univariate volatility models ...............................................................................................51
9.1 Motivation and overview ........................................................................................................ 51
9.2 Different types of volatility ...................................................................................................... 51
9.3 ARCH and GARCH models, their varieties and motivation ........................................................ 52
9.4 Estimation and diagnostics ...................................................................................................... 56
2
Empirical Finance summary
, 9.5 Forecasting and evaluation...................................................................................................... 58
Part X – Extra: Overview tables ........................................................................................................61
10.1 OLS Assumptions ................................................................................................................... 61
10.2 Linear probability model overview ........................................................................................ 62
10.3 Various models overview ...................................................................................................... 62
3
Empirical Finance summary
, Part I – Linear regression
1.1 Explanation of linear models and transformation
Most simple model to explain a relationship between certain factors:
𝑌𝑖 = 𝛽0 + 𝛽1 𝑋𝑖 + 𝛽2 𝑋𝑖 + 𝜖𝑖
This is also called the ‘’Classical Linear Regression Model’’ (CLRM), and it consist of
the following main components:
Dependent variable Is observed and random
Parameters Are fixed & non-random, however they are not known,
therefore we estimate them.
Regressors Are observed and could be random or non-random
Error term Is unobserved and random
Anything that depends on data will be (a) random (variable) and will have
distributional properties.
This linear relationship can also be described through vector notation:
First standard notation: 𝑦𝑖 = 𝑥𝑖′ 𝛽 + 𝜖𝑖
where the beta’s and regressors are stored in column vectors:
𝛽0 1
𝛽 = (𝛽1 ) , 𝑥𝑖 = (𝑋…1 ) Note that the Xi vector needs to be transposed in order to
….
βi 𝑋𝑖
make matrix multiplication possible.
The second standard notation stacks all the observations i = 1, …., 3010:
𝑦 = 𝑋𝛽 + 𝜖
A linear regression can be done when the model is linear in the parameters, e.g.:
𝑊𝑎𝑔𝑒𝑖 = 𝛽0 + 𝛽1 𝐸𝑑𝑢𝑐𝑖 + 𝜖𝑖
4
Empirical Finance summary
