BUSINESS ANALYTICS
CONTENTS
Asymptotic normality 11
INTRODUCTION 2 Functional form and qualitative information 12
Ordinary Least Squares (OLS) 2 Qualitative binary information 12
Econometrics 2 Ordinal information 12
Nonlinear regression functions 12
Probability theory 4
The expected value 4 Instrumental variable estimation (A4) 14
Variance 4 Two stage least squares (TSLS) 14
Covariance and Correlation 4 IV assumptions 14
Variance of sums 4 Variance of the IV estimator 15
Conditional expectations 5 Using the IV estimator 15
Law of Iterated Expectations (LIE) 5 Checking instrument validity 15
Finite sample properties 5
Unbiasedness 5 Heteroskedasticity (A5) 17
Sampling variance of an estimator 5 Detecting heteroskedasticity 17
Relative Efficiency 5 Heteroskedasticity-robust inference after OLS 17
The (weak) Law of Large Numbers (LLN) 6 Alternatives to heteroskedasticity-robust statistics 18
Consistency 6
Asymptotic normality: Central limit theory (CLT) 6 Stationary time series models 19
Autocovariance and Autocorrelation in time series
OLS estimation 7 data 19
Multiple regression analysis 7 Stationarity and weak dependence 19
Interpretation of OLS coefficients 7 Properties of AR(1) models 20
Algebraic facts about OLS 7 Bayes Information Criterion (BIC) and Akaike
CLM assumptions which are necessary for OLS 8 Information Criterion (AIC) 21
Autoregressive Distributed Lag (ADL) models 21
Hypothesis testing 9
Quality of the test 9 Non-stationary time series models 22
Additional hypothesis tests 9 Unit root process for AR(1) models 22
Test for the unit root in an AR(1) model 23
Asymptotic properties of the OLS estimator 11 Volatility clustering and ARCH 24
Consistency 11
,INTRODUCTION
Business analytics The ability of firms and organizations to collect, analyze, and act on data.
It makes use of
1. Data mining, detecting new patterns and relationships
2. Experiments, test economic decisions
3. Statistical and quantitative analyses, investigate the cause of events
4. Predictive analysis, predict future events.
There are two important estimation problems to consider this course:
1. Prediction. What will happen?
Using a formula to predict a dependent variable based on the observed values of independent variables.
2. Causal inference. Why will this happen?
Determine whether a particular independent variable affects the dependent variable, and estimate the magnitude
of the effect.
A linear model 𝒚 = 𝜷𝟎 + 𝜷𝟏 𝒙𝟏 + ⋯ + 𝜷𝒌 𝒙𝒌 + 𝒖
It consists of the outcome (𝒚), the regressors (𝒙𝟏 , … , 𝒙𝒌 ), and the true parameters (𝜷𝟎 , … , 𝜷𝒌 ).
𝒖~𝑵(𝟎, 𝝈𝟐 𝑰) is the error term of the model.
OLS obtains the estimates of the true parameters which minimize the sum of squared residuals.
It has two goals, which correspond to the estimation problems:
1. Predictive modelling. Estimate the conditional mean 𝐸( 𝑦 ∣ 𝑥 ):
̂
𝑬(𝒚 ∣ 𝒙) = 𝜷̂𝟎 + 𝜷
̂ 𝟏 𝒙𝟏 + ⋯ + 𝜷
̂ 𝒌 𝒙𝒌
2. Casual estimation, Interest in a particular 𝛽𝑗 :
̂
𝜹𝑬(𝒚|𝒙)
̂𝒋
=𝜷
𝜹𝒙𝒋
ORDINARY LEAST SQUARES (OLS)
Important for OLS is the assumption of zero conditional mean 𝑬( 𝒖 ∣ 𝒙 ) = 𝟎: 𝑥 and 𝑢 are independent
𝑬(𝒙 ∣ 𝒚) = 𝑬(𝒙𝜷 + 𝒖 ∣ 𝒙) = 𝒙𝜷 + 𝑬(𝒖 ∣ 𝒙)
1. Prediction, the goal is to find a regression line which fits the data as close as possible.
𝐸(𝑢 ∣ 𝑥) is not important, prediction focusses on what is observable, which this factor is not.
2. Causal estimation, the goal is to determine a particular 𝛽̂𝑗
The causal interpretation of 𝛽̂𝑗 fails if 𝐸( 𝑢 ∣∣ 𝑥𝑗 ) ≠ 0, as this will cause issues when defining the partial derivative.
In this case you will return a biased estimate of 𝛽𝑗
ECONOMETRICS
Econometrics Use statistical methods to estimate economic relationships, test economic theories, and to evaluate
and implement government and business policy.
The goal is to infer that one variable has a causal effect on another variable
The ceteris paribus analysis is done to test the effect of 𝒙𝒊 on 𝑦𝑖 while holding all other factors fixed.
A problem is that the available data is mostly the observational data
A solution to this issue, impose assumptions to simulate a ceteris paribus analysis.
Restrict the relationship between 𝒙𝒋 and 𝑦 using the zero conditional mean assumption.
Omitted variables might interfere with the ability to determine causal relationships. These variables can affect both 𝒚𝒄
and 𝒙𝒄 and it ends up in the error term 𝒖𝒄 as it is not possible to observe the variable.
The estimated regression coefficient b is estimated using:
, 𝒄𝒐𝒗(𝒙, 𝒚) 𝒄𝒐𝒗(𝒙, 𝒙𝒃 + 𝒖) 𝒄𝒐𝒗(𝒙, 𝒖)
̂=
𝒃 = =𝒃+
𝒄𝒐𝒗(𝒙, 𝒙) 𝒄𝒐𝒗(𝒙, 𝒙) 𝒄𝒐𝒗(𝒙, 𝒙)
If 𝑐𝑜𝑣(𝑥, 𝑢) = 0, the coefficient is unbiased.
A regression based on variables which are affected by the omitted variables can over- or underestimate the effect. No
causal effect can be found due to the bias from the omitted variable. If the omitted variable is randomized, it is still
possible to determine a causal effect, as the variable does not introduce a bias.
This bias is a problem for causal estimation. It can be an issue for prediction if including the unobserved variables
improves the prediction.
Bias can be remedied by adding predictors that truly capture any unobserved factors that are correlated with 𝑥 and 𝑦.
