Review of Mathematics and Statistics

Outline Introduction Motivation Unit Synopsis Topic 1: Concepts

ETF3600 Quantitative Models for BusinessResearch

Lecture 1: Introduction & Review

March 1, 2013


Introduction

Motivation

Unit Synopsis

Topic 1: ConceptsBasic MathematicsBasic Statistics


ETF3600/5600 - Quantitative Models for BusinessResearch

Lecturer: Dr. Kompal Sinha [[email protected]]

Lecture Room: CA H/H235

Monday 4:00pm - 6:00pm.

Tutorial

Tutorials: Monday 3:00-4:00pmCA K/K101


Quantitative Models and Business Research

“...[marketing] research is the collection, processing, and analysis of

information on topics relevant to marketing. It begins with problem

definition and ends with a report and action recommendations. ”

- Lehmann et al (1998)

Collect data ⇒ Collate data ⇒ Analyse data ⇒ Interpretfinding & Decision making

Data collection and storage techniques have advanced inrecent times.

Growing market competition required analysing these hugedatabases:

Data analysis is more than number crunching.For effective policy making it is important to efficientlytranslate technical information to effective decision making.


Quantitative Models for Business Research

Decision making = Expert Judgement + Quantitative Analysis

Decision making = ω1 Expert Judgement + ω2 QuantitativeAnalysisω1 + ω1 = 1

Quantitative analysis provide basic quantitative concepts andskills that form the base of knowledge essential toquantitative-decision-making professionals in businessenvironment.


Quantitative Models for Business ResearchWhy study ETF 3600/5600

“.There is no such thing as qualitative data. Everything is either 0 or 1. ”

- Fred Kerlinger

Traditional regression tools ”simple regression models” usefulwhen response variable is continuous or measured atcontinuous intervals: GDP, Sales, profits.Many salient variables in business research, social science,biomedical science are not ”continuous”, i.e., they are eitherqualitative or limited in their range:

Revealed preference data: sales and brand choiceCategorical: ”yes” or ”no”; ”employed” or ”unemployed”These variables are limited in their range because of someunderlying stochastic choice mechanism:

”agree”, ”disagree”, ”uncertain”;”poor”, ”good” ”excellent”

The general regression models are inappropriate and givemisleading answers - poor implications and ineffectivedecisions


Quantitative Models for Business Research - Unit Synopsis


Quantitative Models for Business Research - Unit Synopsis

This unit will discuss models that are appropriate when thedependent variable is binary, ordinal, nominal, counted,censored, truncated, latent.

Using E-Views to analyse data.

Theoretical concepts with empirical applications for themodels:

The nature of modelType of data for which it is relevantType of information one can get from estimating it:

Probability of given valueExpected valueMarginal effectsOdds RatioDiscrete changeInterpretation


Lecture Topics

Week 2: Review of regression analysis;Week 3: 3.1 Introduction to maximum likelihood

3.2 Models with binary dependent variable (1)Introduction, Linear Probability Model

Week 4: Models with binary dependent variable (2)4.1 Logit model4.2 Probit model.

Week 5: Models with binary dependent variable (3)5.1 Probit model (cont’)5.2 Inference.

Week 6: Models with binary dependent variable (4)6.1 Latent variable for binary dep. variable.Model with ordered multinomial dependent variable6.2 The ordered probit model

Week 7: 7.1 The ordered probit model (cont’)Models with unordered multinomial dependent variable (1)7.2 Introduction: Logit model for multiple choices


Lecture Topics

Week 8: Models with unordered multinomial dependent variable (2)8.1 Logit model for multiple choices (cont’)8.2 Post estimation analysis

Week 9: Models with unordered multinomial dependent variable (3)9.1 Example for conditional logit model9.2 Post estimation analysis

Week 10: Models for count data (1)10.1 The model using Poisson distribution10.2 The problems of truncation and censoring

Week 11: Models for count data (2)11.1 A test for overdispersion11.2 The negative binomial and the zero modified countmodels

Week 12: The Tobit regression model (if time permits)Summary revisionInformation on Examination


Studying this unitHow to study this unit

Text Prescribed text: The prescribed text below is available fromthe bookshop and the library

1 Long, J.S. (1997), Regression Models for Categorical andLimited Dependent Variables, SAGE Publications, London.(referred to as Long)

Recommended texts: Useful reference texts include1 Franses, P. H. and R. Paap (2003), Quantitative Models in

Marketing Research, Cambridge University press: Cambridge.(referred to as FP)

2 Powers, D.A. and Y. Xie (2000), Statistical Methods forCategorical Data Analysis, Academic Press, London. (referredto as Powers)

Come to lectures

Prepare for tutorials

Attend tutorials

Ask for help


Assessment Criteria∗†

Inclass Test 5%Assignments:Assignment I 12%Assignment II 18%Final examination 65%

† The final exam performance is the hurdle requirement for thisunit and where you fail the unit solely because of failure to satisfy

the hurdle requirement a final mark of 45 will be returned.∗ Please read unit outline for details.


ETF3600/5600: Quantitative Models for BusinessResearch

Topic 1: Some fundamental conceptsReading: Wooldridge Appendix A-C,

Franses and Paap Appendices A.1 and A.2Hill, Griffiths and Lim Appendices A and B.

References Wooldridge, J.M. (2006), Introductory Econometrics: AModern Approach, Thomson Higher Education: USA.Hill, R.C., W.E. Griffiths and G. Lim (2008), Principles ofEconometrics, WileySons: USAFranses, P. H. and R. Paap (2003), Quantitative Models inMarketing Research, Cambridge University press: Cambridge.


Topic 1: Some fundamental conceptsOutline

1 Basic Mathematics

1.1.1 Linear function1.1.2 Nonlinear function1.1.3 Derivatives1.1.4 Optimization1.1.5 Matrices1.1.6 Elasticity

2 Basic Statistics

2.1.1 Random Variable2.2.2 Probability distribution and density function (pdf)2.2.3 Cumulative distribution function (cdf)2.2.4 Normal distribution2.2.5 Standard logistic distribution


Basic Mathematics1.1.1 Linear function

A linear relationship between two variables

y = α + βx (1)

where α = intercept, β = slope

β =∆y

∆x(2)

where ∆ is a very small change

b =∂y

∂x(3)


Basic Mathematics.1.1.2 Non linear function

1.1.2 A nonlinear function is used frequently are quadraticfunction, cubic function and higher order function.

Quadratic : y = α + βx + γx2 (4)

Cubic : y = α + βx + γx2 + φx3 (5)

For a given point on the curve, the slope is the slope of thetangent to the line at that point.

It is calculated by finding ∂y∂x for a given x.

A tangent line that is steeper has higher slope.

For a nonlinear function, slope changes for different values ofx.


Basic Mathematics1.1.2 Non linear function (2)

Some Useful functions

y = β1 + β21

x(6a)

ln(y) = β1 + β2ln(x) (6b)

ln(y) = β1 + β2x (6c)

y = β1 + β2ln(x) (6d)

For useful nonlinear functions see Hill, Griffiths and Lim page 471.


Basic Mathematics1.1.3: Slope of a curve and differentiation

Considery = f (x) (7)

First derivative is = dydx = y

′is derivative of y with respect to

x.The dy

dx is the original notation used by Leibniz; the y′

is theLagrange’s notation.

The process of finding dydx is called differentiation.

Second derivative is to take derivative twice or y′′

.

d

dx(

dy

dx) =

d2y

dx2(8)

First derivative gives the slope and second derivative gives thechange in the slope when x changes.


Basic MathematicsSome Differentiation Rules

Product rule (Leibniz rule): Consider y = uv

d

dx(uv) = u.

dv

dx+ v .

du

dx(9)

Quotient rule: Consider y = uv

d

dx

u

v=

1

v2∗ (v

du

dx− u

dv

dx) (10)

Reciprocal rule: Consider y = 1/f (x)

y ′(x) =−1

[f (x)]2df (x)

dx(11)

Addition and Subtraction rule: Consider y(x) = u(x)± v(x)

dy

dx=

du

dx± dv

dx(12)

Power rule: Consider y = xn

dy/dx = nxn−1 (13)


Basic MathematicsDifferentiation Rules: Constant and Exponential function

Derivative of a constant is zero:Consider y = k

dy

dx=

dk

dx= 0 (14)

Exponential Function: Consider y = ex

dy

dx=

dex

dx= ex (15)

Consider y = kef (x)

dy

dx= k

def (x)

dx= kef (x)

df (x)

dx(16)


Basic MathematicsSome Differentiation Rules: Logarithmic function

Logarithmic FunctionConsider y = ln(x)

dy

dx=

dln(x)

dx=

1

xdx =

1

x(17)

Consider y = lnf (x)

dy

dx=

1

f (x)

df (x)

dx(18)


Basic MathematicsOptimization

Optimization requires finding the extreme point (maximumand minimum value) of a quantity, or finding when maximumand minimum occur.

What to minimize costwant to maximize revenue.

Optimization method:

An extreme point for a curve is where dydx = 0

An extreme point can be a maximum or a minimum

Maximum if :d2y

dx2≤ 0 (19)

Minimum if:d2y

dx2≥ 0 (20)


Basic MathematicsMatrices

Matrices are rectangular arrays of numbers or symbols. Thedimension of a matrix is stated as the number rows bynumber of columns.

Identity matrix is a square matrix in which every element iszero except those on the main diagonals whose values are one.

The transpose of a matrix is the matrix obtained by writingthe row of any matrix as columns.

Matrix addition or subtraction: To add or subtract two ormore matrices all matrices must have exactly the samedimensions. All elements of the two matrices can be added asany scalar.

Matrix multiplication: Matrix A multiply by matrix B as AB isonly possible if number of rows in A is the same as number ofcolumns in B.


Basic MathematicsElasticity

How change in one variable affects another variable i.e., theresponsiveness of one variable with respect to another.Consider two variables:

y = f (x) (21)

elasy ,x =%∆y

%∆x=

y

x

∂y

∂x=∂lny

∂lnx(22)


Basic Statistics (Wooldridge Appendix B)Random variable: A variable is a random variable if the valueit may take is random or unpredictable or uncertain.Discrete versus continuous random variable:

A discrete random variable has a finite number of possiblevalues.A continuous random variable has a continuum of possiblevalues.

Probability distribution is a list of all possible values of therandom variable and their corresponding probabilities.Probability distribution can be presented by

1 table (only for discrete random variable),2 equation or3 graph.

Consider k possible values x1, x2,K ..., xi ,K , xk . LetPr(x = xi ) be probability that x = xiProbability must satisfy two criteria:

1 0 ≤ Pr(x = xi ) ≤ 1 and2 Pr(x = xi ) = 1 .


Measures in Quantitative Statistics

Mean

E (x) =k∑

i=1

Pr(x = xi ) = µ (23)

VarianceVar(x) = E(xi − E (x))2 =

∑ki=1(xi − E (x))2Pr(xi ) = σ2

Standard Deviationσ =√

Var (24)


2.2.2/2.2.3 Density Functions

PDF Probability distribution is known as probability densityfunction (pdf)

f (x) = Pr(x = xi ) (25)

If X is a continuous r.v then pdf of X is the function f (x)such that for two numbers

P(a ≤ X ≤ b) =

∫ b

af (x)dx (26)

CDF Cumulative distribution function (cdf) is related to pdf.

F (x) = Pr(x ≤ xi ) =

∫ x

−∞f (s)ds (27)

The cdf curve relates the range of possible values of x and theprobability that Pr(x ≤ xi )

The curve starts from zero when x is small and ends at 1when x is large. CDF has S shape.


2.2.4 Normal Distribution

A normal random variable is a continuous random variablethat can take on any values.

The pdf curve has bell shape and symmetric around the mean.−∞ < x <∞

f (x) =1√

2πσ2exp[−(x − µ)2/2σ2] (28)

where E (x) = µ, var(x) = σ2 and π =3.14159

The cdf is defined as

F (x) = Φ(x) =

∫ x

−∞(2π)−1/2exp(−t2/2)dt (29)


Standard Logistic Distribution

The pdf is:

f (x) = λ(x) =exp(x)

(1 + exp(x))2(30)

where (−∞ ≤ x ≤ ∞)

The cdf is:

F (x) = Λ(x) =exp(x)

1 + exp(x)(31)

Review of Mathematics and Statistics

Documents

cambridge university press

discrete random variable

binary dependent variable

dx dx dx

continuous random variable

random variable

count data

linear function