Regression analysis

Regression analysis
Developed by
Dr.Ammara Khakwani

Regression:
means “to move backward”,
“Return to an earlier time or stage”
Technically it means
● “The relationship between the mean
value of a random variable and the
corresponding values of one or more
independent variables.”

Formulation of an equation
“The analysis or measure of the
association between one variable
(dependent variable) and one or more
variables ( the independent variable)
usually formulated in an equation in
which the independent variables have
parametric co-efficient which may
enable future values of the dependent
variable to be predicted.”

Regression analysis is concerned
• Regression analysis is largely concerned
with estimating and/or predicting the
(population) mean value of the dependent
variable on the basis of the known or fixed
values of the explanatory variables. The
technique of linear regression is an
extremely flexible method for describing
data. It is a powerful flexible method that
defines much of econometrics. The word
regression, has stuck with us, estimating,
predictive line.

Suppose we want to find out some variables of
interest Y, is driven by some other variable
X. Then we call Y the dependent variable and
X the independent variable. In addition,
suppose that the relationship between Y and
X is basically linear, but is inexact: besides its
determination by X, Y has a random
component µ, which we call the ‘disturbance’
or ‘error’. The simple linear model is
Y = β1+β2X +µi
Where, β1,β2, are parameters the y-intercept
and, the slope of the relationship.

• Helpful for:
• ● Manager making a hiring decision.
• ● Executive can arrive at sales forecasts for a company
• ● Describe relationship between two or more variables
• ● Find out what the future holds before a decision can be made
• ● Predict revenues before a budget can be prepared.
• ● Change in the price of a product and consumer demand for the
product.
• ● An economist may want to find out
• ◘ The dependence of personal consumption expenditure on after-
tax. It will help him in estimating the marginal propensity to
consume, a dollar’s worth of change in real income.
• ● A monopolist who can fix the price or output but not both may
want to find out the response of the demand for a product to
changes in price.

• A labor economist may want to study, the rate of change
of money wages in relation to the unemployment rate.
• ● From monetary Economics, it is known that, other
things remaining the same, the higher the rate of inflation
(п) the lower the proportion (k) of their income that
people would want to hold in the form of money. A
quantitative analysis of this relationship will enable the
monetary economist to predict the amount to predict the
amount of money as a proportion of their income that
people would want to hold at various rate of inflation.

Why to study:
• Tools of regression analysis and correlation analysis have been developed to study and measure the statistical
relationship that exists between two or more variables.
• ● In regression analysis, an estimating, or predicting, equation is developed to describe the pattern or functional
nature of the relationship that exists between the variables.
• ● Analyst prepares an estimating (or regression) equation to make estimates of values of one variable from given
values of the others.
• ● Our concern is with predicting
• ◘ The key-idea behind regression analysis is the statistical dependency of one variable, the dependent variable (y)
on one or more other variables the independent variables (x).
• ◘ The objective of such analysis is to estimate and predict the (mean) average value of the dependent variable on
the basis of the known or fixed values of the explanatory variables
• ◘ The success of regression analysis depends on the availability of the appropriate data.
• ◘ In any research, the researcher should clearly state the source of data used in analysis, their definitions, there
method of collection and any gaps or omissions in the data.
• ◘ Its often necessary to prepare a forecast – an explanations of what the future holds before a decision can be
made. For example, it may be necessary to predict revenues before a budget can be prepared. These predication
become easier if we develop the relationship between the variables to be predict and some other variables relating
to it.
• ◘ Computing regression (estimating equation) and then using it to provide an estimate of the value of the
dependent variable (response) y when given one or more values of the independent or explanatory variables(s) x.
• ◘ Computing measures that show the possible errors that may be involved in using the estimating equation as a
basis for forecasting.
• ◘ Preparing measures that show the closeness or correlation that exists between variables.

What does it provide?
• This method provides an equation
(Model) for estimating; predicting the
average value of dependent variable
(y) from the known values of the
independent variable.
• y is assumed to be random and x are
fixed.

Types of Regression Relationship
• The RELATION between the expected value
of the dependent variable and the
independent variable is called a regression
relation. It is called a simple or two-variable
regression. If the dependency of one variable
is studied on two or more independent
variable is called multiple regression. When
dependency is represented by a straight line
equation, the regression is said to be linear
regression. Otherwise it is Curvilinear.

Example of a model
• Consider a situation where a small ball is
being tossed up in the air and then we measure its
heights of ascent hi at various moments in time ti.
Physics tells us that, ignoring the drag, the
relationship can be modeled as
Yi = β1+β2Xi+µi
• where β1 determines the initial velocity of the
ball, β2 is proportional to the standard gravity,
and µI is due to measurement errors. Linear
regression can be used to estimate the values
of β1 and β2 from the measured data. This model
is non-linear in the time variable, but it is linear in
the parameters β1and β2; if we take
regressor xi = (xi1, xi2) = (ti, ti2), the model takes
on the standard form

Applications of linear regression
• Linear regression is widely used in
biological, behavioral and social
sciences to describe possible
relationships between variables. It
ranks as one of the most important
tools used in these disciplines.

Correlation:
• In statistics, dependence refers to any
statistical relationship between two random
variables or two sets
of data. Correlation refers to any of a broad
class of statistical relationships involving
dependence. Formally, dependence refers to
any situation in which random variables do
not satisfy a mathematical condition
of probabilistic independence

Pearson correlation coefficient
• In loose usage, correlation can refer to any
departure of two or more random variables
from independence, but technically it refers
to any of several more specialized types of
relationship between mean values. There
are several correlation coefficients, often
denoted ρ or r, measuring the degree of
correlation. The most common of these is
the Pearson correlation coefficient

Correlation Coefficient
• The population correlation coefficient
ρX,Y between two random
variables X and Y with expected
valuesμX and μY and standard
deviations σX and σY is defined as:
• where E is the expected
value operator, cov means covariance,
and, corr a widely used alternative
notation for Pearson's correlation.

where x and y are the sample means of X and Y,
and sx and sy are the sample standard deviations ofX and Y.
If x and y are results of measurements that contain measurement error,
the realistic limits on the correlation coefficient are not −1 to +1 but a
smaller range

Regression Versus Correlation:
• They are closely related but conceptually
different in correlation analysis we measure
the strength or degree of linear association
between two variables. But in regression
analysis we try to estimate or predict the
average value of one variable on the basis of
the fixed values of other variables. Examples
in correlation we can measure the relation
between smoking and lung cancer in
regression we can predict the average score
on a statistics exam by knowing a student
score on a mathematical exam.

Regression Versus Correlation:
• They have some fundamental differences.
Correlation theory is based on the assumption of
randomness of variable and regression theory has
assumption that the dependent variable is stochastic
and the explanatory variables are fixed or non
stochastic. In correlation there is no difference
between dependent and explanatory variables.
• In correlation analysis, the purpose is to measure
the strength or closeness of the relationship
between the variables.
• ♦ What is the pattern of existing relationship

• The statistical relationship between
the error terms and the regressor
plays an important role in
determining whether an estimation
procedure has desirable sampling
properties such as being unbiased
and consistent.
IMPORTANT TO NOTE

Trend line
• A trend line represents a trend, the long-term
movement in time series data after other
components have been accounted for. It tells
whether a particular data set (say GDP, oil prices
or stock prices) have increased or decreased over
the period of time. A trend line could simply be
drawn by eye through a set of data points, but
more properly their position and slope is
calculated using statistical techniques like linear
regression. Trend lines typically are straight lines,
although some variations use higher degree
polynomials depending on the degree of curvature
desired in the line.

Trend line uses
• Trend lines are sometimes used in business
analytics to show changes in data over time. Trend
lines are often used to argue that a particular
action or event (such as training, or an advertising
campaign) caused observed changes at a point in
time. This is a simple technique, and does not
require a control group, experimental design, or a
sophisticated analysis technique. However, it
suffers from a lack of scientific validity in cases
where other potential changes can affect the data.

Scatter Diagram:
• To find out whether or not a relationship
between two variables exists, we plot given
data independent and dependent variables
using x-axis for independent regression
variable. Y-axis for dependent variable such
a diagram is called a Scatter Diagram or a
Scatter Plot. If a point shows tendency to a
straight line it is called regression line and if
it shows curve it is called regression curve. It
also shows relationship

Data Graphs:
• Let us assume that a logical
relationship may exist between two
variables.
• To support further analysis we use a
graph to plot the available data. This is
called Scatter Diagram.
• X= Independent or explanatory
• Y= Dependent or response.

Purpose of Diagram:
• To see if there is a useful relationship
between the two variables.
• Determine the type of equation to use
to describe the relationship.

Example of Model Construction:
• Keynes’s consumption function.
• Consumption=C
• Income= X
• C= F(X)
• Consumption and income cannot be connected by any simple
deterministic relationship.
• Linear Model
• C= α + β X
• It is hopeless to attempt to capture every influence in the
relationship so to incorporate the inherent randomness in the
real world counterpart
• C= f(X, ε)
• ε =Stochastic element
• C= α + β X + ε
• Empirical counterpart to Keynes’s theoretical Model.

Example II: Earnings and Education
relationship
• Higher level of education is associated with higher income.
• Simple regression Model is
• Earnings = β1+ β2 education + ε
• [old people have higher income regardless of education]
• And if age and education are positively correlated.
• Han regression Model will associate all the observed increases
in education.
• Than with age effects
• Earnings = β1 + β2 education + β3 age + ε
• We also observe than income that income tends to rise less
rapidly in the later years than in early so to accommodate this
possibility
• Earnings = β1 + β2 education + β3 age + age² + ε
• β3 = +ev
• = -ev

Uses or unique effect of Y
• A fitted linear regression model can be
used to identify the relationship
between a single predictor variable
xj and the response variable y when all
the other predictor variables in the
model are “held fixed”. Specifically, the
interpretation of βj is the
expected change in y for a one-unit
change in xj when the other covariates
are held fixed. This is sometimes called
the unique effect of x j on y.

Simple and multiple linear
regression
• The very simplest case of a
single scalar predictor variable x and a
single scalar response variable y is
known as simple linear regression.
• The extension to multiple
and/or vector-valued predictor
variables (denoted with a capital X) is
known as multiple linear regression.

General linear models
• The general linear model considers the
situation when the response
variable Y is not a scalar but a vector.
Conditional linearity of E(y|x) = Bx is
still assumed, with a
matrix B replacing the vector β of the
classical linear regression model.
Multivariate analogues of OLS and
GLS have been developed.

Heteroscedasticity models
• Various models have been created that
allow for heteroscedasticity, i.e. the
errors for different response variables
may have different variances. For
example, weighted least squares is a
method for estimating linear regression
models when the response variables may
have different error variances, possibly
with correlated errors.

Generalized linear models
• Generalized linear models (GLM's) are a
framework for modeling a response
variable y that is bounded or discrete.
This is used, for example:
• when modeling positive quantities
• when modeling categorical data
• when modeling ordinal data,

Some common examples of
GLM's are:
• Poisson regression for count data.
• Logistic regression and Probit
regression for binary data.
• Multinomial logistic
regression and multinomial
probit regression for categorical data.
• Ordered probit regression for ordinal
data.

Single index models
• allow some degree of nonlinearity in
the relationship between x and y, while
preserving the central role of the linear
predictor β′x as in the classical linear
regression model. Under certain
conditions, simply applying OLS to
data from a single-index model will
consistently estimate β up to a
proportionality constant.

Hierarchical linear models
• Hierarchical linear models (or multilevel
regression) organizes the data into a
hierarchy of regressions, for example
where A is regressed on B, and B is
regressed on C. It is often used where the
data have a natural hierarchical structure
such as in educational statistics, where
students are nested in classrooms,
classrooms are nested in schools, and
schools are nested in some administrative
grouping such as a school district.

Errors-in-variables
extend the traditional linear regression
model to allow the predictor
variables X to be observed with error.
This error causes standard estimators
of β to become biased. Generally, the
form of bias is an attenuation, meaning
that the effects are biased toward zero.

procedures have been developed
for parameter estimation
• A large number of procedures have been
developed for parameter estimation and
inference in linear regression. These
methods differ in computational simplicity
of algorithms, presence of a closed-form
solution, robustness with respect to heavy-
tailed distributions, and theoretical
assumptions needed to validate desirable
statistical properties such
as consistency and asymptotic efficiency.

Some of the more common estimation techniques for
linear regression
• Least-squares estimation and related
techniques
Ordinary least squares (OLS)
Generalized least squares (GLS)
Percentage least squares
 Iteratively reweighed least squares (IRLS)
Instrumental variables regression (IV)
Total least squares (TLS)

Maximum-likelihood estimation and
related techniques
Ridge regression,
Least absolute deviation (LAD)
regression
Adaptive estimation.

Epidemiology
• Early evidence relating tobacco smoking to
mortality and morbidity came from observational
studies employing regression analysis. For
example, suppose we have a regression model in
which cigarette smoking is the independent
variable of interest, and the dependent variable is
life span measured in years. Researchers might
include socio-economic status as an additional
independent variable, to ensure that any observed
effect of smoking on life span is not due to some
effect of education or income. However, it is never
possible to include all possible confounding
variables in an empirical analysis.

Example
• For example, a hypothetical gene might increase
mortality and also cause people to smoke more.
For this reason, randomized controlled trials are
often able to generate more compelling evidence
of causal relationships than can be obtained
using regression analyses of observational data.
When controlled experiments are not feasible,
variants of regression analysis such
as instrumental variables regression may be used
to attempt to estimate causal relationships from
observational data.

Finance
• The capital asset pricing model uses
linear regression as well as the concept
of Beta for analyzing and quantifying
the systematic risk of an investment.
This comes directly from the Beta
coefficient of the linear regression
model that relates the return on the
investment to the return on all risky
assets.

Economics
• Linear regression is the predominant
empirical tool in economics. For
example, it is used to predict
consumption spending fixed
investment spending, inventory
investment, purchases of a country's
exports spending
on imports the demand to hold liquid
assets labor demand and labor supply

Environmental science
• Linear regression finds application in a
wide range of environmental science
applications. In Canada, the
Environmental Effects Monitoring
Program uses statistical analyses on
fish and benthic surveys to measure the
effects of pulp mill or metal mine
effluent on the aquatic ecosystem

Simple Linear Model:
• The correlation co-efficient may indicate that two variables are
associated with one another but it does not give any idea of the
kind of relationship involved.
• Hypothesize one variable (Dependent variable) is determined
by other variable known as explanatory variables, independent
variable or regressor.The hypothesized mathematical
relationship linking them is known as the regression model.If
there is one regressor, it is described as a simple regression
model. If there are two or more regressor it is described as
regressor it is described as a multiple regression.
We would not expect to find an exact relationship between two
economic variables, the relationship is not exact by explicitly
including it a random factor known as the disturbance term

Simple Regression Model:
• Yi =β1 + β2Xi +€i
• Has two component β1 , β2Xi
• where β1 and β2 are fixed quantities
known as parameters. The value of
explanatory variable
• the disturbance Ui
yi
yi

Components of model
• Dependent Variable
• is the variable to be estimated. It is plotted
on the vertical or y-axis of a chart is
therefore identified by the symbol y.
• Independent variable Predictand,
regressand, response, explanatory variable,
• is the one that presumably exerts a influence
on or explains variations in the dependent
variable. It is plotted on x-axis that why
denoted by X. It is also called regressor,
Predicator regression variable and
explanatory variable

We Must Know:
Two Things:
• Value of y-Intercept =a when X is
equal to zero we can read on y-axis.
• Measuring a change of one unit of the
X-variable
• Measuring the Corresponding change
in Yon the y-axis
• Dividing the change inby the change
in X.

Deterministic and Probabilistic:
• Let us consider a set of n pairs of observations (Xi,Yi). If the relation
between the variables is exactly linear, then the mathematical equation
describing the linear relation is
• Yi= a+ bYi
• Where, a= value of Y when x=0 , = Intercept
• b= Change in Y for one unit change in X , Slope of
line
• It is a deterministic Model.
• C= f(X)
• Consumption function
• Y= a + b X
• (Area = п r²)
• But in some situations it is not exact we get what is called non-
deterministic or probabilistic Model.
• = a+ b+ = Unknown Random Errors

Simple Linear regression Model:
• We assume Linear relationship holds between and
= α+ β+ = Fixed, Predetermined values
• = Observations drawn from pop
• = Error components
• α, β = Parameters
• α= Intercept
• β= Slope Reg-co-efficient
• β= +ve, -ve based upon the direction of the relationship between X and
Y.
• Further more we assume.
• E () =0
• => E(Y) ----------- X is a st. line
• Var () = σ²
• ε ~ N(0, σ²)
• E (,) =0 cov=0
• E(X,) are independent of each other.

Multiple Linear Regression
Model:
• It is used to study the relationship between a dependent variable and one or
more independent variables.
• The form of model is
• Y= f (X1+X2+X3+…+Xk) + ε
• = X1 β1 + X2 β2 + X3 β3 +……+ βkXk + ε
• Y= Dependent or explained variable
• X1-------Xk = Independent or Explanatory Variable
• f( X1+X2+X3+…+Xk) = Population regression equation of y on X1-------Xk
• Y= Sum (Deterministic part+ Random Part )
• Y= Regressand
• Xk = Regressors, Covariates
• For example we take a demand equation
• Quantity = β1 + Price× β2 +Income× β3 + ε
• Inverse demand equation
• Price= + × quantity + × income +u
• ε, u = Disturbance, Because it disturb the model.
• Because we cannot hope to capture every influence.

What table is showing:
• Output for a time period in dozens of units (Y).
• Aptitude test results for eight employees (X)
• ♦ It is a sample small of 8 emp.
• Q#1 If the test does what is supposed to do.
• Q#2 Employees with higher scores will be among the higher
producers.
• ♦ Every point on diagram represents each employees
• C= (X, Y) Pairs of observations
• F= (X, Y)
• ♦ They are making a path in straight line.
• ♦ So there is a linear relationship.
• ♦ (+ev) direct relationship

Ordinary Least Square (OLS):
Estimator:
• Is one of the simplest methods of linear
regression. The goal of OLS is to
closely fit a function with the data. It
does so by minimizing the sum of
squared errors from the data. We are
not trying to minimize the sum of
absolute errors, but rather the sum of
squared errors.

Linear regression model and
assumptions:
1. Model Specification:
2. Homoscedasticity and non-
autocorrelation:
3. 1) Linearity There is no exact linear relationship among any of the
independent variables in the model.
• (Identification Condition)
• Exogeneity of independent variables

From monetary Eco nomics
• it is known that, other things remaining the
same, the higher the rate of inflation (п) the
lower the proportion (k) of their income that
people would want to hold in the form of
money. A quantitative analysis of this
relationship will enable the monetary
economist to predict the amount to predict
the amount of money as a proportion of their
income that people would want to hold at
various rate of inflation.

LS OR OLS:
• The principle of least square consists of determining
the values for the unknown parameters that will
minimize the sum of squares of errors (or residuals).
Where errors are defined as the differences between
observed values and the corresponding values
predicted or estimated by the fitted model equation.
• The parameters values thus determine will give the
least sum of the square of errors and are known as
least squares estimates.
• It gives us the least sum of the squares of errors and
are known as least squares estimates.

Method of ordinary least square
(OLS):
• It is one of the econometric methods
that can be used to derive estimates of
the parameters of economic
relationships from statistical
observations.

Advantages of OLS
• 1) It is a fairly simple as compared with other
econometric techniques.
• 2) This method is used in a wide range of economic
relationships.
• 3) It is still one of the most commonly employed
methods in estimating relationships in econometric
models.
• 4) The mechanics of least square are simple to
understand.
• 5) OLS is an essential component of most other
econometric techniques.
• 6) Appealing mathematically as compared to other
methods.

• 7) It is one of the most powerful and
popular methods of regression
analysis.
• 8) They can be easily computed.
• 9) They are point estimators. Each
estimator will provide only a single
(point).
• 10) Once the OLS estimates are
obtained from the sample data the
sample regression line can be easily
obtained.

Model Specification
• Economic study does not specify whether the
supply should be studied with a single – equation
model or with simultaneous - equation.
• We choose to start our investigation with a single –
equation Model.
• Economic theory is not clear about the
mathematical form (linear or non linear)
• We start by assuming that the variables are related
with the simplest possible mathematical form that
the relationship between quantity and price is
linear of the form.
• Y= a+ b X

Example:
• Quantity supplied of a commodity and its price.
• When price raises the quantity of the commodity supplied
increases
• Step I:
• Specification of the supply model.
• i.e.
• Dependent Variable regressand] = quantity supply
• Explanatory variable regressor] = Price
• Y= f(X)
• Y= β1 + β2 X + [Variation in] = [Explained
variation] + [Unexplained Variation]
• β1, β2 = Parameters of supply function our aim is to obtain ,
• = Due to methods of collecting and processing statistical
information.

Assumptions
• Weak exogeneity.
• Linearity.
• Constant variance
(aka homoskedasticity
heteroscedasticity
Independence of errors.
Lack of multicollinearity

Regression analysis

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