Introduction to two variable regression model

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Introduction to two variable regression model

• What is Econometrics?
• Literal meaning is measurement in economics.
• Econometrics is concerned with the empirical
  determination of economic laws.
• Eg Keynesian Consumption theory
• Inherent in econometric analysis are
• Economic theory (qualitative)
• Mathematics to obtain quantitative results
• Statistics to infer from the above results
• Econometrics is different from
• Mathematical economics
• Economic statistics

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Steps involved in the formulation of econometric
models

• Economic or Financial Theory (Previous
  Studies)
• Formulation of an Estimable Mathematical Model
• Econometric model of theory
• Collection of Data
• Model Estimation
• Is the Model Statistically
  Adequate?
• No Yes
• Reformulate Model Interpret
  Model
• Use for control or
  policy purpose

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Mathematical Vs Econometric model

• Mathematical model converts theory into equation
  form with dependent variable on the LHS and
  explanatory variable(s) on the RHS
• Mathematical model assumes that there is an exact
  or deterministic relationship between dependent
  and explanatory variables
• Econometric model includes a random/stochastic/dis
  turbance variable to control for all the
  variables not included in the model
• This disturbance variable has to fulfill well
  defined statistical properties for model to hold

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Mathematical Vs Econometric model

• Mathematical Notation for the straight line -
  dont ask why!
• YabX
• Mathematical economist captures
  Consumption-Income relationship thru above
  equation where Y is consumption, X is Income
• Econometrics Literature includes a residual term
• Yi a bXi ei
• The residual term indicates that consumption may
  depend on factors other than income as well which
  is not included in the model. For eg price
  level, fashion, no. of members in household etc
• Here a, b are called parameters which regression
  estimates and X and Y are variables for which
  data is collected
• parameters a, b may be denoted as b1, b2 or any
  other way. These are all equivalent. Various
  texts use different notations. We simply have to
  live with this inconsistency.

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Linear Mathematical modelA Graphic
Interpretation

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The linear econometric model

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The Mathematical Interpretation The Meaning of
the Regression Parameters

• a the intercept
• the point where the line crosses the Y-axis.
• (the value of the dependent variable when all of
  the independent variables 0)
• b the slope
• the increase in the dependent variable per unit
  change in the independent variable. (also known
  as the 'rise over the run')

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The Error Term

• No model can predict behavior perfectly.
• Vagueness of theory
• Data unavailability
• Randomness of human behaviour
• Wrong functional form
• So we must add a component to adjust or
  compensate for the errors in prediction.
• Having fully described the linear model, the rest
  of the semester (as well as several more) will be
  spent of the error.

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Types of Econometrics

• Theoretical and Applied
• Each of the above maybe Classical or Bayesian
  approach
• We take a classical approach in this course
• Most common tool is the regression analysis
• Linear or nonlinear in variable
• Linear or nonlinear in parameter
• Single or multiple variable
• Single or multiple equation
• We start with the simplest Single linear
  Regression

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Regression Analysis

• Literal meaning of regress means to fall back
  to move back to average. First used by Francis
  Galton.
• In econometrics regression analysis is concerned
  with the study of dependence of dependent
  variable on one or more explanatory variables,
  with a view to estimating population mean value
  of former in terms of the known values of the
  latter
• Correlation measures strength of association
  between two variables. Regression indicates
  direction of causation. Difference also between
  way 2 variables treated
• The direction of causation in regression is
  provided by theory and not by the statistical
  method itself

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Nature of the variables

• The dependent variable is considered to be a
  stochastic variable while explanatory variables
  are deterministic or fixed
• A stochastic or random variable is one that can
  take any set of values with a given probability.
  In repeated sampling various values come up with
  different probability
• Eg If you survey 10 households with R10,000
  income to study their consumption expenditure,
  each house will giv you a different amount
  (expenditure is stochastic variable bcos given
  the income you can attach probability to the
  expenditure of the household)
• Income here is fixed (explanatory variable) and
  consumption is stochastic (dependent variable)

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Regression and probability

• Condition means can also be estimated from
  conditional probabilities

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Population regression line

• A PRL is simply the locus of the conditional
  means of the dependent variable for the fixed
  values of the explanatory variable(s).
• Population regression function is otherwise
  called conditional expectation function
• E(Y/Xi)f(Xi)
• If we assume the above is a linear function, then
• E(Y/Xi)a bXi
• Where a and b are unknown regression coefficients

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• Function E(Y/Xi)f(Xi) may be non-linear
  depending on the nature of relationship between
  variables. For eg Q and MC
• Theory says MC curve is U-shaped not linear. This
  s captured by
• E(Y/Xi)ab1Xb2X2
• Rel between Q and TC curve is 'S' shaped
• E(Y/Xi)ab1Xb2X2b3X3
• They fall under Linear regression because they
  are non-linear only in variables, they are linear
  in parameters.
• Only models with non-linear parameters are said
  to be non-linear regression

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Sample regression function

• Practically economists have to deal with sample
  data and derive conclusions regarding the
  population from the available sample.
• This means that unlike in PR when there were
  multiple set of Y values for each X, in SRF we
  have just one value of Y for each value of X.
• SRF is written as
• YabXu
• indicates that it is an estimator of the
  respective parameter
• How SRF is used to estimate PRF is taken up in
  next class

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Types of Data and Notation

• There are 3 types of data which econometricians
  might use for analysis
• 1. Time series data
• 2. Cross-sectional data
• 3. Panel data, a combination of 1. 2.
• The data may be quantitative (e.g. exchange
  rates, stock prices, number of shares
  outstanding), or qualitative (e.g. day of the
  week).
• Examples of time series data
• Series Frequency
• GNP or unemployment monthly, or quarterly
• government budget deficit annually
• money supply weekly
• value of a stock market index as transactions
  occur

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Types of Data and Notation (contd)

• Examples of Problems that Could be Tackled Using
  a Cross-Sectional Regression
• - The relationship between company size and the
  return to investing in its shares
• - The relationship between a countrys GDP level
  and the probability that the government will
  default on its sovereign debt.
• Panel Data has the dimensions of both time series
  and cross-sections, e.g. the daily prices of a
  number of blue chip stocks over two years.
• It is common to denote each observation by the
  letter t and the total number of observations by
  T for time series data, and to to denote each
  observation by the letter i and the total number
  of observations by N for cross-sectional data.

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Types of Data and Notation (contd)

• Examples of Problems that Could be Tackled Using
  a Time Series Regression
• - How the value of a countrys stock index has
  varied with that countrys macroeconomic
  fundamentals.
• - How the value of a companys stock price has
  varied when it announced the value of its
  dividend payment.
• - The effect on a countrys currency of an
  increase in its interest rate
• Cross-sectional data are data on one or more
  variables collected at a single point in time,
  e.g.
• - A poll of usage of internet stock broking
  services
• - Cross-section of stock returns on the NYSE
• - A sample of bond credit ratings for UK banks

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Some Points to Consider when reading empirical
research papers

• 1. Does the paper involve the development of a
  theoretical model or is it merely a technique
  looking for an application, or an exercise in
  data mining?
• 2. Is the data of good quality? Is it from a
  reliable source? Is the size of the sample
  sufficiently large for asymptotic theory to be
  invoked?
• 3. Have the techniques been validly applied?
  Have diagnostic tests for violations of been
  conducted for any assumptions made in the
  estimation of the model?

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Some Points to Consider when reading empirical
research papers

• 4. Have the results been interpreted sensibly?
  Is the strength of the results exaggerated? Do
  the results actually address the questions posed
  by the authors?
• 5. Are the conclusions drawn appropriate given
  the results, or has the importance of the results
  of the paper been overstated?