Econometric Analysis

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Econometric Analysis
Week 9 Limited dependent variables models
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Lecture outline

• Binary choices and Limited Dependent Variables
  models
• The linear probability model
• logit and probit models
• Examples and results using PcGive
• Further comments on censored and truncated
  regression models (Tobit analysis) and sample
  selection bias, multinomial choice models

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References and recommended reading

• Wooldridge, J M (2006) Introductory Econometrics.
  A Modern Approach. (Third Edition) Chapter 17 pp
  582-595
• Greene, W H (2000) Econometric Analysis (Fourth
  Edition) Chapter 19 pp 811-837
• Kennedy, P (2003) A Guide to Econometrics.
  (Fifth Edition) Chapters 15 16
• Dougherty, C (2007) Introduction to Econometrics
  (Third Edition) Chapter 10
• Spector, L C and Mazzeo, M (1980) Probit analysis
  and Economic Education. Journal of Economic
  Education Spring, pp 37-44
• Doornik, J A and Hendry, D F (2006) Empirical
  Econometric Modelling PcGive Vol III, Chapters 5
  6.

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Basics

• on occasions the variable that we are trying to
  explain may be discrete rather than continuous
• in the most basic case it is a binary,
  dichotomous, dummy or qualitative variable in
  other words it can take only one of two values
  0 or 1
• examples (1) in employment/out of employment
  (2) university educated/ not university educated
  (3) pass test/fail test (4) owns home/does not
  own home
• we might wish to explain how observations fall
  into each category for example in the labour
  market case by linking the dependent variable to
  explanatory variables like age, education,
  marital status etc.
• simple OLS regression will not really be
  appropriate here although an early approach was
  the linear probability model which is based on
  OLS regression
• today you are more likely to use either the
  logit (sometimes called logistic) or probit
  models, which make use respectively of the
  logistic distribution or the cumulative normal
  distribution to provide an S shaped curve linking
  the two sets of points
• more advanced work can extend the number of
  values that the limited dependent variable can
  take beyond two for example the five categories
  on a Likert scale so called multinomial choice
  variables - we wont cover these on this unit

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the Linear Probability Model (LPM)

• consider the simple case with one explanatory
  variable X
• in this model the predicted Y value denotes the
  probability that the dependent variable takes a
  value of 1 - so the probability of success
  (Y1) is linearly related to the explanatory
  variable X
• the Y values can only be 0 or 1 so a straight
  line fit through the points, as shown in figure
  1, will result in predicted Y values outside the
  range 0-1
• the residuals will also be heteroskedastic so
  if we do use OLS we should use robust standard
  errors to calculate t values
• R squared has no meaning here whereas in the
  continuous OLS case it is possible for all points
  to lie on the regression line, here they cannot
  as they must lie along one of the horizontal
  lines at 0 and 1

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Binary Response Models logit and probit models

• These models make use of a squash function G
  to ensure that the fitted values lie strictly
  between 0 and 1
• Logit Model G follows a logistic distribution
• Probit Model G follows a cumulative normal
  distribution
• (see Wooldridge or Greene for the full algebraic
  details)
• The models intrinsically non-linear and so they
• are estimated using Maximum Likelihood
• procedures

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The shape of the logit and probit curves
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Partial responses in binary response models

• whereas in the LPM the marginal or partial effect
  of change in one of the Xs on Y (?Y/?Xj) is
  constant, for binary response models it will vary
  over the curve I will give a detailed
  derivation for the logistic function later
• it is sometimes given in results tables as the
  slope - calculated at the mean values of the X
  variables

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Goodness of fit in binary response models

• You sometimes see count R2 which counts the
  proportion of cases correctly predicted this is
  not very helpful, particularly if the split
  between 0 and 1 values for Y in the sample is
  very uneven, where even a naïve model of
  predicting a success for every case would come
  out well.
• An alternative measure called pseudo R-squared
  given. This is calculated as 1 Lur/L0 where
• Lur log-likelihood for the estimated model and
  L0 log-likelihood for a model with an
  intercept only (see Wooldridge pp589-590 and
  Kennedy p267)

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More detail on the logit model

• Lets look at a simple case with just one
  explanatory variable
• the fitted values are kept between the limits of
  0 and 1
• if we write
• then Y?1 as Z ? ? and Y ?0 as Z ? 0

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Yet more on the logit model

• For this function the partial response of Y to a
  change in X1 turns out to be ?1Y(1-Y)
• see proof on separate sheet
• The model is sometimes reformulated as the
  log-odds model with
• see derivation on separate sheet

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Example

• Greene (2000) illustrates the use of logit and
  probit models with a data set from Spector and
  Mazzeo (1980) which concerns the effectiveness of
  a new method of teaching economics. The Spector
  and Mazzeo data has information on the
  performance of 32 students on the principles in
  macroeconomics courses at Iowa University in the
  spring semesters of 1974 and 1975.
• The dependent variable GRADE is an indicator
  of whether or not students passed a test in
  principles of macroeconomics
• The independent variables are
• GPA the students Grade Point Average prior to
  taking the course
• TUCE the result on a pre-entry Test of
  Understanding in College Economics
• PSI an indicator of whether or not the
  student was taught using the new Personalised
  System of Instruction rather than just in
  lectures

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Model formulation in PcGive (1)
Category Models for discrete data Model class
Binary Discrete Choice using PcGive
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Model formulation in PcGive (2)
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Model formulation in PcGive (3)
Now choose the model logit or probit