Qualitative Response Regression Models

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Qualitative Response Regression Models

• Lecture week 8
• Prepared by
• Dr. Zerihun Gudeta Alemu

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The Nature of Qualitative Response Models

• Binary, or dichotomous, variable
• Trichotomous
• Polychotomous
• Differences in result interpretation and
  estimation
• Quantitative dependent variables expected or
  mean value given the values of the regressors.
• Qualitative dependent variables probability
• Is OLS appropriate to measure qualitative
  response variable models? Will there be problem
  of inference, will the conventional measure of
  goodness of fit be of value.
• Approaches to developing a probability model for
  a binary response variable the linear
  probability model, the logit model, and the
  probit model.

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Data for Qualitative Response Model
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The Linear Probability Model (LPM)

• The naming Assume a qualitative dependent
  variable (yes or no response by farm
  households on fertilizer application in crop
  production). The dependent variable takes a
  value of 0 or 1 reflecting ex post the choice
  made by the household, we can think of their
  being an underlying probability that a particular
  household will apply fertilizer or not given
  their level of income, education level, farm
  asset. Hence the name linear probability.
• Estimation method OLS used to estimate LPMs.
• The model Pi E(FERTILIZERi1/Xi) a1 a2
  INCOMEi a3 Education a2 Farmasset ?t
• Result Interpretation the intercept (the
  probability of a household applying fertilizer
  given that income, level of education of
  household head, and farm asset are all zeros).
  The slope (the change in the probability of
  fertilizer application for a unit change in
  income or education level of HHH or asset base of
  the farm household.
• Forecasting The probability of a FHH applying
  fertilizer for a given level of income, education
  level, and asset base).

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The Linear Probability Model (LPM) Cont.

• Problem of Estimation
• The error terms are not normally distributed
  (follow a binomial distribution)
• The error terms are hetroscedastic (can be solved
  with WLS)
• Difficult to interpret R2 ,
• The predicted values of the dependent variable
  (i.e. probability) can be less than zero or
  greater than one which is nonsensical (this can
  be solved using probit and logit models)
• Probabilities increase linearly with the
  explanatory variables (not true at a low and
  high end of an explanatory variable, change in
  the probability is expected to be low) (can be
  solved using logit and probit models which show
  nonlinear relationships)

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Alternative Models

• To be chosen over LPM, probability estimates of
  alternative models must keep probabilities in
  the 0,1 range and must allow nonlinear
  relationships between the explanatory variables
  and the probabilities (i.e. must be asymptotic as
  probabilities approaches 0 and 1).
• This requires that the alternative models be
  based on Cumulative Density Functions (CDF). CDF
  is nothing but the area under the probability
  density function (PDF) to the left of some
  pre-specified values which gives the
  probabilities that the value of a random variable
  falls below the specified values.
• The logistic CDF (the logit model) and the normal
  CDF (the prpbit model) are best alternatives to
  the LPM.

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The Logistic or Logit Model

• The model uses the cumulative logistic function.
• Estimation the model is nonlinear therefore use
  of OLS is not appropriate. Maximum Likelihood
  Estimation (MLE) technique is used instead.
• MLE A numerical optimization algorithm used to
  choose values of the parameters which maximizes
  the likelihood of drawing that particular sample
  of observations.
• The model the dependent variable is assumed to
  follow the logistic distribution. It is based on
  the formula of logistic PDF.
• Pi E(FERTILIZER1/INCOME, EDUCATION, ASSET)
  1/(1e -(?1 ?2INCOME ?3 EDUCATION ?4ASSET))

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The Logistic or Logit Model (CONT)

• Interpretation Slope coefficient (the change in
  the estimated logit for a unit change in a value
  of a given regressor other variables held
  constant. To get meaningful result, take antilogs
  of the slope coefficients (i.e. ea) which gives
  you odds ratio (applying fertilizer to not to
  applying fertilizer) from which the change in the
  probability of applying fertilizer for a unit
  change in say education level can be calculated).
  
• Prediction (plug the estimated coefficient say
  income and its value into the logit function,
  convert the logit value, compare the resulting
  probability with the threshold value such as 0.5
  and predict the outcome 0 or 1.

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The normal or probit model

• The model uses the normal CDF.
• Estimation the model is nonlinear therefore use
  of OLS is not appropriate. Maximum Likelihood
  Estimation (MLE) technique is used instead.
• Difference with logit model the dependent
  variable is assumed to follow the normal
  distribution. Therefore the likelihood function
  is based on the formula for the standard normal
  PDF
• The model Pi P(Fertilizer 1/ INCOME)f(?1
  ?2 INCOMEt)
• Interpretation slope (the effect of a unit
  change in income on the probability that the farm
  household applies fertilizer is given by dPi /
  Dincomei f(?1 ?2 INCOMEi) ?2 . Where f(?1
  ?2 INCOMEt) ?2 is the standard normal probability
  density function evaluated at ?1 ?2 INCOMEt.
  Refer to the normal distribtuion table to get
  f(?1 ?2 INCOMEt) which is the density function
  of the standard normal variable and multiply this
  by the slope coefficient.