Today's Agenda

1
Today's Agenda

• Binary dependent Variable
• Logit- Estimation and Interpretation
• Grouped data
• Individual data
• Probit -Estimation and Interpretation
• Grouped data
• Individual data

2
CDF best describes the S shaped curve of binary
model
Z
3
Logit Model

• Cumulative distribution function of logistic
  distribution is represented as
• The probability of success is given by above.
  The probability of failure is given by
• Thus as Z ranges from to infinity, P ranges
  from 0 to 1 and P is nonlinearly related to Z
  (ie. X), satisfying the requirements of a binary
  model.

4
Odds ratio

• The ratio of the probability of success to
  probability of failure gives the odds ratio in
  favour of success.
• Taking log of odds ratio, we are able to
  linearise the model
• Slope coefficient here tells how the log-odds in
  favour of success change as X changes by 1 unit

5
Estimation of logit model

• MLE estimation
• Requires large samples
• Instead of t statistic, z statistic is used to
  test individual significance of coefficient (Wald
  z test)
• R2 is not meaningful, instead Count R2 or pseudo
  R2 is used
• Count R2 no. of correct predictions/total no. of
  observations
• pseudo R2 1-(Lmax/L0)
• To test null hypothesis that all slope
  coefficients are simultaneously equal to zero,
  instead of F we use Likelihood ratio (LR)
  statistic. LR stat follows chi2 distribution.
• LRChi22(-Log likelihood UR- Log likelihood R)

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Logit Estimation

• Individual data (logit) MLE
• Grouped data (glogit) observed probability is
  available for each value of X
• OLS maybe used with weights to address
  heteroscedasticity. (WLS)
• 1. Find observed probability obtain logit for
  each X as
• Estimate with OLS
• Note that estimation of transformed data is
  undertaken without constant
• Use OLS inferences

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Interpretation

• Take Antilog of the slope coefficient, subtract 1
  from it and multiply result by 100 to get percent
  change in odds for 1 unit change in X.
• Probability of success at given levels of X can
  be estimated by plugging in values of x and
  estimating z values then use formula
• Rate of change of probability ofsucess with unit
  change in X variable is got by
• Change is not linear but depends on levels of
  probability

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eg. income and house ownership

• Data on income(X), Nitotal families with income
  Xi, nifamilies with houses with Xi income
• Pini/Ni LlnPi/(1-Pi) Lweighted L
• For 1 unit increase in income, odds of owning
  house increases by 8.1
• How to estimate probability when X20

9
Probability calculation

• Plugging X20 in estimated equation we obtain
  L-0.09311, dividing by weight (4.2661) we get
  L -0.2226
• Probability of owning house at income20 is 49

10
Rate of change of probability

• Rate of change is not constant and depends on the
  level of Income from where you are calculating
• 0.07862(0.5056)(0.4944)
• 0.01965
• Rate of change of probability when income
  increases by one unit from the level of 20 units
  is 1.9

11
Try out..

• Given WLS estimate
• Find the probability of owning house at X40 and
  the rate of change in probability when increases
  by 1 unit from this level
• Also given weight2 at X40.

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Logit model for individual data

• WLS will not work! MLE to be used..

13
Logit model for individual data

• MLE inferences Wald test, LR and count
  Rsq/pseudo Rsq
• Eg impact of personalised system of instruction
  on students. Y1, if gradeA otherwise Y0
• PSI is also a dummy here
• -13.02132.826GPA0.095TUCE2.378PSI
• Interpretation
• Antilog of PSI slope (10.789) gives odds of
  getting A grade if PSI1. With PSI the odds of
  getting A is 11 times higher
• Probability of getting A Plug-in values of Xs in
  estimated MLE equation, calculate z value and use
  formula
• Rate of change of probability

14
Try out...

• Estimated logit
• Find impact of X1 on odds ratio
• Find probability when X121.29, X20.42
• Find change in probability when X121.29, X20.42
  and X1 increases by 1 unit

15
Probit Model

• Another CDF popularly used to model dichotomous
  model is is normal CDF. MLE estimation of normal
  CDF for dichotomous model is called as probit or
  normit
• A probit model is defined as
• XiB has a normal distribution and so interpreting
  probit results requires thinking in terms of Z
  (standard normal distribution)
• Interpretation
• Inferences are same as logit
• Probability is obtained by plugging in values of
  X and looking up CDF table
• Change in probability with 1 unit change in X

16
Grouped Probit

• OLS can be used for grouped probit
• After estimating probability, Index Ii from CDF
  is to be estinated
• This invoves finding z from P (normal equivalent
  deviate or normit)
• Excel com normsinv(P)
• I will be negative for P less than 5
• Add 5 to normit to make it Probit
• Normit or Probit can be used as OLS dependent
  variable
• Results do not vary except for the constant term

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Eg

• Estimated probit equation
• Probability when X121.29, X20.42
• 0.0823321.291.5290.42-3.139 -0.7440
• Corresponding CDF probability 0.2284
• Excel command normsdist(Z)
• Change in probability when X1 changes by 1 unit
• F(-0.7440)0.08233
• Normal density function at -0.74400.3025
• stata com normalden(z)
• .30250.08233
• 0.0249

18
Try out...

• Probit estimation of Impact of income on house
  ownership
• -1.020.05 X
• What is the probability of owning a house if
  income6 and by how much will probability
  increase if income increases by 1 unit.
• Normal cdf P values of -0.72 0.2357
• Normal distribution density value of
  -0.720.307851

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Summarising Slope interpretation

• LRM slope coefficient measures change in average
  value of Y for a unit change in X, with all other
  variables held constant
• LPM slope coefficient measures change in
  probability of an event occurring for a unit
  change in X, with all other variables held
  constant
• Logit slope coefficient gives change in log of
  odds for a unit change in X. Rate of change in
  probability is given by
• Probit Z table value of coefficient gives
  probability of success if X1 Rate of change in
  probability is given by
• Where f( ) is the density function of the
  standard normal variable

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Logit-Probit relationship

• Probit coeff 1.81 Logit coeff
• Logit coeff 0.55Probit coeff
• LPM coeff0.25logit coeff (except constant)
• LPM coeff0.25logit coeffconstant (for constant)

21
Some additional Variants

• Tobit modelcombination of 0 and continuous
  variable (expenditure on housing and income)
• Multinomial logitWhen dependent variable is
  discrete and polytomous (mode of transport)
• Ordinal logit When dependent variable can be
  ordered (performance grades)
• Nested logit Models were decision taking is done
  in stages in which decisions in later stages are
  limited by those made earlier