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Limited Dependent Variables  Often there are occasions where we are interested in explaining a dependent variable that has only limited measurement 

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Presentation on theme: "Limited Dependent Variables  Often there are occasions where we are interested in explaining a dependent variable that has only limited measurement "— Presentation transcript:

1 Limited Dependent Variables  Often there are occasions where we are interested in explaining a dependent variable that has only limited measurement  Frequently it is even dichotomous.

2 Examples  War(1) vs. no War(0)  Vote vs. no vote  Regime change vs. no change

3 These are often Probability Models  E.g. Power disparity leads to war: Where Y t is the occurrence (or not) of war, and X t is a measure of power disparity  We call this a Linear Probability Model

4 Problems with LPM Regression  OLS in this case is called the Linear Probability Model  Running regression produces some problems Errors are not distributed normally Errors are heteroskedastic Predicted Ys can be outside the bounds required for probability

5 Logistic Model  We need a model that produces true probabilities  The Logit, or cumulative logistic distribution offers one approach.  This produces a sigmoid curve.  Look at equation under 2 conditions: X i = +∞ X i = -∞

6 Sigmoid curve

7 Probability Ratio  Note that  Where

8 Log Odds Ratio  The logit is the log of the odds ratio, and is given by:  This model gives us a coefficient that may be interpreted as a change in the weighted odds of the dependent variable

9 Estimation of Model  We estimate this with maximum likelihood  The significance tests are z statistics  We can generate a Pseudo R 2 which is an attempt to measure the percent of variation of the underlying logit function explained by the independent variables  We test the full model with the Likelihood Ratio test (LR), which has a χ 2 distribution with k degrees of freedom

10 Neural Networks  The alternate formulation is representative of a single-layer perceptron in an artificial neural network.

11 Probit  If we can assume that the dependent variable is actually the result of an underlying (and immeasurable) propensity or utility, we can use the cumulative normal probability function to estimate a Probit model  Also, more appropriate if the categories (or their propensities) are likely to be normally distributed  It looks just like a logit model in practice

12 The Cumulative Normal Density Function  The normal distribution is given by:  The Cumulative Normal Density Function is:

13 The Standard Normal CDF  We assume that there is an underlying threshold value (I i ) that if the case exceeds will be a 1, and 0 otherwise.  We can standardize and estimate this as

14 Probit estimates  Again, maximum likelihood estimation  Again, a Pseudo R2  Again, a LR ratio with k degrees of freedom

15 Assumptions of Models  All Y’s are in {0,1} set  They are statistically independent  No multicollinearity  The P(Y i =1) is normal density for probit, and logistic function for logit

16 Ordered Probit  If the dependent variable can take on ordinal levels, we can extend the dichotomous Probit model to an n-chotomous, or ordered, Probit model  It simply has several threshold values estimated  Ordered logit works much the same way

17 Multinomial Logit  If our dependent variable takes on different values, but they are nominal, this is a multinomial logit model

18 Some additional info  The Modal category is good benchmark  Present % correctly predicted This can be calculated and presented. This, when compared to the modal category, gives us a good indication of fit.

19 Stata  Use Leadership Change data (1992 cross section) 1992-Stata (1992 cross section)1992-Stata

20 Test different models  Dependent variable Leadership change Examine distribution tables ledchan1 Independent variables Try different Try corr and then (pwcorr)

21 Try the following regress ledchan1 grwthgdp hlthexp illit_f polity2 logit ledchan1 grwthgdp hlthexp illit_f polity2 logistic ledchan1 grwthgdp hlthexp illit_f polity2 probit ledchan1 grwthgdp hlthexp illit_f polity2 ologit ledchan1 grwthgdp hlthexp illit_f polity2 oprobit ledchan1 grwthgdp hlthexp illit_f polity2 mlogit ledchan1 grwthgdp hlthexp illit_f polity2 tobit ledchan1 grwthgdp hlthexp illit_f polity2, ul ll

22 Tobit  Assumes a 0 value, and then a scale  E.g., the decision to incarcerate 0 or 1 (Imprison or not) If Imprison, than for how many years?

23 Other models  This leads to many other models Count models & Poisson regression Duration/Survival/hazard models Censoring and truncation models Selection bias models


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