## Introduction to Modern Econometrics Using Stata

**Assignment 3**

*Example 17.1 In book woold*

*An Introduction to Modern Econometrics Using Stata.pdf Page 250*

*A Handbook of Statistical Analyses Using Stata, Second Edition: Chapter-6*

*Microeconometrics Using Stata.pdf:Chapter-14*

*Stata Primer: Chapter-8*

Your task is to examine the determinants of voluntary contributions to public radio using survey data on approximately listeners and contributors. The data for this case study is contained in the file npr1.txt. Your Assignment should include five sections: (1) conceptual framework, (2) specification of the econometric model, (3) description of the data, (4) discussion of estimation results (5) Marketing plan for NPR based on your results.

**Conceptual Framework and Specification of the Econometric Model**

Our main task is to construct a model that help us in predicting whether an individual is going to contribute to public radio. For this we observe that amongst the variables listed in dataset the following variables seem to be mostly related to the decision of donation:

- age of listener in years
- station listening time in quarter hours per week

- Listener income in dollars

- Whether individual has a master’s degree or higher
- Whether individual has a bachelor’s but not a Masters.
- Whether individual has some college but no degree

- Whether individual has a high school degree
- Whether individual is female

- Whether individual is white
- Whether individual is retired

This is because Contributions of others, Estimated number of public radio listeners in market, Total population / land area of MSA (Metropolitan Statistical Area) and the city have no relation with individuals decision to donate.

city City where station is located

Age age of listener in years

Listen station listening time in quarter hours per week

gother Contributions of others (total contributions from all sources other than the contribution made by individual i) in dollars.

Tlisten Estimated number of public radio listeners in market

Popdens Total population / land area of MSA (Metropolitan Statistical Area) Donate Dummy equal to one if individual made donation

Income Listener income in dollars

Pgrad Dummy =1 if individual has a masters degree or higher Cgrad Dummy =1 if individual has a bachelor’s but not a Masters. Scol Dummy =1 if individual has some college but no degree Hsgrad Dummy =1 if individual has a high school degree

Female Dummy =1 if individual is female White Dummy =1 if individual is white Retired Dummy =1 if individual is retired

We will use the three models to predict the possibility of donation by an individual:

- Linear Probability model where we have the following mathematical model:

Donate=β_{0} + β_{1} age + β_{2} listen+ β_{3} income+ β_{4} Pgrade+ β_{5} Cgrade+ β_{6} Scol+ β_{7} Hsgrad+ β_{8} Female+ β_{9} white + β_{10} retired,

Here the slope β_{j} ,j=1,2…10 is the predicted change in the probability of donation when the corresponding predictor increases by one unit, β_{0} is the predicted probability of donation when all preceptors have zero value while y is the predicted probability of success.

The linear probability model has the drawback that for certain values of the predictors we may have values of predicted probability less than zero or greater than unity which is absurd as probability always lies between 0 and 1. Also non-normality of error terms, heteroscedastic variances of eroor are some other problems. There is one problem that probability cannot linearly related to any predictor as otherwise it may take the value of predicted probability greater than unity. There is one remedy to these problems that we should retain those values of predictors for which probability does not crosses the bound but this may result into loss of some useful information from predictors.

- Logit (logistic regression) model where we have the following mathematical model:

P(Y=1/**x**)=

G(β_{0} + β_{1} age + β_{2} listen+ β_{3} income+ β_{4} Pgrade+ β_{5} Cgrade+ β_{6} Scol+ β_{7} Hsgrad+ β_{8} Female+ β_{9} white + β_{10} retired),

=G(β_{0} + **xβ**)

Here G(.) is the cumulative distribution function of standard logistic random variable defined by G(z)=exp(z)/[1+exp(z)], **x** in bold is the set of all predictors and

**Xβ**= β_{0} + β_{1} age + β_{2} listen+ β_{3} income+ β_{4} Pgrade+ β_{5} Cgrade+ β_{6} Scol+ β_{7} Hsgrad+ β_{8} Female+ β_{9} white + β_{10} retired

This model ensures the probability to take values in between 0 and 1 including them but not beyond these limits

- Probit model where we have the following mathematical model:

P(Y=1/**x**)=

G(β_{0} + β_{1} age + β_{2} listen+ β_{3} income+ β_{4} Pgrade+ β_{5} Cgrade+ β_{6} Scol+ β_{7} Hsgrad+ β_{8} Female+ β_{9} white + β_{10} retired),

=G(β_{0} + **xβ**)

Here G(.) is the cumulative distribution function of the standard normal variable defined by G(z)= , and **x** in bold is the set of all predictors and

**Xβ**= β_{0} + β_{1} age + β_{2} listen+ β_{3} income+ β_{4} Pgrade+ β_{5} Cgrade+ β_{6} Scol+ β_{7} Hsgrad+ β_{8} Female+ β_{9} white + β_{10} retired

This model ensures the probability to take values in between 0 and 1 including them but not beyond these limits

There is not much difference in the shape and limits of the predicted probabilities given by the logit and probit model except the difference in the cumulative distribution function used in both the models. However, the results are similar but parameters are not directly comparable in both models.

**Description of the Data**

Create a table of summary statistics. Provide a brief description of the variables used in your study and discuss the summary statistics.

The table of summary statistics is as follows from where we observe that 33.91% of the individuals in the sample are contributing to public radio by making donations, 38.97% hold masters’ degree or higher, 24.07% hold bachelor’s degree but not masters, 22.49% hold some college degree but nor degree, 11.71% has a high school degree, 47.49% are females (others male), 93.99% are white and 14.42% are retired persons.

Also, mean age of the individuals in the sample is 45.66 years with minimum age 18 years and maximum 93 years. The average income of the individuals in our sample is $46086.11 with minimum income $2500 and maximum income $184488.4.

Summary statistics | |||||

Variable | Obs | Mean | Std. Dev. | Min | Max |

donate | 3731 | 0.339051 | 0.473451 | 0 | 1 |

age | 3731 | 45.65854 | 14.99618 | 18 | 93 |

listen | 3731 | 33.85205 | 40.97719 | 1 | 346 |

income | 3731 | 46086.11 | 36773.26 | 2500 | 184488.4 |

pgrad | 3731 | 0.389708 | 0.487749 | 0 | 1 |

cgrad | 3731 | 0.240686 | 0.427557 | 0 | 1 |

scol | 3731 | 0.224873 | 0.417554 | 0 | 1 |

hsgrad | 3731 | 0.117127 | 0.321614 | 0 | 1 |

female | 3731 | 0.47494 | 0.499439 | 0 | 1 |

white | 3731 | 0.939963 | 0.237588 | 0 | 1 |

retired | 3731 | 0.144197 | 0.351337 | 0 | 1 |

**Results**

Table with three sets of results: (1) Linear Probability Model, (2) Logit Model, and (3) Probit Model is as follows from where we observe that all the three models have same sign of coefficients for all the predictors of donation. Also Percentage Correctly Predicted is almost same for all the three models and R-squared too does not vary much for all the models.

Now looking at the sign of the coefficients we observe that all the variables have positive impact on probability of donation. Only listen, income, pgrad, cgrad, female and white have significant impact on donation. As listening time increases it is obvious that individual has more tendencies to make donation. Also, if the income increases we expect the individual to make some contribution by donation. Similar is the case that when a person is more educated either masters(Pgrade) or college(Cgrade ) then he is more knowledgeable to know th ebenefit of donation and make some contribution. Donations made by females pertain to human behaviour and females are more concerned about public life. White persons are also having greater chance of making donation due to prosperity.

Coefficients with Standard errors in Parenthesis, *p<.05, **p<.01, ***p<.001 | |||

Dependent Variable: donate | |||

Independent Variables | LPM (OLS) | Logit (MLE) | Probit (MLE) |

age | 0.0006475 (.0006152) | 0.003941 (0.0033092) | 0.0023397 (0.0019769) |

listen | .0027042*** (.0001954) | 0.012884*** (0.00096) | 0.0076851*** (0.0005473) |

income | 0.00000222*** (0.00000023) | 0.00001030*** (0.00000105) | 0.000006260*** (0.000000623) |

pgrad | .2194713*** (.0372072) | 1.358775*** (0.3174161) | 0.7745163*** (0.1702777) |

cgrad | .1412591*** (.0378002) | 0.996871** (0.3210875) | 0.5539602** (0.1725862) |

scol | .0643043 (.0369369) | 0.575906 (0.3213468) | 0.3077896 (0.1721771) |

hsgrad | .0447288 (.0380576) | 0.450842 (0.3308597) | 0.2290902 (0.1777501) |

female | .061558*** (.0146819) | 0.309075*** (0.0756224) | 0.1856696*** (0.0451472) |

white | .0976322*** (.0266236) | 0.598563** (0.1834121) | 0.3479993** (0.1042219) |

retired | .049004 (.026371) | 0.228809 (0.1375911) | 0.1478797 (0.0819513) |

Constant | -.1518637** (.0482515) | -3.52899*** (0.3828398) | -2.080306*** (0.2093689) |

Percentage Correctly Predicted
| 69.63 | 69.66 | 69.77 |

Log-Likelihood Value
| ___ | -2137.9406 | -2137.4369 |

Pseudo R-squared | 0.131 | 0.1052 | 0.1054 |

For LPM we have following:

donate_predict | |||

donate | 0 | 1 | Total |

0 | 2,246 | 220 | 2,466 |

1 | 913 | 352 | 1,265 |

Total | 3,159 | 572 | 3,731 |

So Percentage Correctly Predicted=(2246+352)*100/3731=69.63%

Table of marginal effects for logit and probit models is as follows from where we observe that

For logit model:

- corresponding to a unit increase in station listening time in quarter hours per week the individual has 0.28% higher probability of making donation.
- corresponding to a unit increase in income the individual has 0. 000225% higher probability of making donation.

- individual having masters degree or higher has 30.26% higher probability of making donation.

- individual has a bachelor’s but not a Masters has 23.11% higher probability of making donation.
- Females has 6.76% higher probability of making donation.
- Whites have 11.69% higher probability of making donation.

For Probit model:

- corresponding to a unit increase in station listening time in quarter hours per week the individual has 0.27% higher probability of making donation.
- corresponding to a unit increase in income the individual has 0. 000225% higher probability of making donation.

- individual having masters degree or higher has 28.22% higher probability of making donation.

- individual has a bachelor’s but not a Masters has 20.80% higher probability of making donation.
- Females has 6.68% higher probability of making donation.
- Whites have 11.49% higher probability of making donation.

Marginal Effects with SE in parenthesis, *p<.05, **p<.01, ***p<.001 | ||

Independent Variables | Logit (MLE) | Probit (MLE) |

age | 0.0006804 (0.00072) | 0.0008409 (0.00071) |

listen | 0.0028129*** (0.00021) | 0.002762*** (0.0002) |

income | 0.00000225*** (<.000001) | 0.00000225*** (<.000001) |

pgrad | 0.302615*** (0.0692) | 0.2822845*** (0.0.06119) |

cgrad | 0.2311024** (0.07608) | 0.2080118** (0.0661) |

scol | 0.131605 (0.07582) | 0.1141814 (0.0654) |

hsgrad | 0.1036414 (0.07912) | 0.0852101 (0.06798) |

female | 0.0675822*** (0.01652) | 0.0667931*** (0.01623) |

white | 0.1169545** (0.03126) | 0.1148938** (0.03098) |

retired | 0.0513111 (0.03161) | 0.0543233 (0.0307) |

**Marketing Implications and Conclusions**

We conclude that only individual having masters degree or, individual has a bachelor’s but not a Masters, Females and Whites have higher probability of making donation.

Hence NPR should focus on these strata of individuals for donations.