LI Econometrics
Problem Set Week 8
1. An economist studying happiness is interested in the determinants of people’s well-being and life satisfaction, as measured by a composite happiness score. They collected data from a random sample of 600 individuals employed in Birmingham and estimated the following model:
ln(Happyi
) = α + β1 ln(Incomei) + β2 Femalei + β3Marriedi
+ β4 SOCIi + β5 SOCIIi + β6 SOCIIIi + β7 SOCIVi + β8Educi + β9Agei + ϵi (1)
where Happyi
is the happiness score (out of 100), Incomei
is annual income, Femalei
is a dummy equal to one if i is female, Marriedi
is a dummy equal to one if i is married, Educi
is the number of years of education, and Agei
is age.
The sequence of dummy variables SOC are the Standard Occupational Classifiers used by the Office for National Statistics (ONS). SOCI indicates professionals, SOCII managers, SOCIII skilled workers and SOCIV semi-skilled workers. SOCV is the omitted category, indicating unskilled workers.
Estimating this equation by OLS yielded RSS = 6.37 and ESS = 2.72.
a) Excluding the SOC variables from the model and re-estimating yielded ESS = 2.48. Test the joint significance of the SOC variables.
b) The model is re-estimated separately for the 278 men and the 322 women in the sample. Es-timating these two models yielded RSS of 3.13 and 3.02, respectively. Conduct a Chow test for a structural break between the two groups. Carefully write down the models you are estimat-ing and the null hypothesis you are testing. What do you conclude about the determinants of happiness for the two groups?
c) Adding three multiplicative dummy variables to model 1 – Femalei×ln(Incomei), Femalei×Educi and Femalei × Agei – reduced the RSS to 6.18. Test the joint significance of these additional multiplicative variables.
d) Compare the model estimated in c) with the two models estimated in b). Write down the restrictions you must impose on the models in b) to produce the model in c), and then test these restrictions.
2. An insurance company is interested in the relationship between the level of insurance premiums and the value of claims. They are particularly interested in the effect of a new premium structure introduced at the beginning of 2011. They use quarterly data from 1983:1 to 2010:4 (i.e. before the introduction of the new premium structure) to estimate:
Claimst = β0 + β1 Premiumt + β2 Q1t + β3 Q2t + β4 Q3t + ϵt (2)
where Claimst
is the value of insurance claims per insured person, Premiumt
is the average premium, and Qjt are three quarterly dummies indicating the first, second and third quarter of the year (the fourth quarter is omitted as baseline).
These are the results they obtained:
The RSS was 0.057, and the T SS was 4.140.
The company then collected additional data for the first eight quarters after the introduction of the new premium structure. Suppose we add eight dummy variables to model 2, with D2011:1,t taking a value of one for the first quarter of 2011, D2011:2,t taking a value of one for the second quarter of 2011, and so on, with one dummy for each of the eight new quarters. This new model was re-estimated with the same data from 1983:1 to 2010:4 plus the new data from 2011:1 to 2012:4. The coefficients and standard errors on the eight dummy variables were as follows:
a) What is the interpretation of each of the eight coefficients we have just estimated?
b) In the model we have just estimated, what would be the coefficient and standard error on the variable Premiumt? What would the RSS be for the new model? Provide an explanation for your answers.
c) Estimating model 2 using the complete data from 1983:1 to 2012:4 (without the eight dummies), gave an RSS of 0.065. Perform. a predictive failure test to check whether the old model 2 still accurately predicts claims under the new premium structure.