A behaviouralistis trying to build a predictive model for individuals'happiness based on socio-economic factors as well as three variables related to their personal life: (1) friendships, (2) health and (3) altruism. They are interested in learning about the effects of each of these on happiness but are also concerned about the out-of-sample performance of their model.

They undertake a large questionnaire survey on = 200individuals and obtain an index of happiness () and a vector of = 10essential socio- economic variables () including wages, education, profession and so on. Finally, they have a large set of = 150survey question responses with 50 questions each related to friendships (1), health (2) and altruism (3) which they collect into the vector = [1, 2, 3].

They are unsure of how best to use their data in a predictive model and come up with four ideas, outlined below. You should write down each model and critically evaluate the benefits and drawbacks of the model and estimation procedure with respect to their ability to measure the effect of the key variables above, and their potential out-of-sample performance.

a) A linear regression model ofon the combined set of variables=[,]estimated by ordinary least squares (OLS)

b) A linear regression model ofon = [,]estimated by LASSO

c) A linear regression model ofon a set of factors estimated from=[, ]with the regression estimated by OLS and the factors estimated by principal components (PCA)

d) A linear regression model ofon a set of factors estimated from. The variablesare a subset ofobtained by finding the 'best 30'predictors from LASSO estimation of a regression ofon