Exercise 4.1 You are given a simple linear regression model y = b0 + b1x + e e X i.i.d.N(0,e2 n In ) observed

Exercise 4.1 You are given a simple linear regression model y = b0

+ b1x + e e X ≈ i.i.d.N(0,σe2 n In ) observed at certain level of aggregation

(e.g. regions). Suppose further that data are aggregated at a higher level of aggregation (e.g. countries). Call yˉ the dependent variable at this level of aggregation, similarly for x and

e. Suppose further that we have m countries and n1,n2 ,...,nm regions respectively in each country. Let us now specify the model at the higher level of aggregation as: y = b0 + b1x + e .

Prove that even if the regression model is homescedastic at a lower level of aggregation, this property is generally lost at a higher level of aggregation. [Hint: Use the aggregation matrix mGn such that y = Gy and similarly for the other variables].