From Henderson and Quandt, Microeconomic Theory, Pp. 27-31, 2^nd Edition, McGraw-Hill, 1971. Another good source is Layard and Walters, Microeconomic Theory, Pp. 123-143, McGraw-Hill, 1978. In what follows p_i is the price of the i^th good, q_i , and y is income.

1. The uncompensated, own-price elasticity of demand is . For the first ratio we usually substitute the sample means for the respective variables. The second term is the appropriate slope coefficient from the appropriate regression.
   
2. The uncompensated, cross-price elasticity of demand is . Again, the first term is approximated by the sample means. The second term is an appropriate slope coefficient .
   
3. The income elasticity of demand is . Yet again, the first term is approximated by the sample means. The second term is a slope coefficient.
   
4. Define the share of income spent on the i^th good as . You can assume that p and y are exogenous. Define the compensated own-price elasticity as the percentage change in quantity demanded for a small percentage change in price, while restoring the consumer's real income just enough to leave his utility unchanged. From the Slutsky equation we can show that the uncompensated, own-price elasticity of demand is a linear function of the compensated, own-price elasticity and the income elasticity. That is, . You know the uncompensated, own-price elasticity and the income elasticity and can solve for the unobservable compensated elasticity. Since this is directly derived from the Slutsky equation, it also holds for i=j.
   
5. The relationship between the compensated, own- and cross-price elasticities is . (Note that there is a typo in H&Q at (2-19). Compare it to their (2-39). This mistake is in both the 1^st and 2^nd editions.) Thus there is another easy way to solve for the compensated cross price elasticity. What does this 'law' tell you about the compensated elasticities in the two good case? What does it tell you about substitutes in the many good case?
6. Some other useful facts about elasticities: , the share weighted average of the income elasticities is one. And , for the i^th good the sum of the uncompensated own and cross elasticities is equal to the income elasticity.
   
7. For households in 18^th century England (largely agrarian with the bulk of the population working as subsistence tenant farmers), income and prices were largely out of their control and are therefore exogenous. In all of the above cases we have linear combinations of random variables. Constructing test statistics is then quite straightforward.