Let two data sets (x,y) and (x, yi) be related by x = c, + c2x,...

 

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Let two data sets (x,y) and (x, yi) be related by x = c, + c2x, and yi = c, + c,y for all i = 1, .,n. This means that the only differences between the two data sets are the location and the scale of measurement. Such data transformations are often applied in economic studies. For example, y; may be the total variable production costs in y, may be the total variable production costs in dollars of a firm in month i and y the total production costs in million of dollars. Then c, are the total fixed costs (in millions of dollars) and c, 10-6. (a) Suppose that y, & x; satisfy the linear relation y, = a + Bx, V i = 1,.,n. Show that this implies that the relationship between y & xj is y = a' +B*xi with Bc,C Bc. ai = , + C,a -- C2 & B 1.5 marks (b) Suppose the OLS estimators of a & B are, respectively, a & ß. Using your results from part (a), provide the OLS estimators of a" & B", say â* & B". (c) To illustrate, consider the following simple linear regression model: Yi = a + Bx, + u (1) where: y = excess returns on an index of stocks in the sector of consumer goods in percentages X = excess returns on an overall stock market index in percentages. Model (1) has been estimated with monthly sample data from January 1980 to December 1999. Some EViews output follows: Dependent Variable: Y Method: Least Squares Date: 04/29/20 Time 15:40 Sample 1980MO1 1999M12 Included observations: 240 Variable Sid. Error 1 Stafistic Prob Coefficient 0 447481 0.362943 0.075386 1.232924 15 53500 02188 1171128 0.0000 Use these results to provide the estimates of a' & B" for the model: yi = a' + B'xi + uj where the transformed data are the actual excess returns rather than the percentages. Comment. Let two data sets (x,y) and (x, yi) be related by x = c, + c2x, and yi = c, + c,y for all i = 1, .,n. This means that the only differences between the two data sets are the location and the scale of measurement. Such data transformations are often applied in economic studies. For example, y; may be the total variable production costs in y, may be the total variable production costs in dollars of a firm in month i and y the total production costs in million of dollars. Then c, are the total fixed costs (in millions of dollars) and c, 10-6. (a) Suppose that y, & x; satisfy the linear relation y, = a + Bx, V i = 1,.,n. Show that this implies that the relationship between y & xj is y = a' +B*xi with Bc,C Bc. ai = , + C,a -- C2 & B 1.5 marks (b) Suppose the OLS estimators of a & B are, respectively, a & ß. Using your results from part (a), provide the OLS estimators of a" & B", say â* & B". (c) To illustrate, consider the following simple linear regression model: Yi = a + Bx, + u (1) where: y = excess returns on an index of stocks in the sector of consumer goods in percentages X = excess returns on an overall stock market index in percentages. Model (1) has been estimated with monthly sample data from January 1980 to December 1999. Some EViews output follows: Dependent Variable: Y Method: Least Squares Date: 04/29/20 Time 15:40 Sample 1980MO1 1999M12 Included observations: 240 Variable Sid. Error 1 Stafistic Prob Coefficient 0 447481 0.362943 0.075386 1.232924 15 53500 02188 1171128 0.0000 Use these results to provide the estimates of a' & B" for the model: yi = a' + B'xi + uj where the transformed data are the actual excess returns rather than the percentages. Comment.