This section may be concluded by proving a version of

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This section may be concluded by proving a version of the Cauchy–Schwarz inequality that establishes the bounds on Corr ( x, y ) = C ( x, y ) / V ( x ) V ( y ), which is the coefficent of the correlation of x and y . Consider the variance of the prediction error E { y E ( y ) } − β { x E ( x ) } 2 = V ( y ) 2 βC ( x, y ) + β 2 V ( x ) 0 . Setting β = C ( x, y ) /V ( x ) gives V ( y ) 2 { C ( x, y ) } 2 V ( x ) + { C ( x, y ) } 2 V ( x ) 0 . whence V ( x ) V ( y ) ≥ { C ( x, y ) } 2 . It follows that { Corr ( x, y ) } 2 1 and, therefore, that 1 Corr ( x, y ) 1. 6
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Empirical Regressions. Imagine that we have a sample of T observations on x and y which are ( x 1 , y 1 ) , ( x 2 , y 2 ) , . . . , ( x T , y T ). Then we can calculate the following empirical or sample moments: ¯ x = 1 T T t =1 x t , ¯ y = 1 T T t =1 y t , S 2 x = 1 T T t =1 ( x t ¯ x ) 2 = 1 T T t =1 ( x t ¯ x ) x t = 1 T T t =1 x 2 t ¯ x 2 , S xy = 1 T T t =1 ( x t ¯ x )( y t ¯ y ) = 1 T T t =1 ( x t ¯ x ) y t = 1 T T t =1 x t y t ¯ x ¯ y, It seems reasonable that, in order to estimate α and β , we should replace the moments in the formulae of (iii) and (vi) by the corresponding sample moments. Thus the estimates of α and β are ˆ α = ¯ y ˆ β ¯ x, ˆ β = ( x t ¯ x )( y t ¯ y ) ( x t ¯ x ) 2 . The justification of this estimation procedure, which is know as the method of moments, is that, in many of the circumstances under which the sample is liable to be generated, we can expect the sample moments to converge to the true moments of the bivariate distribution, thereby causing the estimates of the parameters to converge likewise to their true values. Often there is insufficient statistical regularity in the processes generating the variable x to justify our postulating a joint probability density function for x and y . Sometimes the variable is regulated in pursuit of an economic policy in such a way that it cannot be regarded as random in any of the senses accepted by statistical theory. In such cases, we may prefer to derive the estimators of the parameters α and β by methods which make fewer statistical assumptions about x . When x is a non stochastic variable, the equation y = α + + ε is usually regarded as a functional relationship between x and y that is subject to the effects of a random disturbance term ε . It is commonly assumed that, in all instances of this relationship, the disturbance has a zero expected value and a variance which is finite and constant. Thus E ( ε ) = 0 and V ( ε ) = E ( ε 2 ) = σ 2 . Also it is assumed that the movements in x are unrelated to those of the disturbance term. 7
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The principle of least squares suggests that we should estimate α and β by finding the values which minimise the quantity S = T t =1 ( y t ˆ y t ) 2 = T t =1 ( y t α x t β ) 2 . This is the sum of squares of the vertical distances—measured parallel to the y - axis—of the data points from an interpolated regression line. Differentiating the function S with respect to α and setting the results to zero for a minimum gives 2 ( y t α βx t ) = 0 , or, equivalently, ¯ y α β ¯ x = 0 . This generates the following estimating equation for α : α ( β ) = ¯ y β ¯ x. (viii) Next, by differentiating with respect to β and setting the result to zero, we get 2 x t ( y t α βx t ) = 0 . (ix) On substituting for α from (vii) and eliminating the factor 2, this becomes x t y t x t y β ¯ x ) β x 2 t = 0 ,
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  • Spring '12
  • D.S.G.Pollock
  • Probability theory, probability density function, yj

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