# On applying the expectations operator to these

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On applying the expectations operator to these equations, we find that (17) E ( ˜ β 1 ) = β 1 + ( X 1 X 1 ) 1 X 1 X 2 β 2 , since E { ( X 1 X 1 ) 1 X 1 ε } = ( X 1 X 1 ) 1 X 1 E ( ε ) = 0. Thus, in general, we have E ( ˜ β 1 ) = β 1 , which is to say that ˜ β 1 is a biased estimator. The only circum- stances in which the estimator will be unbiased are when either X 1 X 2 = 0 or β 2 = 0. In other circumstances, the estimator will suffer from a problem which is commonly described as omitted-variables bias . The Regression Model with an Intercept Now consider again the equations (18) y t = α + x t. β + ε t , t = 1 , . . . , T, which comprise T observations of a regression model with an intercept term α and with k explanatory variables in x t. = [ x t 1 , x t 2 , . . . , x tk ]. These equations can also be represented in a matrix notation as (19) y = ια + + ε. Here, the vector ι = [1 , 1 , . . . , 1] , which consists of T units, is described alter- natively as the dummy vector or the summation vector, whilst Z = [ x tj ; t = 1 , . . . T ; j = 1 , . . . , k ] is the matrix of the observations on the explanatory vari- ables. Equation (19) can be construed as a case of the partitioned regression equation of (1). By setting X 1 = ι and X 2 = Z and by taking β 1 = α , β 2 = β z in equations (4) and (9), we derive the following expressions for the estimates of the parameters α , β z : (20) ˆ α = ( ι ι ) 1 ι ( y Z ˆ β z ) , 3

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TOPICS IN ECONOMETRICS (21) ˆ β z = Z ( I P ι ) Z 1 Z ( I P ι ) y, with P ι = ι ( ι ι ) 1 ι = 1 T ιι . To understand the effect of the operator P ι in this context, consider the follow- ing expressions: (22) ι y = T t =1 y t , ( ι ι ) 1 ι y = 1 T T t =1 y t = ¯ y, P ι y = ι ¯ y = ι ( ι ι ) 1 ι y = [¯ y, ¯ y, . . . , ¯ y ] .
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