Dustmann and Preston:Racial and Economic Factors in Attitudes to Immigration Our estimation strategy now proceeds in three stages.In the first stage, we estimate the reduced form coefficient matrix I =TT2]'.We do that by estimating the coefficients of each equation in (4)and (5)(corresponding to the rows of r)separately by independent (ordered)probits.Due to the discrete nature of the dependent variables,we can only estimate their ratios to the standard deviations of the associated error components. In the second stage we obtain the parameters in X.Again,a normali- sation assumption is required.We adopt the identifying normalisation that the diagonal elements in X and in X are such as to make the diagonal ele- ments of e equal to unity.To estimate e,we take each pairing of questions successively,and estimate the corresponding off-diagonal component of e by bivariate maximum likelihood.We fix the coefficients of the two equations concerned at the values in I estimated in the previous stage.4 Finally,in the third step we estimate the parameters in M,A and X using ninimum distance estimation and the restrictions∑22=∑e+M∑，M'and 12=+MA'.To do so,we make the following assumptions.First,we assume that each of our indicator questions in z*is indicative of one and only one factor.For instance,the three questions on the labour market are assumed to be affected only through the labour market channel,the three questions on welfare concerns only through the welfare channel,and so on.This means that we assume that MM'is a block diagonal matrix,with only one non-zero element in each row of M.Second,we assume that all correlation between responses to these questions (conditional on the regressors Xi)is accounted for by the factor structure,which implies diagonality of the matrix.Notice that we allow for correlation between the factors,since is not required to be diagonal.Finally,we set the diagonal elements of o to unity,which is simply a normalising assumption. Given these assumptions,there is sufficient information in >22 from the con- ditional correlations between responses within blocks to identify all elements of M.5 Having identified M,the off-diagonal elements of Xo are identified without 4Consider for instance the responses to the first two immigration questions,with the latent two equation model being yii Xi71+e1i and y2=Xiy2+e2i.We estimate the covariance Cov(eli,2i)using a bivariate probit likelihood,where we fix Y1 and Y2 at the estimates obtained in the first stage,Y and 72. 5Remembering the particular structure of MEM',suppose that the ith diagonal block has gi elements.Then there are gi(qi-1)/2 off-diagonal elements in the corresponding block of S22 from which to identify them.This is sufficient if gi 3,which is so for each block in our case.This is not to say that the condition is necessary since there is also identifying information in the elements of off-diagonal blocks. Published by The Berkeley Electronic Press,2007 7Our estimation strategy now proceeds in three stages. In the Örst stage, we estimate the reduced form coe¢ cient matrix ￾ = [￾1j￾2] 0 . We do that by estimating the coe¢ cients of each equation in (4) and (5) (corresponding to the rows of ￾) separately by independent (ordered) probits. Due to the discrete nature of the dependent variables, we can only estimate their ratios to the standard deviations of the associated error components. In the second stage we obtain the parameters in  . Again, a normali￾sation assumption is required. We adopt the identifying normalisation that the diagonal elements in u and in w are such as to make the diagonal ele￾ments of  equal to unity. To estimate  , we take each pairing of questions successively, and estimate the corresponding o§-diagonal component of  by bivariate maximum likelihood. We Öx the coe¢ cients of the two equations concerned at the values in ￾ estimated in the previous stage.4 Finally, in the third step we estimate the parameters in M, and v using minimum distance estimation and the restrictions 22 = w + M v M0 and 12 = uw +M v 0 . To do so, we make the following assumptions. First, we assume that each of our indicator questions in z is indicative of one and only one factor. For instance, the three questions on the labour market are assumed to be a§ected only through the labour market channel, the three questions on welfare concerns only through the welfare channel, and so on. This means that we assume that MM0 is a block diagonal matrix, with only one non-zero element in each row of M. Second, we assume that all correlation between responses to these questions (conditional on the regressors Xi) is accounted for by the factor structure, which implies diagonality of the w matrix. Notice that we allow for correlation between the factors, since v is not required to be diagonal. Finally, we set the diagonal elements of v to unity, which is simply a normalising assumption. Given these assumptions, there is su¢ cient information in 22 from the con￾ditional correlations between responses within blocks to identify all elements of M. 5 Having identiÖed M, the o§-diagonal elements of v are identiÖed without 4Consider for instance the responses to the Örst two immigration questions, with the latent two equation model being y 1i = Xi 1 + 1i and y 2i = Xi 2 + 2i . We estimate the covariance Cov(1i ; 2i) using a bivariate probit likelihood, where we Öx 1 and 2 at the estimates obtained in the Örst stage, ^1 and ^2 . 5Remembering the particular structure of MM0 , suppose that the ith diagonal block has qi elements. Then there are qi (qi￾1)=2 o§-diagonal elements in the corresponding block of 22 from which to identify them. This is su¢ cient if qi  3, which is so for each block in our case. This is not to say that the condition is necessary since there is also identifying information in the elements of o§-diagonal blocks. 7 Dustmann and Preston: Racial and Economic Factors in Attitudes to Immigration Published by The Berkeley Electronic Press, 2007