Appendix B BASIC MATRIX OPERATIONS IN PROC IML B.1 ASSIGNING SCALARS

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1 Appendix B BASIC MATRIX OPERATIONS IN PROC IML B.1 ASSIGNING SCALARS Scalars can be viewed as 1 1 matrices and can be created using Proc IML by using the statement x¼scalar_value or x¼{scalar_value}. As an example, the statements x¼14.5 and x¼{14.5} are the same and both store the value 14.5 in x. We can also store character values as the commands name¼james and hello¼hello World illustrate. The stored values in the variables can easily be determined by using the print command in Proc IML. For example to view the values in the variables x, name, and hello use the command Print x name hello. B.2 CREATING MATRICES AND VECTORS As mentioned in Appendix A, it is easy to create matrices and vectors in Proc IML. The command A¼{2 4, 3 1} will create the matrix A ¼ Each row of the matrix is separated by a comma. That is, each row of the above command yields a row vector. For instance, the command A¼{ } creates the row vector A ¼ ½ Š. If we separate each entry in the row vector by a comma, we will get a columnvector. As an example, the command A¼{1,2,3,4} creates the column vector A ¼ Applied Econometrics Using the SAS Ò System, by Vivek B. Ajmani Copyright Ó 2009 John Wiley & Sons, Inc. 249

2 250 APPENDIX B BASIC MATRIX OPERATIONS IN PROC IML These commands can easily be extended to create matrices consisting of character elements. For example, the command A¼{a b, c d} will create the matrix A ¼ a b c d B.3 ELEMENTARY MATRIX OPERATIONS B.3.a Addition/Subtraction of Matrices For two conformable matrices, A and B, their sum can be computed by using the command C¼A þ B, where the sum is stored in C. Changing the addition to a subtraction yields the difference between the two matrices. B.3.b Product of Matrices For two conformable matrices, A and B, the element by element product of the twois givenby the command C¼A#B. For example, consider the two matrices A ¼ 1 2 and B ¼ The element by element product of these two is given by C ¼ The product of the two matrices is given by using the command C¼A*B. In the above example, the product is C ¼ The square of a matrix is given by either of the following commands C¼A##2 or C¼A*A. Of course, we can use these commands to raise a matrix to any power (assuming that the product is defined). B.3.c Kronecker Products The Kronecker product of two matrices A and B can be obtained by using the command A@B. For example, let A ¼ and B ¼ Then, the command C¼A@B will produce C ¼ B.3.d Inverses, Eigenvalues, and Eigenvectors As shown in Appendix A, the inverse of a square matrix A can be computed by using the command C¼inv(A). Eigenvalues and eigenvectors can be computed easily by using the commands C¼eigval(A) or C¼eigvec(A).

3 APPENDIX B BASIC MATRIX OPERATIONS IN PROC IML 251 B.4 COMPARISON OPERATORS The max(min) commands will search for the maximum(minimum) element of any matrix or vector. To use these commands, simply type C¼max(A) or C¼min(A). For two conformable matrices (of the same dimension), we can define the elementwise maximums and minimums. Consider matrices A and B, which were given in (Section B.3.b). The command C¼A<>B will find the elementwise maximum between the two matrices. In our example, this will yield C ¼ The command C¼A><B will yield the elementwise minimum between the two matrices. In our example, this is simply A. B.5 MATRIX-GENERATING FUNCTIONS B.5.a Identity Matrix The command Iden¼I(3) will create a 3 3 identity matrix. B.5.b The J Matrix This is a matrix of 1 s. The command J¼J(3,3) will create a 3 3 matrix of 1 s. This command can be modified to create a matrix of constants. For example, suppose that we want a 3 3 matrix of 2 s. We can modify the above command as follows J¼J(3,3,2). B.5.c Block Diagonal Matrices Often, we will have towork with block diagonal matrices. A block diagonal matrix can be created by using the command C¼block (A 1,A 2,...) where A 1,A 2...are matrices. For example, for the A and B matrices defined earlier, the block diagonal matrix C is given by C ¼ B.5.d Diagonal Matrices The identity matrix is a matrix with 1 s on the diagonal. It is easy to create any diagonal matrix in Proc IML. For instance, the command C¼diag({1 2 4}) will create the following diagonal matrix C ¼ Given a square matrix, the diag command can be used to extract the diagonal elements. For example, the command C¼diag({1 2,3 4}) will create the following matrix C ¼ B.6 SUBSET OF MATRICES Econometric analysis using Proc IML often involves extracting specific columns (or rows) of matrices. The command C¼A[,1] will extract the first column of the matrix A, and the command R¼A[1,] will extract the first row of the matrix A. B.7 SUBSCRIPT REDUCTION OPERATORS Proc IML can be used to easily calculatevarious row- and column-specific statistics of matrices. As an example, consider the 3 3 matrix defined by the command A¼{0 1 2, 5 4 3, 7 6 8}. Column sums of this matrix can be computed by using the command

4 252 APPENDIX B BASIC MATRIX OPERATIONS IN PROC IML Col_Sum¼A[ þ,]. Using this command yields a row vector Col_Sum with elements 12, 11, and 13. The row sums of this matrix can be computed by using the command Row_Sum¼A[, þ ]. We can also determine the maximum element in each column by using the command Col_Max¼A[<>,]. Using this command yields a row vector Col_Max with elements 7, 6, and 8. The command Row_Min¼A[,><] will yield the minimum of each row of the matrix. The column means can be calculated by using the command Col_Mean¼A[,]. Using this command, yields the row vector Col_Mean with elements 4, 3.67, and The command Col_Prod¼A[#,] results in a row vector Col_Prod that contains the product of the elements in each column. In our example, the result is a row vector with elements 0, 24, and 48. We can easily extend this command to calculate the sums of squares of each column. This is calculated by using the command Col_SSQ¼A[##,]. The result is a row vector Col_SSQ with elements 74, 53, and 77. B.8 THE DIAG AND VECDIAG COMMANDS The Proc IML Diag command create a diagonal matrix. For example, if A ¼ 1 3 ; 2 4 then the command B¼Diag(A) results in a diagonal matrix B whose diagonal elements are the diagonal elements of A. That is, B ¼ This command is useful when extracting the standard errors of regression coefficients from the diagonal elements of the variance covariance matrices. If a column vector consisting of the diagonal elements of A is desirable, then one can use the VecDiag function. As an example, the command B¼VecDiag(A) results in B ¼ 1 4 B.9 CONCATENATION OF MATRICES There are several instances wherewe have a need to concatenate matrices. A trivial case is wherewe need to append a column of 1s to a data matrix. Horizontal concatenation can be done by using jj, while vertical concatenation can be done by using ==. For example, consider the following matrices A ¼ and B ¼ The command AjjB gives the matrix whereas the command A==B gives the matrix ; B.10 CONTROL STATEMENTS Several Proc IML routines given in this book make use of control statements. For example, we made use of control statements when computing MLE estimates for the parameters. These statements were also used when computing estimates through iterative procedures.

5 DO-END Statement The statements following the DO statement are executed until a matching END statement is encountered. DO Iterative Statement The DO iterative statements take the form DO Index=start TO end; IML statements follow END; For example, the statements DO Index=1 to 5; Print Index; END; will print the value of INDEX for each iteration of the DO statement. The output will consist of the values of INDEX starting from 1 through 5. IF-THEN/ELSE Statement These statements can be used to impose restrictions or conditions on other statements. The IF part imposes the restriction and the THEN part executes the action to be taken if the restrictions are met. The ELSE portion of the statement execute the action for the alternative. For example, the statements IF MAX(A)<30 then print Good Data ; ELSE print Bad Data ; evaluate the matrix A. If the maximum element of the matrix is less than 30, then the statement Good Data is printed, else the statement Bad Data is printed. B.11 CALCULATING SUMMARY STATISTICS IN PROC IML Summary statistics on the numeric variables stored in matrices can be obtained in Proc IML by using the SUMMARY command. The summary statistics can be based on subgroups (e.g., Panel Data) and can be saved in matrices for later use. As an example, consider the cost of US airlines panel data set from Greene (2003). The data consist of 90 observations for six firms for The following SAS statements can be used to summarize the data by airline. The option opt(save) saves the summary statistics by airline. The statements will retrieve and save the summary statistics in matrices. The names of the matrices are identical to the names of the variables. The statement print LnC produces the means and standard deviations for the six airlines for the variable LnC. The first column contains the means, whereas the second column contains the standard deviations. We have found this command useful when programming the Hausman Taylor estimation method for panel data models. The resulting output is given in output B.1. proc import out=airline datafile="c\temp\airline" dbms=excel Replace; getnames=yes; run; Data airline; set airline; LnC=log(C); LnQ=Log(Q); LnPF=Log(PF); run; proc iml; use airline; summary var {LnC LnQ LnPF} class {i} stat{mean std} opt{save}; print LnC; run; APPENDIX B BASIC MATRIX OPERATIONS IN PROC IML 253

6 254 APPENDIX B BASIC MATRIX OPERATIONS IN PROC IML I Nobs Variable MEAN STD LNC LNQ LNPF LNC LNQ LNPF LNC LNQ LNPF LNC LNQ LNPF LNC LNQ LNPF LNC LNQ LNPF All 90 LNC LNQ LNPF LNC OUTPUT B.1. Summary statistics of three variables for each airline.

x = 12 x = 12 1x = 16

x = 12 x = 12 1x = 16 2.2 - The Inverse of a Matrix We've seen how to add matrices, multiply them by scalars, subtract them, and multiply one matrix by another. The question naturally arises: Can we divide one matrix by another?