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1 Regression as Moment Structure

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2 Regression Equation Y = X + v Observable Variables Y z = X Moment matrix YY YX = YX XX Moment structure = 2 XX + vv XX = XX XX Parameter vector = ( , XX, vv )’

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3 Sample: z 1, z 2,..., z n n iid Sample Moments S = n -1 z i z i ’ s yy s yx S = s yx s xx Fitting S to = Estimator S close to 3 moment equations s yy = 2 XX + vv s yx = XX s xx = XX with 3 (unknown) parameters Parameter estimates = (s yx /s xx, s XX, s yy - ( ) 2 s XX )’ is the same as the usual OLS estimate of ^ ^ ^ ^ ^ ^

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4 Regression Equation Y = x + v X = x + u Observable Variables Y z = X Moment structure = 2 XX + vv XX = XX XX + uu Parameter vector = ( , XX, vv, uu )’ new parameter

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5 Sample: z 1, z 2,..., z n n iid Sample Moments S := n -1 z i z i ’ s yy s yx S = s yx s xx Fitting S to = Estimator = S close to 3 moment equations s yy = 2 xx + vv s yx = xx s xx = xx + uu with 4 (unknown) parameters Parameter estimates = ?? is the same as the usual OLS estimate of ^ ^^ ^ ^

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6 The effect of measurement error in regression x Y X u v Y = (X -u)+ v = X + (v - u) = X + w, where w = v - u Note that w is correlated with X, unless u or equals zero So, the classical LS estimate b of is neither ubiased, neither consistent. In fact, b ---> YX / XX = xx / XX )= k k is the so called Fiability coefficient (reliability of X). Since 0 k 1 b suffers from downward bias

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7 Regression Equation Y = x 1 + x p x p + v X k = x k + u k Observable Variables b = S XX -1 S XY does not converge to b* := (S XX - uu ) -1 S XY In multiple regression Examples with EQS of regression with error in variables Using suplementary information to assessing the magnitude of variances of errors in variables.

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8 Path analysis & covariance structure Example with ROS data

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9 Sample covariance matrix ROS92 ROS93ROS94 ROS95 ROS ROS ROS ROS Mean: n = 70 ROS92 ROS93 ROS94 F b1b2b3 SEM: bj = ? It is a valid model ?

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10 Calculations b 1 b 2 = b 1 b 3 = b 2 b 3 = b 1 b 2 /b 1 b 3 = b 2 /b 3 = 29.56/ > b 2 =.978b = b 2 b 3 = b 3 (.978b 3 ) --> b 3 2 = 31.09/.978 b 3 = 5.64 In the same way, we obtain b 1 =5.34 b 2 =5.52 Model test in this case is CHI2 = 0, df = 0

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11 Fitted Model R92R94R93 F CHI2 = 0, df =

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12 /TITLE FACTOR ANALYSIS MODEL (EXAMPLE ROS) /SPECIFICATIONS CAS=70; VAR=4; /LABEL V1=ROS92; V2=ROS93; V3=ROS94; V4=ROS95; /EQUATIONS V1 = *F1 + E1; V2 = *F1 + E2; V3 = *F1 + E3; /VARIANCES F1 = 1.0; E1 TO E3 = *; /COVARIANCES /MATRIX /END

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13 ROS92 =V1 = 5.359*F E ROS93 =V2 = 5.516*F E ROS94 =V3 = 5.637*F E VARIANCES OF INDEPENDENT VARIABLES E D E1 -ROS *I I I I I I I I E2 -ROS *I I I I I I I I E3 -ROS *I I I I I I I I

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14 … with the help of EQS RESIDUAL COVARIANCE MATRIX (S-SIGMA) : ROS92 ROS93 ROS94 V 1 V 2 V 3 ROS92 V ROS93 V ROS94 V CHI-SQUARE = BASED ON 0 DEGREES OF FREEDOM STANDARDIZED SOLUTION: ROS92 =V1 =.631*F E1 ROS93 =V2 =.917*F E2 ROS94 =V3 =.827*F E3

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15 one - factor four- indicators model R93R95R94 F ** * CHI2 = ?, df = ? p-value = ? R92 * ****

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16 … with the help of EQS /TITLE FACTOR ANALYSIS MODEL (EXAMPLE ROS) ! This line is not read /SPECIFICATIONS CAS=70; VAR=4; /LABEL V1=ROS92; V2=ROS93; V3=ROS94; V4=ROS95; /EQUATIONS V1 = *F1 + E1; V2 = *F1 + E2; V3 = *F1 + E3; V4 = *F1 + E4; /VARIANCES F1 = 1.0; E1 TO E4 = *; /COVARIANCES /MATRIX /END

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17 ROS92 =V1 = 4.998*F E ROS93 =V2 = 4.837*F E ROS94 =V3 = 6.417*F E ROS95 =V4 = 5.393*F E VARIANCES OF INDEPENDENT VARIABLES E D E1 -ROS *I I I I I I I I E2 -ROS *I I I I I I I I E3 -ROS *I I I I I I I I E4 -ROS *I I I I I I … with the help of EQS

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18 RESIDUAL COVARIANCE MATRIX (S-SIGMA) : RESIDUAL COVARIANCE MATRIX (S-SIGMA) : ROS92 ROS93 ROS94 ROS95 V 1 V 2 V 3 V 4 ROS92 V ROS93 V ROS94 V ROS95 V CHI-SQUARE = BASED ON 2 DEGREES OF FREEDOM PROBABILITY VALUE FOR THE CHI-SQUARE STATISTIC IS STANDARDIZED SOLUTION: ROS92 =V1 =.631*F E1 ROS93 =V2 =.917*F E2 ROS94 =V3 =.827*F E3

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19 Fitted Model R93R95R94 F CHI2 = 6.27, df = 2 p-value =.043 R

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20 /TITLE FACTOR ANALYSIS MODEL (EXAMPLE ROS) /SPECIFICATIONS CAS=70; VAR=4; /LABEL V1=ROS92; V2=ROS93; V3=ROS94; V4=ROS95; /EQUATIONS V1 = *F1 + E1; V2 = *F1 + E2; V3 = *F1 + E3; V4 = *F1 + E4; /VARIANCES F1 = 1.0; E1 TO E4 = *; /COVARIANCES /CONSTRAINTS (V1,F1)=(V2,F1)=(V3,F1)=(V4,F1); /MATRIX /END

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21 … estimation results ROS92 =V1 = 5.521*F E ROS93 =V2 = 5.521*F E ROS94 =V3 = 5.521*F E ROS95 =V4 = 5.521*F E CHI-SQUARE = BASED ON 5 DEGREES OF FREEDOM PROBABILITY VALUE FOR THE CHI-SQUARE STATISTIC IS

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22... EQS use an iterative optimization method ITERATIVE SUMMARY PARAMETER ITERATION ABS CHANGE ALPHA FUNCTION

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23 Exercise: a)Write the covariance structure for the one - factor four- indicators model b) From the ML estimates of this model, shown in previous slides, compute the fitted covariance matrix. c) In relation with b), compute the residual covariance matrix Note: For c), use the following sample moments: ROS92 ROS93ROS94 ROS95 ROS ROS ROS ROS Mean: n = 70

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24 one - factor four- indicators model with means R93R95R94 F ** * CHI2 = ?, df = ? p-value = ? R92 * **** 1 * * * *

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25 /TITLE FACTOR ANALYSIS MODEL (EXAMPLE ROS data) /SPECIFICATIONS CAS=70; VAR=4; ANALYSIS = MOMENT; /LABEL V1=ROS92; V2=ROS93; V3=ROS94; V4=ROS95; /EQUATIONS V1 = *V999+ *F1 + E1; V2 = *V999+ *F1 + E2; V3 = *V999+ *F1 + E3; V4 = *V999+ *F1 + E4; /VARIANCES F1 = 1.0; E1 TO E4 = *; /COVARIANCES /CONSTRAINTS ! (V1,F1)=(V2,F1)=(V3,F1)=(V4,F1); /MATRIX /MEANS /END

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26 ROS92 =V1 = 6.270*V *F E ROS93 =V2 = 7.350*V *F E ROS94 =V3 = *V *F E ROS95 =V4 = 8.800*V *F E VARIANCES OF INDEPENDENT VARIABLES E D E1 -ROS *I I I I I I I I E2 -ROS *I I I I I I I I E3 -ROS *I I I I I I I I E4 -ROS *I I I I I I

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