1 Regression as Moment Structure

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 )’

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  ^ ^ ^ ^ ^ ^

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

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  ^ ^^ ^ ^

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

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.

8 Path analysis & covariance structure Example with ROS data

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 ?

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

11 Fitted Model R92R94R93 F CHI2 = 0, df =

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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