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Regression as Moment Structure

Regression as Moment Structure. Regression Equation Y = b X + v Observable Variables Y z = X Moment matrix s YY s YX S = s YX s XX Moment structure S = S(q) b 2 s XX + s vv bs XX

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Regression as Moment Structure

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  1. Regression as Moment Structure

  2. Regression Equation Y = b X + v Observable Variables Y z = X Moment matrix sYY sYX S = sYX sXX Moment structure S = S(q) b2sXX +svv bsXX S = bsXXsXX Parameter vector q = (b, sXX, svv )’

  3. Sample: z1, z2, ..., zn n iid • Sample Moments • S = n-1S zi zi’ • syy syx • S = • syx sxx • Fitting S to S = S(q) • Estimator q S close to S = S(q) • 3 moment equations • syy= b2sXX +svv • syx= bsXX • sxx= sXX • with 3 (unknown) parameters • Parameter estimates • q = (syx/sxx, sXX, syy - (b )2sXX )’ ^ ^ ^ ^ b is the same as the usual OLS estimate of b ! ^ ^

  4. Regression Equation Y = b x + v X = x + u Observable Variables Y z = X Moment structure S = S(q) b2sXX +svv bsXX S = bsXXsXX + suu Parameter vector q = (b, sXX, svv , suu )’ new parameter

  5. Sample: z1, z2, ..., zn n iid • Sample Moments • S := n-1zi zi’ • syy syx • S = • syx sxx • Fitting S to S = S(q) • Estimator q = S close to S = S(q ) • 3 moment equations • syy= b2sxx +svv • syx= bsxx • sxx= sxx + suu • with 4 (unknown) parameters • Parameter estimates • q = ?? ^ ^ ^ ^ b is the same as the usual OLS estimate of b ! ^

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

  7. In multiple regression Regression Equation Y = b1x1 + b2x2...+ b pxp+ v Xk = xk + uk Observable Variables b = SXX-1SXY does not converge to b b* := (SXX - Quu)-1 SXY 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 ROS93 ROS94 ROS95 ROS92 72.07 ROS93 29.5636.21 ROS94 30.2131.0946.51 ROS95 27.63 24.04 35.19 46.62 Mean: 6.27 7.35 10.02 8.80 n = 70 SEM: bj = ? It is a valid model ? F b1 b2 b3 ROS92 ROS93 ROS94

  10. Calculations b1b2= 29.56 b1b3= 30.21 b2b3= 31.09 b1b2/b1b3 = b2/b3 = 29.56/30.21--> b2 = .978b3 31.09 = b2b3= b3 (.978b3) --> b32= 31.09/.978 b3 = 5.64 In the same way, we obtain b1=5.34 b2=5.52 Model test in this case is CHI2 = 0, df = 0

  11. Fitted Model 1 F 5.34 5.52 5.64 R92 R93 R94 43.34 5.80 14.74 CHI2 = 0, df = 0

  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 72.07 29.56 36.21 30.21 31.09 46.51 27.63 24.04 35.19 46.62 /END

  13. ROS92 =V1 = 5.359*F1 + 1.000 E1 .974 5.504 ROS93 =V2 = 5.516*F1 + 1.000 E2 .650 8.482 ROS94 =V3 = 5.637*F1 + 1.000 E3 .753 7.482 VARIANCES OF INDEPENDENT VARIABLES ---------------------------------- E D --- --- E1 -ROS92 43.347*I I 8.205 I I 5.283 I I I I E2 -ROS93 5.789*I I 3.924 I I 1.475 I I I I E3 -ROS94 14.736*I I 4.693 I I 3.140 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 1 0.000 ROS93 V 2 0.000 0.000 ROS94 V 3 0.000 0.000 0.000 CHI-SQUARE = 0.000 BASED ON 0 DEGREES OF FREEDOM STANDARDIZED SOLUTION: ROS92 =V1 = .631*F1 + .776 E1 ROS93 =V2 = .917*F1 + .400 E2 ROS94 =V3 = .827*F1 + .563 E3

  15. one - factor four- indicators model F * * * * R92 R93 R94 R95 * * * * CHI2 = ?, df = ? p-value = ?

  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 72.07 29.56 36.21 30.21 31.09 46.51 27.63 24.04 35.19 46.62 /END

  17. … with the help of EQS ROS92 =V1 = 4.998*F1 + 1.000 E1 .966 5.175 ROS93 =V2 = 4.837*F1 + 1.000 E2 .622 7.779 ROS94 =V3 = 6.417*F1 + 1.000 E3 .653 9.833 ROS95 =V4 = 5.393*F1 + 1.000 E4 .710 7.590 VARIANCES OF INDEPENDENT VARIABLES ---------------------------------- E D --- --- E1 -ROS92 47.090*I I 8.437 I I 5.581 I I I I E2 -ROS93 12.810*I I 2.775 I I 4.616 I I I I E3 -ROS94 5.332*I I 3.017 I I 1.767 I I I I E4 -ROS95 17.536*I I 3.682 I I 4.763 I I

  18. Fitted Model F 4.84 6.42 5.40 4.99 R92 R93 R94 R95 47.10 12.81 5.33 17.54 CHI2 = 6.27, df = 2 p-value = .043

  19. /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 72.07 29.56 36.21 30.21 31.09 46.51 27.63 24.04 35.19 46.62 /END

  20. … estimation results ROS92 =V1 = 5.521*F1 + 1.000 E1 .528 10.450 ROS93 =V2 = 5.521*F1 + 1.000 E2 .528 10.450 ROS94 =V3 = 5.521*F1 + 1.000 E3 .528 10.450 ROS95 =V4 = 5.521*F1 + 1.000 E4 .528 10.450 CHI-SQUARE = 12.425 BASED ON 5 DEGREES OF FREEDOM PROBABILITY VALUE FOR THE CHI-SQUARE STATISTIC IS 0.02941

  21. ... EQS use an iterative optimization method ITERATIVE SUMMARY PARAMETER ITERATION ABS CHANGE ALPHA FUNCTION 1 21.878996 1.00000 1.39447 2 5.741889 1.00000 0.43985 3 2.309283 1.00000 0.19638 4 0.477505 1.00000 0.18079 5 0.147232 1.00000 0.18014 6 0.056361 1.00000 0.18008 7 0.014530 1.00000 0.18007 8 0.005784 1.00000 0.18007 9 0.001423 1.00000 0.18007 10 0.000598 1.00000 0.18007

  22. Exercise: a) Write the covariance structure for the one - factor four- indicators modelb) 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 ROS93 ROS94 ROS95 ROS92 72.07 ROS93 29.5636.21 ROS94 30.2131.0946.51 ROS95 27.63 24.04 35.19 46.62 Mean: 6.27 7.35 10.02 8.80 n = 70

  23. one - factor four- indicators model with means F 1 * * * * * * * * R92 R93 R94 R95 * * * * CHI2 = ?, df = ? p-value = ?

  24. /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 72.07 29.56 36.21 30.21 31.09 46.51 27.63 24.04 35.19 46.62 /MEANS 6.27 7.35 10.02 8.80 /END

  25. ROS92 =V1 = 6.270*V999 + 4.998*F1 + 1.000 E1 1.022 .966 6.135 5.175 ROS93 =V2 = 7.350*V999 + 4.837*F1 + 1.000 E2 .724 .622 10.146 7.779 ROS94 =V3 = 10.020*V999 + 6.417*F1 + 1.000 E3 .821 .653 12.204 9.833 ROS95 =V4 = 8.800*V999 + 5.393*F1 + 1.000 E4 .822 .710 10.706 7.591 VARIANCES OF INDEPENDENT VARIABLES ---------------------------------- E D --- --- E1 -ROS92 47.092*I I 8.437 I I 5.582 I I I I E2 -ROS93 12.810*I I 2.775 I I 4.616 I I I I E3 -ROS94 5.332*I I 3.017 I I 1.767 I I I I E4 -ROS95 17.535*I I 3.682 I I 4.763 I I

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