Slide 10.1 Structural Equation Models MathematicalMarketing Chapter 10 Structural Equation Models In This Chapter We Will Cover The theme of this chapter

Slide 10.Slide 10.11Structural Structural

Equation ModelsEquation ModelsMathematicalMathematicalMarketingMarketing

Chapter 10 Structural Equation ModelsIn This Chapter We Will Cover

The theme of this chapter is causation. We have already looked at one class of causal models,confirmatory factor analysis. We will now look at a more general formulation including

Path analysis

All-y models

Nonrecursion

Path models with latent variables

Second order factor models

Models that include means



Key Terminology

Endogenous

Exogenous

Path analysis

Causal model

Structural equation model

Covariance structure model



ζΓxByy

The Basic SEM Model

p

2

1

q

2

1

pq2p1p

q22221

q11211

p

2

1

2p1p

p221

p112

p

2

1

x

x

x

y

y

y

0

0

0

y

y

y

vector of pendogeneousvariables

regressioncoefficients for endogeneousvariables onother endogeneousvaribles

regression coefficients for endogeneousvariables onexogeneousvariables

specificationerror

exogeneousvariables



Some Basic Assumptions

E(y) = 0

E(x) = 0

Cov(x, ) = 0

0|| BI

V(x) = E(xx) =

V() = E() = .

Some Definitions



Reduced Form Coefficients vs. Structural Coefficients

ζBIΓxBIy

ζΓxyBI

ζΓxByy

ζΓxByy

11 )()(

)(

eGxy



What Are the Model's Implications for V(y)?

))((E)(E eGxeGxyyΣ

.)(E)(E)(E)(E eeGxeeGxGxGx

ζBIGxeGx 1)(E)(E

1)()(E BIζxG

E(yy) = GE(xx)G + E(ee)



Proceeding With the V(y)

E(yy) = GE(xx)G + E(ee)

11

1111

)]([)(

)()()()(

BIΨΓΓΦBI

BIΨBIBIΓΓΦBI

G = (I – B)-1 E(xx) = V(x) =

e = (I – B)-1

V() =



Another Important Piece: Cov(x, y)

1))((E)(E

E

)(E)(E

BIζxGxx

exGxx

eGxxyx e = (I – B)-1

1)( BIΓΦ0GΦ



The Structure of the Covariance Matrix for All the Variables

ΦBIΓΦ

BIΨΓΓΦBI

ΣΣ

ΣΣ

1

11

xxxy

yy

)(

)]([)(



A Simple Causal Model

Intention to Purchase

y1

Purchasing Behavior

y2

Perceived Costx2

Attractiveness of Product

x1

DescriptionVariable



Graphical Representation of Path Models

x2

y1 y2

x1

21

11

12

Boxes are manifest variables Circles are latent variables Unlabeled arrows are error Labeled single headed arrows are causal paths Double headed arrows are covariances

21



The Equations for the Sample Model

21212

12121111

yy

xxy

ζΓxByy

2

1

2

11211

2

1

212

1

x

x

00y

y

0

00

y

y

ΦSx

ζ

xx

2221

11

22

11

)(V

0)(V



We Do Not Really Need x and y Variables

Rewriting the model slightly

y = By + x +

x = 0y + Ix + 0

Now define

x

yz

I0

ΓBG

0

ζe

eGzz

so the model is



We Can Also Consolidate the Covariance Matrices

eGzz

Now for our model

AΦ0

e

)(V

We have the following covariance matrices for the exogenous factors



So We Don't Need So Many Matrices

Instead of having x and y we can get by with one set of variables: z

Instead of having B and we can get by with just G

Instead of having and we can get by with just A



In What Sense Are These Causal Models?

y3y2 y1

y2

y1

y3

The "same" arrow is missing in both.



Both Models Have 1 DF and That DF Implies the Same Restriction

Both causal diagrams require only that the partial covariance 23·1 = 0

where 23·1 is the Cov(e2, e3) and e2 and e3 are defined as the errors in

y2 = y1 + e2 and

y3 = y1 + e3

Assuming that this constraint is met, that just means we have failed to reject Ho. Of course that doesn't mean we have proven it.



Regression Is a Special Case of a Causal Model

x2 y1

x1

21

11

12

x3

1332

31

What are the degrees of freedom for this model?

Can we reject the causal hypothesis?



Multivariate Regression Is Also a Special Case

x2

y1

x1

21

11

12

x3 23

32

31

y2

22

21

13

12

The same questions apply here



Recursive Systems

A recursive system is characterized by V() = diagonal, and by the fact that it is possible to arrange the y variables so that B is lower (or upper) triangular

x2

y1x1

y2 x2

y1x1

y2

Two Nonrecursive Examples



Structural Equation Models with Latent Variables

y = y +

x = x +

= B + +

Measurement models

Structural equation models



Some of the Assumptions of the Model

Cov (, ) = 0

Cov (, ) = 0

Cov (, ) = 0

Cov (, , ) = 0

Diag (B) = 0

| I – B| 0



Naming Some More Matrices

y = y +

x = x +

= B + +

V() =

V() =

V() =

V() =



An Example Longitudinal Model with Latent Variables

y1

21

1

121

y2

y3

42

1

232

y4

y5

63

1

343

y6

y7

84

1

4

y8



The Matrices for the Model

8

7

6

5

4

3

2

1

4

3

2

1

84

63

42

21

8

7

6

5

4

3

2

1

000

1000

000

0100

000

0010

000

0001

y

y

y

y

y

y

y

y

4

3

2

1

4

3

2

1

43

32

21

4

3

2

1

000

000

000

0000



Variance Matrices for Exogenous Factors

88

22

11

00

00

00

Θ

44

33

22

11

000

000

000

000

Ψ

The Matrices for the Model



The Second Order Factor Analysis Model

y = y + = +

εζΓξΛy y

V() =

V() =

V() =

ΘΛΨΓΓΦΛyyy

)(V



Path Diagram of Second Order Factor Analysis

y2y1 y3

211

1 2

y5y4 y6

3 4

y7 y8

1

1 1 142 63 84

1 413121



Models with Structured Means

x = x + x +

y = y + y +

= + B + +

E(x) = x + x

E(y) = y + y (I – B)-1 ( + )

E() = (I - B)-1( + )

x measurement model. Includes x0

y measurement model

SEM

= E()



Details on the Model

1 = 10 + 0

δ

1

ξ

η

νΛ0

ν0Λ

x

y

xx

yyε

0

1

1101

κξ

ζ

κ

α

ξ

η

00

000

0ΓB

ξ

η



A Sequence of Hypotheses for Multiple Group Analysis

)2(

y

)1(

y0:H ΛΛ

)2(

y

)1(

y0:H νν

)2()1(

0:H ΘΘ

)2()1(

0:H αα

)2()1(

0:H ΨΨ

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Slide 10.1 Structural Equation Models MathematicalMarketing Chapter 10 Structural Equation Models In This Chapter We Will Cover The theme of this chapter