1

University of Essex

Estimation of a general structural equation latent variable autoregressive distributed

lag model with an application to UK micro-consumption function

Dario Cziráky

September 2002

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Abstract This paper generalises the structural equation model with latent variables (LISREL), to an

estimable structural autoregressive distributed lag model and subsequently applies these

methods to a known empirical problem in economics. The existing approaches to the use of

latent variable models with time dimension data are briefly reviewed and some estimation and

specification problems are outlined. A general structural autoregressive distributed lag model

with latent variables is developed and two general estimation procedures are proposed. We

describe how generalised instrumental variable estimation and three-stage least squares

methods can be applied to estimate dynamic latent variable models and subsequently describe

their asymptotic properties. All discussed estimation methods are asymptotically distribution

free and thus require no specific assumptions about the joint density of the modelled

variables. Additionally, a simple identification procedure is developed which can be used in

both static and dynamic latent variable models. The described methods are then applied to

estimation of a latent consumption function model that takes into account dynamic structure

of the modelled relationships by using a large micro-economic panel data set.

JEL Classification: C3, C22, C23, E2

Keywords: Dynamic latent variable models, structural equations, systems estimators, panel

data

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1. Introduction 1.1 Introduction to the general problem area The general structural equation model with latent variables (LISREL), developed by Karl

Jöreskog in the early seventies (Jöreskog, 1973; Jöreskog and Goldberger, 1975; see also

Keesling, 1972 and Wiley, 1973) has been extensively applied to cross-sectional data for the

last three decades. The significance of the LISREL model lies in its integration of factor-

analytic measurement models (typically employed in psychometrics) and standard

econometric simultaneous (structural) equation models. However, an estimable dynamic

generalisation of the LISREL model was not developed for a structural autoregressive

distributed lag model with latent variables, which is the most suitable form needed for most of

the time series and panel models in micro and macroeconomics (e.g., consumption function

estimation), political science (e.g., time series relationship between democracy and economic

development), sociology (e.g., various household panel models), etc. Cross-sectional

applications of LISREL models in economics are relatively scarce, and applications of various

dynamic extensions are exceptionally rare, mainly due to the lack of appropriate estimation

methods. This is unfortunate because latent variables are just as important in dynamic

economic models as they are in psychometrics, thus the availability of appropriate methods

could very well advance the application of LISREL-type models in economic and other

dynamic social science research areas. For some examples of typical economic applications

see in. al. Bagozzi (1980), Baldini and Cherubini (1998), Brumm and Cloninger (1995), Fritz

(1986), Gerpott, et al. (2001), Lewbel (1996), McFatter (1987), Pindyck and Rotemberg

(1993), Pung and Staelin (1983).

The purpose of this paper is to develop an estimable generalisation of the classical LISREL

model to dynamic autoregressive distributed lag (ADL) models and subsequently apply these

methods to a known empirical problem in economics. The paper is organised as follows.

Section 1.1 introduces the basic elements of the classical LISREL model demonstrating some

issues and problems in modelling data containing a time dimension. Section 2 develops a

generalised ADL-LISREL model and derives estimable (observed-form) equations, and

subsequently describes estimation methods capable of estimating such models. Section 3

describes an alternative efficient method of estimation based on three-stage least squares

technique, and also briefly discusses issues of estimation with dynamic panel data. Section 4

develops a general identification procedure for dynamic latent variable models and Section 5

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applies the described methods to the analysis of the consumption-income relationship using

British microeconomic household panel data (British Household Panel Survey).

1.2 The static LISREL model and time-dimension data The Jöreskog (LISREL) model is specified with three matrix equations, namely the (vector)

structural equation given in the form of a standard econometric simultaneous system:

ζΓξBηαη +++= η (1.1)

and two measurement equations; for latent exogenous variables:

δξΛαx ++= xx (1.2)

and for latent endogenous variables:

εηΛαy ++= yy (1.3)

where 1m×∈η and 1g ×∈ξ denote latent endogenous and exogenous variables,

respectively; x and y are vectors of the observed variables that serve as measurable indicators

of the unobserved latent exogenous and endogenous variables, respectively; α, B, Г, and Λ

are the coefficient matrices and ζ, δ, and ε are the latent errors in structural and measurement

parts of the model (see Bollen, 1989; Jöreskog, et al. 2000; Kaplan, 2000). Eqs. (1.1-1.3)

allow very general specification of various linear structural models including a limited class

of time-dimension (TD) models. The problems arise in estimation of a more general class of

time series and panel models using typical LISREL covariance structure approach—while it is

simple to formulate any type of TD generalisation of the structural model given in Eq. (1.1),

the estimation with standard maximum likelihood covariance structure approach will not be

possible in many cases either because the model cannot be formulated as a LISREL model or

because it does not have a closed form covariance structure. Nevertheless, there are several

relevant extensions of the LISREL model that focus on particular types of time-dimension

data. Most of these models were developed for a specific kind of panel data known in

psychology and sociology as “repeated measurement” data. Other types of data, including

pure time series and classical econometric panels were also given attention in the literature.

One of the first TD generalisations for time-series and panel data was the simplex model

(Guttman, 1954; Anderson, 1960; Jöreskog, 1970, Jöreskog, 1979), which is a form of a first

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order autoregressive (AR) model. Additional work and further dynamic extensions of these

and similar models were discussed by Jöreskog and Sörbom (1979), McDonald (1985),

McDonald (1999), Rogosa (1979), and Rogosa and Willett (1985). Some generalisations to

time series models and application of LISREL-type covariance structures to autoregressive

moving average (ARMA) models were developed by West and Hepworth (1991) and Du Toit

and Browne (2001). An example of dynamic factor analysis and an attempt to merge

structural equation modelling with time series data is given in Hershberger, et al. (1994),

Hershberger, et al. (1996), Molenaar (1985), Nesselroade, et al. (2001), and Wood and Brown

(1994). Similar dynamic generalisation of LISREL-type of models is described in McArdle

(1988), McArdle and Anderson (1990), McArdle and Bell (2000), Rovine and Molenaar

(1998a, 1988b), and Willett and Sayer (1996). Latent curve modelling is yet another TD

generalisation focused on repeated measurement data and individual time-specific differences

(Du Toit, 1979; Rogosa, et al. 1982; Willett, 1988; Muthén, 1991; Bryk and Raudenbush,

1992; Willett and Sayer, 1994; Chou, et al. 1998; Curran and Muthén, 1999; Muthén and

Curran, 1997; McArdle and Bell, 2000). Curran and Bollen (2001) attempted to combine the

AR and latent curve models and integrate both approaches within standard LISREL

framework. Another recent extension and dynamic generalisation for repeated measurement

data is the latent differences score modelling described in McArdle (2001), McArdle (1991),

and McArdle and Hamagami (1992; 1996).

However, the mentioned approaches are specific in several respects in terms of their

general applicability to data generated by stochastic processes (pure time series) and even

within the restricted class of cross-section time series data (panels) there are some unresolved

problems in both specification and testing (see Bast and Reitsma, 1997; Curran, 2000; Kenny

and Campbell, 1989; Marsh, 1993; and Rogosa and Willett, 1985).

The above outlined literature applies LISREL covariance structure models primarily on

already mentioned repeated measurement (longitudinal) data. Such data are made out of

observations on N individuals repeatedly taken T times. If T is some regular time interval

(e.g., year, quarter, month) it is more customary to call such data a “panel”. The currently

existing models that incorporate factor-analytic type of latent variables (as opposite to the use

of LISREL covariance structures to fit observed stochastic data such as ARMA structures,

e.g., Du Toit and Browne, 2001), as a rule, use repeated measurement data which can be a

repeated longitudinal survey carried out at distant time points (e.g, repeated after several years

or decades) or regular (e.g., annual) survey data collected on the same individuals

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(households, firms or countries). The former case indeed lends itself neatly to standard

LISREL-type modelling where the previous period values can be treated as separate variables

and values on all N individuals taken at T = t can be, for example, regressed on N-valued

vector of the observation (taken on the same individuals) in some time T = t – i. Latent curves,

latent difference score and most other time-dimension applications of LISREL models use

this type of data. However, analysing typical panel data this way might be rather problematic.

To illustrate this point suppose we have a panel with N individuals taken at T time points. The

input data matrix in standard structural equation modelling software programmes will operate

on the matrix of the form:

{ }1,2,...,1,2,...,

11 12 1

21 22 2

1 2

i Nt T

T

Tit

N N NT

y y yy y yy

y y y==

=

(1.4)

Note that the matrix in Eq. (1.4) contains data on N individuals (rows) repeated across T time

points (columns). A simplex model (Jöreskog, 1970) of the form 1it it ity yτ ρ ε−= + + might be

fitted to such data. This model is the simplest dynamic formulation that is often used in

applied research in conjecture with LISREL-type covariance structures. However, important

differences exist between such an approach when applied to data matrix from Eq. (1.4) and

what is used in standard (econometric) panel data models. Suppose we wish to estimate the

simplest case, 1it it ity y −= +ρ ε with ordinary least squares (OLS), and further suppose that T =

3. Applying OLS to data stacked in the matrix from Eq. (1.4) allows us to estimate separately (1)

2 1 2i A i iy y= +ρ ε and (1)3 2 3i A i iy y= +ρ ε , which will produce the AR(1) coefficient estimates

( ) ( )(1) 22 1 11 1

ˆ N NA i i ii i

y y y= =

= ∑ ∑ρ , and ( ) ( )(1) 23 2 21 1

ˆ N NB i i ii i

y y y= =

= ∑ ∑ρ (1.5)

Note that, in general, (1) (2)

A A≠ρ ρ and that all standard errors and fit statistics will be based on

sample size N, and the covariance matrix will be (3 × 3) and of the general form:

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21 1 2 1 3

1 1 1

22 1 2 2 3

1 1 1

23 1 3 2 3

1 1 1

N N N

i i i i ii i i

N N N

i i i i ii i iN N N

i i i i ii i i

y y y y y

y y y y y

y y y y y

= = =

= = =

= = =

=

∑ ∑ ∑

∑ ∑ ∑

∑ ∑ ∑

S (1.6)

On the other hand, typical panels would use data stacked in NT × k matrices, where k is the

number of variables. For the simplest case of k = 1, the data matrix (i.e., vector) will have the

form:

{ } [ ]1,2,...,1,2,...,

11 12 1 21 22 2 1 2i Nt T

it T T N N NTy y y y y y y y y y==

′= (1.7)

It follows that the AR(1) coefficient from the model 1it it ity yτ ρ ε−= + + will be estimated by

OLS as:

( ) ( )(2) 21 11 1 1 1

ˆ N T N Tit it iti t i t

y y y− −= = = == ∑ ∑ ∑ ∑ρ (1.8)

Clearly, Eqs. (1.5) and (1.8) produce generally different estimates, the first being based on

sample size N and the second on sample size NT. From the stochastic point of view, only the

estimator given in Eq. (1.8) is correct because the dynamics relate to dynamic properties of a

an underlying stochastic process that generated T observations for each of the N individuals.

This issue can be masked by allowing for time-varying AR(1) coefficients and using

structural equations framework. Still, the used covariance matrix will be of the form given in

Eq. (1.6) and the resulting estimator will be biased if the correct form of data matrix is the one

from Eq. (1.7). It is frequently found in the applied literature that LISREL panel models with

T > 2 do not fit well. This brings us to an important special case. When T = 2 we have

1 2 11 1 1

N T Nit it i ii t i

y y y y−= = ==∑ ∑ ∑ therefore (1) (2)ˆ ˆ=ρ ρ and the estimators based on data matrices

in Eqs. (1.5) and (1.8) collapse to the same estimator, namely the one given in Eq. (1.5).

The next section develops a general dynamic formulation that aims at capturing dynamics in

a standard econometric sense by treating model coefficients as time-invariant. Therefore, the

estimation of multiple AR(1) coefficients (between each pair of time points) will be avoided.

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2. Dynamic ADL-LISREL model with latent variables 2.1. Dynamic generalisation of the LISREL model The widely used approach to model time-dimension data within the classical LISREL

framework is to treat observations taken at different time points as distinct variables in the

model (see Jöreskog, et al. 2000). The simplest example is the above discussed simplex model

1it it ity y −= +ρ ε with T = 3, which can be formulated using standard LISREL notation as:

1 1 1

2 21 2 2

3 32 3 3

0 0 00 0

0 0

η η ζ η = ρ ⋅ η + ζ η ρ η ζ

and 1 1 1

2 2 2

3 3 3

1 0 00 1 00 0 1

yyy

η ε = ⋅ η + ε

η ε

(2.1)

This model has a simple covariance structure given by:

[ ] ( )( ) ( )( ) ( )( )( ) ( ) ( ) ( )

( ) ( )

1 1

1 1 1 1

1 1

E E E

E

− −

− − − −

− −ε

′ ′′= = + + = − + − + ′ ′ ′ ′ ′ ′= − − + − + − +

′= − − +

Σ yy Λη ε Λη ε I B ζ ε I B ζ ε

I B ζζ I B I B ζε εζ I B εε

I B Ψ I B Θ

(2.2)

where Ψ ≡ E[ζζ΄] and Θε ≡ E[εε΄]. Assuming multivariate normality of y = (y1, y2, y3), the

model parameters (i.e., ρ1, ρ2, ζ1, ζ2, ζ3, ε1, ε2, and ε3) can be obtained by minimisation of the

likelihood function given by { }1 3MLF ln tr ln−= + − −Σ SΣ S , which produces (Gaussian)

maximum likelihood estimates.

Note that the above example assumed that we have observations on N individuals across

three time points. What would happen if we had T observations on a single individual (or a

country), taken over T time points where T is large? Obviously, the employed specification

and estimation approach would fail to model the autoregressive process, which is the simplest

case of a dynamic time series model. Note that in this case 1T×∈y and thus ∈S , i.e., a

scalar, 211 1

(1 ) N Titi t

NT y −= =∑ ∑ .

Furthermore, the model specification in Eq. (2.1) is odd and the natural way to specify a

dynamic LISREL model is 0 0

p qt η j t- j j t- j tj j= == + + +∑ ∑η α B η Γ ξ ζ , which is a general

autoregressive distributed lag (ADL) structural equation model with latent variables. An

example of ADL-type specifications is given in Kaplan, et al. (2001) who used a similar

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specification to estimate the impact of a possible dynamic dis-equilibrium effect on cross-

sectional estimates.

However, the problem with structural equation ADL models with latent variables is in

estimation. Namely, the estimation methods based on maximum likelihood optimisation of the

multivariate Gaussian (discrepancy) likelihood function { }1 ( )F ln tr ln p q−= + − − +Σ SΣ S

requires derivation of the implied covariance structure, i.e., the model-implied covariance

matrix Σ which, in terms of the static LISREL notation of Eqs. (1.1)-(1.3), is given by:

1 1 1

1

( ) ( )[( ) ] ( )[( ) ]

yy yx

xy xx

y y y x

x y x x

− − −ε δε

−εδ δ

=

′ ′ ′ ′ ′ − + − + − += ′ ′ ′ ′− + +

Σ ΣΣ

Σ Σ

Λ I B ΓΦΓ Ψ I B Λ Θ Λ I B ΓΦΛ ΘΛ ΦΓ I B Λ Θ Λ ΦΛ Θ

(2.3)

To see why a closed-form covariance structure for a general ADL(p, q) cannot be derived in

the form given in Eq. (2.3) note that the a general structural ADL model with latent variables

in the form:

∑ ∑= =

+++=p

j

q

jtjt-jjt-jηt

0 0

ζξΓηBαη (2.4)

contains, in addition to α, B and Г, matrices from the original Jöreskog’s model, additional

p + q matrices B1, B2, Bp, Г1, Г2, …, Гp. If we suppose that the measurement model is time

invariant we can retain the usual specification of the measurement models for xt and yt and

append the structural part of the model from Eq. (2.4) with:

ttxxt δξΛαx ++= (2.5)

and

ttyyt εηΛαy ++= (2.6)

The matrix equations (2.4)-(2.6) provide full specification of a general structural ADL model

with latent variables and directly extend the original LISREL model to time series and panel

cases. The retained assumption behind this model is that latent errors follow a white noise

process and are thus serially uncorrelated. Note that Eq. (2.4) does not require specification of

lagged latent variables as separate variables; rather each vector containing all modelled and

exogenous latent variables is written for each included lag separately, with separate

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coefficient matrix. Also note that that expression (2.4) allows for different lag length of

different latent variables (i.e., elements of η and ξ vectors) by appropriate specification of Bj

and Гj matrices (e.g., zero elements).

2.2. Observed-form specification (OFS) The latent variable specification in Eqs. (2.4)-(2.5) cannot be directly estimated due to the

presence of unobserved latent components. To solve this problem and enable estimation of the

model parameters we re-write the latent variable model in terms of the observed variables and

latent errors only following Bollen (1995; 1996; 2001). Bollen developed such specification

to enable a non-parametric 2SLS estimation of standard (cross-sectional) LISREL models

with the aim of achieving greater robustness to misspecification and non-normality. We will

show that his approach can be used to re-write the latent variable model in the observed form

specification (OFS) and subsequently estimate all model parameters (except latent error

terms) by generalised instrumental variables estimation (GIVE) methods.

The OFS uses the fact that in the measurement model for each latent variable one loading

can be fixed to one without loss of generality. Using this trick we can re-write the

measurement models for xt and yt as:

1t 1t

(x) (x)t t2t 2 t2 2

= = + +

0 Ix δx ξx δα Λ (2.7)

and

1t 1t(y) (y)t t

2 t 2 t2 2

= = + +

0 Iy εy ηy εα Λ (2.8)

Note that the observed indicators with unit loadings were placed in the top part of the vectors

for xt and yt and thus the upper part of the lambda matrix is an identity matrix. Having divided

xt into xt1 and xt2 note that for xt1 it holds that:

tttttt 1111 δxξδξx −=⇒+= (2.9)

and, similarly, for yt1 we can replace the latent variable with its unit-loading indicators:

tttttt 1111 ε−=η⇒ε+η= yy (2.10)

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It is now possible to use the relation in Eqs. (2.9) and (2.10) to re-write the measurement

models for xt:

)(

)(

1(x)221

(x)2

(x)2

211(x)2

(x)22

ttt

tttt

δΛδxΛα

δδxΛαx

−++=

+−+= (2.11)

and for yt:

)(

)(

1(y)221

(y)2

(y)2

211(y)2

(y)22

ttt

tttt

εΛεyΛα

εεyΛαy

−++=

+−+= (2.12)

Following the same principle it is possible to re-write the structural part of the model using

definitions (2.9) and (2.1) as follows:

∑ ∑= =

+−+−+=−p

j

q

jtjt-jt-jjt-jt-jηtt

0 0111111 )()( ζδxΓεyBαεy (2.13)

Separating the observed part of the model from latent errors we obtain:

q

1 η j 1t- j 1 - 1 1 - 1 -0 j 0 0 0

p p q

t j t j t t j t j j t jj j j= = = =

= + + + + − −

∑ ∑ ∑ ∑y α B y Γ x ζ ε B ε Γ δ (2.14)

with the measurement model for latent endogenous variables:

(y) (y) (y)2 2 2 1 2 2 1( )t t t t= + + −y α Λ y ε Λ ε (2.15)

and the measurement model for latent exogenous variables:

(x) (x) (x)2 2 2 1 2 2 1( )t t t t= + + −x α Λ x δ Λ δ (2.16)

Aside of the specific structure of the latent error terms, the Eq. (2.14)-(2.16) present a

classical simultaneous or structural equation system with observed variables. However, it can

be seen that the OFS of a dynamic LISREL model cannot be treated in the same way as

standard econometric simultaneous equation system. The reason for this lies in the

formulation of the LISREL measurement model. Inheriting the traditional factor-analytic

assumption of exogeneity of latent factors from psychometrics, it his original formulation of

the model (which is retained here), Jöreskog incorporated the same assumption into the

formulation of his general model. Since we merely replaced latent variables with their (unit-

loading) indicators in Eqs. (2.14)-(2.15), it follows that the OFS equations contain no truly

exogenous variables, which is caused by the specific structure of the latent error terms. The

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estimation of the model will thus require the use of an instrumental variable (IV) technique, or

some other member of the generalised method of moments (GMM) class of estimators. We

will describe a generalised IV (GIVE) technique capable of estimating the OFS equations

consistently and efficiently by applying model-implied instruments (i.e., not assuming

availability of instruments that are not already included in the model) in form of lagged

indicators of latent variables. In the following text we will retain standard time series notation

avoiding at this point notational complications characteristic for panel data.

3. Estimation of the OFS system 3.1. Full-sample specification

Estimation of the OFS equations aims at consistent and, possibly, efficient estimation of the

structural and measurement-model parameters. Its down-side is that latent errors cannot be

directly estimated. Therefore, ignoring the specific structure of the measurement error terms

we write the structural part of the model as:

∑ ∑= =

+++α=p

0j

q

0jtj-tjj-tjηt 1111 uxΓyBy (3.1)

and measurement models as:

ttt 21

(y)2

(y)22 uyΛαy ++= (3.2)

and:

ttt 31(x)2

(x)22 uxx +Λ+α= (3.3)

For the purpose of estimation we turn to the full-sample notation. Define 1 1 1( , )j jt jt -k=Y Y Y

and 1 1 1( , )j jt jt -k=X X X . The “j” subscripts refer to the jth equation where there are m

individual y1 equations, n of y2 equations and h of the x2 equations (see Table 1 for more

details on this notation). Using full-sample notation, the OFS equations can be written as:

jjjjjjj 111

(y)11 uγXβYαy +++=

j(y)jjtjj 21

(y)22 uλYαy ++= (3.4)

jjjtjj 3(x)

1(x)22 uλXαx ++=

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It is assumed that there are max(p, q) available pre-sample observations (see Hamilton, 1994

for more details).

Table 1. The OFS notation for variables and parameters

=

)1(

)1(1

)1(0

1

jT

j

j

j

y

yy

y

=

)(2

)(21

)(20

2

jT

j

j

j

y

yy

y

=

)(2

)(21

)(20

2

jT

j

j

j

x

xx

x

=

−−−−−−

−−−

−−−

−−−−−−

)1((12)(11))1(1

(12)1

(11)1

)1((12)(11)

)1(2

(12)2

(11)2

)1(1

(12)1

(11)2

)1(2

(12)2

(11)2

)1(1

(12)1

(11)1

)1(0

(12)0

(11)0

)1(1

(12)1

(11)1

)1((12)(11))1(1

(12)1

(11)1

)1(0

(12)0

(11)0

1

mpTpTpT

mTTT

mTTT

mppp

mm

mppp

mm

mppp

mm

j

yyyyyyyyy

yyyyyyyyyyyyyyyyyyyyyyyyyyy

Y

=

−−−−−−

−−−

−−−

−−−−−−

)1((12)(11))1(1

(12)1

(11)1

)1((12)(11)

)1(2

(12)2

(11)2

)1(1

(12)1

(11)2

)1(2

(12)2

(11)2

)1(1

(12)1

(11)1

)1(0

(12)0

(11)0

)1(1

(12)1

(11)1

)1((12)(11))1(1

(12)1

(11)1

)1(0

(12)0

(11)0

1

gqTqTpT

gTTT

gTTT

gqqq

gg

gqqq

gg

gqqq

gg

j

xxxxxxxxx

xxxxxxxxxxxxxxxxxxxxxxxxxxx

X

=

)1((12)(11)

)1(2

(12)2

(11)2

)1(1

(12)1

(11)1

)1(0

(12)0

(11)0

1

mTTT

m

m

m

jt

yyy

yyyyyyyyy

Y

(11) (12) (1 ) (11) (12) (1 )1 1 1 p p p

(11) (12) (1 ) (11) (12) (1 )0 0 0 1 1 1 p(11) (12) (1 ) (11) (12) (1 )

1 2 1 1 2 2 2 p

(11) (12) (1 ) (11) (12) (11 1 1 p p

m m

m mp p

m mjt k p p

mT T T T T p T

y y y y y yy y y y y yy y y y y y

y y y y y y

− − − − − −

− − −

− − − −

− − − − − −

=Y

)m

=

)1((12)(11)

)1(2

(12)2

(11)2

)1(1

(12)1

(11)1

)1(0

(12)0

(11)0

1

mTTT

m

m

m

jt

xxx

xxxxxxxxx

X

(11) (12) (1 ) (11) (12) (1 )1 1 1

(11) (12) (1 ) (11) (12) (1 )0 0 0 1 1 1(11) (12) (1 ) (11) (12) (1 )

1 2 1 1 2 2 2

(11) (12) (1 ) (11) (12) (11 1 1

g gq q q

g gq q q

g gjt k q q q

gT T T T q T q T q

x x x x x xx x x x x xx x x x x x

x x x x x x

− − − − − −

− − −

− − − −

− − − − − −

=X

)g

) , ... , , , ... , , ... , , , , ... , ,( )1((12)(11))1(1

(12)1

(11)1

)1(0

(12)0

(11)0 ′= m

pppmm

j ββββββββββ

) , ... , , , ... , , ... , , , , ... , ,( )1((12)(11))1(1

(12)1

(11)1

)1(0

(12)0

(11)0 ′= g

qqqgg

j γγγγγγγγγγ

) , ... , ,( )2((22)(21)(y) ′= nyjyjyjj λλλλ

) , ... , ,( )2((22)(21)(x) ′= hxjxjxjj λλλλ

14

The actual estimation of the model proceeds with specification of individual equations.

Individual structural equations can be specified as:

jt

g

k

q

i

kit

ki

m

k

p

i

kit

kijj uxyy 1

1 0

)1()1(

1 0

)1()1((y)11 ∑∑∑∑

= =−

= =− +γ+β+α= (3.5)

Similarly, the individual measurement equations are in the form:

jt

m

k

ktjkjj uyy 2

1

)1((y)2

(y)22 +λ+α= ∑

=

(3.6)

for the endogenous part, and:

jt

g

k

kjkij uxx 3

1

)1(t

(x)2

(x)22 +λ+α= ∑

=

(3.7)

for the exogenous part. This completes the specification issues, as any ADL system with

latent variables can be formulated in the form of Eq. (3.5)-(3-6). It remains to show that the

available instruments (i.e., certain lags of modelled variables) can assure consistent

estimation. The issue of the choice of instruments is also discussed in Bollen (1996; 2001),

however he does not discuss this issue in the context of dynamic models and thus has no

lagged instruments on disposal. The following discussion takes into account the specific

structure of the OFS system and the implications derived from the composition of the latent

errors. This (known) composition of the latent error terms and their implied relation with the

observed components of the model, as a consequence of the latent structure, presents the

major difference between OFS LISREL equations and classical econometric models.

Specifically, it is not possible to simply assume the availability of external instrumental

variables that satisfy some general conditions such as being uncorrelated with the errors and

correlated with the repressors. Rather, it will be necessary to show under which conditions the

lagged modelled variables can serve as valid instruments in the estimation of the OFS

equations.

3.2. Consistency conditions and instrumental variables The standard consistency conditions needed for the validity of instrumental variables (see

Judge, et al. 1985; Davidson and MacKinnon, 1993) can be stated in terms of the data matrix

X defined as ( )j j, ,=X ι Y X where 1 1 1( , )j jt jt -k=Y Y Y and 1 1 1( , )j jt jt -k=X X X are defined as

15

in Table 1. Now let Z be the matrix of valid instruments defined as ( )1 2 1 2* * *, , , ,= *Z Y Y X X (see

Table 2 for specific details). We state the general conditions for these instruments in terms of

the joint matrices X and Z though, in practice, only subsets of these matrices will be used in

estimated models. It is generally necessary that ( ) ( )1 1

Tplim lim ZZT T− −

→∞′ ′= =Z Z Z Z Σ and also

that ( ) ( )1 1plim lim ZXTT T− −

→∞′ ′= =Z X Z X Σ , where ΣZZ and ΣZX are positive definite matrices.

These conditions will generally hold for the case of lagged instruments given they satisfy

certain stochastic conditions. In addition we assume homoscedastic residuals, i.e.,

IijjiE σ=′ ][ uu and, specially, 0uZ =′ ][ iE . The last condition must be taken into special

consideration since the structure of the OFS model itself implies endogeneity of all modelled

variables. This essentially implies that Z will include (specific) lags of the modelled variables

only.

Table 2. The OFS instruments

=

−−−−−−−−−−−−−−−−−−

−−−−−−−−−−−−−−−−−−

−−−−−−−−−−−−−−−−−−

−−−−−−−−−−−−−−−−−−

)1((12)(11))1(2

(12)2

(11)2

)1(1

(12)1

(11)1

)1(2

(12)2

(11)2

)1(22

(12)22

(11)22

)1(12

(12)12

(11)12

)1(1

(12)1

(11)1

)1(21

(12)21

(11)21

)1(11

(12)11

(11)11

)1((12)(11))1(2

(12)2

(11)2

)1(1

(12)1

(11)1

1

mapTapTapT

mpTpTpT

mpTpTpT

mapapap

mppp

mppp

mapapap

mppp

mppp

mapapap

mppp

mppp

*

yyyyyyyyy

yyyyyyyyyyyyyyyyyyyyyyyyyyy

Y

=

−−−−−−−−−−−−

−−−−−−

−−−−−−−−−

−−−−−−−−−−−−

)2((22)(21))2(2

(22)2

(21)2

)2(1

(22)1

(21)1

)2(2

(22)2

(21)2

)2(0

(22)0

(21)0

)2(1

(22)1

(21)1

)2(1

(22)1

(21)1

)2(1

(22)1

(21)1

)2(0

(22)0

(21)0

)2((22)(21))2(2

(22)2

(21)2

)2(1

(22)1

(21)1

2

nbpTbpTbpT

nTTT

nTTT

nbpbpbp

nn

nbpbpbp

nn

nbpbpbp

nn

*

yyyyyyyyy

yyyyyyyyyyyyyyyyyyyyyyyyyyy

Y

=

−−−−−−−−−−−−−−−−−−

−−−−−−−−−−−−−−−−−−

−−−−−−−−−−−−−−−−−−

−−−−−−−−−−−−−−−−−−

)1((12)(11))1(2

(12)2

(11)2

)1(1

(12)1

(11)1

)1(2

(12)2

(11)2

)1(22

(12)22

(11)22

)1(12

(12)12

(11)12

)1(1

(12)1

(11)1

)1(21

(12)21

(11)21

)1(11

(12)11

(11)11

)1((12)(11))1(2

(12)2

(11)2

)1(1

(12)1

(11)1

1

gcqTcqTcqT

gqTqTqT

gqTqTqT

gcqcqcq

gqqq

gqqq

gcqcqcq

gqqq

gqqq

gcqcqcq

gqqq

gqqq

*

xxxxxxxxx

xxxxxxxxxxxxxxxxxxxxxxxxxxx

X

=

−−−−−−−−−−−−

−−−−−−

−−−−−−−−−

−−−−−−−−−−−−

)2((22)(21))2(2

(22)2

(21)2

)2(1

(22)1

(21)1

)2(2

(22)2

(21)2

)2(0

(22)0

(21)0

)2(1

(22)1

(21)1

)2(1

(22)1

(21)1

)2(1

(22)1

(21)1

)2(0

(22)0

(21)0

)2((22)(21))2(2

(22)2

(21)2

)2(1

(22)1

(21)1

2

hdqTdqTdqT

hTTT

hTTT

hdqdqdq

hh

hdqdqdq

hh

hdqdqdq

hh

*

xxxxxxxxx

xxxxxxxxxxxxxxxxxxxxxxxxxxx

X

16

To assure the consistency of IV estimators we will need to make the following assumptions

about the stochastic properties of the observed variables.

Assumption: For stochastic processes {yt} and {xt} suppose that:

A1. t,yE ijijt ∀µ= (y)][

A2. t,xE ijijt ∀µ= (x)][

A3. t,yyE efijwrefwtef,ijrtij, ∀γ=µµ −−−

)((y)(y) )]-)(-[(

A4. t,xxE efijwrefwtef,ijrtij, ∀δ=µµ −−−

)((x)(x) )]-)(-[(

A5. t,xyE efijwrefwtef,ijrtij, ∀ψ=µµ −−−

)((x)(y) )]-)(-[(

A6. ∞<ψ∞<δ∞<γ ∑∑∑∞

=

∞

=

∞

= 0

(.)

0

(.)

0

(.)

kk

kk

kk ,,

These assumptions essentially imply stationary data, and are more likely to be satisfied in

panel models then in typical time series applications. Thus, the non-stationary case requires

further research. We will also need the following two lemmas:

Lemma 1. Let wt be a covariance-stationary process with finite fourth moments and absolutely summable autocovariances. Then the sample mean satisfies:

w

..T

tt µ

T

sm→∑

=1

w1 ,

where m.s. indicates convergence in mean square. Proof Omitted. See Hamilton (1994), Proposition 7.5, p. 188.

Lemma 2. Let {yt} and {xt} be stochastic processes satisfying Assumption 1. Then the

following convergence results hold:

(i) (y)

0

][1ijijt

T

t

p

stij, yEyT

µ=→∑=

−

(i) 2(y))(0

2

0

2 )(][1ij

ijijt

T

t

p

stij, yEyT

µ+γ=→∑=

−

(iii) (y)(y))(

0

][1efij

ijefwrwtij,rtij,

T

t

p

wtef,rtij, yyEyyT

µµ+γ=→ −−−=

−−∑

17

(iv) (x)

0

][1ijijt

T

t

p

stij, xExT

µ=→∑=

−

(v) 2(x))(0

2

0

2 )(][1ij

ijijt

T

t

p

stij, xExT

µ+δ=→∑=

−

(vi) (x)(x))(

0

][1efij

ijefwrwtij,rtij,

T

t

p

wtef,rtij, xxExxT

µµ+δ=→ −−−=

−−∑

(vii) ( ) (y) (x), , , , -

0

1 [ ]T p

ijefij t r ef t w ij t r ef t w ij efr w

ty x E y x

Tψ µ µ− − − −

=

→ = +∑

Proof See Appendix 1.

Note that similar convergence results can be stated for panel data where N can go to infinity

instead of T or a sequential limit approach is taken where N goes to infinity followed by T,

sequentially.

Proposition 1. Let ( )1 1j j, ,=X ι Y X where 1 1 1( , )j jt jt -k=Y Y Y and 1 1 1( , )j jt jt -k=X X X . Let Z be

a matrix of valid defined as ( )1 2 1 2* * *, , , ,= *Z Y Y X X . We also assume that Iuu ijjiE σ=′ ][ . Then

the following results hold:

(i) ZZ1 ΣZZ =

′

Tplim

(ii) ZX1plimT ′ =

Z X Σ

(iii) 0uZ =′ ][ iE

Proof See Appendix 1.

The above results allow consistent GIVE (or 2SLS) estimation of the OFS equations using

the available, model-implied (lagged) instruments contained in Z, which thus denotes the set

of all available eligible instruments that do not originate from outside of the modelled data. It

must be mentioned that nothing precludes availability of valid instruments that are not merely

lags of the modelled variables. However, the nature of structural equation models with latent

variables casts doubt that such variables would be available. In any case, valid variables will

satisfy the same conditions, but we have shown that available instruments already might exist

in the used data in forms of lagged values not already included in the model.

18

3.3. Consistent generalised instrumental variable estimation of the OFS equations Formulation and estimation of OFS system of equations requires reliance on specific

structure and status of all variables in the model. This structure is determined by the latent

formulation and makes specification of the OFS equations rather complex. In order to derive

generalised instrumental variable estimators (GIVE) for the OFS equations, we start from the

system of equations given in Eq. (3.4) and write it by positioning its matrix and vector

elements in the way that will facilitate the use of more concise notation:

jjjtjj

jjjtjj

jjjjjjj

3(x)

1(x)22

2(y)

1(y)22

111(y)11

uλXαx

uλYαy

uγXβYαy

++=

++=

+++=

(3.8)

We are now able to simplify our notation by stacking all of the right-hand-side variables of

each of the three parts of the system (3.8) by making the following definitions:

1 1 1( , , )j j j≡W ι Y X , 2 1( , )j j t≡W ι Y , 3 1( , )j j t≡W ι X , (y) (y)1 1( , )j j j j' ' , ' ′≡δ α β γ , (y) (y)

2 2( ,j j '≡δ α

(y)2 )j ' ′λ , and (x) (x) (x)

2 2 2( , )j j j' ' ′≡δ α λ . It is now possible to re-write the system in Eq. (3.8) in a

simpler, more concise notation as

jjjj 1

(y)111 uδWy +=

jjjj 2(y)222 uδWy += (3.9)

jjjj 3(x)232 uδWx +=

Appropriate matrix of instruments Z need not contain all available eligible instruments, but it

needs to have at least as many of them as there are endogenous variables in each equation.

Therefore, in principle there is nothing that would prevent the matrix of instruments Z to

differ across different (individual) equations of the system (3.9). For simplicity we will

assume that Z is correctly specified. Aside of the above outlined general validity conditions

for the column vectors of Z (i.e., particular instruments), we will also consider a formal test

for empirical validity of chosen instruments. At this point we proceed in defining the GIVE

estimator.

19

First, by premultiplying each part of the system by Z we obtain matrix equations

jjjj 1(y)111 uZδWZyZ ′+′=′ , jjjj 2

(y)222 uZδWZyZ ′+′=′ , and jjjj 3

(x)232 uZδWZxZ ′+′=′ . Now

we define usual GIVE estimators for coefficient vectors (y) (y) (x)1 2 2j j j

ˆ ˆ ˆ, ,δ δ δ as:

(y) -1 -11 1 1 1 1[ ( ) ] ( )j j j j j

ˆ ′ ′ ′ ′ ′ ′=δ W Z Z Z Z W W Z Z Z Z y (3.10)

jjjjj 2-1

22-1

2(y)2 )(])([ yZZZZWWZZZZWδ ′′′′′′= (3.11)

jjjjj 2-1

33-1

3(x)2 )(])([ xZZZZWWZZZZWδ ′′′′′′= (3.12)

It is easy to show that the GIVE estimators given in Eq. (3.10)-(3.12) are consistent

estimators of the unknown coefficient vectors (y) (y) (x)1 2 2j j j, ,δ δ δ . To show this note that:

ijijijijijijˆ uZZZZWWZZZZWδδ ′′′′′′+= -1-1(*)(*) )(])([ (3.13)

Taking probability limits we obtain:

1(*) (*) -1

-1

(*)

1 1 1ˆplim ( ) plim plim ( ) plim

1 1 1 plim plim ( ) plim

(i

ij ij ij ij

ij ij

ij W

T T T

T T T

− ′ ′ ′= + ⋅

′ ′ ′⋅ ⋅

= +

δ δ W Z Z Z Z W

W Z Z Z Z u

δ Σ 1 -1 1

(*)

)

j ij ijZ ZZ ZW W Z ZZ

ij

− − ⋅

=

Σ Σ Σ Σ 0

δ

(3.14)

The above results holds for each of the vectors (y) (y) (x)

1 2 2j j jˆ ˆ ˆ, ,δ δ δ , where superscripts (y, x) were

replaced by asterisks, and subscripts (1, 2) by i. For computational purposes, the GIVE

estimators, using specific OFS notation, are given in full notation as follows. First, the three

sets of coefficient vectors in the structural part of the model are estimated by:

11 1 1 1

1 1 11 1 1

1 1 1 1 1 11 1 1

1 1 1 1 1

( ) ( ) ( ) ( )( ) ( ) ( ) ( )( ) ( ) ( )

j j j j

j j j j j j j

j j j j j j

ˆˆ

ˆ

−− − − −η

− − − −

− − −

′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′= ′ ′ ′ ′ ′ ′ ′ ′ ′

α ι Z Z Z Z ι ι Z Z Z Z Y ι Z Z Z Z X ι Z Z Z Z yβ Y Z Z Z Z ι Y Z Z Z Z Y Y Z Z Z Z X Y Z Z Zγ X Z Z Z Z ι X Z Z Z Z Y X Z Z Z Z X

11

11 1( )

j

j j−

′ ′ ′ ′

Z yX Z Z Z Z y

(3.15)

Though, using GIVE estimation, which is not a systems estimator, allows separate estimation

of individual equations, and thus separate estimation of the structural part of the model, while

ignoring the measurement model, which might be justified by possible misspecification of

20

some equations in the measurement model (see Bollen, 1996; 2001; 2002, regarding

robustness of limited information estimators such as GIVE or 2SLS to misspecification in

some parts of the complete LISREL model).

The GIVE estimators of the measurement model are estimated by:

1(y) 1 1 1

2 1 2(y) 1 1 12 1 1 1 1 2

( ) ( ) ( )( ) ( ) ( )

j j t j

j j t j t j t j t j

ˆ−− − −

− − −

′ ′ ′ ′ ′ ′ ′ ′ ′ = ′ ′ ′ ′ ′ ′ ′ ′ ′

α ι Z Z Z Z ι ι Z Z Z Z Y ι Z Z Z Z yλ Y Z Z Z Z ι Y Z Z Z Z Y Y Z Z Z Z y

(3.16)

and:

1(x) 1 1 12 1 2(x) 1 1 12 1 1 1 1 2

( ) ( ) ( )( ) ( ) ( )

j j t j

j j t j t j t j t j

ˆ−− − −

− − −

′ ′ ′ ′ ′ ′ ′ ′ ′ = ′ ′ ′ ′ ′ ′ ′ ′ ′

α ι Z Z Z Z ι ι Z Z Z Z X ι Z Z Z Z xλ X Z Z Z Z ι X Z Z Z Z X X Z Z Z Z x

(3.17)

The distribution of these estimators does not depend on the assumption that modelled data

come from a multivariate normal distribution and, thus, GIVE estimators of an ADL LISREL

model are distribution free. This is an advantage over the maximum likelihood estimator of

the static LISREL model, and therefore, GIVE can prove to be more robust to both

misspecification of certain parts of the model (i.e., misspecification of one OFS equation will

not necessarily affect coefficients of other equations since these are estimated separately using

a limited information estimator), and to departure from normality. These are the advantages

particularly advocated by Bollen (1996; 2001) who used such estimators for the case of a

static LISREL model.

The asymptotic distribution of the GIVE estimators is normal and it can be derived by

noting that:

′

′

′

′

′

′=

−

ijijijijijij TTTTTTˆT uZZZZWWZZZZWδδ 1)(111)(11)-( 1-

11-(*)(*)

(3.18) If we assume that:

) ,(1ZZΣ0uZ ij

d

ij NT

σ→

′ (3.19)

we can conclude that the asymptotic distribution of the ADL LISREL coefficients is:

))( ,()-( 1-ZW

1ZW

(*)(*)ijij ZZij

d

ijij NˆT ΣΣΣ0δδ −σ→ (3.20)

21

The asymptotic covariance matrix -1 -1[ ( ) ]ij ij ijˆ ′ ′ ′σ W Z Z Z Z W can be estimated by

(*)-1 -1[ ( ) ]

ijˆ ij ij ij

ˆ ˆ ′ ′ ′= σδ

Σ W Z Z Z Z W , where 1 1 (*) (*)( ) ( )ij ij ij ij ij ij ij ij ijˆ ˆˆ ˆ ˆT T− −′ ′σ = = − −u u y W δ y W δ .

The empirical validity of instrumental variables, as opposite to their model-implied

eligibility, is empirically testable.

The validity of the choice of the instrumental variables can be tested by the Sargan’s (1964)

chi-square test. Applied to the OFS equations, the Sargan’s test can be calculated as:

-1 (*) -1 (*)

21

( ) ( )ij ij ij ij ij ij

appij ij

ˆ ˆX ( d )

ˆ ˆT −

′′ ′ ′ ′ ′ ′ − ′

y Z Z Z Z y δ W Z Z Z Z W δu u

∼ (3.21)

where d is the number of over-identifying instruments, assumed to be independent of the

equation error. It is important to note that selection of the IV’s purely on the basis of the

model-implied eligibility without testing their empirical validity can result in considerable

bias in the estimated coefficients. As the choice of instruments affects consistency of GIVE

estimates, an error in IV selection will result in estimators that will not be robust to mis-

specification. Therefore, testing for the validity of IV’s, which can be performed by Sargan’s

test should be an important part in empirical estimation of dynamic latent variable models

discussed in this paper.

3.4. Three-stage least squares (3SLS) estimation of the OFS system So far we have described a consistent limited information estimation procedure that can be

used to estimate coefficients of a dynamic structural equation model with latent variables.

This was achieved by employing the basic idea from Bollen (1995; 1996; 2001) and

extending it to the dynamic case. The advantages of this approach such as robustness to mis-

specification and to non-normality are common to both static and dynamic models. It down-

side is that it lacks a counterpart of the popular overall-fit statistic widely used in maximum-

likelihood framework. An additional weakness is that GIVE or 2SLS estimator is not

necessarily asymptotically efficient. A full-information efficient estimator for simultaneous

equation systems is developed by Zellner and Theil (1962). This estimator is generally known

as the three-stage least squares (3SLS) estimator and in its general form it might be employed

in the estimation of dynamic ADL LISREL models. The main requirement for the use of

3SLS is to have a common matrix of instruments (Z) that can be applied to all equations in

the OFS system. If such instruments could be found then the 3SLS will be more efficient then

22

Bollen’s 2SLS even in the static case. However, due to the unavailability of lagged variables,

the existence of a common matrix Z is doubtful in the static case, but for dynamic models it

might be made out of instruments that are eligible for all equations in the system.

The 3SLS is a system estimator and can be obtained by staking all equation (premultiplied

by the common IV matrix Z) in a single matrix equation:

′

′′

′′

′

+

′

′′

′′

′

=

′

′′

′′

′

(x)2

(x)21

(y)2

(y)21

(y)1

(y)11

(x)2

(x)21

(y)2

(y)21

(y)1

(y)11

3

31

2

21

1

11

2

21

2

21

1

11

h

n

m

h

n

m

h

n

m

h

n

m

uZ

uZuZ

uZuZ

uZ

δ

δδ

δδ

δ

WZ

WZWZ

WZWZ

WZ

xZ

xZyZ

yZyZ

yZ

(3.22)

The matrix equation (3.22) can be simplified by defining:

11

121

231

3

t

m

n

h

≡

WW

WX

WW

W

,

11

1

21

2

21

2

m

n

h

≡

y

yy

yyx

x

,

(y)11

(y)1(y)21

(y)2(x)21

(x)2

m

n

h

≡

δ

δδ

δδδ

δ

, and

(y)11

(y)1(y)21

(y)2(x)21

(x)2

m

n

h

≡

u

uu

uuu

u

(3.23)

Using definitions (3.23) we can re-write Eq. (2.23) as uZIδXZIyZI ~~~~ )()()( ′⊗+′⊗=′⊗ .

Note that dim( ) ( 1) ( 1)p m q g n h = + + + + +X . It is also possible to write a compact

expression for the asymptotic covariance matrix as:

][ ]))()([(])()[(

ZZΣZIIΣZIZIuuZI

′⊗=′′⊗⊗′⊗=′′⊗′′⊗

EE~~E

(3.24)

We can now derive the 3SLS estimator for the OFS equations as:

yZZZZΣXXZZZZΣX

yZIZZΣZIXXZIZZΣZIXδ~~~~

~~~~~

])([}])([{

)]()([)(})(])([)({1-11-1-1

-11-1-11

′′⊗′′′⊗′=

′⊗′⊗′′⊗′⊗′⊗′′⊗′=−−

−−

(3.25)

The matrix equation (3.25) enables direct, non-iterative estimation of all structural parameters

and measurement model coefficients, jointly. This procedure, given the Z matrix is valid for

23

all equations in the system, yields consistent and efficient estimators of the parameters of a

dynamic LISREL model.

Finally we briefly discuss computation of the residual covariance matrix Σ. First note that:

1 1 1 2 1 3

( ) ( )1 1 2 1 2 2 2 3

3 1 3 2 3 3

( ) m n h m n hE E + + × + +

′ ′ ′ ′ ′ ′ ′= = ∈

′ ′ ′

u u u u u uΣ u u u u u u u u

u u u u u u (3.26)

The individual scalar elements of the block elements of Σ are calculated as follows. We have

(y) (y)1 1 1 1{ ( )} m m

i jE u u ×′ = ∈u u with typical element (yy) 1 (y) (y)11 1 i 1 1 i 1( ) ( )ij j j j j j jT − ′σ = − −y W δ y W δ ;

similarly we have (y) (y) (y) (y)1 2 1 2 2 1 2 1{ ( )} { ( )}m n n m

i j j iE u u E u u× ×′ ′= ∈ ⇒ = ∈u u u u with typical

element given by (yy) 1 (y) (y)12 1 1 1 2 2 2( ) ( )ij j j j j j jT − ′σ = − −y W δ y W δ .

Note that the (1, 3) block element of the matrix in Eq. (3.26) is merely a transpose of the

block (3, 1), thus its individual elements can be estimated in the same way, namely (y) (x) (x) (y)

1 3 1 2 3 1 2 1{ ( )} { ( )}m h h mi j j iE u u E u u× ×′ ′= ∈ ⇒ = ∈u u u u which has a typical element given by

(yx) 1 (y) (x)13 1 1 1 2 3 2( ) ( )ij j j j j j jT − ′σ = − −y W δ x W δ . Finally, for the remaining two blocks, we have

(y) (y)2 2 2 2{ ( )} n n

i jE u u ×′ = ∈u u where scalar elements can be estimated by

(yy) 1 (y) (y)22 2 2 2 2 2 2( ) ( )ij j j j j j jT − ′σ = − −y W δ y W δ and (x) (x)

3 3 2 2{ ( )} h hi jE u u ×′ = ∈u u with typical element

estimated by (xx) 1 (x) (x)22 2 3 2 2 3 2( ) ( )ij j j j j j jT − ′σ = − −x W δ x W δ .

Consistency of the 3SLS estimator, when applied to specific OFS models, can be shown in a

similar way as is usually shown for classical simultaneous equation systems. However, in the

OFS case, similarly to the GIVE case, it will depend on the assumed properties of the model-

implied (lagged) instruments. Consistency of this estimator can be shown by noting that:

uZZZZΣXXZZZZΣXδ

yZZZZΣXXZZZZΣXδ~ˆ~~ˆ~~

~ˆ~~ˆ~~̂

])([}])([{

])([}])([{1-11-1-1

1-11-1-1

′′⊗′′′⊗′+=

′′⊗′′′⊗′=−−

−−

(3.27)

Proposition 1 (i) now implies that ( )1 1 1 1 1

ZZplim plim( ( ) ) ZZˆT T− − − − −′ ′= ⇒ ⊗ = ⊗Z Z Σ Σ Z Z Σ Σ

and by (ii) it follows that:

24

*

*1 1

1 1 1 1

Z ZY ZY ZX ZX ZX

1 1plim plim ( , , )T T

1 plim ( , , , , )T

( , , , , )t t-i t t-i

j j

j j,t i jt j,t i

ι

− −

′ ′ ′ ′=

′ ′ ′ ′ ′=

= =

Z X Z ι Z Y Z X

Z ι Z Y Z Y Z X Z X

Σ Σ Σ Σ Σ Σ

(3.28)

Using the results (iv) of Proposition 1 and expanding the Kronecker products we get the

following convergence results:

(y)11

(y)1

(y)21

(y)2

(x)31

(x)3

plim

plim

plim( )plim

plim

plim

plim

m

n

h

T

T

TT

T

T

T

′ ′

′ ′⊗ = = ′ ′

′

Z u

Z u

Z uI Z u 0

Z u

Z u

Z u

(3.29)

The Eq. (3.28) further implies that:

11

1

21

2

31

3

( )plim

plim

plim

plim

plim

plim

plim

m

n

h

T

T

T

T

T

T

T

′⊗ =

′

′ ′

′ ′ ′

I Z X

Z W

Z W

Z W

Z W

Z W

Z W

(3.30)

Therefore we have specifically:

1

1 1 1 ( ) ( ) ( ) ( ) ( ) ( )Z ZY ZY ZX ZX ZW

( , , )plim plim ( , , , , )

T T t t-i t t-i

j j j j j j j j jι

′ ′ = = =

Z W Z ι Y XΣ Σ Σ Σ Σ Σ (3.31)

and:

25

2

2 1 ( ) ( ) ( )Z ZY ZW

( , )plim plim ( , )

T T t

j jt j j jι

′ ′ = = =

Z W Z ι YΣ Σ Σ (3.32)

Finally it follows that:

1

3 1 ( ) ( ) ( )Z ZX ZW

( , )plim plim ( , )

T T t

j jt j j jι

′ ′ = = =

Z W Z ι XΣ Σ Σ (3.33)

We can now derive the probability limit for the matrix equation (3.30) as:

1

1

2

2

3

3

(1)

(m)

(1)

(n)

(1)

(h)

( )plim

ZW

ZW

ZW

ZX

ZW

ZW

ZW

T

′⊗ = =

Σ

Σ

ΣI Z X Σ

Σ

Σ

Σ

(3.34)

Using the above results, it follows that:

1 -1 -1 1 -1

1 -1 -1 1 -1

ˆ ˆ ˆplim ( ) plim{ [ ( ) ] } [ ( ) ]ˆ ˆ plim{ ( ) [ ( ) ]( ) } ( ) [ ( ) ]( )

( ) plim [plim(T

− −

− −

′ ′ ′ ′ ′ ′= + ⊗ ⊗

′ ′ ′ ′ ′ ′ ′ ′ ′ ′= + ⊗ ⊗ ⊗ ⊗ ⊗ ⊗

′ ′ ′⊗= +

δ δ X Σ Z Z Z Z X X Σ Z Z Z Z u

δ X I Z Σ Z Z I Z X X I Z Σ Z Z I Z u

X I Zδ-1

1 -1

1 -1

1 1 -1 1 1

( )ˆ ( ) )] plim

( ) ( )ˆ plim [plim( ( ) )] plim

{ ( ) } ( )

ZZ ZZXZ ZX XZ

TT

TT T

−

−

− − − −

′⊗ ′⊗

′ ′ ′ ′⊗ ⊗ ′⋅ ⊗ = + ⊗ ⊗ ⋅

=

I Z XΣ Z Z

X I Z I Z uΣ Z Z

δ Σ Σ Σ Σ Σ Σ Σ 0

δ

Therefore, the 3SLS estimates are consistent if applied to the OFS model. It is also possible to

show that 3SLS is asymptotically more efficient then the GIVE estimator, though given we

are using the same model specification in both cases the proof of asymptotic efficiency is

equivalent to that for the ordinary (observed) system of equations (see Schmidt, 1976; Judge,

et al. (1985).

26

The 3SLS estimator, just like the GIVE estimator, is distribution-free in the sense that it is

asymptotically normally distributed, given merely the assumption of Gaussian disturbances

and no distributional assumptions about the modelled variables (as opposite to Gaussian

covariance structure based static LISREL method which generally requires modelled

variables to be multivariate Gaussian). To see this, note that:

uZIZZΣZIX

XZIZZΣZIXδδ~TTˆ~T

~TTˆ~T~~̂T

)(])([)(

})(])([)({(211-11

1-11-11

′⊗′⊗′′⊗′⋅

′⊗′⊗′′⊗′=−−−−

−−−

(3.35)

Assuming Gaussian disturbances, ),()(),( 121 ZZΣ0uZIIΣ0u ′⊗′⊗⇒⊗ −− TN~~TN~~ ,

therefore asymptotic normality immediately follows:

)])([,(( 1-XZ

1-ZZ

1-ZX ~~N~~~̂T ΣΣΣΣ0δδ ⊗− (3.36)

Note that -11-1-1

XZ-1ZZ

-1ZX ])([])([ XZZZZΣXΣΣΣΣ ~)(ˆ~

~~ ′′⊗′≈⊗ − and ZX~X~Z ΣΣ =′ . It is worth emphasising that the above results apply only if the IV matrix Z is valid for all

equations of the system. Practically, this means that for estimation of the jth equation there

might be eligible instruments that are not eligible for estimation of the jth equation. Formally,

Z that is eligible for the entire OFS system contains the intersection of the rows of Zj, i.e.,

instruments for each jth equation in the system. If there are enough such instruments (given

sufficiently long time span of the data, there will always be enough lagged variables to satisfy

this requirement) such matrix Z can be constructed so to enable identification of each

equation in the system and consistent 3SLS estimation. However, as already mentioned,

model-implied validity might by misleading if the model itself is mis-specified, thus empirical

testing of IV’s validity is essential.

3.5 Dynamic panel estimation The methods discussed so far are sufficient for the analysis of pure time series (given some

stationarity assumptions are satisfied), while the use of panel data will require consideration

of some further estimation issued.

When a panel model is dynamic common estimators are inconsistent and inapplicable

because the presence of lagged endogenous variables induces dependence in the error terms

due to existence of the individual-effect error components (see Baltagi, 2001 for more

details). In the panel data, assuming simple one-way error component model we have:

27

it it it it i it,= + = +y X β ε ε µ v (3.37)

where the error term combines individual effects and random effects, thus if lags of

endogenous variable are included in the model individual components of the errors will

induce serial dependence since if yit is a function of µi, it follows that yit-1 is a function of µi as

well, hence yit will be correlated with εit.

The estimation of the model in Eq. (3.37) requires differencing that will remove the

individual components of the error term (see Arellano and Bond, 1991):

1

1 1 ( ) ( )

it it it

it it it it

it it

−

− −

∆ = −= + − += ∆ + ∆

y y yX β ε X β εX β v

(3.38)

If panel data are used for the estimation of the structural ADL model of Eq. (2.4), the

structural model becomes:

( )

0 0

p qη

it i j it- j j it- j it it ζi itj j

,η= =

= + + + = +∑ ∑η α B η Γ ξ ζ ζ µ v (3.39)

Similarly, the measurement models will also include individual error components:

(x)it ix x it it it i it, δ= + + = +x α Λ ξ δ δ µ v (3.40)

and:

(y)it iy y it it it i it, ε= + + = +y α Λ η ε ε µ v (3.50)

Such models can be estimated in differenced form, i.e.,

0 0

p q

it j it- j j it- j itj j= =

∆ = ∆ + ∆ + ∆∑ ∑η B η Γ ξ v (3.51)

using IV-type of estimators (or more generally specific cases of the generalised method of

moments estimators). The issue of IV selection and composition of the Z matrix would be

significantly complicated by the use of Arellano and Bond (1991) estimators. Arellano and

Bond (1991) define valid instruments in respect to their lag characteristics, i.e., different time

period are assigned different lagged instruments, with the exceptions of strictly exogenous

regressors, which are valid for all periods. The IV matrix Z needed for estimation of the Eq.

28

(3.51) should be constructed, following Arellano and Bond (1991) in such way that different

time periods have, as instruments, different lags of the appropriate lagged variables.

The Arellano and Bond (1991) method of constructing IV matrices for dynamic panels can

adopted for estimation of the panel OFS equations using a GIVE-type estimator. Moreover,

this procedure can be extended to 3SLS estimation if a unique IV matrix (for all equations in

the system) could be specified (see Arellano and Bover, 1995).

4. Identification Identification of LISREL-type models is known to be a rather problematic issue. An early

discussion of this topic can be found already in Wiley (1973), but a simple and

straightforward procedure still does not exist. On the other hand, identification is well defined

and straightforward in classical econometric simultaneous equation systems, and a similar

approach can be developed for the OFS equations.

A simple procedure that uses only the coefficient matrices from the latent specification for

identifying the OFS estimation equations will be developed here. The following technique

provides sufficient conditions for identification of all equations in the systems.

Proposition 2. Given a dynamic LISREL model

0 0

p qt η j t- j j t- j tj j= == + + +∑ ∑η α B η Γ ξ ζ with

the measurement model given by ttxxt δξΛαx ++= and ttyyt εηΛαy ++= , define:

(y)0 2( )

n

h

′′− −

=

I B Λ 0K 0 I 0

0 0 I, and

( ) ( )2 2

1

2

( )0 2

1

y x

p

x

q

η − − − ′− ′− ′−=

′′ − − ′−

′−

α α αB 0 0B 0 0

B 0 0GΓ 0 ΛΓ 0 0

Γ 0 0

Then, the jth equation of the system will be identified iff

1−++≥

hnmrank j GK

R (4.1)

29

where Rj is the zero-one selection matrix having one’s in places of omitted variables and one

row for each omission. Where if the equality holds the equation is exactly identified,

otherwise it is overidentified.

Proof See Appendix 1.

A corollary to Proposition 2 is that unless:

1}{ −++≥ hnmrank jR (4.2) the jth equation is surely not identified. The condition (4.2) is necessary for identification,

while condition (4.1) is, itself, sufficient.

Therefore, it is possible to use these rules to check for identification of each individual

equation. Note that it is perfectly possible for some equations to be identified while others

might not be so. The relevance of this approach lies in its ability to check for identification of

the model that is specified in latent form and thus save us the trouble of deriving the OFS

equations, and additionally, it is generally valid for the GIVE/2SLS procedures whether

applied to static or dynamic models.

5. Application to a dynamic panel model of UK micro consumption function 5.1 The income-consumption puzzle and latent variables The relationship between consumption expenditure and personal disposable income is one

of the most researched topics in both macro and microeconomics. The cornerstone of most of

the debates has been the relationship between consumption and income in relation to the

permanent income hypothesis (Friedman, 1957) and the life-cycle hypothesis (Modigliani and

Ando, 1963; Modigliani, 1986; for an extensive review see Deaton, 1992). In brief, this

theoretical framework predicts a relationship between permanent income (annuity of the life-

cyle income) and consumption, but does not predict strong relationship between current

income (and thus changes in current income) and consumption. Simply, rational consumers

should not respond to windfall gains and temporary income increase in increased

consumption, rather their consumption should be smooth across the life-cycle, which is

achieved by borrowing when income is low and repaying the debts when income increases

(e.g., later in life). The problem, however, is that most of the empirical tests with both micro

30

and macro data reject the permanent income hypothesis insofar they find strong and

statistically significant relationship between current income and consumption. Such empirical

finding is known as excess sensitivity of consumption (Davidson, et al. 1978; Hayashi, 1982;

Campbell and Mankiw, 1989, 1990, 1991; Deaton, 1992; Browning and Lusardi, 1996;

Madsen and McAleer, 2000, 2001).

There have are several possible theoretical explanations for the finding of excess sensitivity

of consumption in the literature. These explanations can be classified into three main groups,

the liquidity constraint approach (Flavin, 1981; Hubbard and Judd, 1986; Jappelli and Pagano,

1989; Scheinkman and Weiss, 1986), the uncertainty hypothesis (Blanchard and Fischer,

1989; Zelds, 1989, Deaton, 1991; Aiyagari, 1994; Muellbauer and Lattimore, 1995; Carroll,

1997, Ludvigson and Paxson, 2001; Hahm and Steigerwarld, 1999; Gourinchas and Parker,

2002) and the behavioural life-cycle hypothesis (Madsen and McAleer, 2001).

The common theme in all these and similar approaches is an attempt to bridge the gap

between the theoretical predictions of the permanent income theory and overwhelming

empirical rejection of its major claims. While there can be numerous theoretical and empirical

reasons for this discrepancy there is one issue that particularly concerns latent variables

methodology and the problem of imperfect measurement. Namely, it is well known that true

consumption is unobserved (merely, certain expenditures can be observed), and income is

notoriously measured with error. Modelling the relationship between consumption and

income as latent variables, each measured with certain number of observed indicators, offers

possible new insights into the decades-old debate.

An important question is what happens to the income elasticity (coefficient of income in the

regression of consumption on income) if the consumption function is estimated as a latent-

variable model and not by simple regression technique where, instead of incorporating

factor-analytic type of measurement models for consumption and income, their observed

indicators are simply summed? Namely, can we bridge the gap between the theoretical

predictions of permanent income hypothesis and empirical findings of excess sensitivity of

consumption? Such finding can be obtained if the (current) income elasticity from a latent

variable model is found to be smaller in magnitude and statistically less significant then the

one in the model where truly latent variables are obtained by simple summing of their

available indicators (naturally, using same indicators in both type of models).

There are numerous other possibilities that can be incorporated into the model when

LISREL-type of methods are used, e.g., we can model unobserved liquidity constraints by

either directly incorporating latent liquidity constraints into the model or by splitting the

31

sample to constrained and unconstrained consumers and using multi-group estimation.

Various other complications can be further considered, but these will not be pursued in this

paper.

The statistical explanation of excess sensitivity finding might rest in the effect of

contemporaneous correlations among income and consumption indicators on the relationship

between income and consumption itself. Concretely, if various observed indicators of income

(e.g., labour, non-labour, investment incomes, etc.) are mutually correlated, and perhaps form

a latent factor, and similarly, if consumption indicators (e.g., expenditure on food, housing,

durable goods, etc.) also form a latent factor, and there is a true relationship between the two

(unobserved) factors, then what happens if, instead of estimating a (true) latent model, we

estimate the relationship between income and consumption that are calculated as simple

(unweighted sums of their indicators? A likely possibility is that such relationship will be

incorrect and possibly exaggerated, which would offer an explanation for the excess

sensitivity finding, at least to some degree beyond of what is already know about this

extensively researched topic.

Recently Larsen (2002) proposed a new way of estimating latent total consumption in a

household aiming at improving the accuracy of permanent income studies. He noted that,

while the sum of individual expenditures (in a household) is an unbiased estimator of latent

total household consumption, it is also in-optimal as such sum is an unweighted sum of

components that contain measurement error. It can be added that not all expenditures are

always reported, thus even if we accept to operate with an unweighted sum, such variable

might still not resemble total household consumption expenditure, and a similar thing holds

for income (where non-reporting of some types of incomes is a well known problem in

household surveys). Larsen (2002) derived an alternative estimator of total household

consumption, based on latent variable methods, that is unbiased and variance minimising.

Essentially, by estimating a latent-type of model for household consumption, Larsen (2002)

derived weights for various considered types of consumption expenditures (including also

non-expenditure indicators). This line of research extended the previous efforts of estimating

household consumption more precisely though still relying on total purchase expenditure (see

Kay, Keen and Morris, 1984; Aasness, et al. 1993, 1995).

Empirical studies that attempt to model the income-consumption relationship using latent

variable techniques are scarce in the literature. There were few attempts to use the LISREL-

type of methods for this problem, primarily due to dynamic nature of the income-consumption

relationship and inability of the typical covariance structure based models to handle data with

32

pronounced dynamic component. An example of such an attempt is made by Ventura and

Sattora (1998) who, using Spanish household data, estimated life-cycle effects on some

product expenditures using LISREL methodology, but only with two years of data.

The model and estimation methods developed in this paper enable estimation of dynamic

latent consumption function models allowing for all previously considered aspects (see

Larsen, 2002 for a review), across any number of years. Additionally, it is possible to

compare the results from a latent model with those from standard models that treat

consumption and income as sums of their respective components and see if the income

elasticity and other parameters differ across the two models.

We will demonstrate these methods by estimating a latent consumption function model that

incorporates possible liquidity constraints effects using micro data from the British Household

Panel Study survey and incorporating data for all 10 available waves.

5.2 The data The data for this empirical analysis comes from the British Household Panel Survey

(Taylor, et al. 2001), which is an exceptionally rich source of individual and household level

data. At present, 10 waves (i.e., years) of data are available and under some constraints, all

years can be merged into a joint panel. The available variables on consumption expenditure

and types of income, as well as potential liquidity constraints indicators vary across waves,

and as our primary purpose is to illustrate dynamic latent variable modelling using data with

pronounced time-series dimension we compromise by using only those variables that were

available across all 10 waves.

Table 3. BHPS variables used in the model BHPS code Description HSIZE Number of persons in household XPHSD1 Housing payments required borrowing XPHSD2 Housing payments required cutbacks XPHSDB Been 2+ months late with housing payment XPHSDF Problems paying four housing over the year XPFOOD Total weekly food and grocery bill XPHSG Gross monthly housing costs QFACHI Highest academic qualification HGEMP Employment status IYRL Annual labour income FIYRNL Annual non-labour income FIYRI Annual investment income SAVED Amount saved each month

33

Specifically, we are forced to give up otherwise relevant durable expenditure data that are

available only for the last four waves. The variables that we use in the model (with original

BHPS codes) are shown in Table 3 (note that by the BHPS convention the variable codes are

prefixed by wave identifiers a, b, … , j; we drop these prefixes because we pool the data into a

single file). Household data (expenditures) were first spread onto individual level, and

subsequently combined with the individual level income data, thus creating all-individual data

files for all waves. Finally, wave-specific files were merged into a joint panel for all

individuals across all waves in the “long format”, meaning the first individual in the sample is

recorded on each time point followed by second individual, etc. In this analysis we do not

address the issues of missing values and attrition as these are not the focus of investigation but

we note that we used data on 3,324 individuals that had no missing values (for the variables in

Table 3), thus a panel with NT observations amounted to 33,240 observations. Needless to

say, such sample size and the pronounced time-series dimension would pose practically

irresolvable challenge for classical static and simplex types of LISREL models. Methods

capable of handling such large data sets would pass a major “endurance” test. Note that all

methods developed and discussed in this paper are non-iterative and thus promise far greater

stability and computing power then covariance structure based maximum likelihood

approaches.

Data transformations used to create substantively relevant quantities are described in Table

4. We create indicators for three (assumed) latent variables, namely consumption, income and

liquidity constraints. Latent approach promises to account for both imperfectly observed

variables and measurement error, and due to data specifics, we are surely facing both.

Table 4. Data transformation and variable names

Symbol Description Transformation (using BHPS variable codes) Ft Annual personal food

expenditure 52*(XPFOOD / HSIZE)

Ht Annual personal housing costs 12*(XPHSG / HSIZE) Lt Annual labour income IYRL NLt Annual non-labour,

non-investment income FIYRNL – FIYRI

It Annual investment income FIYR St Annual personal savings 12*(SAVED) Bt Cumulative credit repayment

problem YPHSD1 + XPHSD2 + XPHSDB + XPHSDF + HGEMPa

EDt Highest level of academic education

GFACHI

a HGEMP was recoded so that 1 = unemployed; 0 = employed

34

The variable transformations allowed us to compute all quantities on annual, individual

level. The “Cumulative credit repayment problem” variable was created in an attempt to take

most of the data-based information on possible credit constraints; it sums indicators (0 and 1)

of several types of credit repayment problems, thus cumulating to total number of credit

difficulties (see Table 3). Personal savings is another variable frequently used in liquidity

constraints modelling (Hayashi, 1982, 1985a, 1985b), where individuals (or households) wth

positive saving rates are assumed to be liquidity un-constrained, which is a bit streched notion

of liquidity constraints. Additionally, when combined with other likely liquidity constraints

indicators, savings creates troubles as it is not measured on the same scale (e.g., credit

repeyment problems are 0-1 dummies, and savigs is a continuous variable). For these reasons

we simply treat liquidity constraints as a latent variable jointly measured by observed (with

error) savings and cummulative credit repayment problem (later one being treated as if it was

continuous, though from the the strict point of view it is a count variable with finite number of

levels; the reason for this is, naturally, unavailability of dynamic latetent variable estimation

methods for ordinal variables, at least in the cosidered methodological framework). The

computer code (SPSS syntax) for all data manipulations is given in Appendix 2. Note that

values for all variables were created on the same (annual) level, thus variables originally

recorded on monthly level were multiplied by 12 (XPHSG, SAVED) and the weekly food

expenditure variable (XPFOOD) was multiplied by 52. The expenditures were further divided

by household size (HSIZE) to obtain an approximate estimate for individuals.

5.3 Specification of the model We specify the model as a general structural autoregressive distributed lag model with latent

variables, described in Section 2.1. Fig. 1 shows the path diagram for the model, relating

current values of the modelled variables (with subscript t) with past values (with superscript

t – i) of the same (autoregressive part) or different (distributed lag part) variables. Note that

we do not use double subscripting with “i,t”, as usual for panel models, and to simplify

already complex notation we only use t subscripts. The path diagram in Fig. 1 follows

standard LISREL-type notational conventions, though we note two specific differences. First,

the usual causal arrows in the measurement model were omitted as we cannot and do not wish

to imply any causality in the measurement part of our model, and second, we omit errors

(delta’s and upsilon’s) from the path diagram, as these are not explicitly modelled and

estimated.

35

Figure 1. Conceptual path diagram for the latent variable model

The model assumes that current consumption, modelled as a latent variable, depends on

current income (also latent), previous period consumption and previous period income.

Simultaneously, current income depends on the last period (previous year) income and

education (assumed to be measured without error for simplicity; note that we assume

education, Et, to be a latent variable perfectly measured with a single indicator EDt). Finally,

the liquidity constraints (assumed, of course, latent) are directly incorporated into the model

and assumed to depend on previous period consumption (logically, we expect that excessive

spending in one year causes greater degree of liquidity constraints in the following year), and

36

on previous period and current income. The structural part of the model, describing the

relationships among latent variables is given, in matrix notation, as:

[ ](0) (1) (1) (1)

13 11 12 13 1(0) (0) (1) (1)21 23 22 23 1

(1)33 1 31

0 0 00 0 0

0 0 0 0 0

t LQ t t LQ

t C t t t C

t Y t t Y

LQ LQ LQC C C EY Y Y

α β β β β ζα β β β β ζα β γ ζ

−

−

−

= + ⋅ + ⋅ + ⋅ +

(5.1) There are three measurement models, for latent consumption, income and liquidity constraints

variables. The measurement model, using specification of Eq. (2.7) is given by:

( )

( )

( )

( )( )41

( )( )52

( )( )63( )( )73

1 0 00 1 00 0 1

0 00 00 00 0

yt StS

yt FtF

yt t LtL

yyt t BtB

yyt t HtH

yyt NLtNL

yyt II

SFL LQB CH YNLI

εαεαεα

λ εαλ εα

λ εαλ εα

= + ⋅ + t

(5.2)

Note that we can add , 0t t EDt EDtED E δ= + = for completeness, but this will not be necessary

in the estimation of the OFS equations, where EDt will be treated as an observed exogenous

regressor. Following derivation of the Eqs. (2.14)-(2.16), we can re-write the model in the

specific OFS form. The OFS form for the structural model is thus given by:

[ ](0) (1) (1) (1)

13 11 12 13 1 11(0) (0) (1) (1)21 23 22 23 1 12

(1)33 1 31 13

0 0 00 0 0

0 0 0 0 0

t S t t t

t F t t t t

t L t t t

S S S uF F F ED uL L L u

α β β β βα β β β βα β γ

−

−

−

= + ⋅ + ⋅ + ⋅ +

(5.3) and the OFS for the measurement model is given by:

( )( )2141

( )( )2252

( )( )2363

( )( )2473

0 00 00 00 0

yyt tB

tyyt tH

tyyt tNL

tyyt tI

B uS

H uF

NL uL

I u

λαλα

λαλα

= + ⋅ +

(5.4)

The Eqs. (5.3) and (5.4), given they are identified, can now be estimated with GIVE (2SLS)

and 3SLS methods, using theoretically eligible and empirically valid instrumental variables.

37

5.4 Identification analysis In order to uniquely estimate all coefficients in each equation, the model must be identified.

The identification analysis follows the rules described in Section 4, noting that single-

equation GIVE can be applied on individual equations that are identified, while other,

possible unidentified equations can remain not estimated. 3SLS is a system estimator and

requires identification of the entire system. As we have implied, the identification can proceed

by using merely LISREL representation of the model without having to derive the OFS

equations. First note that the relevant coefficient matrices are given by:

[ ]

(0)21

( )2 0

(0) (0)13 23

(1)11(1) (1)

1 12 22 0 31(1) (1) (1)

13 23 33

( )41

( ) ( )2 52

1 0, , ( ) 0 1 0 ,

1

0 00 , 0 0 ,

0 0 00 0 00 0

BLQ

HyC

NLY

I

y

y y

η

αα β

αα

αα β β

α

ββ β γβ β β

λλ

λ

− ′= = − = − −

− ′ ′− = − − − = − − − −

−′− = −

−

α α I B

B Γ

Λ( ) ( )63 73

y yλ

−

Using the definition from the Eq. (4.3) it can be shown that the K matrix is given by:

(0) ( )21 41

( )52

(0) (0) ( ) ( )13 23 63 73

1 0 0 0 00 1 0 0 0 0

1 0 00 0 0 1 0 0 00 0 0 0 1 0 00 0 0 0 0 1 00 0 0 0 0 0 1

y

y

y y

β λλ

β β λ λ

− − − − − − − =

K

Similarly, the G matrix, following Eq. (4.4) is given by:

(1)11(1) (1)

12 22(1) (1) (1)

13 23 33

31

0 0 0 0 0 00 0 0 0 0

0 0 0 00 0 0 0 0 0

LQ C Y B H NL Iα α α α α α αββ ββ β β

γ

− = − − − − − −

G

38

We can now derive the identification conditions by computing the rank of the matrix (4.1)

fore each individual equation in the system. To simplify these derivations and facilitate

exposition, we write the OFS equations in the scalar notation (writing each variable in its

specific place, which is indicated by column headings in the top row) as follows:

1 1 1(0) (1) (1) (1)

13 11 1 12 1 13 1(0) (0) (1) (1)21 23 22 1 23 1

(1)33 1 31

( )41

( )52

( )63

73

1 t t t t t t t t t t t

t LQ t t t t

t C t t t t

t Y t ty

t B ty

t H ty

t NL t

t I

S F L B H NL I S F L EDS L S F LF S L F LL L EDB SH FNL LI

− − −

− − −

− −

−

− − − − −− − − − −− − −− −− −− −− −

α β β β βα β β β βα β γα λα λα λα λ

11

12

13

21

22

23( )

24

t

t

t

t

t

ty

t t

uuuuuu

L u

=======

Note that the first equation (for St) has first, second, fourth, fifth, sixth, seventh, and eleventh

variable omitted, thus the selection matrix R1 will have “1” in these positions, where each

position occupies one row only. Derivation of the Rj matrices for the remaining equations

follows the same simple procedure, and for clarity these matrices are shown bellow:

1 2

3

0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 00 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 00 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0, ,0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 00 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 00 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1

1 0 0 0 0 0 0 0 0 0 0 00 1 0 0 0

= =

=

R R

R 4

0 1 0 0 0 0 0 0 0 0 0 00 0 1 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 00 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 00 0 0 0 1 0 0 0 0 0 0 0 , 0 0 0 0 0 0 1 0 0 0 0 00 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 00 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 00 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 00 0 0 0 0 0 0 0 0 1 0 0 0 0 0

=

R ,

0 0 0 0 0 0 0 0 1

39

5 6

1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 00 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 00 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 00 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0

,0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 00 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 1 0 00 0 0 0 0 0 0 0 0 0 1 00 0 0 0 0 0 0 0 0 0 0 1

= =

R R

7

,0 0 0 1 0 0 0

0 0 0 0 0 0 0 0 0 1 0 00 0 0 0 0 0 0 0 0 0 1 00 0 0 0 0 0 0 0 0 0 0 1

1 0 0 0 0 0 0 0 0 0 0 00 1 0 0 0 0 0 0 0 0 0 00 0 0 1 0 0 0 0 0 0 0 00 0 0 0 1 0 0 0 0 0 0 00 0 0 0 0 1 0 0 0 0 0 00 0 0 0 0 0 0 0 1 0 0 00 0 0 0 0 0 0 0 0 1 0 00 0 0 0 0 0 0 0 0 0 1 00 0 0 0 0 0 0 0 0 0 0 1

=

R

Using the above Rj, K, and G matrices, we can derive the identification conditions for the

first equation as: ( )32

1

31

0 1 0 0 0 00 0 0 1 0 0 00 0 0 0 1 0 00 0 0 0 0 1 00 0 0 0 0 0 10 0 0 0 0 0

yλ

γ

−

= −

KR G ,

which has a rank 6 (m + n + h – 1 = 6) and we conclude that the first equation is (exactly)

identified. Similarly, the second equation is identified because

2

(1)11

31

0 0 0 1 0 0 00 0 0 0 1 0 00 0 0 0 0 1 00 0 0 0 0 0 1

0 0 0 0 0 00 0 0 0 0 0β

γ

= − −

KR G

has also rank 6. We can proceed in this fashion for the remaining equations showing that each

is exactly identified with identification condition matrix of rank 6. For the third equation we

have:

40

(0) ( )21 41

( )52

3

(1)11(1) (1)

12 22

1 0 0 0 00 1 0 0 0 00 0 0 1 0 0 00 0 0 0 1 0 00 0 0 0 0 1 00 0 0 0 0 0 1

0 0 0 0 0 00 0 0 0 0

y

yβ λ

λ

ββ β

− − − = − − −

KR G

For the fourth equation the rank is also 6:

( )52

(0) (0) ( ) ( )13 23 63 73

4(1)

11(1) (1)

12 22(1) (1) (1)

13 23 33

31

0 1 0 0 0 01 0 0

0 0 0 0 1 0 00 0 0 0 0 1 00 0 0 0 0 0 1

0 0 0 0 0 00 0 0 0 0

0 0 0 00 0 0 0 0 0

y

y yλ

β β λ λ

ββ ββ β β

γ

− − − − −

= − − − − − −

−

KR G

And also for the firth, sixth and seventh equations:

(0) ( )21 41

(0) (0) ( ) ( )13 23 63 73

5(1)

11(1) (1)

12 22(1) (1) (1)

13 23 33

31

1 0 0 0 01 0 0

0 0 0 1 0 0 00 0 0 0 0 1 00 0 0 0 0 0 1

0 0 0 0 0 00 0 0 0 0

0 0 0 00 0 0 0 0 0

y

y yβ λ

β β λ λ

ββ ββ β β

γ

− − − − − −

= − − − − − −

−

KR G

(0) ( )21 41

( )52

6(1)

11(1) (1)

12 22(1) (1) (1)

13 23 33

31

1 0 0 0 00 1 0 0 0 00 0 0 1 0 0 00 0 0 0 1 0 00 0 0 0 0 0 1

0 0 0 0 0 00 0 0 0 0

0 0 0 00 0 0 0 0 0

y

yβ λ

λ

ββ ββ β β

γ

− − −

= − − − − − −

−

KR G

41

(0) ( )21 41

( )52

7(1)

11(1) (1)

12 22(1) (1)

13 23

31

1 0 0 0 00 1 0 0 0 00 0 0 1 0 0 00 0 0 0 1 0 00 0 0 0 0 1 0

0 0 0 0 0 00 0 0 0 00 0 0 0 0

0 0 0 0 0 0

y

yβ λ

λ

ββ ββ β

γ

− − −

= − − − − −

−

KR G

This concludes the identification of the model and we proceed with the estimation. We will

compare the estimates of both GIVE and 3SLS methods by selecting empirically and

theoretically acceptable instruments.

5.5 Estimation results We estimate the Eqs. (5.3) and (5.4) fist separately using single-equation GIVE technique

and then the 3SLS systems estimation. The advantage of separate GIVE estimation of each

equation in the system is twofold. Firstly, it is possible to use different instruments for

different equations (i.e., we can specify an IV matrix Zj for each j). This is perhaps more

important to cross-sectional models (where lagged variables are generally not available) then

to dynamic models, though, we must be aware that not all model-implied (lagged) eligible

instruments are equally good and the issue of empirical testing (e.g., Sargan test) is a rather

important one. Secondly, it might be the case that only part of the model (e.g., structural part)

is identified and can be estimated. In such case we need single-equation methods to estimate

only the identified part of the model while the un-identified part is not estimated (note,

however that estimation of partially identified systems might affect distribution of the

estimators; see Phillips and Choi, 1992).

The 3SLS, on the other hand, estimates the entire system, using a single IV matrix Z. Thus,

the 3SLS estimates will be generally more efficient meaning their standard errors will be

smaller and the coefficient estimates will be more precise. However, if the model is not

correctly specified and thus some of the used instruments are in fact ineligible (correlated

with the error terms) there might be a difference between 3SLS and GIVE estimates, given

GIVE used different (assuming correct) IV matrix for each estimated equation. Thus, large

discrepancy between GIVE and 3SLS results might indicate model mis-specification.

42

Table 5 shows the IV-validity test results (Sargan, 1964) for individual equations estimated

by GIVE methods using varying Zj matrices. The equations in levels are estimated for

demonstrative purposes only; due to the panel nature of the used BHPS data, it was necessary

to estimate differenced equations using appropriately constructed IV matrices (see Arellano

and Bond, 1991). Though differences as well as lags can be used as instruments (given they

are appropriately selected from among the set of eligible instruments for each equation), we

used lagged variables only (see Arellano, 1989 for more details on problems caused by the

use of differences for instruments in simple error-component models).

Table 5. GIVE estimation and validity of instruments tests

Equation Instruments X2 d.f. (0) (1) (1) (1)13 11 1 12 1 13 1 11t LQ t t t t tS L S F L u− − −= α + β + β + β + β + Bt-2, Ht-2, NLt-2, It-2,

St-2, Ft-2 184.020 5

(0) (1) (1) (1)13 11 1 12 1 13 1 11t t t t t tS L S F L v− − −∆ = β ∆ +β ∆ +β ∆ +β ∆ + ∆ Bt-3, Ht-3, EDt 0.687 2

(0) (0) (1) (1)21 23 22 1 23 1 12t C t t t t tF S L F L u− −= α + β + β + β + β + Bt-2, Ht-2, NLt-2 7.335 1

(0) (0) (1) (1)21 23 22 1 23 1 12t t t t t tF S L F L v− −∆ = β ∆ + β ∆ + β ∆ + β ∆ + ∆ Bt-3, Ht-3, EDt 0.253 1

(1)33 1 31 13t Y t t tL L ED u−= α + β + γ + Bt-2, Ht-2, Ft-2 3.251 3

(1)33 1 31 13t t t tL L ED v−∆ = β ∆ + γ + ∆ Bt-3, Ht-3, Ft-3 1.253 3

(y)41 21t B t tB S u= α + λ + Ht-2, Ft-2 10.514 1

(y)41 21t t tB S v∆ = λ ∆ + ∆ NLt-2, It-2, EDt 9.167 2

(y)52 22t H t tH F u= α + λ + Bt-2, St-2 8.072 1

(y)52 22t t tH F v∆ = λ ∆ + ∆ Bt-2, Ht-2, NLt-2, It-2,

St-2, Ft-2, Lt-2, EDt 2.771 7

(y)63 23t NL t tNL L u= α + λ + Ht-2, St-2 14.506 1

(y)63 23t t tNL L v∆ = λ ∆ + ∆ Bt-2, Ht-2, EDt 1.379 2

(y)73 23t I t tI L u= α + λ + Ht-2, EDt 0.054 1

(y)73 23t t tI L v∆ = λ ∆ + ∆ Bt-2, Ht-2, EDt 2.007 2

The first column of Table 5 shows OFS equations in levels and differences. Note that

constant term cannot be estimated directly in the differenced model, and as intercepts carry

no special substantive importance in this model we do not attempt to recover them; similarly

we do not report intercept estimates for the levels equations. The second column of Table 5

shows which particular variables were used for estimation. The selection was based

primarily on minimisation of Sargan (1964) validity of instruments test and the fact that

selected (best) instruments differ across equation asks for caution with the 3SLS estimates

that will rely on a joint IV matrix. Such jointly valid IV matrix should be made out of the

instruments that are common to each equation (according to validity results that accompany

43

GIVE estimates). Due to the individual-effects bias characteristic for panel data the

estimates of the levels equations are biased. We report correct estimates together with the

incorrect ones in Table 6 allowing for comparison and evaluation of the possible magnitude

of error if the fact that panel data is used is ignored. It is obvious that these two sets of

estimates differ to large degree indicating that we would get rather erroneous results with

levels estimates.

Table 6. Coefficient estimates 2SLS 3SLS Levels Differences Levels Differences

Coeff.. Estim. (S.D.) Estim. (S.D.) Estim. (S.D.) Estim. (S.D.)

(0)13β 0.0222 (0.0138) -0.091 (0.0267) 0.2392 (0.0073) -0.110 (0.0079) (1)11β 0.3438 (0.0059) -0.342 (0.0059) 0.2303 (0.0055) -0.341 (0.0057) (1)12β 0.1453 (0.0198) 0.0254 (0.0287) 0.1310 (0.0176) 0.0531 (0.0273) (1)13β 0.0091 (0.0099) -0.017 (0.0047) -0.1341 (0.0053) -0.015 (0.0021) (0)21β 0.2693 (0.0507) -0.071 (0.4430) 0.0321 (0.0039) -0.004 (0.0025) (0)23β 0.0005 (0.0054) 0.1553 (0.0921) -0.0113 (0.0021) 0.0973 (0.0016) (1)22β 0.5364 (0.0134) -0.317 (0.0908) 0.5766 (0.0049) -0.337 (0.0057) (1)23β -0.0102 (0.0039) 0.0239 (0.0121) 0.0075 (0.0015) 0.0112 (0.0004) (1)33β 0.7162 (0.0042) -0.169 (0.0006) 0.6928 (0.0042) -0.167 (0.0060)

31γ 310.8400 (22.3690) 118.12 (12.374) 302.3880 (21.6740) 77.057 (11.9410) (y)41λ -0.0003 (0.0000) -0.000 (0.0000) -0.0003 (0.0000) -0.000 (0.0002) (y)52λ 1.0127 (0.1719) 0.1933 (0.2763) 0.3472 (0.0302) 0.1037 (0.0720) (y)63λ -0.1188 (0.0091) 0.3368 (0.0699) -0.1374 (0.0028) 0.3201 (0.0116) (y)73λ -0.0052 (0.0044) 0.0349 (0.0261) 0.0287 (0.0013) 0.0163 (0.0053)

The difference between GIVE (single-equation) and 3SLS (system) estimates (columns 4

and 8 of Table 6) is, on the other hand, far smaller and we can notice that standard errors, on

average, decreased in the 3SLS case, which would be expected if the model is correctly

specified and if 3SLS instruments are jointly appropriate for each equation in the system.

The used instruments for 3SLS are all those used in estimation of the individual equations,

thus to some degree we can cite the acceptable result of the reported Sargan’s X2 test in

support to the selection of the joint IV matrix for the entire system.

Looking at the individual coefficient estimates it is possible to conclude that most

coefficients are well determined with small standard errors. The 3SLS estimate of the

income elasticity coefficient ( (0)23β ) is 0.0973 with standard deviation of 0.0016, thus current

income significantly affects current consumption but its magnitude is relatively within the

predictions of the permanent income theory (see Deaton, 1992). The attempt to model the

44

degree of liquidity constraints and its influence on relationship between consumption and

income provided little new insight in this well researched topic. Namely, the efforts to

construct and model a liquidity constraints variable that includes cumulative credit

repayment problem measure, though conceptually promising, resulted in poor statistical

results; the loading of Bt ( (y)41λ ) turned out to be insignificant, thus effectively all that we

have in the liquidity constraints measurement model is personal savings, which however,

has small (though significant) negative effect on consumption (higher savings, in return,

result in smaller consumption). A significant negative effect of the LQt-1 ( (1)11β ) suggest

cyclical saving pattern, i.e., savings is lower in the current period if it was unusually high in

the previous period and vice versa. The meaning of the 31γ coefficient can best be explained

as the increase in income (in ₤) for each additional year of education. Note how 31γ changes

across different specifications; the consistent estimates (GIVE) and 3SLS affect this

coefficient more then other coefficients because of the inclusion/exclusion of the constant

term.

Table 7 shows correlation matrix of the residuals (from all equations in the system). There

appears to be no notable correlation left among the residuals from different equations in the

estimated system.

Table 7. Correlation of the residuals (estimation in differences) ∆S ∆F ∆L ∆B ∆H ∆NL ∆H∆S 1.0000 ∆F 0.0930 1.0000 ∆L 0.0434 0.0189 1.0000 ∆B -0.0201 -0.0359 -0.0390 1.0000 ∆H -0.0074 -0.0082 0.0073 0.0080 1.0000 ∆NL -0.0453 -0.0123 -0.0751 0.0252 -0.0009 1.0000 ∆H 0.0099 0.0083 -0.0027 -0.0027 -0.0011 0.0081 1.0000

We can now wonder how is the estimated dynamic latent variable model different in its

insights and results to one estimated with proxy variables. By “proxy variable” we mean a

constructed variable (such as total consumption expenditure or total personal income) that

replaces the unobserved true variable, as typically used in classical consumption research.

To see this, we construct a classical econometric model of consumption using proxies

obtained by summing the indicators of the above modelled latent variables and subsequently

estimating an observed model where latent variables are replaced with their proxies while

all other aspects (such as dynamics) were left unchanged. The only difference is that in the

45

latent model we included credit repayment problem variable that is measured on different

scale from savings variable (and so it cannot be added to savings variable), thus we only

include savings in this model and note that credit repayment problems did not have

significant loading coefficient in the latent model. The conceptual path diagram is identical

to the one for dynamic latent model but without the measurement models (see Fig. 2).

Figure 2. Conceptual path diagram: proxy variable model

Such model can be estimated using classical econometric techniques (e.g., 3SLS applied

directly to proxy variables) and it is specified (in differences) as: (0) (1) (1) (0)23 22 1 23 1 21 1

(1)33 1 31 2

(0) (1) (1) (1)13 11 1 12 1 13 1 3

t t t t t t

t t t t

t t t t t t

C Y C Y S v

Y Y ED v

S Y S C Y v

− −

−

− − −

∆ = β ∆ + β ∆ + β ∆ + β ∆ + ∆

∆ = β ∆ + γ + ∆

∆ = β ∆ +β ∆ +β ∆ +β ∆ + ∆

using “~” symbol to indicate (summed) proxies. Thus, we obtain the estimates of the same

structural coefficients, which enable us to compare the coefficients of interest across the two

46

models. Using 3SLS method (similar results are obtained with maximum likelihood

estimation), we obtain the following estimates (standard errors are in the parentheses):

1 1 1 1

1 2

1 1

(0.0167) (0.005) (0.004) (0.002)

(0.006) (13.045)

(0.008) (0.007) (0.0346)

0 176 0 431 0 043 0 007

0 242 147 5

0 019 0 365 0.049 0 013

t t t t t it

t t it it

t t t t

C . Y . C . Y . S v

Y . Y . ED v

S . Y . S C .

− − −

−

− −

∆ = ∆ − ∆ + ∆ + ∆ + ∆

∆ = − ∆ + + ∆

∆ = − ∆ − ∆ + ∆ + 1 3

(0.005)c

7 t itY v−∆ + ∆

Most of the coefficients from the proxy variable model are close to those from the structural

part of the latent variable model, with noticeable difference in the income coefficient in the

consumption equation. Namely, the (0)23β coefficient increased from 0.097 to 0.176 indicating

that inappropriate aggregation of consumption expenditure and personal income sources (by

simple summation), at least in this specific case, inflated the value of the income elasticity

coefficient causing excessive sensitivity result as a possible statistical artefact. We can also

note that lagged consumption effect in the consumption equation is larger in magnitude (in

absolute values 0.431 vs. 0.337 in the latent model), and a similar thing happened to the

coefficient of lagged income in the income equation (abs. 0.242 vs. 0.167 in the latent model).

Therefore, it is quite possible that excess sensitivity of consumption is, at least partly,

caused by the effect of specific (proxy) measurement of the latent consumption and income

variables commonly used in the bulk of the consumption function models.

Contemporaneously correlated variables that are, themselves, indicators of a (true) latent

variable, if summed (instead of measured by a factor analytic-type of measurement models),

therefore, might introduce spurious correlation between proxy latent variables.

6. Discussion The methods for estimation of structural autoregressive distributed lag models with latent

variables, described in this paper, enabled us to estimate a latent consumption function model

taking into account dynamic structure of the modelled relationships by using a large micro-

economic panel data set. Furthermore, we used these methods to model consumption-income

relationships with direct incorporation of (conjectured) liquidity constrains measurement

model. It should be pointed out that if, in the estimated consumption function model, true

weights of the observed indicators are not equal, and if these indicators measure a single

47

latent variable, then treating their weights as equal and summing them (thus subsequently

using proxy-latent variables in a structural model) is a form of misspecification. What our

findings here add to this is that the use of such proxies can further cause spurious

relationships by inflating the coefficients among proxies as opposite to coefficients in the

structural model of (factor-analytically) measured latent variables. The application to a well-

known problem in economics showed that this effect might be at least a partial cause for the

known discrepancy between theoretical expectations and empirical findings in testing

permanent income consumption function models. A similar analysis might be conducted

using pure time series data (macroeconomic series), which is an interesting director for further

empirical research.

The scope of the presented methods was limited to stationary cases and it assumed

coefficient constancy across time. Important directions for further research should focus on

modelling non-stationary variables, where non-stationarity relates not only to the observed

indicators (which is a well researched topic) but also to the latent variables which would

introduce the concept of latent cointegration or cointegration among latent variables.

Additionally, time-varying coefficients can be considered and this will closely follow the

common issue of factorial-invariance testing (across time) commonly used in psychometric

and sociometric research. Finally, note that all methods discussed above are asymptotically

distribution free and thus do not depend on any particular distributional assumptions

regarding observed variables and their joint density. While such methods are growingly

popular do to their greater robustness to various model misspecifications and non-normality

of data, maximum likelihood methods, if applied to the OFS equations, could deliver also

overall fit measures and various likelihood-ratio test statistics. Such statistics are very popular

in cross-sectional LISREL models and mainly for this reason these models are most

frequently estimated by (Gaussian) maximum likelihood methods. Maximum likelihood

methods can be developed in classical econometric fashion by applying full or limited

information maximum likelihood on the OFS equations (or the entire system), which would

be a different approach then typical covariance-structure maximum likelihood that requires a

closed-form covariance structure. However, the expense of going in this direction would be in

seeking refuge in, often unfeasible, distributional assumptions.

48

Acknowledgments I wish to acknowledge valuable support of the Summer School in Social Science Data

Analysis of the University of Essex for carrying out this research. I am particularly grateful to

the UK Data Archive and staff of the Institute for Social and Economic Research for

providing me with the BHPS data and kind assistance in their use.

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Appendix 1: Proofs Proof of Lemma 2 Results (i) and (iv) follow directly from assumption A1 and Lemma 1. To

prove (ii) note that the convergence in mean square in Lemma 1 implies that:

( )

( ) 2 2 ( ) ( )2

1 1

22 ( )

1

1 1( ) ( 2 )

1

T Ty y y

ijt r ij ijt r ijt r ij ijt t

Ty

ijt r ijt

E y E y yT T

E yT

µ µ µ

µ

− − −= =

−=

− = − + = −

∑ ∑

∑ (A1.1)

where for r = w, this is equal to γ0, thus:

( )22 ( ) ( )0

1

1 Tij y

ijt r ijt

E yT

γ µ−=

= + ∑ (A1.2)

and similarly for (v):

( )

( ) 2 2 ( ) ( )2

1 1

22 ( )

1

1 1( ) ( 2 )

1

T Tx x x

ijt r ij ijt r ijt r ij ijt t

Tx

ijt r ijt

E x E x xT T

E xT

µ µ µ

µ

− − −= =

−=

− = − + = −

∑ ∑

∑ (A1.3)

57

( )22 ( ) ( )0

1

1 Tij x

ijt r ijt

E xT

δ µ−=

⇒ = + ∑ (A1.4)

establishing (v).

To show (iii), the above is easily generalised for higher moments by noting that:

( ) ( ) ( ) ( ), , , ,

1 1

( ) ( ) ( )

1 1( )( )T T

y y y yij t r ij ef t w ef ij t r ef t w ij ef

t t

ijef y yij efr w

E y y E y yT T

µ µ µ µ

γ µ µ

− − − −= =

−

− − = − = −

∑ ∑ (A1.5)

and similarly for (vi):

( ) ( ) ( ) ( ), , , ,

1 1

( ) ( ) ( )

1 1( )( )T T

x x x xij t r ij ef t w ef ij t r ef t w ij ef

t t

ijef x xij efr w

E x x E x xT T

µ µ µ µ

δ µ µ

− − − −= =

−

− − = − = −

∑ ∑ (A1.5)

Finally, for (vii) we have:

( ) ( ) ( ) ( ), , , ,

1 1

( ) ( ) ( )

1 1( )( )T T

y x y xij t r ij ef t w ef ij t r ef t w ij ef

t t

ijef x xij efr w

E y x E y xT T

µ µ µ µ

ψ µ µ

− − − −= =

−

− − = − = −

∑ ∑ (A1.5)

Q.E.D.

Proof of Proposition 1 To show that ( )1

ZZplim T − ′ =Z Z Σ , where ZZ ≠Σ 0 . Define a general

matrix of all eligible instruments ( )1 2 1 2* * *, , , ,≡ *Z Y Y X X as in Table 2. First, we have:

* * * * * * * *

1 1 1 2 1 1 1 2* * * * * * * *2 1 2 2 2 1 2 2* * * * * * * *1 1 1 2 1 1 1 2* * * * * * * *2 1 2 2 2 1 2 2

′ ′ ′ ′

′ ′ ′ ′ ′ = ′ ′ ′ ′ ′ ′ ′ ′

Y Y Y Y Y X Y XY Y Y Y Y X Y XZ ZX Y X Y X X X XX Y X Y X X X X

(A1.6)

The upper-left block of ′Z Z is given by (see Table 2):

(1 ) (1 ) (1 ) (1 ) (1 ) (1 ) (1 )1 1 1 1 2 1

(1 ) (1 ) (1 ) (1 ) (1 ) (1 ) (1 )* * (1 ) (1 ) (1 )2 2 1 2 2 1

1 1 1 2

(1 ) (1 )

j j j j j j jp p p p p p p a

j j j j j j jj j jp p p p p p p a

p p p a

j jp a p a p

− − − − − − −

− − − − − − −− − −

− − −

′ ′ ′ ′

′ ′ ′ ′ ′ = = ′ ′

y y y y y y y

y y y y y y yY Y y y y

y y y (1 ) (1 ) (1 ) (1 ) (1 )1 2j j j j j

p a p p a p a− − − −

′ ′ y y y y

(A1.7)

58

Notice that, e.g., upper-left (block) element of matrix (A1.7) can be expanded as:

(11) (12) (1 )1 1 1(11) (11) (11) (11)

1 1 1 2 1 1 (11) (1 1 1 1(12) (12) (12) (12)

(1 ) (1 ) 1 1 1 2 1 11 1

(1 ) (1 ) (1 ) (1 )1 1 1 2 1 1

mp p p

p p p T pp p

j j p p p T pp p

m m m mp p p T p

y y yy y y y

y yy y y y

y y y y

− − − − − −− − − − − − − −

− − − −− − − − − − − −

− −

− − − − − − − −

′ =

y y

( )

( )

12) (1 )1 1

(11) (12) (1 )2 1 2 1 2 1

(11) (12) (1 )1 1 1

2(11) (11) (12) (11) (1 )1 1 1 1 1

1 1 1

2(12) (11) (12)1 1 1

1

mp

mp p p

mT p T p T p

T T Tm

t p t p t p t p t pt t t

T

t p t p t pt

yy y y

y y y

y y y y y

y y y

− −

− − − − − −

− − − − − −

− − − − − − − − − −= = =

− − − − − −=

=

∑ ∑ ∑

∑

( )

(12) (1 )1 1

1 1

2(1 ) (11) ( 1) (12) (1 )1 1 1 1 1

1 1 1

T Tm

t p t pt t

T T Tm m m

t p t p t p t p t pt t t

y y

y y y y y

− − − −= =

− − − − − − − − − −= = =

∑ ∑

∑ ∑ ∑

(A1.8)

Taking probability limits we have:

( )

( )

2(11) (11) (12) (11) (1 )1 1 1 1 1

1 1 1

2(12) (11) (12)(1 ) (1 ) 1 1 1

1 1 1 1

1 1 1plim plim plim

1 11 plim plim pplim

T T Tm

t p t p t p t p t pt t t

T T

j j t p t p t pp p t t

y y y y yT T T

y y yT TT

− − − − − − − − − −= = =

− − − − − −− − = =

′ =

∑ ∑ ∑

∑ ∑y y

( )

(12) (1 )1 1

1

2(1 ) (11) ( 1) (12) (1 )1 1 1 1 1

1 1 1

1lim

1 1 1plim plim plim

Tm

t p t pt

T T Tm m m

t p t p t p t p t pt t t

y yT

y y y y yT T T

− − − −=

− − − − − − − − − −= = =

∑

∑ ∑ ∑(A1.9)

We can now apply the convergence results implied by Lemma 2 obtaining:

( )( )

2(11) (11) (12) (11) (1 )1 1 1 1 1

2(12) (11) (12) (12) (1 )(1 ) (1 ) 1 1 1 1 1

1 1

(1 ) (11)1 1

1plim

mt p t p t p t p t p

mj j t p t p t p t p t p

p p

mt p t p t

E y E y y E y y

E y y E y E y yT

E y y E y

− − − − − − − − − −

− − − − − − − − − −− −

− − − −

′ =

y y

( )2( 1) (12) (1 )1 1 1

(11) ( ) 2 (11,12) ( ) ( ) (11,1 ) ( ) ( )0 11 0 11 12 0 11 1

(12,11) ( ) ( ) (12) ( ) 2 (12,1 ) ( ) ( )0 12 11 0 12 0 12 1

(1 ,11) (0 1

( )( )

m mp t p t p

y y y m y ym

y y y m y ym

mm

y E y

γ µ γ µ µ γ µ µγ µ µ γ µ γ µ µ

γ µ

− − − − − −

+ + ++ + +=

+ ) ( ) (1 ,12) ( ) ( ) (1 ) ( ) 211 0 1 12 0 1( )y y m y y m y

m mµ γ µ µ γ µ

+ +

(A1.10)

which gives the required result for the upper-left block element of * *

1 1′Y Y . However, note that

all blocks of matrix (A1.9) can be written in the general form:

59

(11) (12) (1 )(11) (11) (11) (11)

1 2 (11) (1 1(12) (12) (12) (12)

(1 ) (1 ) 1 2

(1 ) (1 ) (1 ) (1 )1 2

mp w p w p w

p r p r p r T p rp w p w

j j p r p r p r T p rp r p w

m m m mp r p r p r T p r

y y yy y y y

y yy y y y

y y y y

− − − − − −− − − − − − − −

− − − −− − − − − − − −

− −

− − − − − − − −

′ =

y y

12) (1 )1

(11) (12) (1 )2 2 2

(11) (12) (1 )

(11) (11) (11) (12) (11) (1 )

1 1 1

(12) (11)

1

mp w

mp w p w p w

mT p w T p w T p w

T T Tm

t p r t p w t p r t p w t p r t p wt t tT

t p r t p w tt

yy y y

y y y

y y y y y y

y y y

− −

− − − − − −

− − − − − −

− − − − − − − − − − − −= = =

− − − −=

=

∑ ∑ ∑

∑ (12) (12) (12) (1 )

1 1

(1 ) (11) ( 1) (12) (1 ) (1 )

1 1 1

T Tm

p r t p w t p r t p wt t

T T Tm m m m

t p r t p w t p r t p w t p r t p wt t t

y y y

y y y y y y

− − − − − − − −= =

− − − − − − − − − − − −= = =

∑ ∑

∑ ∑ ∑

(A1.11)

and thus, from Lemma 2, it follows that block elements of * *

1 1′Y Y converge to:

(11) (11) (11) (12) (11) (1 )

(12) (11) (12) (12) (12) (1 )(1 ) (1 )

(1 )

1plim

mt p r t p w t p r t p w t p r t p w

mj j t p r t p w t p r t p w t p r t p w

p r p w

mt p r t

E y y E y y E y y

E y y E y y E y yT

E y y

− − − − − − − − − − − −

− − − − − − − − − − − −− −

− − −

′ =

y y

(11) ( 1) (12) (1 ) (1 )

(11) ( ) 2 (11,12) ( ) ( ) (11,1 ) ( ) ( )11 11 12 11 1

(12,11) ( ) ( ) (12) ( ) 2 (12,1 )12 11 12

( )

( )

m m mp w t p r t p w t p r t p w

y y y m y ymr w r w r w

y y y mr w r w r w

E y y E y y

γ µ γ µ µ γ µ µγ µ µ γ µ γ µ

− − − − − − − − −

− − −

− − −

+ + +

+ + +=

( ) ( )12 1

(1 ,11) ( ) ( ) (1 ,12) ( ) ( ) (1 ) ( ) 21 11 1 12 1( )

y ym

m y y m y y m ym m mr w r w r w

µ

γ µ µ γ µ µ γ µ− − −

+ + +

(A1.12)

Therefore, a typical scalar element of * *

1 1′Y Y has a non-zero probability limit of the general

form: ( )1 (1 ) (1 ) (1 ,1 ) ( ) ( )1 11

plim T i j i j y yt p r t p w i jr wt

T y y γ µ µ−− − − − −=

= +∑ .

The second block element of ′Z Z , i.e., * *1 2′Y Y is given by:

(1 ) (1 ) (2 ) (1 ) (2 ) (1 ) (2 )

1 1 1 1 2 1

(1 ) (1 ) (2 ) (1 ) (2 ) (1 ) (2 )* * (2 ) (2 ) (2 )2 2 1 2 2 1

1 2 1 2

(1 ) (1 ) (21

j j j j j j jp p p p p b

j j j j j j jj j jp p p p p b

p b

j j jp a p a

− − − − − − − −

− − − − − − − −− − − −

− − −

′ ′ ′ ′

′ ′ ′ ′ ′ = = ′ ′

y y y y y y y

y y y y y y yY Y y y y

y y y ) (1 ) (2 ) (1 ) (2 )2

j j j jp a p a p b− − − − −

′ ′ y y y y

(A1.13)

Similarly to derivation of (A1.12), we can write a typical scalar element of * *

1 2′Y Y as

(1 ) (2 )1

T i jt p r t wt

y y− − −=∑ , and therefore, ( )1 (1 ) (2 ) (1 ,2 ) ( ) ( )1 21

plim T i j i j y yt p r t w i jp r wt

T y y γ µ µ−− − − − −=

= +∑ .

60

Following the same principle and applying again Lemma 2, we can write typical elements

of each of the blocks of the matrix (A1.6) as:

( )* * 1 (1 ) (1 ) (1 ,1 ) ( ) ( )1 1 1 11

: plim T i j i j y yt p r t p w i jr wt

T y y γ µ µ−− − − − −=

′ = +∑Y Y

( )* * * * 1 (1 ) (2 ) (1 ,2 ) ( ) ( )1 2 2 1 1 21

& : plim T i j i j y yt p r t w i jp r wt

T y y γ µ µ−− − − − −=

′ ′ = +∑Y Y Y Y

( )* * 1 (2 ) (2 ) (2 ,2 ) ( ) ( )2 2 2 21

: plim T i j i j y yt r t w i jr wt

T y y γ µ µ−− − −=

′ = +∑Y Y

( )* * * * 1 (1 ) (1 ) (1 ,1 ) ( ) ( )1 1 1 1 1 11

& : plim T i j i j y xt p r t q w i jp r q wt

T y x ψ µ µ−− − − − − − −=

′ ′ = +∑Y X X Y

( )* * * * 1 (2 ) (1 ) (2 ,1 ) ( ) ( )2 1 1 2 2 11

& : plim T i j i j y xt r t q w i jr q wt

T y x ψ µ µ−− − − − −=

′ ′ = +∑Y X X Y

( )* * 1 (2 ) (2 ) (2 ,2 ) ( ) ( )2 2 2 21

: plim T i j i j y xt r t w i jr wt

T y x ψ µ µ−− − −=

′ = +∑Y X

( )* * 1 (1 ) (1 ) (1 ,1 ) ( ) ( )1 1 1 11

: plim T i j i j x xt q r t q w i jr wt

T x x δ µ µ−− − − − −=

′ = +∑X X

( )* * * * 1 (1 ) (2 ) (1 ,2 ) ( ) ( )1 2 2 1 1 21

& : plim T i j i j x xt q r t w i jr q wt

T x x δ µ µ−− − − − −=

′ ′ = +∑X X X X

( )* * 1 (2 ) (2 ) (2 ,2 ) ( ) ( )2 2 2 21

: plim T i j i j x xt r t w i jr wt

T x x δ µ µ−− − −=

′ = +∑X X

This proves (i) by showing that ZZ ≠Σ 0 , as required.

To show that (ii) holds define ( )1 1j j, ,=X ι Y X where 1 1 1( , )j jt jt -k=Y Y Y and

1 1 1( , )j jt jt -k=X X X . Let Z be defined as above. Therefore, the matrix ′Z X is given by:

* * * * *

1 1 1 1 1 1 1 1 1

* * * * *2 2 1 2 1 2 1 2 1

* * * * *1 1 1 1 1 1 1 1 1

* * * * *2 2 1 2 1 2 1 2 1

jt jt k jt jt k

jt jt k jt jt k

jt jt k jt jt k

jt jt k jt jt k

− −

− −

− −

− −

′ ′ ′ ′ ′

′ ′ ′ ′ ′ ′ =

′ ′ ′ ′ ′ ′ ′ ′ ′ ′

Y ι Y Y Y Y Y X Y X

Y ι Y Y Y Y Y X Y XZ X

X ι X Y X Y X X X X

X ι X Y X Y X X X X

(A1.6)

Now, using the same technique and applying Lemma 2, we derive typical elements of the

blocks of ′Z X specified in Eq. (A1.6) and show that they have non-zero probability limits.

Namely, we have:

( )( )( )( )

* 1 (1 ) ( )1 11

* 1 (2 ) ( )2 21

* 1 (1 ) ( )1 11

* 1 (2 ) ( )2 21

: plim

: plim

: plim

: plim

T i yt p r it

T i yt r it

T i xt q r it

T i xt r it

T y

T y

T x

T x

µ

µ

µ

µ

−− −=

−−=

−− −=

−−=

′ =

′ =

′ =

′ =

∑

∑

∑

∑

Y ι

Y ι

X ι

X ι

61

( )( )( )

* 1 (1 ) (1 ) (1 ,1 ) ( ) ( )1 1 1 11

* 1 (2 ) (1 ) (2 ,1 ) ( ) ( )2 1 2 11

* 1 (1 ) (1 ) (1 ,1 ) ( ) ( )1 1 1 11

* 12 1

: plim

: plim

: plim

: plim

T i j i j y yjt t p r t i jp rt

T i j i j y yjt t r t r i jt

T i j i j x yjt t q r t i jq rt

jt t

T y y

T y y

T x y

T x

γ µ µ

γ µ µ

ψ µ µ

−− − −=

−−=

−− − −=

−

′ = +

′ = +

′ = +

′

∑

∑

∑

Y Y

Y Y

X Y

X Y ( )(2 ) (1 ) (2 ,1 ) ( ) ( )2 11

T i j i j x yr t r i jt

y ψ µ µ−== +∑

( )( )( )

* 1 (1 ) (1 ) (1 ,1 ) ( ) ( )1 1 1 11

* 1 (2 ) (1 ) (2 ,1 ) ( ) ( )2 1 2 11

* 1 (1 ) (1 ) (1 ,1 ) ( ) ( )1 1 1 11

: plim

: plim

: plim

T i j i j y yjt k t p r t w i jp r wt

T i j i j y yjt k t r t w i jr wt

T i j i j x yjt k t q r t w i jq r wt

T y y

T y y

T x y

γ µ µ

γ µ µ

ψ µ µ

−− − − − − −=

−− − − −=

−− − − − − −=

′ = +

′ = +

′ = +

∑

∑

∑

Y Y

Y Y

X Y

( )* 1 (2 ) (1 ) (2 ,1 ) ( ) ( )2 1 2 11

: plim T i j i j x yjt k t r t w i jr wt

T x y ψ µ µ−− − − −=

′ = +∑X Y

( )( )( )

* 1 (1 ) (1 ) (1 ,1 ) ( ) ( )1 1 1 11

* 1 (2 ) (1 ) (2 ,1 ) ( ) ( )2 1 2 11

* 1 (1 ) (1 ) (1 ,1 ) ( ) ( )1 1 1 11

* 12 1

: plim

: plim

: plim

: plim

T i j i j y xjt t p r t i jp rt

T i j i j y xjt t r t r i jt

T i j i j x xjt t q r t i jq rt

jt t

T y x

T y x

T x x

T x

ψ µ µ

ψ µ µ

δ µ µ

−− − −=

−−=

−− − −=

−

′ = +

′ = +

′ = +

′

∑

∑

∑

Y X

Y X

X X

X X ( )(2 ) (1 ) (2 ,1 ) ( ) ( )2 11

T i j i j x xr t r i jt

x δ µ µ−== +∑

( )( )( )

* 1 (1 ) (1 ) (1 ,1 ) ( ) ( )1 1 1 11

* 1 (2 ) (1 ) (2 ,1 ) ( ) ( )2 1 2 11

* 1 (1 ) (1 ) (1 ,1 ) ( ) ( )1 1 1 11

: plim

: plim

: plim

T i j i j y yjt k t p r t w i jp r wt

T i j i j y yjt k t r t w i jr wt

T i j i j x xjt k t q r t w i jq r wt

T y x

T y x

T x x

ψ µ µ

ψ µ µ

δ µ µ

−− − − − − −=

−− − − −=

−− − − − − −=

′ = +

′ = +

′ = +

∑

∑

∑

Y X

Y X

X X

( )* 1 (2 ) (1 ) (2 ,1 ) ( ) ( )2 1 2 11

: plim T i j i j x xjt k t r t w i jr wt

T x x δ µ µ−− − − −=

′ = +∑X X

This proves (ii) by showing that ZX ≠Σ 0 , as required. Finally, to prove (iii) let uj = (u1, u2, u3), following notation of Eqs. (3.1)-(3.3). Thus, we

require that [ ] [ ] [ ]1 2 3E E E′ ′ ′= = =Z u Z u Z u 0 . Using definitions from Eqs. (2.13)-(2.15) this

implies:

[ ]

[ ] ( )[ ] ( )

1 1 1 - 1 -0 0

(y)2 2 2 1

(x)3 2 2 1

p q

t t j t j j t jj j

t t

t t

E E

E E

E E

= =

′ ′= + − −

′ ′= − ′ ′= −

∑ ∑Z u Z ζ ε B ε Γ δ

Z u Z ε Λ ε

Z u Z δ Λ δ

(A1.7)

Using the definition of Z (Table 2) we have:

62

[ ]

*1*2

1 1 1 - 1 -*0 01

*2

* * * *1 1 1 1 1 - 1 1 -

0 0

* * * *2 2 1 2 1 - 2 1 -

0 0

* *1 1 1

p q

t t j t j j t jj j

p q

t t j t j j t jj jp q

t t j t j j t jj j

t

E E

E

= =

= =

= =

′

′ ′ = + − − ′ ′

′ ′ ′ ′+ − −

′ ′ ′ ′+ − −=

′ ′+

∑ ∑

∑ ∑

∑ ∑

YYZ u ζ ε B ε Γ δXX

Y ζ Y ε Y B ε Y Γ δ

Y ζ Y ε Y B ε Y Γ δ

X ζ X ε * *1 1 - 1 1 -

0 0

* * * *2 2 1 2 1 - 2 1 -

0 0

1 1 1 1 1 - 1 1 -0 0

2 2 1 2

p q

t j t j j t jj jp q

t t j t j j t jj j

p q

t p r t t p r t t p r j t j t p r j t jj j

t r t t r t t r j

E

= =

= =

− − − − − − − −= =

− − −

′ ′− − ′ ′ ′ ′+ − −

′ ′ ′ ′+ − −

′ ′ ′+ −=

∑ ∑

∑ ∑

∑ ∑

X B ε X Γ δ

X ζ X ε X B ε X Γ δ

Y ζ Y ε Y B ε Y Γ δ

Y ζ Y ε Y B 1 - 2 1 -0 0

1 1 1 1 1 - 1 1 -0 0

2 2 1 2 1 - 2 1 -0 0

p q

t j t r j t jj jp q

t q r t t q r t t q r j t j t q r j t jj jp q

t r t t r t t r j t j t r j t jj j

−= =

− − − − − − − −= =

− − − −= =

′− ′ ′ ′ ′+ − − ′ ′ ′ ′+ − −

=

∑ ∑

∑ ∑

∑ ∑

ε Y Γ δ

X ζ X ε X B ε X Γ δ

X ζ X ε X B ε X Γ δ

0

(A1.8)

Q.E.D.

Proof of Proposition 2 We show that a general structural equation latent variable ADL

model equations can be written in the form uWGVK =+ . Then the result follows using a

known result (see, e.g. Judge, et al., 1985) that: [ ( , ) ] 1jrank M′ ′ ′ ≥ −R K G , where M is the

number of the left-hand side variables, i.e., number of the equations in the system and where

Rj is the selection matrix for the jth equation, therefore M = m + n +h.

First, Eqs. (3.1)-(3.3) can be written as:

1 0 1 1 1 1 1 0 1 1 1 1 1 1t

(y) (y)2 2 2 1 2

(x) (x)2 2 2 1 3

... ...t t t p t p t t q t q

t t t

t t t

η − − − −= + + + + + + + + +

= + +

= + +

y α B y B y B y Γ x Γ x Γ x u

y α Λ y u

x α Λ x u

(A1.8)

Rearranging by moving endogenous variables on the left hand side we get:

63

0 1 1 1 1 1 0 1 1 1 1 1 1t

(y) (y)2 1 2 2 2

(x) (x)2 2 2 1 3

( ) ... ...t t p t p t t q t q

t t t

t t t

η − − − −− − − − − − − − − =

− − − =

− + =

I B y α B y B y Γ x Γ x Γ x u

Λ y y α u

x α Λ x u

(A1.7)

which can be written in a single matrix expression as:

1 1

1 2

1 2 0 10 1 1t(y) (y)

12 2 2 2(x) (x)

2 312 2

1 1

1

( )

t

t

p qt

t pn t t

h t tt

t

t q

η

−

−

−

−

−

− − − − − − − − − − − = −

ιyy

α B B B Γ Γ ΓI B 0 0 y uyΛ I 0 y α 0 0 0 0 0 0 u

0 0 I x uxα 0 0 0 Λ 0 0x

x

(A1.9) Transposing the Eq. (A1.9) we get:

[ ]

[ ]

(y)0 2

1 2 2

( ) ( )2 2

1

2

1 1 1 2 1 1 1 1 1 1t 2 3( )

0 2

1

( )t t t n

h

y x

pt t t p t t t q t tx

q

η

− − − − −

′′− − +

− − − ′−

′− ′− = ′′− − ′− ′−

I B Λ 0y y x 0 I 0

0 0 I

α α αB 0 0B 0 0

B 0 0ι y y y x x x u u uΓ 0 ΛΓ 0 0

Γ 0 0

(A1.10) which can be now written in a full-sample notation (see Table 1):

( ) ( )2 2

1

2(y)

0 2

1 2 2 1 1 1( )

0 2

1

( )

y x

pj j j n jt k jt jtx

h

q

η

−

− − − ′−

′− ′′− − ′− − = ′′− − ′− ′−

α α αB 0 0B 0 0

I B Λ 0B 0 0y y x 0 I 0 ι Y X X u

0 0 I Γ 0 ΛΓ 0 0

Γ 0 0

64

(A1.11) Now define: 1 2 2( , ) jt jt jt,≡V y y x , 1 1 1( )jt jt k jt k, , ,− −≡ ιW X X Y ,

(y)0 2( )

n

h

′′− −

≡

I B Λ 0K 0 I 0

0 0 I, and

( ) ( )2 2

1

2

(x)0 2

1

y x

p

q

η ′− ′− ′−=

′′ − − ′−

′−

α α αB 0 0B 0 0

B 0 0GΓ 0 ΛΓ 0 0

Γ 0 0

Therefore the system of equations (A1.10) is given by uWGVK =+ , as required.

Q.E.D.

Appendix 2: Programme syntax A2.1 SPSS syntax for data manipulation Syntax for sub-sample selection (household-level variables), omitting the wave replications

and noting that code is repeated by using appropriate wave-specific identifiers (a, b, c, d, e, f,

g, h, i, j) is given by:

get file 'c:\bhps\ahhresp.sav'/keep ahid ahhsize axphsd1 axphsd2 axphsdb axphsdf axpfood axphsg.

save out = 'c:\bhps\ahhresp2.sav'....

get file 'c:\bhps\jhhresp.sav'/keep jhid jhhsize jxphsd1 jxphsd2 jxphsdb jxphsdf jxpfood jxphsg.

save out = 'c:\bhps\jhhresp2.sav'.

Similarly, sub-sample selection syntax for individual-level variables is given by: get file 'c:\bhps\aindresp.sav'

/keep ahid pid aqfachi ahgemp afiyrl afiyrnl afiyri asaved.save out = 'c:\bhps\aindresp2.sav'.

.

.

.get file 'c:\bhps\jindresp.sav'

/keep jhid pid jqfachi jhgemp jfiyrl jfiyrnl jfiyri jsaved.save out = 'c:\bhps\jindresp2.sav'.

65

Household-level data can be spread to individual-level by the following programme:

match files file = 'c:\bhps\aindresp2.sav'/table = 'c:\bhps\ahhresp2.sav' /by ahid.

sort cases by pid.save out = 'c:\bhps\aresp.sav'.

.

.

.match files file = 'c:\bhps\jindresp2.sav'

/table = 'c:\bhps\jhhresp2.sav' /by jhid.sort cases by pid.save out = 'c:\bhps\jresp.sav'.

The above given syntax will produce separate files for each individual wave (i.e., year)

retaining relevant variables and matching household level information with individual-level

data, all placed on the lowest, individual, record type. Though it is possible to immediately

create a panel-structured data set (in SPSS language called “long-format”), it is advisable to

handle all missing data problems and insure that retained individuals are only those present in

all 10 waves in the “wide-format” (SPSS term for parallel placement of subsequent

observations on the same variables). The following syntax will produce such wide-form

(single) data set:

match files file = "c:\bhps\aresp.sav" /in=w1/file = "c:\bhps\bresp.sav" /in=w2/file = "c:\bhps\cresp.sav" /in=w3/file = "c:\bhps\dresp.sav" /in=w4/file = "c:\bhps\eresp.sav" /in=w5/file = "c:\bhps\fresp.sav" /in=w6/file = "c:\bhps\gresp.sav" /in=w7/file = "c:\bhps\hresp.sav" /in=w8/file = "c:\bhps\iresp.sav" /in=w9/file = "c:\bhps\jresp.sav" /in=w10

/by=pid.

save out = 'c:\bhps\bhps_wide.sav'/keep = w1 w2 w3 w4 w5 w6 w7 w8 w9 w10

ahid ahhsize axphsd1 axphsd2 axphsdb axphsdf axpfood axphsg pid aqfachiahgemp afiyrl afiyrnl afiyri asaved

.

.

.jhid jhhsize jxphsd1 jxphsd2 jxphsdb jxphsdf jxpfood jxphsg pid jqfachijhgemp jfiyrl jfiyrnl jfiyri jsaved.

Note that we omit listing variables with all wave identifiers to save space. Individual wave-

identifiers can be used to drop individuals that did not stay in the sample for all 10 years, and

this can be obtained by the following command:

filter off.use all.

66

select if (w1 = 1 & w2 = 1 & w3 = 1 & w4 = 1 & w5 = 1 & w6 = 1 & w7 = 1 &w10 = 1).execute.

From wide-form data set one can directly convert to long-form (which is needed for our

analysis (see Taylor, et al. 2001) using simple programming code, but this might create

problems with very large data sets. Though more tedious in terms of programme length, it

would be recommended, nevertheless, to split the wide-form to wave-individual files, which

will not contain only those individuals that are present in the sample for all 10 years of

survey. This is achieved by the following programme:

get file 'c:\bhps\bhps_wide.sav'

/keep ahid ahhsize axphsd1 axphsd2 axphsdb axphsdf axpfood axphsg pid aqfachi ahgemp afiyrl afiyrnl afiyri asaved.save out = 'c:\bhps\apanel.sav'.

.

.

.get file 'c:\bhps\bhps_wide.sav'

/keep jhid jhhsize jxphsd1 jxphsd2 jxphsdb jxphsdf jxpfood jxphsg pid jqfachi jhgemp jfiyrl jfiyrnl jfiyri jsaved.save out = 'c:\bhps\jpanel.sav'. Finally, we merge these files in the needed long-form (panel) data set by running the

following programme (again, omitting repeated specification for waves b-i): get file 'c:\bhps\apanel.sav'/keep ahid ahhsize axphsd1 axphsd2 axphsdb axphsdf axpfood axphsg pid aqfachi ahgemp afiyrl afiyrnl afiyri asaved.

compute wave=1.

rename var

ahid=hidahhsize=hhsizeaxphsd1=xphsd1axphsd2=xphsd2axphsdb=xphsdbaxphsdf=xphsdfaxpfood=xpfoodaxphsg=xphsgaqfachi=qfachiahgemp=hgempafiyrl=fiyrlafiyrnl=fiyrnlafiyri=fiyriasaved=saved.

save out = "c:\temporary\apaneltemp.sav"....

get file 'c:\bhps\jpanel.sav'

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/keep jhid jhhsize jxphsd1 jxphsd2 jxphsdb jxphsdf jxpfood jxphsg pid jqfachi jhgemp jfiyrl jfiyrnl jfiyri jsaved.

compute wave=1.

rename var

jhid=hidjhhsize=hhsizejxphsd1=xphsd1jxphsd2=xphsd2jxphsdb=xphsdbjxphsdf=xphsdfjxpfood=xpfoodjxphsg=xphsgjqfachi=qfachijhgemp=hgempjfiyrl=fiyrljfiyrnl=fiyrnljfiyri=fiyrijsaved=saved.

save out = "c:\temporary\jpaneltemp.sav".

add file file = 'c:\temporary\apaneltemp.sav'/file = 'c:\temporary\bpaneltemp.sav'/file = 'c:\temporary\cpaneltemp.sav'/file = 'c:\temporary\dpaneltemp.sav'/file = 'c:\temporary\epaneltemp.sav'/file = 'c:\temporary\fpaneltemp.sav'/file = 'c:\temporary\gpaneltemp.sav'/file = 'c:\temporary\hpaneltemp.sav'/file = 'c:\temporary\ipaneltemp.sav'/file = 'c:\temporary\jpaneltemp.sav'

sort cases by pid, wave.

A2.2 PcGive/PcFiml code The estimation batch syntax that can be run from OxMetrix software, namely PcGive and

PcFiml, is given below (see Hendry and Doornik, 1999a; 1999b for more details and

programming reference). Assuming that the working file containing the data is opened in the

working data base of GiveWin (here named “BHPS_PcGive_NM.xls”) and that the appropriate

lags and differences were created, the following command syntax will reproduce the results

presented in the paper (to obtain GIVE results for differenced data drop the “Constant” and

replace levels by differences on the “Y” line; see note on creating differences in the section on

3SLS estimation below).

Single-equation (GIVE) syntax: module("PcGive");usedata("BHPS_PcGive_NM.xls");

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system{

Y = S, L;Z = Constant, S_1, F_1, L_1;A = B_2, H_2, NL_2, I_2;

}estsystem("IVE", 3, 1, 26736, 1); module("PcGive");usedata("BHPS_PcGive_NM.xls");system{

Y = F, S, L;Z = Constant, F_1, L_1;A = B_2, H_2, NL_2;

}estsystem("IVE", 3, 1, 26736, 1);

module("PcGive");usedata("BHPS_PcGive_NM.xls");system{

Y = L;Z = Constant, L_1, ED;

}estsystem("IVE", 3, 1, 26736, 1);

module("PcGive");usedata("BHPS_PcGive_NM.xls");system{

Y = B, S;Z = Constant;A = H_2, F_2;

}estsystem("IVE", 3, 1, 26736, 1);

module("PcGive");usedata("BHPS_PcGive_NM.xls");system{

Y = H, F;Z = Constant;A = B_2, S_2;

}estsystem("IVE", 3, 1, 26736, 1);

module("PcGive");usedata("BHPS_PcGive_NM.xls");system{

Y = NL, L;Z = Constant;A = H_2, S_2;

}estsystem("IVE", 3, 1, 26736, 1);

module("PcGive");usedata("BHPS_PcGive_NM.xls");system{

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Y = I, L;Z = Constant, F_1, L_1;A = H_2, ED;

}estsystem("IVE", 3, 1, 26736, 1);

3SLS (system) estimation code: module("PcFiml");usedata("BHPS_PcGive_NM.xls");system{

Y = S, F, L, B, H, NL, I;Z = S_1, F_1, L_1, ED, S_2, F_2, L_2, B_1, B_2, H_1, H_2, NL_1, NL_2,

I_1, I_2, Constant;

}estsystem("OLS", 2, 1, 26736, 1);model{

S = L, S_1, F_1, L_1, Constant;F = S, L, F_1, L_1, Constant;L = L_1, ED, Constant;B = S, Constant;H = F, Constant;NL = L, Constant;I = L, Constant;

}estmodel("3SLS", 0);

Defining dS=S-S_1, dF=F–F_1, etc. the code for the differenced model is given by:

module("PcFiml");usedata("BHPS_PcGive_NM.xls");system{

Y = dS, dF, dL, dB, dH, dNL, dI;Z = dS_1, dF_1, dL_1, ED, S_2, F_2, L_2, B_2, H_2, NL_2, I_2;

}estsystem("OLS", 2, 1, 26736, 1);model{

dS = dL, dS_1, dF_1, dL_1;dF = dS, dL, dF_1, dL_1;dL = dL_1, ED;dB = dS;dH = dF;dNL = dL;dI = dL;

}estmodel("3SLS", 0);