Estimation of Parameters in the Presence of Model misspecification

and Measurement Error

P. A. V. B. Swamy, George S. Tavlas, Stephen G. HallAnd

George Hondroyiannis

Introduction

Most econometric relationships are subject to at least the following three problems

•Measurement Error•The true functional form is unknown•Omitted variables

While there are specific techniques or procedures that deal with these problems one at a time there is no single approach which works for them all.

This paper proposes a new estimation strategy which potentially deals with all forms of model misspecification simultaneously.

Unlike some non-parametric techniques the proposed method is feasible with relatively small data sets

It also offers a new interpretation of regression coefficients in the presence of misspecification and a formal representation of the biases which result

2. The Interpretation of Model Coefficients

Consider the relationship between an endogenous variable and K-1 of its determinants ….

Where in particular K-1 may be only a subset of the full set of determinants so that we have omitted variables . In addition we have measurement error = + , = +

And we may be estimating the wrong functional form

*ty

*1tx *

1,K tx

ty *ty 0tv jtx *

jtx jtv

What is the econometrician trying to achieve?

Our answer to this is to derive an estimate of the partial derivative of with respect to , and to test hypothesis about this.

If you want to estimate the true model then there really is no alternative to specifying it correctly.

However if you are only interested in partial effects then we offer another way forward.

*ty

*jtx

Consider the following time varying parameter model

= + + … +

All potential misspecification is captured in the time varying coefficients which offer a completeExplanation of y.

Now the key question is what are the stochastic assumptions about the TVCs

ty 0t 1 1t tx 1, 1,K t K tx

The correct stochastic assumptions about the TVC comes from an understanding of the misspecification which drives the time variation.

Notation and assumptionsLet denote the total number of determinants of y, this can not generally be known, but in general m>k-1.Now let and j=1…K-1 and g=k…m be the true coefficients on the underlying model, where the parameters vary because of either a non-linear functional form or truly changing parameters

tm

*0t *

jt *gt

Now for g=k… let denote the intercept and j=1…K-1 denote the other coefficients of the regression of on …

tm *0gt

*jgt

*gtx *

1tx *1,K tx

Then we can establish the following representation

Theorem 1.

= + +

And

= j=1…k-1

The first term is the true variation, the second the measurement effect, the third the omitted variables.

0t *0t * *

0

tm

gt gtg K

0tv

jt jt* * * * * *

jt

v

x

t tm m

jt gt jgt jt gt jgtg K g K

And if we estimate a standard fixed coefficient regression model then the error term comprises all these effects.

This means that the error term can not be independent of the included Xs (as it contains them). It is also impossible for valid instruments to exist in this example as if a variable is correlated with the included X variables it must be correlated with the errors (as the error contains the included Xs)

Theorem 2:For j=1…K-1 the component of is the direct or ‘biased free’ effect of on with all the other determinants of held constant, and it is unique.

The direct effect will be constant if the relationship between y and all the Xs is linear and time invariant.

This is a useful interpretation of standard regression coefficients, which emphasises their potential biases.

*jt jt

*jtx *

ty*ty

To make this approach useful as an estimation strategy we must have some way of identifying the bias part of the TVC.

= + +

Assumption 1 Each coefficient of (1) is linearly related to certain drivers plus a random error,

jt0j

1

1

p

jd dtd

z

jt

Assumption 2: For j=1,…,K-1, the set of P-1 coefficient drivers and the constant term divides into two disjoint subsets S1 and S2 so that + has the same pattern of time variation as and + has the same pattern of time variation as the sum of the last two terms on the RHS of (3) over the relevant estimation and forecasting periods.

So we assume the drivers identify the bias component. This is like the dual of IV

0j1

jd dtd Sz

*jt

2jd dtd S

z jt

Assumption 3, The K-vector = Of errors in (4) follows the stochastic equation

= +

Assumption 4, The regressor of (1) is conditionally independent of its coefficient given the coefficient drivers for all j and t

t 0 1 1,( , ,..., )t t K t

t 1t tu

jtxjt

A vector formulation of the model

Where

And =

Then

or

t t ty x

t t tz

0 1,0 1jd j K d p

( ) Longt t t t ty z x x

Longz xy X D

Theorem 3Under Assumptions 1-4

E =

And

Var =

Where is the covariance matrix of

( | )zy X LongzX

( | )zy X 2x u xD D

2u

3. Identification and Consistent Estimation of TVC

The fixed coefficient vector is identified if . has full column rank. A necessary condition for this is that T>Kp.

The errors are not identified

Thus assumptions 1-4 make all the fixed parameters of the model identifiable.

This does not happen if we assume random walk TVCs

LongzX

Estimation under assumptions 1-4

IV estimation works under very extreme assumptions, we argue that in the presence of general mis-specification IV can not work

Estimation under 1-4Why IV can not workFrom Theorem 1

= + +

And

= j=1…k-1And so if we estimate fixed parameters the second and third term go into the error. An instrument can not be correlated with x and uncorrelated with the error.

0t *0t * *

0

tm

gt gtg K

0tv

jt jt* * * * * *

jt

v

x

t tm m

jt gt jgt jt gt jgtg K g K

Practical EstimationThe TVC Model can be estimated by an iteratively rescaled generalized least squares (IRSGLS) method developed in Chang, Swamy, Hallahan and Tavlas (2000).

Or alternatively it can be specified in state space form and estimated by maximum likelihood.

Choice of coefficient drivers

At present this is somewhat arbitrary as in the case of selecting instrumental variables. However some guidelines… Coefficient drivers should

Be correlated with omitted variables or structural change

Be correlated with any policy change

Selecting a subset of the drivers

The bias can be removed by selecting a subset of the drivers and removing their effect. How do we choose this subset (S1 and S2):

Theory often suggests a sign or value for the true parameter. Coefficient drivers which give coefficients outside this range should be rejected

Beyond that we have the following sugestion

3.6 Bayesian averaging of coefficient drivers.

Each possible set or subset of drivers may be seen as a different model of the bias component. If we assign a prior probability to each admissible model then we can compute the prior density for each M

=

And then average over these

=

( | , )z ip y X M Long( | , , , ) ( , | )d dLong

Long Longz i ip y X M p M

z i( | X of all )p y M ( | , ) ( )z i ii

p y X M P M

By Bayes Theorem the Posterior probability of M is

=

And the posterior for is

=

i( | , of all )i zP M y X M( ) ( | , )

( | of all )i z i

z i

P M p y X M

p y X M

jd

( | , , )jd z ip y X MLong-jd

Long

( | , , , ) ( , | )d d

( | , , , ) ( , | )d d

Longjd

Long

Long Longz i i

Long Longz i i

p y X M p M

p y X M p M

A Bayes estimator of is then

=

+

*jt

*jtE( | , of all )z iy X M j0[ ( | , of all ){E( | , , )i z i z i

i

P M y X M y X M

1

jdE( | , , ) }]i

z i dtd S

y X M z

ConclusionsAll models are subject to a range of sources of bias

We offer a new representation of this bias

This implies a potentially powerful way of estimating consistent effects

It also offers a fundamental criticism of instrumental variables in practise

An Example

A PORTFOLIO BALANCE APPROACH TO EURO-AREA MONEY DEMAND

IN A TIME-VARYING ENVIRONMENT Stephen G.Hall

Leicester University, Bank of Greece and NIESRGeorge Hondroyiannis

Bank of Greece P.A.V.B. Swamy

U.S. Bureau of Labor StatisticsGeorge S. TavlasBank of Greece

1. Introduction

Money has played an important role in the setting of monetary policy within the Euro area since the inception of the ECB

A reference value of 4.5% growth in M3 has been set but rarely achieved

This has raised concerns about

• The possibility of a monetary overhang

• The reference path itself

• The stability of the money demand function

0

2

4

6

8

10

12

80 82 84 86 88 90 92 94 96 98 00 02 04 06

M3 growth inflation

Figure 2Annualized Inflation Rate and M3 Growth

Reference value 4.5%in effect sinceJanuary 1999

Between 2001 and 2003 a number of attempts were made to repair the money demand function but these all proved unstable beyond 2003 (Beyer, Fischer, and von Landesberger,

2007; Fischer, Lenza, Pill, and Reichlin, 2007)

This paper considers money demand from a portfolio balance perspective and emphasizes the role of wealth. Unfortunately there are not good data for euro wide wealth.

Our view is that there is a stable but complex money demand function which really requires much better wealth data to uncover

In the absence of this good data we propose to use two techniques to support our contention

A structural VEC approach augmented by a range of effects to proxy what we believe has been happening to wealth

A TVC approach which is designed to reveal structural underlying parameters which are being obscured by a range of econometric problems including omitted variables, measurement error and misspecified functional form

2. Theoretical and empirical underpinnings

The basic portfolio theory implies an equilibrium relationship of the form

Assuming rate of return homogeneity

And assuming log linearity and reparameterising gives

),,,( eeem prprwyfpm

),,()( me rrwyfpm

tme

ttt urraywayaapm )()()( 3210

2.3 Existing modelsThe current ‘workhorse’ model used by the ECB has the

following long run

This is known to be unstable since 2001 but the model currently in use still uses this long run as no stable alternative has been found

The income elasticity of 1.3 suggests omitted wealth effects with wealth growing faster than income.

Re-estimating this model using ECB data to 2006:3 gives the following results

tmi

tt rrykpm )(1.13.1)(

Which clearly shows no cointegration

We can see why in terms of velocity in the next picture

( )i mr r( )l mr r( )i mr r( )i mr r( )l mr r( )i mr r (m-p), y, , (m-p), y, (m-p), y, , (m-p), y,

Table 1

Johansen Co-integration Tests of Long-Run Money Demand, ECB “Workhorse” Model: Sample 1980:Q1 – 2006:Q3

Variables included Rank=0 Rank≤1 Rank≤2 Rank≤3 Co-integration

Maximum Eigenvalue

13.04 10.95 2.90 0.42 No

9.82 2.61 0.37 - No

Trace Statistic

27.31 14.27 3.32 0.42 No

12.80 2.98 0.37 - No

Notes: A VAR model of order two is used.

.1

.2

.3

.4

.5

.6

80 82 84 86 88 90 92 94 96 98 00 02 04 06

Figure 1Log of Income Ve locity

3. Data and Empirical Results

So given the lack of good data on wealth and its components (especially perhaps house prices) we construct a set of proxy variables.

A stock market index

An HP filtered version of the stock market index

A split trend starting in 2001 (ST1)

A complex dummy capturing stock market effects between 1988 and 1994 (ST2) (German unification, the hard EMS etc)

The change in oil prices was also used in the VAR to help with the explanation of inflation

Table 2

Johansen Co-integration TestsLong-Run Demand for Money in Euro Area: Sample 1980:Q1-2006:Q3

VAR of order 2, Variables: (m-p), y, (w-y)and six exogenous variables

Maximum Eigenvalue

Null Alternative Eigenvalue CriticalValues

95% 99%

r=0 r=1 29.50*** 20.97 25.52

r<=1 r=2 17.63** 14.07 18.63

r<=2 r=3 13.10*** 3.76 6.65

Trace Statistic

Null Alternative Trace CriticalValues

95% 99%

r=0 r>=1 60.23*** 29.68 35.65

r<=1 r>=2 30.73*** 15.41 20.04

r<=2 r>=3 13.10*** 3.76 6.65

We then apply 9 identifying restrictions to allow us to give a structural interpretation to the vectors for money, income and the wealth to income ratio. Here is the money demand CV

t(m-p) ty t(w-y) tst1 t-1hp(w-y) tst2

VEC Model Estimation

(a) Co-integrating Equation

1.000 -0.829 0.248 -0.02 -0.574 -0.0028

(-5.55) (6.35 (3.0) (-2.39) (-1.2)

The Dynamic Money demand function

ECTt-1(m-p) t-2(m-p) t-1y

t-2y t-1(w-y) t-2(w-y)tp t-1oil e m

t-1(r r )

unrestricted model

-0.034(-2.23)

0.551(5.73)

-0.103(-1.14)

-0.055(-0.71)

0.127(1.64)

-0.004(-0.63)

-0.001(-0.18)

0.146(4.7)

0.004(1.69)

-0.001(-0.08)

Parsimonious model

-0.034(-2.27)

0.0552(5,82)

-0.117(-1.35)

- 0.121(1.62)

- - 0.148(4.9)

0.004(1.69)

-0.001(0.84)

Recursive estimates of the long run coefficients

1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007

-1.0

-0.5

0.0

y

1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007

0.2

0.4

0.6

0.8 w-y

Stability tests

1990 1995 2000 2005

0.0

0.5

1.0

1.5 1-step aheadm-p 5%

1990 1995 2000 2005

0.5

1.0

1-step aheadsystem 5%

1990 1995 2000 2005

0.5

1.0predictive failurem-p 5%

1990 1995 2000 2005

0.5

1.0

predictive failure

system 5%

1990 1995 2000 2005

0.25

0.50

0.75

1.00break-pointm-p 5%

1990 1995 2000 2005

0.50

0.75

1.00

break-pointsystem 5%

CUSUM and CUSUMSQ tests

-12

-8

-4

0

4

8

12

02Q3 03Q1 03Q3 04Q1 04Q3 05Q1 05Q3 06Q1 06Q3

CUSUM 5% Significance

-0.4

0.0

0.4

0.8

1.2

1.6

02Q3 03Q1 03Q3 04Q1 04Q3 05Q1 05Q3 06Q1 06Q3

CUSUM of Squares 5% Significance

So a highly stable money demand function once we have allowed for these other factors which amount to wealth proxies.

3.2 TVC results

Next we present the results for the TVC estimation.

We investigate the following model

Where

Which implies the long run is our primary interest

tttttt ywayaapm )()( 210

jtptjptjjjt zza ...110

The coefficient drivers are all the variables making up the dynamic VEC money demand function.

The equation for the coefficient allows us to break up the evolution in the coefficients into that caused by simultaneous equation bias or omitted variable bias and then remove this.

2R

Table 4

TVC Estimation of Long-Run Money Demand for Euro-Area

Variables Total effects(1)

Bias-free effects (2)

•Constant -9.425***[-5.63]

-7.239***[-1.66]

•y 1.274***[11.38]

1.113***[14.86]

•w-y -0.022[-1.13]

0.373***[15.66]

0.99 0.99

.

• This is the average effect, the next three slides show the time variation in the total coefficient and the bias free coefficient

• The interesting thing here is that in each case the bias free coefficients are remarkably stable

Conclusion

We have argued that the demand for money is a stable function but that it requires more than the usual small group of variables to produce a good model

In particular we believe there is a strong need for good wealth data.

We support this view by demonstrating a stable VEC model with effects proxying wealth and by showing that the TVC technology reveals stable underlying long run elasticity's