Dougherty С. Introduction to Econometrics, 3Ed

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SIMULTANEOUS EQUATIONS

ESTIMATION

If you employ OLS to estimate the parameters of an equation that is embedded in a simultaneous

equations model, it is likely that the estimates will be biased and inconsistent and that the statistical

tests will be invalid. This is demonstrated in the first part of this chapter. The second part discusses

how these problems may be overcome by using instrumental variables estimation.

10.1 Simultaneous Equations Models: Structural and Reduced Form Equations

Measurement error is not the only possible reason why the fourth Gauss–Markov condition may not be

satisfied. Simultaneous equations bias is another, and it is best explained with an example.

Suppose that you are investigating the determinants of price inflation and wage inflation. We will

start with a very simple model that supposes that

, the annual rate of growth of prices, is related to

the annual rate of growth of wages, it being hypothesized that increases in wage costs force prices

upwards:

(10.1)

At the same time

is related to

and

, the rate of unemployment, workers protecting their real

wages by demanding increases in wages as prices rise, but their ability to do so being the weaker, the

higher the rate of unemployment (

< 0):

(10.2)

and

are disturbance terms.

By its very specification, this simultaneous equations model involves a certain amount of

circularity:

determines

in the first equation, and in turn

helps to determine

in the second. To

cut through the circularity we need to make a distinction between

endogenous

and

exogenous

variables. Endo- and exo- are Greek prefixes that mean within and outside, respectively. Endogenous

variables are variables whose values are determined by the interaction of the relationships in the

model. Exogenous ones are those whose values are determined externally. Thus in the present case

and

are both endogenous and

is exogenous. The exogenous variables and the disturbance terms

ultimately determine the values of the endogenous variables, once one has cut through the circularity.

The mathematical relationships expressing the endogenous variables in terms of the exogenous

SIMULTANEOUS EQUATIONS ESTIMATION

variables and disturbance terms are known as the reduced form equations. The original equations that

we wrote down when specifying the model are described as the structural equations. We will derive

the reduced form equations for p and w. To obtain that for p, we take the structural equation for p and

substitute for w from the second equation:

p =

w + u

(

p +

U + u

) + u

(10.3)

Hence

(1 –

)p =

U + u

(10.4)

and so

223211

−

++++

uuU

(10.5)

Similarly we obtain the reduced form equation for

(

) +

(10.6)

Hence

(1 –

)

(10.7)

and so

23121

αα

−

++++

uuU

(10.8)

Exercise

10.1*

A simple macroeconomic model consists of a consumption function and an income identity:

where

is aggregate consumption,

is aggregate investment,

is aggregate income, and

is a

disturbance term. On the assumption that

is exogenous, derive the reduced form equations for

and

SIMULTANEOUS EQUATIONS ESTIMATION

10.2 Simultaneous Equations Bias

In many (but by no means all) simultaneous equations models, the reduced form equations express the

endogenous variables in terms of all of the exogenous variables and all of the disturbance terms. You

can see that this is the case with the price inflation/wage inflation model. In this model, there is only

one exogenous variable, U. w depends on it directly; p does not depend it directly but does so

indirectly because it is determined by w. Similarly, both p and w depend on u

, p directly and w

indirectly. And both depend on u

, w directly and p indirectly.

The dependence of w on u

means that OLS would yield inconsistent estimates if used to fit

equation (10.1), the structural equation for p. w is a stochastic regressor and its random component is

not distributed independently of the disturbance term u

. Similarly the dependence of p on u

means

that OLS would yield inconsistent estimates if used to fit (10.2). Since (10.1) is a simple regression

equation, it is easy to analyze the large-sample bias in the OLS estimator of

and we will do so.

After writing down the expression for

OLS

b , the first step, as usual, is to substitute for p. Here we

have to make a decision. We now have two equations for p, the structural equation (10.1) and the

reduced form equation (10.5). Ultimately it does not matter which we use, but the algebra is a little

more straightforward if we use the structural equation because the expression for

OLS

b decomposes

immediately into the true value and the error term. We can then concentrate on the error term.

(w)

wuwww

wuw

Var

),(Cov

)(Var

),(Cov),(Cov),(Cov

)(Var

)],([Cov

)(Var

),(Cov

OLS

(10.9)

The error term is a nonlinear function of both u

and u

(remember that w depends on both) and it

is not possible to obtain an analytical expression for its expected value. Instead we will investigate its

probability limit, using the rule that the probability limit of a ratio is equal to the probability limit of

the numerator divided by the probability limit of the denominator, provided that both exist. We will

first focus on plim Cov(u

, w). We need to substitute for w and again have two choices, the structural

equation (10.2) and the reduced form equation (10.8). We choose (10.8) because (10.2) would

reintroduce p and we would find ourselves going round in circles.













−













++++

−

),(Cov plim),(Cov plim

),(Cov plim])[,(Cov plim

)(

,Cov plim),(Cov plim

3121

23121

ppwp

pwpp

uuuu

Uuu

uuUuwu

αα

(10.10)

Cov(u

, [

]) is 0 since [

] is a constant. plim Cov(u

, U) will be 0 if U is truly

exogenous, as we have assumed. plim Cov(u

, u

) will be 0 provided that the disturbance terms in the

structural equations are independent. But plim Cov(u

, u

) is nonzero because it is plim Var(u

) and

the limiting value of the sample variance of u

is its population variance,

. Hence

SIMULTANEOUS EQUATIONS ESTIMATION

),(Cov

lim

σα

−

(10.11)

Now for plim Var(













−













−

121

Var

lim

Var

lim)Var(

lim

αα

uuU

(10.12)

since

121

αα

−

is an additive constant. So













−

),(Cov

lim2

),(Cov

lim2),(Cov

lim2

)Var(

lim)Var(

lim

)1(

)Var(

lim

233

uUuU

uuU

ααα

αα

(10.13)

Now if

and

are independently distributed, the limiting values of the three covariance

terms are 0. The limiting values of the variance terms are the corresponding population variances.

Hence

()

222

)1(

)Var(

lim

uuU

σασσα

−

(10.14)

Thus

222

OLS

)1(

lim

uuU

σασσα

σα

−+=

(10.15)

and so

OLS

is an inconsistent estimator of

The direction of simultaneous equations bias depends on the structure of the model being fitted.

Can one say anything about it in this case? Variances are always positive, if not 0, so it depends on

the sign of (1 –

). Looking at the reduced form equation (10.8), it is reasonable to suppose that

will be negatively influenced by

. Since it is also reasonable to suppose that

is negative, one may

infer that (1 –

) is positive. Actually, this is a condition for equilibrium in this model. Consider

the effect of a change

∆

. In view of (10.2), its immediate effect is to change

, in the opposite

direction, by an amount

∆

. Looking at (10.1), this in turn changes

by an amount

∆

Returning to (10.2), this causes a secondary change in

∆

, and hence, returning to (10.1),

a secondary change in

equal to

232

αα

∆

. Returning again to (10.2), this causes a further change

equal to

αα

∆

. The total change in

will therefore be

∆

222

...)1(

++++∆

(10.16)

SIMULTANEOUS EQUATIONS ESTIMATION

and this will be finite only if

< 1.

A Monte Carlo Experiment

This section reports on a Monte Carlo experiment that investigates the performance of OLS and, later,

IV when fitting the price inflation equation in the price inflation/wage inflation model. Numerical

values were assigned to the parameters of the equations as follows:

p = 1.5 + 0.5w + u

(10.17)

w = 2.5 + 0.5p – 0.4U + u

(10.18)

U was assigned the values 2, 2.25, increasing by steps of 0.25 to 6.75. u

was generated as a normal

random variable with 0 mean and unit variance, scaled by a factor 0.8.

The disturbance term u

is not

responsible for bias when OLS is used to fit the price inflation equation and so, to keep things simple,

it was suppressed. Each replication of the experiment used a sample of 20 observations. Using the

expression derived above, plim

OLS

b is equal to 0.87 when the price inflation equation is fitted with

OLS. The experiment was replicated 10 times with the results shown in the Table 10.1.

It is evident that the estimates are heavily biased. Every estimate of the slope coefficient is above

the true value of 0.5, and every estimate of the intercept is below the true value of 1.5. The mean of

the slope coefficients is 0.96, not far from the theoretical plim for the OLS estimate. Figure 10.1

shows how the bias arises. The hollow circles show what the relationship between p and w would

look like in the absence of the disturbance terms, for 20 observations. The disturbance term u

alters

the values of both p and w in each observation when it is introduced. As can be seen from the reduced

form equations, it increases p by an amount u

/(1 –

) and w by an amount

/(1 –

). It

follows that the shift is along a line with slope 1/

. The solid circles are the actual observations, after

has been introduced. The shift line has been drawn for each observation. As can be seen, the

overall effect is to skew the pattern of observations, with the result that the OLS slope coefficient is a

compromise between the slope of the true relationship,

, and the slope of the shift lines, 1/

. This

can be demonstrated mathematically be rewriting equation (10.15):

ABLE

10.1

Sample a s.e.(a) b s.e.(b)

1 0.36 0.39 1.11 0.22

2 0.45 0.38 1.06 0.17

3 0.65 0.27 0.94 0.12

4 0.41 0.39 0.98 0.19

5 0.92 0.46 0.77 0.22

6 0.26 0.35 1.09 0.16

7 0.31 0.39 1.00 0.19

8 1.06 0.38 0.82 0.16

9 –0.08 0.36 1.16 0.18

10 1.12 0.43 0.69 0.20

SIMULTANEOUS EQUATIONS ESTIMATION

Figure 10.1





































−+=

222

OLS

)1(

lim

uuU

σασσα

σα

σασσα

σσα

σασσα

σα

σασσα

σα

(10.19)

plim

OLS

is thus a weighted average of

and 1/

, the bias being proportional to the variance of

Exercises

10.2*

In the simple macroeconomic model

described in Exercise 10.1, demonstrate that OLS would yield inconsistent results if used to fit

the consumption function, and investigate the direction of the bias in the slope coefficient.

10.3

A researcher is investigating the impact of advertising on sales using cross-section data from

firms producing recreational goods. For each firm there are data on sales,

, and expenditure on

advertising,

, both measured in suitable units, for a recent year. The researcher proposes the

following model:

01234

SIMULTANEOUS EQUATIONS ESTIMATION

S =

A + u

A =

S + u

where u

and u

are disturbance terms. The first relationship reflects the positive effect of

advertising on sales, and the second the fact that largest firms, as measured by sales, tend to

spend most on advertising. Give a mathematical analysis of what would happen if the

researcher tried to fit the model using OLS.

10.3 Instrumental Variables Estimation

As we saw in the discussion of measurement error, the instrumental variables approach may offer a

solution to the problems caused by a violation of the fourth Gauss–Markov condition. In the present

case, when we fit the structural equation for p, the fourth Gauss–Markov condition is violated because

w is not distributed independently of u

. We need a variable that is correlated with w but not with u

and does not already appear in the equation in its own right. The reduced form equation for w gave us

some bad news – it revealed that w was dependent on u

. But it also gives us some good news – it

shows that w is correlated with U, which is exogenous and thus independent of u

. So we can fit the

equation using U as an instrument for w. Recalling that, for simple regression analysis, the

instrumental variables estimator of the slope coefficient is given by the covariance of the instrument

with the dependent variable divided by the covariance of the instrument with the explanatory variable,

the IV estimator of

is given by

),(Cov

(10.20)

We will demonstrate that it is consistent. Substituting from the structural equation for p,

),(Cov

),(Cov),(Cov),([Cov

),(Cov

])[,(Cov

2121

uUwUU

uwU

(10.21)

since the first covariance in the numerator is 0 and the second is equal to

Cov(U, w). Now plim

Cov(U, u

) is 0 if U is exogenous and so distributed independently of u

. plim Cov(U, w) is nonzero

because U is a determinant of w. Hence the instrumental variable estimator is a consistent estimate of

Table 10.2 shows the results when IV is used to fit the model described in Section 10.2. In

contrast to the OLS estimates, the IV estimates are distributed around the true values, the mean of the

estimates of the slope coefficient (true value 0.5) being 0.37 and of those of the intercept (true value

1.5) being 1.69. There is no point in comparing the standard errors using the two approaches. Those

for OLS may appear to be slightly smaller, but the simultaneous equations bias renders them invalid.

SIMULTANEOUS EQUATIONS ESTIMATION

ABLE

10.2

OLS IV

Sample b

s.e.(b

) b

s.e.(b

) b

s.e.(b

) b

s.e.(b

)

1 0.36 0.39 1.11 0.22 2.33 0.97 0.16 0.45

2 0.45 0.38 1.06 0.17 1.53 0.57 0.53 0.26

3 0.65 0.27 0.94 0.12 1.13 0.32 0.70 0.15

4 0.41 0.39 0.98 0.19 1.55 0.59 0.37 0.30

5 0.92 0.46 0.77 0.22 2.31 0.71 0.06 0.35

6 0.26 0.35 1.09 0.16 1.24 0.52 0.59 0.25

7 0.31 0.39 1.00 0.19 1.52 0.62 0.33 0.32

8 1.06 0.38 0.82 0.16 1.95 0.51 0.41 0.22

9 –0.08 0.36 1.16 0.18 1.11 0.62 0.45 0.33

10 1.12 0.43 0.69 0.20 2.26 0.61 0.13 0.29

In this example, IV definitely gave better results than OLS, but that outcome was not inevitable.

In Table 10.2 the distribution of the OLS estimates of the slope coefficient is more concentrated than

that of the IV estimates. The standard deviation of the estimates (calculated directly from the

estimates, ignoring the standard errors) is 0.15. For the IV estimates it is 0.20. So if the bias had been

smaller, it is possible that OLS might have yielded superior estimates according to a criterion like the

mean square error that allows a trade-off between bias and variance.

Underidentification

If OLS were used to fit the wage inflation equation

w =

p +

U + u

(10.22)

the estimates would be subject to simultaneous equations bias caused by the (indirect) dependence of p

on u

. However, in this case it is not possible to use the instrumental variables approach to obtain

consistent estimates, and the relationship is said to be underidentified. The only determinant of p,

apart from the disturbance terms, is U, and it is already in the model in its own right. An attempt to

use it as an instrument for p would lead to a form of exact multicollinearity and it would be impossible

to obtain estimates of the parameters. Using the expression in the box, we would have

),(Cov),(Cov)(Var),(Cov

UUUpUpU

UUwUUwU

UZUpUpZ

UZwUUwZ

−

(10.23)

and the numerator and denominator both reduce to 0.

SIMULTANEOUS EQUATIONS ESTIMATION

However, suppose that the rate of price inflation were hypothesized to be determined by the rate

of growth of the money supply, m, as well as the rate of growth of wages, and that m were assumed to

be exogenous:

p =

w +

m + u

(10.24)

The reduced form equations become

2323211

−

+++++

uumU

(10.25)

2323121

αα

−

+++++

uumU

(10.26)

U may be used as an instrument for w in the price inflation equation, as before, and m can be used as

an instrument for p in the wage inflation equation because it satisfies the three conditions required of

an instrument. It is correlated with p, by virtue of being a determinant; it is not correlated with the

disturbance term, by virtue of being assumed to be exogenous; and it is not already in the structural

equation in its own right. Both structural equations are now said to be exactly identified, exact

identification meaning that the number of exogenous variables available as instruments (that is, not

already in the equation in their own right) is equal to the number of endogenous variables requiring

instruments.

Instrumental Variables Estimation in a Model with Two Explanatory Variables

Suppose that the true model is

Y =

+ u

that X

is not distributed independently of u, and that Z is being used as an instrument for

. Then the IV estimators of

and

are given by

),(Cov),(Cov)(Var),(Cov

33232

333

XZXXXXZ

XZYXXYZ

−

),(Cov),(Cov)(Var),(Cov

),(Cov),(Cov),(Cov),(Cov

33232

3223

XZXXXXZ

XXYZXZYX

−

SIMULTANEOUS EQUATIONS ESTIMATION

Overidentification and Two-Stage Least Squares

Next consider the model

p =

w + u

(10.27)

w =

p +

U +

x + u

(10.28)

where x is the rate of growth of productivity. The corresponding reduced form equations are

22423211

−

+++++

uuxU

(10.29)

243121

ααα

αα

−

+++++

uuxU

(10.30)

The wage equation is underidentified because there is no exogenous variable available to act as an

instrument for p. p is correlated with both U and x, but both these variables appear in the wage

equation in their own right.

However the price inflation equation is now said to be overidentified because we have two

potential instruments for w. We could use U as an instrument for w, as before:

),(Cov

(10.31)

Alternatively, we could use x as an instrument:

),(Cov

(10.32)

Both are consistent estimators, so they would converge to the true value, and therefore to each

other, as the sample size became large, but for finite samples they would give different estimates.

Suppose that you had to choose between them (you do not, as we will see). Which would you choose?

The population variance of the first is given by

)(Var

×=

(10.33)

The population variance of the second estimator is given by a similar expression with the

correlation coefficient replaced by that between w and x. We want the population variance to be as

small as possible, so we would choose the instrument with the higher correlation coefficient.