Dougherty С. Introduction to Econometrics, 3Ed

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STOCHASTIC REGRESSORS AND MEASUREMENT ERRORS

An Illustration

The Friedman analysis will be illustrated with a Monte Carlo experiment. It was supposed that a

sample of 20 individuals had permanent incomes 2000, 2100, 2200, ... , 3900. It was also assumed

that each individual had transitory income equal to 200 times a random number drawn from a normal

population with mean 0 and unit variance. Measured income for each individual was the sum of

permanent and transitory income. It was assumed that the true value of

was 0.9, so that permanent

consumption was 0.9 times the corresponding permanent income. There was no provision for a

transitory component for consumption, so measured consumption was equal to permanent

consumption. When measured consumption was regressed on measured income, the result was

(standard errors in parentheses)

= 443 + 0.75

= 0.89 (9.29)

(179) (0.06)

As anticipated, the estimated marginal propensity to consume is below the true value. Indeed, if you

construct a 95

percent confidence interval using the regression results, the true value lies outside it and

would therefore be rejected at the 5

percent significance level. With 18 degrees of freedom, the

critical level of

is 2.10, so the confidence interval would be calculated as

0.75 – 2.10

0.06

≤

0.75 + 2.10

0.06 (9.30)

that is,

0.62

≤

0.88 (9.31)

Therefore you would make a type I error. Actually, the presence of measurement errors makes the

standard error of

, and hence the confidence interval, invalid. A further side effect is that the

constant term, which ought to have been 0 since there was no intercept in the model, appears to be

significantly positive at the 5

percent level. The experiment was repeated nine further times and the

results are summarized in Table 9.1, set A.

clearly gives a downward-biased estimate of the marginal propensity to consume. It is lower

than the true value of 0.90 in nine of the 10 samples. We will check whether the results support the

analysis leading to equation (9.28). In this example

is 40,000, since

has standard deviation

200. We will assume that in large samples

takes the values 2,000, 2,100, … , 3,900 with equal

probability, and hence that

is the variance of these numbers, which is 332,500. Hence in large

samples

would be underestimated by an amount

10.090.011.090.0

000,40500,332

000,40

=×=×

σσ

(9.32)

STOCHASTIC REGRESSORS AND MEASUREMENT ERRORS

ABLE

9.1

Experiments with Measurement Error

Experiment A Experiment B

Sample b

s.e.(

)

s.e.(

)

s.e.(

)

s.e.(

)

1 443 179 0.75 0.06 1001 251 0.56 0.08

2 152 222 0.83 0.07 755 357 0.62 0.11

3 101 222 0.89 0.08 756 376 0.68 0.13

4 195 179 0.83 0.06 668 290 0.66 0.09

5 319 116 0.78 0.04 675 179 0.64 0.06

6 371 200 0.78 0.07 982 289 0.57 0.10

7 426 161 0.74 0.05 918 229 0.56 0.07

8 –146 275 0.93 0.09 625 504 0.66 0.16

9 467 128 0.74 0.04 918 181 0.58 0.06

10 258 153 0.80 0.05 679 243 0.65 0.08

It should be stressed that this is valid only for large samples, and that we are not entitled to say

anything about the behavior of

in small samples. However, in this case we can see that it does in

fact provide a good guide. Looking at the estimates of

in the 10 samples, we see that they appear to

be randomly distributed about 0.80, instead of 0.90, and that there is thus a downwards bias of about

0.10.

A consequence of the underestimation of

is that

is overestimated, appearing to be positive

even though its true value is 0. Indeed in four cases a

test would indicate that it is significantly

different from 0 at the 5

percent significance level. However, in these conditions the

tests are invalid

because the failure of the fourth Gauss–Markov condition to be satisfied causes the standard errors,

and hence the

statistics, to be estimated wrongly.

What would happen if we increased the variance of

, keeping everything else the same? In set

in Table 9.1 the original random numbers were multiplied by 400 instead of 200, so

is now

160,000 instead of 40,000. The error term in expression (9.26) is now equal to 160,000/(332,500 +

160,000), which is 0.32, so we would anticipate that

would tend to (0.9 – 0.32

0.9), which is 0.61,

in large samples. Again, we see that this is a good guide to the actual behavior of

, even though each

sample has only 20 observations. As should be predicted, the estimates of

are even greater than in

set A.

Policy Implications

There are two separate and opposite implications for the multiplier. First, if Friedman is right, a

regression of actual consumption on actual income will yield an underestimate of the marginal

propensity to consume and hence an underestimate of the multiplier. In the illustration in the previous

section, the true value of

was 0.90, so the true value of the multiplier was 10. But in set A the

STOCHASTIC REGRESSORS AND MEASUREMENT ERRORS

estimate of

was tending to 0.80, implying a multiplier of only 5. In set B, it would have been lower

still. The estimate of

was tending to 0.61, giving a multiplier of 2.6.

If the government underestimates the multiplier, it will underestimate the effects of fiscal policy.

For example, an increase in government expenditure intended to reduce unemployment may in fact

lead to an excessive increase in effective demand and an increase in the rate of inflation.

The second implication is that the multiplier applies only to that part of a change in income that is

perceived as permanent, because (according to Friedman) consumption depends only on permanent

income. Thus, if the increase in government expenditure is thought to be temporary, it will not affect

consumption at all (as a first approximation), and the multiplier associated with it will be 1.

These remarks must however be qualified by a consideration of the form in which individuals

hold their savings. We have implicitly assumed so far that they take them in the form of financial

assets (bank deposits, bonds, and so on). However, in the Friedman model, expenditure on consumer

durables is considered to be a form of saving. An increase in transitory income will not be spent on

ordinary consumer goods, but it may be partly saved in the form of purchases of consumer durables,

and the increase in the demand for these will give rise to a multiplier effect. The multiplier for

transitory income may not be so low after all.

This discussion has been confined to consumption theory, the original application of Friedman's

concept of permanent income. But the concept can be, and has been, used in other fields. In

particular, in monetary theory it can be argued that the transactions demand for cash should be related

to permanent, rather than actual, income, and in investment theory it can be argued that the accelerator

should be related to changes in permanent income rather than changes in actual income. The seminal

contributions are Friedman (1959) and Eisner (1967).

Exercises

9.6

In a certain economy the variance of transitory income is 0.5 that of permanent income, the

propensity to consume nondurables out of permanent income is 0.6, and there is no expenditure

on durables. What would be the value of the multiplier derived from a naïve regression of

consumption on income, and what would be the true value?

9.7

In his definition of permanent consumption, Friedman includes the consumption of services

provided by durables. Purchases of durables are classified as a form of saving. In an economy

similar to that in Exercise 9.6, the variance of transitory income is 0.5 that of permanent

income, the propensity to consume nondurables out of permanent income is 0.6, and half of

current saving (actual income minus expenditure on nondurables) takes the form of expenditure

on durables. What would be the value of the multiplier derived from a naïve regression of

consumption on income, and what would be the true value?

9.4 Instrumental Variables

What can be done about measurement errors? If the measurement errors are due to inaccuracy in the

recording of the data, not much. If they arise because the variable being measured is conceptually

different from the true variable in the relationship, the obvious answer is to attempt to obtain more

STOCHASTIC REGRESSORS AND MEASUREMENT ERRORS

appropriate data. Often, however, this is not possible. In the case of Friedman's Permanent Income

Hypothesis, there is no way of obtaining data directly on permanent income since it is a subjective

concept. Sometimes the problem can be sidestepped. Friedman's own approach will be discussed in

Section 12.3. Another technique, known as instrumental variables (IV), will be discussed here. IV is

a major variation on OLS and it will be of great importance when we come to the fitting of models

comprising several simultaneous equations.

Essentially IV consists of semi-replacing a defective explanatory variable with one that is not

correlated with the disturbance term. The discussion will be confined to the simple regression case

+ u

(9.33)

and we shall suppose that for some reason X

has a random component that depends on u

. A

straightforward OLS regression of Y on X would then lead to inconsistent estimates of the parameters.

Suppose, however, that we can think of another variable Z that is correlated with X but not

correlated with u. We shall show that the instrumental variables estimator of

, defined as

),(Cov

(9.34)

is consistent, provided that

, the population covariance of Z and X, is nonzero.

Before doing this, it is instructive to compare

with the OLS estimator, which will be denoted

OLS

b .

),(Cov

)(Var

),(Cov

OLS

(9.35)

since Cov(X, X) and Var(X) are the same thing. The IV estimator, in simple regression analysis, is

obtained by substituting the instrument Z for the X in the numerator and for one X (but not both) in the

denominator.

Substituting for Y from (9.33), we can expand the expression for

b :

),(Cov

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

),(Cov

])[,(Cov

),(Cov

uZXZZ

uXZ

(9.36)

since Cov(Z,

) is 0 (

being a constant), and Cov(Z,

X) is equal to

Cov(Z, X).

We can see therefore that the instrumental variable estimator is equal to the true value plus an

error term Cov(Z, u)/Cov(Z, X). In large samples, the error term will vanish, for

STOCHASTIC REGRESSORS AND MEASUREMENT ERRORS

222

),(Cov plim

plim

=+=+=

(9.37)

provided that we are correct in supposing that

is distributed independently of

. Hence in large

samples

will tend to the true value

Nothing much can be said about the distribution of

in small samples, but as

increases, the

distribution will converge on a normal one with mean

and variance

given by

XZX

×=

(9.38)

where

is the sample correlation between

and

Compare this with the variance of the OLS estimator:

OLS

(9.39)

The difference is that the variance of

is multiplied by the factor 1/

. The greater the correlation

between

and

, the smaller will be this factor, and hence the smaller will be the variance of

We are now in a position to state the three requirements of an instrument:

1. It should be correlated with the variable being instrumented, and the higher the correlation,

the better, provided that the second requirement is satisfied.

2. It should not be correlated with the disturbance term. If it is stochastic, its random

component should be distributed independently of the disturbance term. Otherwise plim

Cov(

) in (9.37) will not be 0. Thus it would not be desirable to use an instrument that is

perfectly correlated with

, even if you could find one, because then it would automatically

be correlated with

as well and you would still obtain inconsistent estimates.

3. It should not be an explanatory variable in its own right.

What should you so if you cannot find an instrumental variable highly correlated with

? Well,

you may wish to stick with OLS after all. If, for example, your criterion for selecting an estimator is

its mean square error, you may find that the OLS estimator is preferable to an IV estimator, despite the

bias, because its variance is smaller.

Example: Use of IV to Fit the Friedman Consumption Function

The pioneering use of IV in the context of the Friedman permanent income hypothesis is Liviatan

(1963). Liviatan had data on the consumption and income of the same 883 households for two

consecutive years. We will denote consumption and income in the first year

and

, and in the

second year

and

Liviatan observed that if Friedman's theory is correct,

can act as an instrument for

Obviously it is likely to be highly correlated with

, so the first requirement of a good instrument is

satisfied. If the transitory component of measured income is uncorrelated from one year to the next, as

STOCHASTIC REGRESSORS AND MEASUREMENT ERRORS

hypothesized by Friedman, Y

will be uncorrelated with the disturbance term in the relationship

between C

and Y

, and so the second condition is satisfied. Finally, C

is likely to be related to Y

rather than Y

, so the third condition is satisfied. The instrumental variable estimator is then given by

),(Cov

(9.40)

Alternatively, one could use C

as an instrument for Y

. It will be highly correlated with Y

, and

therefore with Y

, and also not correlated with the disturbance term in the relationship between C

and

if, as Friedman hypothesized, the transitory components of consumption are uncorrelated.

Similarly one could run regressions using the data for year 2, using Y

and C

as instruments for

. Liviatan tried all four combinations, separating his sample into employees and self-employed. He

found that four of the estimates of the marginal propensity to consume were significantly greater than

those obtained by straightforward OLS at the 1 percent level; one was significantly greater at the 5

percent level, and in the other three cases the difference was not significant, evidence that on the

whole corroborates the permanent income hypothesis. However, the marginal propensity to consume

was generally not as high as the average propensities; therefore, his results did not support the

hypothesis of a unit elasticity of consumption with respect to permanent income, which is implicit in

equation (9.25).

Example: Use of IV to Fit an Earnings Function

In some data sets up to 10 percent of the variance of measured years of schooling is thought to be

attributable to measurement error. Accordingly, the coefficient of schooling in an earnings function

may be underestimated. The regression output gives first the output from an OLS regression of the

logarithm of hourly earnings on years of schooling and the ASVABC score, and then the output from

an IV regression with mother's years of schooling used as an instrument for years of schooling. SM is

likely to be a suitable instrument because it is correlated with S, unlikely to be correlated with the

disturbance term, and unlikely to be a direct determinant of earnings.

. reg LGEARN S ASVABC

Source | SS df MS Number of obs = 570

---------+------------------------------ F( 2, 567) = 57.45

Model | 25.9166749 2 12.9583374 Prob > F = 0.0000

Residual | 127.885218 567 .225547121 R-squared = 0.1685

---------+------------------------------ Adj R-squared = 0.1656

Total | 153.801893 569 .270302096 Root MSE = .47492

------------------------------------------------------------------------------

LGEARN | Coef. Std. Err. t P>|t| [95% Conf. Interval]

---------+--------------------------------------------------------------------

S | .0544266 .0099018 5.497 0.000 .034978 .0738753

ASVABC | .0114733 .0026476 4.333 0.000 .0062729 .0166736

_cons | 1.118832 .124107 9.015 0.000 .8750665 1.362598

------------------------------------------------------------------------------

STOCHASTIC REGRESSORS AND MEASUREMENT ERRORS

. ivreg LGEARN ASVABC (S=SM)

Instrumental variables (2SLS) regression

Source | SS df MS Number of obs = 570

---------+------------------------------ F( 2, 567) = 41.48

Model | 15.5448371 2 7.77241854 Prob > F = 0.0000

Residual | 138.257056 567 .243839604 R-squared = 0.1011

---------+------------------------------ Adj R-squared = 0.0979

Total | 153.801893 569 .270302096 Root MSE = .4938

------------------------------------------------------------------------------

LGEARN | Coef. Std. Err. t P>|t| [95% Conf. Interval]

---------+--------------------------------------------------------------------

S | .121573 .0564962 2.152 0.032 .0106057 .2325403

ASVABC | .0010966 .0090152 0.122 0.903 -.0166106 .0188039

_cons | .7313408 .3455687 2.116 0.035 .0525898 1.410092

------------------------------------------------------------------------------

Instrumented: S

Instruments: ASVABC SM

------------------------------------------------------------------------------

The coefficient of schooling is much larger in the IV regression, suggesting that measurement

error may have led to a downwards bias in its coefficient in the OLS regression. However, note that

its standard error is also much larger than in the OLS regression. This is because the correlation

between S and SM, 0.36, is not very high. It is possible that the difference in the OLS and IV

estimates of the coefficient may be due to chance. We will improve the IV estimation by drawing on a

group of family background variables, instead of just SM, to instrument for S, and then we will

perform a formal test of the difference in the coefficients.

First, however, some practical notes. The example used here is a multiple regression model, and

you should be aware that the expressions for IV coefficients in a multiple regression model are more

complex than those in a simple regression model, in the same way that OLS multiple regression

coefficients are more complex than OLS simple regression coefficients. However, the expressions are

straightforward in a treatment using matrix algebra. A facility for performing IV regressions is a

standard feature of all regression applications. A typical procedure is to state the variables for the

regression in the usual way and append a list of non-instrumented variables and instrument(s) in

parentheses. The output above used version 6 of Stata, which departs from this convention in two

ways. The command for IV regressions is different from that for OLS regressions (“

ivreg

” instead

of “

reg

”), and the list of variables in parentheses is in the form of an equation with the instrumented

variable(s) to the left of the = sign and the instruments to the right of it.

Multiple Instruments

Father's years of schooling, number of siblings, and possession of a library card are other factors that

may be associated with years of schooling of the respondent but are not likely to be direct

determinants of earnings. Thus we have four potential instruments for S and, for reasons that will be

explained in the next chapter, IV estimation is more efficient if they are used as a group rather than

individually. To do this, you include all of them in the list of instruments in the regression command.

The corresponding regression output is shown below.

STOCHASTIC REGRESSORS AND MEASUREMENT ERRORS

. ivreg LGEARN ASVABC (S=SM SF SIBLINGS LIBRARY)

Instrumental variables (2SLS) regression

Source | SS df MS Number of obs = 570

---------+------------------------------ F( 2, 567) = 43.58

Model | 20.3209223 2 10.1604612 Prob > F = 0.0000

Residual | 133.48097 567 .235416173 R-squared = 0.1321

---------+------------------------------ Adj R-squared = 0.1291

Total | 153.801893 569 .270302096 Root MSE = .4852

------------------------------------------------------------------------------

LGEARN | Coef. Std. Err. t P>|t| [95% Conf. Interval]

---------+--------------------------------------------------------------------

S | .1037467 .0422673 2.455 0.014 .0207272 .1867663

ASVABC | .0038515 .0068948 0.559 0.577 -.009691 .0173939

_cons | .8342134 .2686343 3.105 0.002 .3065736 1.361853

------------------------------------------------------------------------------

Instrumented: S

Instruments: ASVABC SM SF SIBLINGS LIBRARY

------------------------------------------------------------------------------

The smaller standard error indicates that there has been a modest gain in efficiency, but it is still

much larger than that for the original OLS regression and it is possible that the difference in the OLS

and IV estimates of the coefficient of S is purely random. We will perform a formal test.

The Durbin–Wu–Hausman Specification Test

Most economic data are subject to some element of measurement error and a recurring issue, as in the

present case, is whether it is potentially serious enough to require the use of IV instead of OLS to fit a

model. It has been shown that if measurement error is serious, OLS estimates will be inconsistent and

IV is to be preferred. However, if there is no measurement error, both OLS and IV will be consistent

and OLS will be preferred because it is more efficient. The Durbin–Wu–Hausman specification test

(sometimes described as the Hausman test: the standard reference is Hausman(1978), but Durbin

(1954) and Wu (1973) made important contributions to its development) can be used in this context to

discriminate between the two possibilities. We will assume that the regression model is given by

Y =

+ ...

+ u (9.41)

where one or more of the explanatory variables are potentially subject to measurement error. Under

the null hypothesis that there is no measurement error, the OLS and IV coefficients will not be

systematically different. The test statistic is based on the differences between the OLS and IV

coefficients (all of them, not just those of the variables potentially subject to measurement error) and it

has a chi-squared distribution with degrees of freedom equal to the number of instrumented variables

under the null hypothesis of no significant difference. Its computation is too complex to be

reproduced here and you would be well-advised to employ a regression application such as Stata

version 6 or EViews version 3 that will do it for you.

We will perform the test for the earnings function example using Stata. We first run the IV

regression of the logarithm of earnings on the ASVABC score, dummy variables for male sex and

black and hispanic ethnicity, and years of schooling instrumented with multiple instruments: mother’s

and father’s years of schooling, number of siblings, and a dummy variable equal to 1 if anyone in the

STOCHASTIC REGRESSORS AND MEASUREMENT ERRORS

family possessed a library card, 0 otherwise, when the respondent was 14. After this we give the

command "

hausman, save

". We then run the corresponding OLS regression and follow with the

command "

hausman, constant sigmamore

". This produces the following output:

. ivreg LGEARN ASVABC MALE ETHBLACK ETHHISP (S=SM SF SIBLINGS LIBRARY)

Instrumental variables (2SLS) regression

Source | SS df MS Number of obs = 570

---------+------------------------------ F( 5, 564) = 23.49

Model | 29.1540126 5 5.83080252 Prob > F = 0.0000

Residual | 124.64788 564 .22100688 R-squared = 0.1896

---------+------------------------------ Adj R-squared = 0.1824

Total | 153.801893 569 .270302096 Root MSE = .47011

------------------------------------------------------------------------------

LGEARN | Coef. Std. Err. t P>|t| [95% Conf. Interval]

---------+--------------------------------------------------------------------

S | .1026815 .0381431 2.692 0.007 .0277615 .1776014

ASVABC | .0025508 .0067323 0.379 0.705 -.0106727 .0157743

MALE | .2280404 .0422514 5.397 0.000 .145051 .3110298

ETHBLACK | -.15289 .0882356 -1.733 0.084 -.3262005 .0204204

ETHHISP | .0463734 .085714 0.541 0.589 -.1219842 .214731

_cons | .7939315 .2347929 3.381 0.001 .3327562 1.255107

------------------------------------------------------------------------------

Instrumented: S

Instruments: ASVABC MALE ETHBLACK ETHHISP SM SF SIBLINGS LIBRARY

------------------------------------------------------------------------------

. hausman, save

. reg LGEARN S ASVABC MALE ETHBLACK ETHHISP

Source | SS df MS Number of obs = 570

---------+------------------------------ F( 5, 564) = 30.63

Model | 32.8416113 5 6.56832227 Prob > F = 0.0000

Residual | 120.960281 564 .214468584 R-squared = 0.2135

---------+------------------------------ Adj R-squared = 0.2066

Total | 153.801893 569 .270302096 Root MSE = .46311

------------------------------------------------------------------------------

LGEARN | Coef. Std. Err. t P>|t| [95% Conf. Interval]

---------+--------------------------------------------------------------------

S | .0618848 .0098386 6.290 0.000 .04256 .0812097

ASVABC | .0093287 .0027721 3.365 0.001 .0038838 .0147737

MALE | .2130222 .0394229 5.404 0.000 .1355886 .2904557

ETHBLACK | -.1019355 .0741871 -1.374 0.170 -.2476523 .0437813

ETHHISP | .0537519 .0841815 0.639 0.523 -.1115956 .2190993

_cons | 1.009459 .1295912 7.790 0.000 .7549185 1.263999

------------------------------------------------------------------------------

. hausman, constant sigmamore

---- Coefficients ----

| (b) (B) (b-B) sqrt(diag(V_b-V_B))

| Prior Current Difference S.E.

---------+-------------------------------------------------------------

S | .1026815 .0618848 .0407967 .0362637

ASVABC | .0025508 .0093287 -.0067779 .0060248

MALE | .2280404 .2130222 .0150182 .0133495

ETHBLACK | -.15289 -.1019355 -.0509546 .045293

ETHHISP | .0463734 .0537519 -.0073784 .0065586

_cons | .7939315 1.009459 -.2155273 .1915801

---------+-------------------------------------------------------------

b = less efficient estimates obtained previously from ivreg.

B = more efficient estimates obtained from regress.

STOCHASTIC REGRESSORS AND MEASUREMENT ERRORS

Test: Ho: difference in coefficients not systematic

chi2( 1) = (b-B)'[(V_b-V_B)^(-1)](b-B)

= 1.27

Prob>chi2 = 0.2606

The chi-squared statistic is 1.27, lower than 3.84, the critical value of chi-squared at the 5 percent

significance level, and so we do not reject the null hypothesis of no difference in the OLS and IV

estimates. We infer that it is safe to use OLS rather than IV, and we are happy to do so because the

OLS standard errors, particularly those of the coefficients of S and ASVABC, are smaller than their IV

counterparts. This is likely to be the correct conclusion. The schooling histories are recorded in great

detail in the NLSY data set and accordingly the measurement error is almost certainly minimal.

The Hausman test can be used in any comparison of OLS and IV estimators where both are

consistent, but OLS more efficient, under a null hypothesis, and OLS in inconsistent under the

alternative hypothesis. We will encounter another application in the next chapter. With the usage of

the test becoming more common, a facility for performing it is becoming a standard feature of

regression applications.

Exercises

9.8

In Exercise 9.1, the amount of labor, L, employed by the firms is also a linear function of

expected sales:

L =

Explain how this relationship might be exploited by the investigator to counter the problem of

measurement error bias.

9.9*

It is possible that the ASVABC test score is a poor measure of the kind of ability relevant for

earnings. Accordingly, perform an OLS regression of the logarithm of hourly earnings on years

of schooling and the ASVABC score using your EAEF data set and an IV regression using SM,

SF, SIBLINGS, and LIBRARY as instruments for ASVABC. Perform a Durbin–Wu–Hausman

test to evaluate whether ASVABC appears to be subject to measurement error.

9.10*

What is the difference between an instrumental variable and a proxy variable (as described in

Section 7.4)? When would you use one and when would you use the other?