Christopher Dougherty EC220 - Introduction to econometrics (chapter 6) Slideshow: variable...

Christopher Dougherty

EC220 - Introduction to econometrics (chapter 6)Slideshow: variable misspecification i: omitted variable bias

Original citation:

Dougherty, C. (2012) EC220 - Introduction to econometrics (chapter 6). [Teaching Resource]

© 2012 The Author

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Available in LSE Learning Resources Online: May 2012

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VARIABLE MISSPECIFICATION I: OMISSION OF A RELEVANT VARIABLE

In this sequence and the next we will investigate the consequences of misspecifying the regression model in terms of explanatory variables.

1

Consequences of variable misspecification

TRUE MODEL

FIT

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uXXY 33221 uXY 221

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221ˆ

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To keep the analysis simple, we will assume that there are only two possibilities. Either Y depends only on X2, or it depends on both X2 and X3.

2


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If Y depends only on X2, and we fit a simple regression model, we will not encounter any problems, assuming of course that the regression model assumptions are valid.

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Correct specification,no problems


Likewise we will not encounter any problems if Y depends on both X2 and X3 and we fit the multiple regression.

4


TRUE MODEL

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uXXY 33221 uXY 221

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In this sequence we will examine the consequences of fitting a simple regression when the true model is multiple.

5


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In the next one we will do the opposite and examine the consequences of fitting a multiple regression when the true model is simple.

6


TRUE MODEL

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TRUE MODEL

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The omission of a relevant explanatory variable causes the regression coefficients to be biased and the standard errors to be invalid.

7



Coefficients are biased (in general). Standarderrors are invalid.


8

In the present case, the omission of X3 causes b2 to be biased by the term highlighted in yellow. We will explain this first intuitively and then demonstrate it mathematically.

uXXY 33221 221ˆ XbbY

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Y

X3X2

direct effect of X2, holding X3 constant

effect of X3

apparent effect of X2, acting as a mimic for X3

2 3

9

The intuitive reason is that, in addition to its direct effect 2, X2 has an apparent indirect effect as a consequence of acting as a proxy for the missing X3.


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Y

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effect of X3


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The strength of the proxy effect depends on two factors: the strength of the effect of X3 on Y, which is given by 3, and the ability of X2 to mimic X3.


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effect of X3


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The ability of X2 to mimic X3 is determined by the slope coefficient obtained when X3 is regressed on X2, the term highlighted in yellow.

12



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3322133221

We will now derive the expression for the bias mathematically. It is convenient to start by deriving an expression for the deviation of Yi about its sample mean. It can be expressed in terms of the deviations of X2, X3, and u about their sample means.

13



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Although Y really depends on X3 as well as X2, we make a mistake and regress Y on X2 only. The slope coefficient is therefore as shown.

14



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We substitute for the Y deviations and simplify.

15

Hence we have demonstrated that b2 has three components.



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To investigate biasedness or unbiasedness, we take the expected value of b2. The first two terms are unaffected because they contain no random components. Thus we focus on the expectation of the error term.

17

X2 is nonstochastic, so the denominator of the error term is nonstochastic and may be taken outside the expression for the expectation.


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18

In the numerator the expectation of a sum is equal to the sum of the expectations (first expected value rule).


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In each product, the factor involving X2 may be taken out of the expectation because X2 is nonstochastic.


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By Assumption A.3, the expected value of u is 0. It follows that the expected value of the sample mean of u is also 0. Hence the expected value of the error term is 0.


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Thus we have shown that the expected value of b2 is equal to the true value plus a bias term. Note: the definition of a bias is the difference between the expected value of an estimator and the true value of the parameter being estimated.



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As a consequence of the misspecification, the standard errors, t tests and F test are invalid.



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. reg S ASVABC SM

Source | SS df MS Number of obs = 540-------------+------------------------------ F( 2, 537) = 147.36 Model | 1135.67473 2 567.837363 Prob > F = 0.0000 Residual | 2069.30861 537 3.85346109 R-squared = 0.3543-------------+------------------------------ Adj R-squared = 0.3519 Total | 3204.98333 539 5.94616574 Root MSE = 1.963

------------------------------------------------------------------------------ S | Coef. Std. Err. t P>|t| [95% Conf. Interval]-------------+---------------------------------------------------------------- ASVABC | .1328069 .0097389 13.64 0.000 .1136758 .151938 SM | .1235071 .0330837 3.73 0.000 .0585178 .1884963 _cons | 5.420733 .4930224 10.99 0.000 4.452244 6.389222------------------------------------------------------------------------------

23

We will illustrate the bias using an educational attainment model. To keep the analysis simple, we will assume that in the true model S depends only on ASVABC and SM. The output above shows the corresponding regression using EAEF Data Set 21.


uSMASVABCS 321

. reg S ASVABC SM



24


We will run the regression a second time, omitting SM. Before we do this, we will try to predict the direction of the bias in the coefficient of ASVABC.

uSMASVABCS 321

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ASVABCASVABC

SMSMASVABCASVABCbE

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. reg S ASVABC SM



25


It is reasonable to suppose, as a matter of common sense, that 3 is positive. This assumption is strongly supported by the fact that its estimate in the multiple regression is positive and highly significant.

uSMASVABCS 321

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ASVABCASVABC

SMSMASVABCASVABCbE

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. reg S ASVABC SM



26


The correlation between ASVABC and SM is positive, so the numerator of the bias term must be positive. The denominator is automatically positive since it is a sum of squares and there is some variation in ASVABC. Hence the bias should be positive.

. cor SM ASVABC(obs=540)

| SM ASVABC--------+------------------ SM| 1.0000 ASVABC| 0.4202 1.0000

uSMASVABCS 321

2322 )(

ASVABCASVABC

SMSMASVABCASVABCbE

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. reg S ASVABC


------------------------------------------------------------------------------ S | Coef. Std. Err. t P>|t| [95% Conf. Interval]-------------+---------------------------------------------------------------- ASVABC | .148084 .0089431 16.56 0.000 .1305165 .1656516 _cons | 6.066225 .4672261 12.98 0.000 5.148413 6.984036------------------------------------------------------------------------------

27

Here is the regression omitting SM.


uSMASVABCS 321

. reg S ASVABC SM


. reg S ASVABC

------------------------------------------------------------------------------ S | Coef. Std. Err. t P>|t| [95% Conf. Interval]-------------+---------------------------------------------------------------- ASVABC | .148084 .0089431 16.56 0.000 .1305165 .1656516 _cons | 6.066225 .4672261 12.98 0.000 5.148413 6.984036------------------------------------------------------------------------------

28

As you can see, the coefficient of ASVABC is indeed higher when SM is omitted. Part of the difference may be due to pure chance, but part is attributable to the bias.


uSMASVABCS 321

. reg S SM


------------------------------------------------------------------------------ S | Coef. Std. Err. t P>|t| [95% Conf. Interval]-------------+---------------------------------------------------------------- SM | .3130793 .0348012 9.00 0.000 .2447165 .3814422 _cons | 10.04688 .4147121 24.23 0.000 9.232226 10.86153------------------------------------------------------------------------------

29

Here is the regression omitting ASVABC instead of SM. We would expect b3 to be upwards biased. We anticipate that 2 is positive and we know that both the numerator and the denominator of the other factor in the bias expression are positive.


2233 )(

SMSM

SMSMASVABCASVABCbE

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uSMASVABCS 321

30

In this case the bias is quite dramatic. The coefficient of SM has more than doubled. The reason for the bigger effect is that the variation in SM is much smaller than that in ASVABC, while 2 and 3 are similar in size, judging by their estimates.


. reg S ASVABC SM


. reg S SM

------------------------------------------------------------------------------ S | Coef. Std. Err. t P>|t| [95% Conf. Interval]-------------+---------------------------------------------------------------- SM | .3130793 .0348012 9.00 0.000 .2447165 .3814422 _cons | 10.04688 .4147121 24.23 0.000 9.232226 10.86153------------------------------------------------------------------------------

uSMASVABCS 321

. reg S ASVABC SM Source | SS df MS Number of obs = 540-------------+------------------------------ F( 2, 537) = 147.36 Model | 1135.67473 2 567.837363 Prob > F = 0.0000 Residual | 2069.30861 537 3.85346109 R-squared = 0.3543-------------+------------------------------ Adj R-squared = 0.3519 Total | 3204.98333 539 5.94616574 Root MSE = 1.963

. reg S ASVABC Source | SS df MS Number of obs = 540-------------+------------------------------ F( 1, 538) = 274.19 Model | 1081.97059 1 1081.97059 Prob > F = 0.0000 Residual | 2123.01275 538 3.94612035 R-squared = 0.3376-------------+------------------------------ Adj R-squared = 0.3364 Total | 3204.98333 539 5.94616574 Root MSE = 1.9865

. reg S SM Source | SS df MS Number of obs = 540-------------+------------------------------ F( 1, 538) = 80.93 Model | 419.086251 1 419.086251 Prob > F = 0.0000 Residual | 2785.89708 538 5.17824736 R-squared = 0.1308-------------+------------------------------ Adj R-squared = 0.1291 Total | 3204.98333 539 5.94616574 Root MSE = 2.2756

31

Finally, we will investigate how R2 behaves when a variable is omitted. In the simple regression of S on ASVABC, R2 is 0.34, and in the simple regression of S on SM it is 0.13.


uSMASVABCS 321




32


Does this imply that ASVABC explains 34% of the variance in S and SM 13%? No, because the multiple regression reveals that their joint explanatory power is 0.35, not 0.47.

uSMASVABCS 321




33


In the second regression, ASVABC is partly acting as a proxy for SM, and this inflates its apparent explanatory power. Similarly, in the third regression, SM is partly acting as a proxy for ASVABC, again inflating its apparent explanatory power.

uSMASVABCS 321

. reg LGEARN S EXP

Source | SS df MS Number of obs = 540-------------+------------------------------ F( 2, 537) = 100.86 Model | 50.9842581 2 25.492129 Prob > F = 0.0000 Residual | 135.723385 537 .252743734 R-squared = 0.2731-------------+------------------------------ Adj R-squared = 0.2704 Total | 186.707643 539 .34639637 Root MSE = .50274

------------------------------------------------------------------------------ LGEARN | Coef. Std. Err. t P>|t| [95% Conf. Interval]-------------+---------------------------------------------------------------- S | .1235911 .0090989 13.58 0.000 .1057173 .141465 EXP | .0350826 .0050046 7.01 0.000 .0252515 .0449137 _cons | .5093196 .1663823 3.06 0.002 .1824796 .8361596------------------------------------------------------------------------------

34

However, it is also possible for omitted variable bias to lead to a reduction in the apparent explanatory power of a variable. This will be demonstrated using a simple earnings function model, supposing the logarithm of hourly earnings to depend on S and EXP.


uEXPSLGEARN 321

. reg LGEARN S EXP



35


If we omit EXP from the regression, the coefficient of S should be subject to a downward bias. 3 is likely to be positive. The numerator of the other factor in the bias term is negative since S and EXP are negatively correlated. The denominator is positive.

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SS

EXPEXPSSbE

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. cor S EXP(obs=540)

| S EXP--------+------------------ S| 1.0000 EXP| -0.2179 1.0000

. reg LGEARN S EXP



36


For the same reasons, the coefficient of EXP in a simple regression of LGEARN on EXP should be downwards biased.

2233 )(

EXPEXP

SSEXPEXPbE

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uEXPSLGEARN 321

. cor S EXP(obs=540)

| S EXP--------+------------------ S| 1.0000 EXP| -0.2179 1.0000

. reg LGEARN S EXP------------------------------------------------------------------------------ LGEARN | Coef. Std. Err. t P>|t| [95% Conf. Interval]-------------+---------------------------------------------------------------- S | .1235911 .0090989 13.58 0.000 .1057173 .141465 EXP | .0350826 .0050046 7.01 0.000 .0252515 .0449137 _cons | .5093196 .1663823 3.06 0.002 .1824796 .8361596

. reg LGEARN S------------------------------------------------------------------------------ LGEARN | Coef. Std. Err. t P>|t| [95% Conf. Interval]-------------+---------------------------------------------------------------- S | .1096934 .0092691 11.83 0.000 .0914853 .1279014 _cons | 1.292241 .1287252 10.04 0.000 1.039376 1.545107

. reg LGEARN EXP------------------------------------------------------------------------------ LGEARN | Coef. Std. Err. t P>|t| [95% Conf. Interval]-------------+---------------------------------------------------------------- EXP | .0202708 .0056564 3.58 0.000 .0091595 .031382 _cons | 2.44941 .0988233 24.79 0.000 2.255284 2.643537

37

As can be seen, the coefficients of S and EXP are indeed lower in the simple regressions.


. reg LGEARN S EXP Source | SS df MS Number of obs = 540-------------+------------------------------ F( 2, 537) = 100.86 Model | 50.9842581 2 25.492129 Prob > F = 0.0000 Residual | 135.723385 537 .252743734 R-squared = 0.2731-------------+------------------------------ Adj R-squared = 0.2704 Total | 186.707643 539 .34639637 Root MSE = .50274

. reg LGEARN S Source | SS df MS Number of obs = 540-------------+------------------------------ F( 1, 538) = 140.05 Model | 38.5643833 1 38.5643833 Prob > F = 0.0000 Residual | 148.14326 538 .275359219 R-squared = 0.2065-------------+------------------------------ Adj R-squared = 0.2051 Total | 186.707643 539 .34639637 Root MSE = .52475

. reg LGEARN EXP Source | SS df MS Number of obs = 540-------------+------------------------------ F( 1, 538) = 12.84 Model | 4.35309315 1 4.35309315 Prob > F = 0.0004 Residual | 182.35455 538 .338948978 R-squared = 0.0233-------------+------------------------------ Adj R-squared = 0.0215 Total | 186.707643 539 .34639637 Root MSE = .58219

38

A comparison of R2 for the three regressions shows that the sum of R2 in the simple regressions is actually less than R2 in the multiple regression.


. reg LGEARN S EXP Source | SS df MS Number of obs = 540-------------+------------------------------ F( 2, 537) = 100.86 Model | 50.9842581 2 25.492129 Prob > F = 0.0000 Residual | 135.723385 537 .252743734 R-squared = 0.2731-------------+------------------------------ Adj R-squared = 0.2704 Total | 186.707643 539 .34639637 Root MSE = .50274

. reg LGEARN S Source | SS df MS Number of obs = 540-------------+------------------------------ F( 1, 538) = 140.05 Model | 38.5643833 1 38.5643833 Prob > F = 0.0000 Residual | 148.14326 538 .275359219 R-squared = 0.2065-------------+------------------------------ Adj R-squared = 0.2051 Total | 186.707643 539 .34639637 Root MSE = .52475

. reg LGEARN EXP Source | SS df MS Number of obs = 540-------------+------------------------------ F( 1, 538) = 12.84 Model | 4.35309315 1 4.35309315 Prob > F = 0.0004 Residual | 182.35455 538 .338948978 R-squared = 0.0233-------------+------------------------------ Adj R-squared = 0.0215 Total | 186.707643 539 .34639637 Root MSE = .58219

39

This is because the apparent explanatory power of S in the second regression has been undermined by the downwards bias in its coefficient. The same is true for the apparent explanatory power of EXP in the third equation.


Copyright Christopher Dougherty 2011.

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The content of this slideshow comes from Section 6.2 of C. Dougherty,

Introduction to Econometrics, fourth edition 2011, Oxford University Press.

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11.07.25

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