Documento 2 - test with one independent variable, dummy variables and homo or hetero error

Y

Small (STR > 20) 657.4 19.4 238

≥ 20)

Large (STR 650.0 17.9 182

   

Difference in means : Y Y 657.4 650.0 7.4

small large

2 2 2 2

S S 19.4 17.9

    

s l

Standard error SE 1.8

n n 238 182

s 1

Summary: regression when X is binary (0/1)

i

β β

Y = + X + u

i 0 1 i i

• β = mean of Y when X = 0

0

• β β

+ = mean of Y when X = 1

0 1

• β = difference in group means, X =1 minus X = 0

1 ˆ



• SE ( ) has the usual interpretation

1

• t-statistics, confidence intervals constructed as usual

• This is another way (an easy way) to do difference-in-means analysis

• The regression formulation is especially useful when we have

additional regressors

Heteroskedasticity and Homoskedasticity, and

Homoskedasticity-Only Standard Errors (Section 5.4)

What…?

1.

2. Consequences of homoskedasticity

3. Implication for computing standard errors

What do these two terms mean?

–

If var(u|X=x) is constant that is, if the variance of the conditional

–

distribution of u given X does not depend on X then u is said to

be homoskedastic. Otherwise, u is heteroskedastic.

Example: hetero/homoskedasticity in the case of a

binary regressor (that is, the comparison of means)

• Standard error when group variances are unequal:

2 2

s s

 

s l

SE n n

s l

• Standard error when group variances are equal:

1 1

 

SE s p n n

s l

  

2 2

( n 1) S ( n 1) S



2 s s l l

Where S (SW, Sect 3.6)

 

p n n 2

s l   

 

“pooled estimator of ” when

2 2 2

S p l s

• Equal group variances = homoskedasticity

• Unequal group variances = heteroskedasticity

Homoskedasticity in a picture:

• β

E(u|X=x) = 0 (β + X is the population regression line)

0 1

• The variance of u does not depend on x

A real-data example from labor economics: average

hourly earnings vs. years of education (data source:

Current Population Survey):

Heteroskedastic or homoskedastic?

The class size data:

Heteroskedastic or homoskedastic?

So far we have (without saying so) assumed

that u might be heteroskedastic.

Recall the three least squares assumptions:

1. E(u|X = x) = 0

= 1,…,n,

2. (X ,Y ), i are i.i.d.

i i

3. Large outliers are rare

Heteroskedasticity and homoskedasticity concern var(u|X=x).

Because we have not explicitly assumed homoskedastic errors, we

have implicitly allowed for heteroskedasticity.

What if the errors are in fact homoskedastic? (1 of 2)

• You can prove that OLS has the lowest variance among estimators that are

a result called the Gauss-Markov

linear in Y… theorem that we will return

to shortly. ˆ



• The formula for the variance of and the OLS standard error simplifies :

1

 

If var(

u | X x ) , then

i i 



var[( X )

u ]

 i x i

var( ) (general formula)

 2 2

n ( )

X

 2

 u (simplification of u is homoscedastic)

 2

n X

ˆ



Note : var( ) is inversely proportional to var( X ): more spread in X

1 ˆ

 

means more information about we discussed this earlier but it is

1

clearer from this formula.

What if the errors are in fact homoskedastic? (2 of 2) ˆ



• Along with this homoskedasticity-only formula for the variance of ,

1

we have homoskedasticity-only standard errors:

Homoskedasticity-only standard error formula:

n

1  ˆ 2

u

 i

1 n 2

ˆ

   

i 1

SE ( ) .

1 n

1 

n  2

( X X )

i

n 

i 1

Some people (e.g. Excel programmers) find the homoskedasticity-only

–

formula simpler but it is wrong unless the errors really are

homoskedastic.

We now have two formulas for standard

ˆ



errors for .

1

• –

Homoskedasticity-only standard errors these are valid only if

the errors are homoskedastic.

• –

The usual standard errors to differentiate the two, it is

–

conventional to call these heteroskedasticity robust standard

errors, because they are valid whether or not the errors are

heteroskedastic.

• The main advantage of the homoskedasticity-only standard

errors is that the formula is simpler. But the disadvantage is that

the formula is only correct if the errors are homoskedastic.

Practical implications… ˆ



• The homoskedasticity-only formula for the standard error of and

1



the “heteroskedasticity-robust” formula differ so in general, you get

different standard errors using the different formulas

.

• Homoskedasticity-only standard errors are the default setting in

–

regression software sometimes the only setting (e.g. Excel). To get

the general “heteroskedasticity-robust” standard errors you must

override the default.

• If you don’t override the default and there is in fact

heteroskedasticity, your standard errors (and t-statistics and

–

confidence intervals) will be wrong typically, homoskedasticity-

only SEs are too small.

Heteroskedasticity-robust standard errors

in STATA

regress testscr str, robust

Regression with robust standard errors Number of obs = 420

F( 1, 418) = 19.26

Prob > F = 0.0000

R-squared = 0.0512

Root MSE = 18.581

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

Robust

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

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

qqqqstr | -2.279808 .5194892 -4.39 0.000 -3.300945 -1.258671

qq_cons | 698.933 10.36436 67.44 0.000 678.5602 719.3057

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

• “,

If you use the robust” option, STATA computes heteroskedasticity-robust

standard errors

• Otherwise, STATA computes homoskedasticity-only standard errors

The bottom line:

• If the errors are either homoskedastic or heteroskedastic and you

use heteroskedastic-robust standard errors, you are OK

• If the errors are heteroskedastic and you use the homoskedasticity-

only formula for standard errors, your standard errors will be wrong

ˆ



(the homoskedasticity-only estimator of the variance of is

1

inconsistent if there is heteroskedasticity).

• The two formulas coincide (when n is large) in the special case

of homoskedasticity

• So, you should always use heteroskedasticity-robust standard

errors.

Some Additional Theoretical Foundations

of OLS (Sections 5.5)

We have already learned a very great deal about OLS: OLS is

unbiased and consistent; we have a formula for heteroskedasticity-

robust standard errors; and we can construct confidence intervals

and test statistics. –

Also, a very good reason to use OLS is that everyone else does

so by using it, others will understand what you are doing. In

effect, OLS is the language of regression analysis, and if you use a

different estimator, you will be speaking a different language.

Still, you may wonder…

• Is this really a good reason to use OLS? Aren’t there other

–

estimators that might be better in particular, ones that might

have a smaller variance?

• Also, what happened to our old friend, the Student t distribution?

–

So we will now answer these questions but to do so we will need

to make some stronger assumptions than the three least squares

assumptions already presented.

The Homoskedastic Normal Regression

Assumptions

These consist of the three LS assumptions, plus two more:

1. E(u|X = x) = 0.

=1,…,n,

2. (X ,Y ), i are i.i.d.

i i ∞, ∞).

4 4

3. Large outliers are rare (E(Y ) < E(X ) <

4. u is homoskedastic 2

5. u is distributed N(0,σ )

• –

Assumptions 4 and 5 are more restrictive so they apply to fewer

cases in practice. However, if you make these assumptions, then

certain mathematical calculations simplify and you can prove

–

strong results results that hold if these additional assumptions

are true.

• We start with a discussion of the efficiency of OLS

Efficiency of OLS, part I: The Gauss-Markov

Theorem (1 of 2)

Under assumptions 1-4 (the basic three, plus homoskedasticity),

ˆ

 has the smallest variance among all linear estimators

1

(estimators that are linear functions of Y ,..., Y ). This is

1 n

the Gauss - Markov theorem .

Comments

• The GM theorem is proven in SW Appendix 5.2

Efficiency of OLS, part I: The Gauss-Markov

Theorem (2 of 2)

ˆ



• is a linear estimator, that is, it can be written as a linear function

1

of Y ,..., Y :

1 n n

 

( X X )

u

i i n

1 

ˆ

 

  



i 1 w u ,

1 1 i i

n

 n

 

2 i 1

( X X )

i



i 1 

( X X )

 i

where w .

i n

1   2

( X X )

i

n 

i 1

• The G-M theorem says that among all possible choices of { w }, the OLS

i

ˆ



weights yield the samllest var( )

1

Efficiency of OLS, part II:

• Under all five homoskedastic normal regression assumptions

ˆ



 

including normally distributed errors has the smallest

1

variance of all consistent estimators (linear or nonlinear

 

functio ns of Y ,..., Y ), as - .

1 n

• –

This is a pretty amazing result it says that, if (in addition to LSA 1-3)

the errors are homoskedastic and normally distributed, then OLS is a

better choice than any other consistent estimator. And because an

estimator that isn’t consistent is a poor choice, this says that OLS

–

really is the best you can do if all five extended LS assumptions

hold. (The proof of this result is beyond the scope of this course and

isn’t in SW – it is typically done in graduate courses.)

Some not-so-good thing about OLS (1 of 2)

–

The foregoing results are impressive, but these results and the

–

OLS estimator have important limitations.

1. The GM theorem really isn’t that compelling:

– The condition of homoskedasticity often doesn’t hold (homoskedasticity

is special)

– –

The result is only for linear e