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Suggested Answers for Problem Set 6 - Econometrics | ECON 30331, Assignments of Introduction to Econometrics

Material Type: Assignment; Class: Econometrics; Subject: Economics; University: Notre Dame; Term: Unknown 1989;

Typology: Assignments

2009/2010

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Suggested Answers for Problem Set 6
ECON 30331
Dan Hungerman
1. Consider the regression 01
yb bxe=+ +
.
A. Write down the formula for the OLS estimate of 1
b
12
()()
ˆ()
ii
i
i
i
x
xy y
bxx
−−
=
B. Using your answer in part A, explain what would happen if you instead estimated the equation
01
**ybbxe=+ +
, where *
y
y
α
= and *
x
x
α
=
, and
α
is a constant. In other words, you take
your y observations and multiply them all by some number
α
, do the same with the x variable,
and then do OLS on the transformed data. Does this affect the OLS estimates 1
ˆ
b or 0
ˆ
b?
It does not affect our estimate of 1
b. Note that if we multiply all of our x variables by
α
, the
mean of x will now be
x
, and similarly for the mean of y. Thus the new OLS estimate for 1
b is
*
12222
22
1
22 2 2 2
(* *)(* *) ( )( ) ( )( )
ˆ(* *) ( ) ( )
()() ()() ()()
ˆ
() () ()
ii i i ii
ii i
ii i
ii i
ii ii ii
iii
iii
iii
x
xy y x x y y xx yy
bxx x x xx
xxyy xxyy xxyy b
xx xx xx
αααα α α
αα α
αα
αα
−−
== =
−−
−− −− −−
== ==
−−
∑∑
∑∑
∑∑
∑∑
The
α
cancels out of the numerator and denominator and the estimate is unchanged. It will
affect our estimate of 0
b, however. The first-order condition for 0
ˆ
b is 01
ˆˆ
bybx=− . With the
transformed data this becomes
**
01 1 1 10
ˆˆ ˆ ˆ ˆˆ
**** ( )bybxybx ybx ybx b
α
αα α
=− =− = = =
This makes sense, since 0
ˆ
b estimates y when x is zero. If we multiply y by
α
, then when x is zero
we will expect y to be
α
-times bigger than it was before.
C. Your study partner Mr. Silly is now confused, because when studying WLS in class we did
some similar transformations of variables and got something different than what we find in part B
above. Why is the answer to part B above different from what we discussed in class?
This is different from WLS for at least two reasons. First, in WLS we typically multiply the data
by a variable, not just a constant. Second, when we estimated WLS & multiply terms in the
regression equation by a variable, we multiplied all terms in the regression by a variable,
including the constant. Here, we did not do that.
pf3
pf4
pf5
pf8

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Suggested Answers for Problem Set 6 ECON 30331

Dan Hungerman

  1. Consider the regression y = b 0 + b x 1 + e.

A. Write down the formula for the OLS estimate of b 1

(^1 )

i^ i^ i

i i

x x y y b x x

B. Using your answer in part A, explain what would happen if you instead estimated the equation

y* = b 0 + b x 1 +e , where y = αyand x* = αx, and α is a constant. In other words, you take

your y observations and multiply them all by some number α , do the same with the x variable,

and then do OLS on the transformed data. Does this affect the OLS estimates bˆ 1 or bˆ 0?

It does not affect our estimate of b 1. Note that if we multiply all of our x variables by α , the

mean of x will now be α x, and similarly for the mean of y. Thus the new OLS estimate for b 1 is

(^1 2 2 2 )

(^2 )

2 2 2 2 2 1

i i^ i^ i i^ i^ i i^ i

i i^ i i^ i i

i i^ i^ i i^ i^ i i^ i

i i^ i i^ i i

x x y y x x y y x x y y b x x x x x x

x x y y x x y y x x y y b x x x x x x

The α cancels out of the numerator and denominator and the estimate is unchanged. It will

affect our estimate of b 0 , however. The first-order condition for bˆ 0 is bˆ 0 = y − b xˆ 1. With the

transformed data this becomes

0 1 1 1 1 0

bˆ^ = y * −b x ˆ^ * = y * −b x ˆ^ * = α y − bˆ^ α x = α( y − b xˆ^ )=αbˆ

This makes sense, since bˆ 0 estimates y when x is zero. If we multiply y by α , then when x is zero

we will expect y to be α -times bigger than it was before.

C. Your study partner Mr. Silly is now confused, because when studying WLS in class we did some similar transformations of variables and got something different than what we find in part B above. Why is the answer to part B above different from what we discussed in class?

This is different from WLS for at least two reasons. First, in WLS we typically multiply the data by a variable, not just a constant. Second, when we estimated WLS & multiply terms in the regression equation by a variable, we multiplied all terms in the regression by a variable, including the constant. Here, we did not do that.

D. Would your answer for B change if this were a multivariate regression? That is, if there were

k different x variables, and we multiplied each of them along with y by a constant α , what would

happen to our OLS estimates?

The answer in part B would still be correct—the coefficients for our x variables would be

unchanged while bˆ 0 would increase by α. To see, this let’s look at the first-order conditions for

the multivariate solution:

0 0 1 1

0 1 1

i^ i^ i^ k^ ik

j (^) i ij i i k ik

for b y b b x b x

for b x y b b x b x j

Now consider a solution analogous to the one in part B—the coefficients for the x variables are

unchanged while bˆ 0 will increase by a factor α. The first order conditions (with the

transformed data) would look like this:

0 0 1 1

0 1 1

i^ i^ i^ k^ ik

j (^) i ij i i k ik

for b y b b x b x

for b x y b b x b x

But these can just be rewritten like this:

0 0 1 1 0 1 1

2 2 0 1 1 0 1 1

i i^ i^ k^ ik^ i i^ i^ k^ ik

j (^) i ij i i k ik (^) j ij i i k ik

for b y b b x b x y b b x b x

for b x y b b x b x x y b b x b x

These are the same first order conditions that we solved before we transformed the data. Thus the

solutions here will be the same as before (with the exception of bˆ 0 , which will increase by a factor α .)

(analytic weights assumed) (sum of wgt is 5.3644e-02)

Source | SS df MS Number of obs = 114 -------------+------------------------------ F( 3, 110) = 9. Model | 230479.881 3 76826.6271 Prob > F = 0. Residual | 908079.872 110 8255.27157 R-squared = 0. -------------+------------------------------ Adj R-squared = 0. Total | 1138559.75 113 10075.75 Root MSE = 90.


price | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- bedrooms | 16.84739 12.28053 1.37 0.173 -7.489728 41. age | -.0786841 .2701835 -0.29 0.771 -.6141244. sq_feet | .0982406 .024969 3.93 0.000 .0487579. _cons | 104.8052 41.59718 2.52 0.013 22.36938 187.


  1. Consider the variables income and pop. The variable income is how much income a person

makes and pop is the population of the country where they live. Further, suppose that for person i living in country c, income depends on pop in the following way:

incomei = α 0 + α 1 popc +ui

where

2 ui ∼ N(0, σu).

A. You would like to estimate the α coefficients in the above equation, but unfortunately you

don’t have individual-level data. All you can find is a country-level dataset, which reports for

each country average income in the population, income (^) c, and the population popc. So you

will have to estimate:

incomec = α 0 + α 1 popc +uc

What is the variance of u c? (Note that u^ cis an average of ui , which is averaged over popc

unique observations. You can assume the covariance between any two ui^ variables is zero.)

Here, u cis the average of income in the population: pop

i^ i c

u u =

. It is like we multiply

each ui by 1/pop and then we add all the ui terms together; if pop is a million then we have a

million of them to add together. If we multiply ui by 1/pop , the product has variance

2 2 σ (^) u/ pop.

So we have pop different terms to add together, and each one has variance

2 2 σ (^) u/ pop. If you

add some independent variables together (where the covariance is zero), the variance of their

sum is just the sum of their variances. So if there were 5 observations, the variance of uc

would be

2 2 5 σ (^) u/ pop.

Here, the number of terms we are adding together is the number of people in the population,

or pop. The variance of uc will be

2 2 2 pop* σ (^) u / pop = σu/ pop.

B. Suppose you wanted to transform the equation in part A to perform a weighted-least-squares

regression where the error term was appropriately homoskedastic. How could you transform

the equation in part A?

You would estimate:

incomec popc = α 0 popc + α 1 popc popc +u c popc

If you multiply u (^) c by popc the resulting variance is simply

2

σ u.

C. What would you type into Stata to perform the regression in part B? Could you just type,

“regress income pop”?

No, that would be zany! Instead we would type this:

regress income pop [weight = pop]

Or even smarter, just to be super conservative:

regress income pop [weight = pop], robust

  1. Suppose we estimated the relationship between the weight of a baby at birth, babyw, with the

weight of a pregnant woman at the end of her pregnancy, mommyw*. Such an estimation might

be useful if a doctor would want to estimate how large a fetus is, based on a pregnant woman’s

weight.

So the equation we want to estimate is this, which we’ll call equation (1)

babyw = a 0 + a mommyw 1 *+e (1)

Let’s suppose that in our dataset we perfectly observe baby’s birthweight, but that mother’s

weight is reported by the women themselves and hence measured with error. So the regression

we actually run is equation (2):

babyw = a 0 + a mommyw 1 +u (2)

True mother’s weight mommyw *is related to the observed mommyw as follows:

mommyw* = b mommy 1 + v (3)

C. Suppose b 1 = 1. What is the expected value of the OLS estimate aˆ 1? (We did something

similar like this back in September when we showed that OLS was unbiased, so your notes

from back then may be helpful. Also, your answer might depend on the properties of the

unobserables in equation (2) & you might want to think about that.)

Now our answer becomes:

2 2 1 1 ˆ (^1 2 2 2 ) i i^ i^ i^ i i^ i i^ i^ i i^ i

i i^ i i^ i i^ i i

a d d u a d d u d u a a d d d d

= = + = +

So, given the womenw values (and thus given di ) the expected value of aˆ 1 will be

(^1 1 )

E( | )

E( ˆ| )

i i^ i

i i

d u womenw a womenw a d

Looking back on equation (2) we can see that the unobservable there can be written as

ui = a v 1 i + ei. We’ll assume as usual that the error term e has no relation to womenw or

womenw*. But the error term v could. If E( vi ( womenwi − womenw)) ≠ 0 , then OLS is

biased.

Note this discussion has focused on bias, not (as in class) consistency. But clearly

measurement error affects OLS’ unbiased property and as with consistency this depends

upon the correlation of the measurement error with the observed values of the x variables.

  1. Suppose that you want to estimate the equation

y = β 0 + β 1 x *+e

But instead of observing the true x* you observe x = x *+v. So the regression you really estimate is

y = β 0 + β 1 x *+u

Let v ∼ N(0,1)and assume x* ∼ N(0,1), and also assume that x* and v are independent (and

thus have zero covariance). Suppose that the true value of β 1 was 4. What is the probability

limit of the OLS estimate of β? What is covariance between the observed x and the error term

u?

We saw in class that

1

1

2

(^1 1 2 ) plim ˆ x

x v

, where the denominator represent the covariance of

the observed x and the error term in the regression that is run. From the question we know that

1

2 1 x

σ = ,

2 σ (^) v = 1 , and β 1 = 4. Thus the OLS estimate of β 1 is 4*(1/2) = 2.

The term u can be expressed as e − β 1 v. If we assume that cov( ,e x ) = 0 , then

cov( ,x e − β 1 v) = −β 1 cov( , )x v. Since x = x +v and by assumption x and v are independent,

this becomes

2 − β 1 cov( , )x v = − β 1 cov( x * + v v, ) = −β 1 var( )v = − β σ 1 v. Plugging in we see that

the covariance is 4.

  1. Your study friend Mr. Silly reasons that if measurement error is correlated with the true x

variable, but uncorrelated with what is observed, than you must have a serious problem since the true x variable is what you care about. Does Mr. Silly’s reasoning make sense or is he being silly again?

Mr. Silly is wrong again. The key is weather measurement error is correlated with what you actually use in the regression. Measurement error that is correlated only with the true x variable, x*, is not a serious concern. (Measurement error in the dependent variable is also not as big a concern.) What matters is what is going on with the variables you are actually using to make your estimates. Unfortunately, it is often the case that in the presence of measurement error we would expect the nature of the error to be different depending on the values of x we observe, and so we would then have to worry about attenuation bias.