Econometrics advanced questions

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2. Ordinary Least Squares (15 points)

Which of the following can cause the usual OLS t-statistics to be invalid?

i) Heteroskedasticity

ii) Including an irrelevant explanatory variable (one whose true regression coefficient is zero).

iii) Omitting an important explanatory variable (one whose true regression coefficient is nonzero).

3. Hypothesis Testing in a Simple Linear Regression Context (20 points)

A researcher with a sample of 62 individuals with similar education but different amounts of training hypothesizes that hourly earnings, EARNINGS, may be related to hours of training, TRAINING, according to the relationship

She is prepared to test the hypothesis against the alternative at both the five-percent and one-percent levels of significance. What should she report

i)

if

ii)

if

iii)

if

iv)

if

5. Heteroskedasticity (15 points)

Consider the following model for real estate values applied to a cross-section of homes:

where PRICE is sales price in thousands of dollars, SQFT is living area in square feet, YARD is yard size in square feet, and POOL is a dummy variable indicating whether the house has a swimming pool.

You suspect that the random error term is heteroskedastic, and that the variance of is proportional to SQFT: . Describe step-by-step how you should use the weighted least squares procedure to take care of the heteroscedasticity problem. Be sure to describe precisely how the original model is transformed and explain why the procedure works (given your assumption about the functional form of ).

6. Serial Correlation (20 points)

Consider the double-log model of farm output

where Q is output, K is capital, L is labor, A is acreage planted, F is the amount of fertilizers used, and S is the amount of seed planted. Using annual data for the years 1949 through 1988, the model was estimated by ordinary least squares. You are concerned about the possibility of serial correlation, and you decide to perform an LM (Lagrange Multiplier) test for first and second order autocorrelation.

a)

The auxiliary regression was estimated (in which the residuals from the original OLS estimation of the model are regressed against a constant, all of the explanatory variables from the original model, and the lagged residuals from the two previous time periods), and the unadjusted was 0.687. Compute the LM test statistic and state its distribution (including the degrees of freedom) under the null hypothesis.

b) Write down the critical value of the test statistic at the five percent level of significance and carry out the test. What do you conclude about serial correlation?

β̂1 =0.30,s.e.(β̂1)=0.12?

ˆ

b

1

=0.30,s.e.(

ˆ

b

1

)=0.12?

β̂1 =0.55,s.e.(β̂1)=0.12?

ˆ

b

1

=0.55,s.e.(

ˆ

b

1

)=0.12?

β̂1 =0.10,s.e.(β̂1)=0.12?

ˆ

b

1

=0.10,s.e.(

ˆ

b

1

)=0.12?

β̂1 = −0.27,s.e.(β̂1)=0.12?

ˆ

b

1

=-0.27,s.e.(

ˆ

b

1

)=0.12?

PRICEi = β0 +β1SQFTi +β2YARDi +β3POOLi +εi

PRICE

i

=b

0

+b

1

SQFT

i

+b

2

YARD

i

+b

3

POOL

i

+e

i

εi

e

i

εi

e

i

var[εi]=σ i 2 =σ 2SQFTi

2

var[e

i

]=s

i

2

=s

2

SQFT

i

2

σ i 2

s

i

2

ln(Qt)= β1 +β2 ln(Kt)+β3 ln(Lt)+β4 ln(At)+β5 ln(Ft)+β6 ln(St)+εt

ln(Q

t

)=b

1

+b

2

ln(K

t

)+b

3

ln(L

t

)+b

4

ln(A

t

)+b

5

ln(F

t

)+b

6

ln(S

t

)+e

t

2

R

EARNINGSi = β0 +β1 ⋅TRAININGi +εi

EARNINGS

i

=b

0

+b

1

×TRAINING

i

+e

i

H0 :β1 =0

H

0

:b

1

=0

Ha :β1 ≠0

H

a

:b

1

¹0