Consider the standard linear regression model y = X

TOULOUSE SCHOOL OF ECONOMICS MASTER 1 - FALL 2016

INTERMEDIATE ECONOMETRICS

TD3 Tests and Restricted LS

Exercise 3.1

Consider the standard linear regression model y = X

β

+ X

β

+ ε . 1. Show that the constrained LS estimate of β = (β

, β

)

subject to the

constraint that β

= c (for some given vector c ) is the OLS regression of y − X

c on X

.

2. Consider the case where X

and X

are univariate. Rewrite the model so that the restriction β

− β

= 1 becomes a single zero restriction (i.e., γ = 0 ).

Exercise 3.2

Resolve the following paradox: The Gauss-markov theorem states that β b is the minimum-variance linear unbiased estimator, whereas the restricted LS β b

is clearly more ecient linear unbiased when the restrictions hold.

Exercise 3.3

We study the relation between the price of a house and its characteristics.

To this end, we consider the following model:

log(price) = β

+ β

sqrf t + β

bdrms + β

sqrf t × bdrms + β

bthrms + ε where price is the selling price of the house, sqrf t is the square footage of house, bdrms is the number of bedrooms and bthrms is the number of bath- rooms.

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1. Derive the marginal impact of the number of bedrooms in the house.

The model has been estimated on a sample of 546 residential houses during between July, August and September 2003 in the area of Windsor, Canada.

Dependent variable: log(price) Estimated param. Std.Error

constant 7.92151 0.21912

sqrf t 0.00027 0.00008

bdrms 0.06491 0.03640

sqrf t × bdrms 0.00056 0.00043

bthrms 0.09039 0.05221

n = 546 , R

= 0.261

2. Compute the estimated marginal impact of the number of bedrooms in the house for sqrf t = 100 and for sqrf t = 200 . Interpret.

3. Does the impact of the number of bedrooms in the house depend on the square footage of the house?

4. Does the hypothesis H

: β

= β

has any economic meaning? Test this as- sumption against H

: β

6= β

at the 5% level, knowing that the estimated covariance between c β

and c β

is 0.00004.

Exercise 3.4

We want to determine if the price paid in auctions for paintings is aected by their size. To address this question, we use a sample of 430 observations (which we assume to be a large sample) of auction prices for Monet paintings, with data on the dimensions of the paintings and other characteristics.

The following model has been estimated by OLS on the sample:

log(price)

= β

+ β

log(area)

+β

ratio

+β

signed

+

for i = 1, . . . , 430 where price is the price paid in auction (in million $ ), area is the area of the painting (height times width, both measured in inches), ratio is the aspect ratio (height divided by width) and signed is a an indicator variable equal to 1 if the painting has been signed by Monet.

Results of the OLS regression are reported in Table 1.

1. What is a condence interval for β

at condence level 95%? Give the formula and the result.

2. Is the variable log(area) signicant? In case it is, give the interpretation of β b

.

3. Does the aspect ratio have a negative impact on the log of the price? You should write the null and alternative hypothesis, give the form of the test statistic, its behavior under the null hypothesis, the formal decision rule at 5% level, and your conclusion based on the above results.

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Table 1: Estimation Results dependent variable: log(price)

_intercept -9.6403

(0.5642)

log(area) 1.3493

(0.0817)

ratio -0.0786

(0.1152)

signed 1.2554

(0.1253)

N 430

Note: Standard errors in parentheses

4. The relative dierence of expected price (in level) between signed and non signed paintings is equal to exp(β

) − 1 . Perform a Wald test at the 5%

level to test H

: exp(β

) − 1 = 0 against H

: exp(β

) − 1 6= 0 . You should give the form of the test statistic, its behavior under the null hypothesis, the formal decision rule, and your conclusion based on the above results.

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