Introduction to Econometrics

Introduction to Econometrics Chapter 3

Ezequiel Uriel Jiménez University of Valencia

Valencia, September 2013

3 Multiple linear regression: estimation and properties 3.1 The multiple linear regression model 3.2 Obtaining the OLS estimates, interpretation of the coefficients, and other characteristics 3.3 Assumptions and statistical properties of the OLS estimators 3.4 More on functional forms 3.5 Goodness-of-fit and selection of regressors. Exercises Appendixes

3 Multiple linear regression: estimation and properties

3.2 Obtaining the OLS estimates, interpretation of the coefficients, and other characteristics

[3]

EXAMPLE 3.1 Quantifying the influence of age and wage on absenteeism in the firm Buenosaires (file absent) absent  1   2 age   3tenure   4 wage  u  = 14.413 - 0.096 age - 0.078 tenure - 0.036 wage absent i i i i (1.603)

(0.048)

R 2 = 0.694

(0.067)

(0.007)

n = 48

EXAMPLE 3.2 Demand for hotel services (file hostel) ln (hostel )  b1 + b2 ln(inc) + b3 hhsize + u  ln( hostel )  -27.36 + 4.442 ln(inc ) - 0.523hhsize i

i

i

R 2 = 0.738 n = 40

EXAMPLE 3.3 A hedonic regression for cars (file hedcarsp) ln( price)  1   2 volume   3 fueleff  u

 ln( pricei )  4.97 + 0.0956volumei - 0.1608 fueleff i R 2 = 0.765 n = 214

3 Multiple linear regression: estimation and properties

3.2 Obtaining the OLS estimates, interpretation of the coefficients, and other characteristics

[4]

EXAMPLE 3.4 Sales and advertising: the case of Lydia E. Pinkham (file pinkham) Vt 1    1 Pt 1  1 2 Pt  2  1 3 Pt  3     ut 1  = 138.7 + 0.3288advexp + 0.7593sales sales t t -1

R 2 = 0.877 n = 53 The sum of the cumulative effects of advertising expenditures on sales:

ˆ1 0.3288   1.3660 1  ˆ 1  0.7593 Periods of time required to reach half of the total effects:

ln(1  0.5) hˆ(0.5)   2.5172 ln(0.7593)

3 Multiple linear regression: estimation and properties

3.3 Assumptions and statistical properties of the OLS estimators

[5]

y

y

xj

xj 2 a) sˆ

big

b)

sˆ 2

small

2 FIGURE 3.1. Influence of sˆ on the estimator of the variance..

3 Multiple linear regression: estimation and properties

3.3 Assumptions and statistical properties of the OLS estimators

[6]

y

y

xj

a)

S 2j small

xj 2 b) S j

big

2 FIGURE 3.2. Influence of S j on the estimator of the variance..

3.4 More on functional forms 3 Multiple linear regression: estimation and properties

Example 3.5 Salary and tenure (file ceosal2)

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 ln( salaryi )  6.246 0.0006 profitsi  0.0440 ceoteni  0.0012 ceoteni2 (0.086)

(0.0001)

(0.0156)

(0.00052)

R 2  0.1976 n  177 Marginal effect of ceoten on salary, expressed in percentage:

 %  4.40  2  0.12ceoten me salary / ceoten

Example 3.6 The marginal effect in a cost function (file costfunc)   29.16 2.316 output  0.0914 output 2  0.0013 output 3 cost i i i i (1.602)

(0.2167)

(0.0081)

(0.000086)

R 2  0.9984 n  11 Marginal cost :

 i  2.316  2  0.0914output  3  0.0013output 2 marcost i i

3.5 Goodness-of-fit and selection of regressors Example 3.7 Selection of the best model (file demand) 3 Multiple linear regression: estimation and properties

Alternative models:

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1)

dairy  1   2 inc  u

2)

dairy  1   2 ln(inc)  u

3)

dairy  1   2 inc   3 punder 5  u

4)

dairy   2 inc   3 punder 5  u

5)

dairy  1   2 inc   3 hhsize  u

6)

ln( dairy )  1   2 inc  u

7)

ln( dairy )  1   2 inc   3 punder 5  u

8)

ln(dairy )   2 inc   3 punder 5  u n = 40

ln( dairy ) = 2.3719

Corrected AIC for model 6)

AICC = AIC + 2ln(Y ) = 0.2794 + 2´ 2.3719=5.0232

3 Multiple linear regression: estimation and properties

3.5 Goodness-of-fit and selection of regressors

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TABLE 3.1. Measures of goodness of fit for eight models. Model number

1

2

3

4

5

6

7

8

Regressand Regressors

dairy intercept inc

dairy intercept ln(inc)

dairy inc punder5

0.4567

0.5531

0.4978

ln(dairy ) intercept inc punder5 0.5986

ln(dairy ) inc punder5

0.4584

dairy intercept Inc househsize 0.4598

ln(dairy ) intercept inc

R-squared

dairy intercept inc punder5 0.5599

Adjusted R-squared

0.4441

0.4424

0.5361

0.5413

0.4306

0.4846

0.5769

-0.7255

Akaike information criterion

5.2374

5.2404

5.0798

5.0452

5.2847

0.2794

0.1052

1.4877

Schwarz criterion

5.3219

5.3249

5.2065

5.1296

5.4113

0.3638

0.2319

1.5721

Corrected Akaike information criterion

5.0232

4.849

6.2314

Corrected Schwarz criterion

5.1076

4.9756

6.3159

-0.6813