Model and Data ECON 35* -- NOTE 3 Tests for Coeffcent Dfferences: Examples. Introducton Sample data: A random sample of 534 pad employees. Varable defntons: W hourly wage rate of employee ; lnw the natural logarthm of W ; S years of schoolng completed by employee ; years of work experence accumulated by employee. a female ndcator varable, f employee s female, otherwse; M a male ndcator varable, f employee s male, otherwse. The Model: A smple log-ln (sem-log wage equaton of the form ln W β + β S + β + u ( Two Groups of Employees: emale and Male The female wage equaton ln W α + α S + α + u,..., N f 45 (. The male wage equaton ln W β + β S + β + u,..., N m 89 (. ECON 35* -- Note 3: Tests for Coeffcent Dfferences: Examples Page of 5 pages
Null and Alternatve Hypotheses. Tests for ull Coeffcent Equalty H : α β and α β and α β H : α β and/or α β and/or α β. (3 coeffcent restrctons Unrestrcted Model Approach : Separate emale and Male Wage Equatons The female wage equaton ln W α + α S + α + u,..., N f 45 (. The OLS sample regresson equaton for females (wth t-ratos s: ln W αˆ + αˆ S + αˆ + û,..., N f 45 K 3 ˆα.33 ˆα.7 ˆα.8854 RSS ( 43.866 (.699 (9.38 (3.868 df ( 45 3 4 The male wage equaton ln W β + β S + β + u,..., N m 89 (. The OLS sample regresson equaton for males (wth t-ratos s: ln W βˆ + βˆ S + βˆ + û,..., N m 89 K 3 ˆβ.6965 ˆβ.93 ˆβ.635 RSS ( 63.474 (4.79 (8.43 (6.69 df ( 89 3 86 The unrestrcted resdual sum-of-squares s: RSS RSS ( + RSS ( 43.866 + 63.474 6.434; df df ( + df ( 4 + 86 58. ECON 35* -- Note 3: Tests for Coeffcent Dfferences: Examples Page of 5 pages
Unrestrcted Model Approach : Pooled ull-interacton Wage Equatons No base group no ntercept coeffcent (t-ratos n parentheses ln W α + αs + α + βm + βm S + βm + u (.,..., N N f + N m 534 RSS 6.434 wth df N K 534 6 58 ˆα.33 ˆα.7 ˆα.8854 (.6 (8.837 (3.643 ˆβ.6965 ˆβ.93 ˆβ.635 (4.478 (8.8 (7. α ˆ ˆ β.3933 αˆ βˆ.44 αˆ βˆ.749 (.65 (.38 (.3 β ˆ ˆ α.3933 βˆ αˆ.44 βˆ αˆ.749 (.65 (.38 (.3 Compare coeffcent estmates of pooled regresson equaton (. wth the separate female and male sample wage equatons: The OLS sample regresson equaton for females s: ln W αˆ + αˆ S + αˆ + û,..., N f 45 K 3 ˆα.33 ˆα.7 ˆα.8854 RSS ( 43.866 (.699 (9.38 (3.868 df ( 45 3 4 The OLS sample regresson equaton for males s: ln W βˆ + βˆ S + βˆ + û,..., N m 89 K 3 ˆβ.6965 ˆβ.93 ˆβ.635 RSS ( 63.474 (4.79 (8.43 (6.69 df ( 89 3 86 ECON 35* -- Note 3: Tests for Coeffcent Dfferences: Examples Page 3 of 5 pages
emales as base group (t-ratos n parentheses ln W α + α S + α + γ M + γ M S + γ M + u (. where γ β α; γ β α; γ β α. RSS 6.434 wth df N K 534 6 58 ˆα.33 ˆα.7 ˆα.8854 (.6 (8.837 (3.643 ˆγ.3933 ˆγ.44 ˆγ.749 (.65 (.38 (.3 rom OLS-SRE (.: β ˆ ˆ α.3933 βˆ αˆ.44 βˆ αˆ.749 (.65 (.38 (.3 Males as base group (t-ratos n parentheses ln W β + β S + β + δ + δ S + δ + u (. where δ α β ; δ α β ; δ α β. RSS 6.434 wth df N K 534 6 58 ˆβ.6965 ˆβ.93 ˆβ.635 (4.478 (8.8 (7. ˆδ.3933 ˆδ.44 ˆδ.749 (.65 (.38 (.3 rom OLS-SRE (.: α ˆ ˆ β.3933 αˆ βˆ.44 αˆ βˆ.749 (.65 (.38 (.3 ECON 35* -- Note 3: Tests for Coeffcent Dfferences: Examples Page 4 of 5 pages
Restrcted Model Same for Approach and Approach Corresponds to the null hypothess that all female and male coeffcents are equal: H : α β and α β and α β n pooled equaton (. γ and γ and γ n pooled equaton (. δ and δ and δ n pooled equaton (. The restrcted model can be wrtten as ether or ln W α + α S + α + u,..., N 534 (. ln W β + β S + β + u,..., N 534 (. The restrcted OLS-SRE for the full sample of 534 observatons s ~ ~ ~ ln W +,,..., N N + N 534 ~ β + βs ~ + β u β.588 ~ ~ β.964 β.75 (4.646 (.6 (6.7 The restrcted resdual sum-of-squares s: RSS 7.66 wth df N K 534 3 53. ECON 35* -- Note 3: Tests for Coeffcent Dfferences: Examples Page 5 of 5 pages
The -Test for Equalty of All Coeffcents Between Males and emales ( RSS RSS ( df df ( RSS RSS ( K K RSS df RSS ( N K [ df, ] ~ df df RSS 7.66 wth df N K 534 3 53 RSS 6.434 wth df N K 534 6 58 Sample value of the -statstc: ( RSS RSS ( df df ( 7.66 6.434.686 3 6.434 58 7.58 RSS df (53 58 6.434 58 Null dstrbuton of : ~ [ K, N ] K 7.58 [3, 58] under H. Decson Rule: At the α percent sgnfcance level. reject H f α [ K, N K ] or p-value for α ;. retan H f < α K, N ] or p-value for > α. [ K Crtcal Values of [3, 58]: at the 5% and % sgnfcance levels. At α.5:.5 [3, 58].6 At α.:. [3, 58] 3.89 P-value for.. Inference: Snce 7.58 > 3.89. [3, 58], reject H at the % sgnfcance level. Snce p-value for. <., reject H at the % sgnfcance level. ECON 35* -- Note 3: Tests for Coeffcent Dfferences: Examples Page 6 of 5 pages
3. Tests for Equalty of Both Slope Coeffcents Null and Alternatve Hypotheses female S coeffcent male S coeffcent α β female coeffcent male coeffcent α β aganst H : α β and α β n pooled equaton (. γ and γ n pooled equaton (. δ and δ n pooled equaton (. H : α β and/or α β n pooled equaton (. γ and/or γ n pooled equaton (. δ and/or δ n pooled equaton (. Unrestrcted Model corresponds to H Any one of the three pooled full-nteracton regresson equatons for lnw. ln W α + αs + α + βm + βm S + βm + u (.,..., N N f + N m 534 ln W α + α S + α + γ M + γ M S + γ M + u (. ln W β + β S + β + δ + δ S + δ + u (. OLS estmaton of any one of the three unrestrcted regresson equatons (., (. or (. on the pooled sample of 534 male and female employees yelds the followng value for the unrestrcted RSS: RSS 6.434 wth df N K 534 6 58. ECON 35* -- Note 3: Tests for Coeffcent Dfferences: Examples Page 7 of 5 pages
Restrcted Model corresponds to H Obtaned by substtutng the two coeffcent restrctons specfed by the null hypothess H nto any one of the three pooled full-nteracton regresson equatons for lnw. In equaton (. no base group: set α β and α β ln W α + α S + α + β M + β M S + β M + u (. ln W α α α α + β S + β + β + β M M + β S + β + β M + β (S + M S + β ( + β ( + M S + β + u + β M S + β + M + β ( + M M M + u + u + u sn ce + M. ln W α + β M + β S + β + u (3. In equaton (. females as base group: set γ and γ ln W α + α S + α + γ M + γ M S + γ M + u (. ln W α + α S + α + γ M + u (3. In equaton (. males as base group: set δ and δ ln W β + β S + β + δ + δ S + δ + u (. ln W β + β S + β + δ + u (3. OLS estmaton of any one of the three restrcted regresson equatons (3., (3. or (3. on the pooled sample of 534 male and female employees yelds the followng value for the restrcted RSS: RSS 8.45 wth df N K 534 4 53. ECON 35* -- Note 3: Tests for Coeffcent Dfferences: Examples Page 8 of 5 pages
The -Test for Male-emale Equalty of Both Slope Coeffcents ( RSS RSS ( df df ( RSS RSS ( K K RSS df RSS ( N K [ df, ] ~ df df RSS 8.45 wth df N K 534 4 53 RSS 6.434 wth df N K 534 6 58 Sample value of the -statstc: ( RSS RSS ( df df ( 8.45 6.434.6 6.434 58 5. RSS df (53 58 6.434 58 Null dstrbuton of : ~ [ df df, ] df 5. [, 58] under H. Decson Rule: At the α percent sgnfcance level. reject H f α [ df df, df] or p-value for α ;. retan H f < α [ df df, df] or p-value for > α. Crtcal Values of [, 58]: at the 5% and % sgnfcance levels. At α.5:.5 [, 58] 3.3 At α.:. [, 58] 4.646 P-value for.76. Inference: Snce 5. > 4.646. [, 58], reject H at the % sgnfcance level. Snce p-value for.76 <., reject H at the % sgnfcance level. ECON 35* -- Note 3: Tests for Coeffcent Dfferences: Examples Page 9 of 5 pages
4. Condtonal Log-Wage Dfferentals Between Males and emales Queston: What s the mean log-wage dfferental between male and female employees wth the same educaton and work experence.e., wth the same values of S and? Condtonal female-male mean log-wage dfferental s: or E ( ln W S,, E( ln W S,, E ( ln W S,, M E( ln W S,, M. The female log-wage equaton s ln W α + α S + α + u,..., N f 45 (. Hence, the condtonal mean log-wage of females wth S years of completed schoolng and years of work experence s: E ( ln W S,, E( ln W S,, M α + αs + α The male log-wage equaton s (4. ln W β + β S + β + u,..., N m 89 (. Hence, the condtonal mean log-wage of males wth S years of completed schoolng and years of work experence s: E ( ln W S,, E( ln W S,, M β + βs + β (4. ECON 35* -- Note 3: Tests for Coeffcent Dfferences: Examples Page of 5 pages
The condtonal female-male mean log-wage dfferental s obtaned by subtractng (4. from (4.: E E E ( ln W S,, E( ln W S,, M α + αs + α ( ln W S,, E( ln W S,, M β + βs + β (4. (4. ( ln W S,, E( ln W S,, α + αs + α β βs β ( α β δ β + ( α βs + ( α + δs + δ where δ α β ; δ α β ; δ α β. Pooled full nteracton log-wage equaton wth males as base group ln W β + β S + β + δ + δ S + δ + u (. where δ α β ; δ α β ; δ α β. OLS estmaton of pooled full nteracton log-wage equaton (. yelds the sample regresson equaton ln W βˆ + βˆ S + βˆ S + û (.* The OLS coeffcent estmates (and t-ratos for (. are: ˆβ.6965 ˆβ.93 ˆβ.635 (4.478 (8.8 (7. ˆδ.3933 ˆδ.44 ˆδ.749 (.65 (.38 (.3 ECON 35* -- Note 3: Tests for Coeffcent Dfferences: Examples Page of 5 pages
The estmate of the condtonal female-male mean log-wage dfferental s Ê ( ln W S,, Ê( ln W S,, ( αˆ ˆ ( ˆ ˆ ˆ βˆ δˆ β + α βs + ( α ˆ ˆ + δs + δ.3933 +.44S.749 (5 Example: The estmated condtonal female-male mean log-wage dfferental for employees wth 6 years of schoolng and years of work experence s obtaned by settng S 6 and n equaton (5: Ê ( ln W S 6,, Ê( ln W S 6,,.3933 +.44S.749.3933 +.44(6.749(.57 Interpretaton: The average wage of female employees wth 6 years of schoolng and years of work experence s approxmately.5 percent less than the average wage of male employees wth the same schoolng and work experence. The varance of the condtonal female-male mean log-wage dfferental s Var ( δˆ S Var(ˆ δ + S Cov(ˆ δ + S Var(ˆ δ + Var(ˆ δ, δˆ + Cov(ˆ δ, δˆ + S Cov(ˆ δ, δˆ The standard error of the condtonal female-male mean log-wage dfferental s smply the square root of the varance: se ( δ ˆ S Var( δˆ S ECON 35* -- Note 3: Tests for Coeffcent Dfferences: Examples Page of 5 pages
Proposton: The condtonal female-male mean log-wage dfferental for employees wth 6 years of schoolng and years of work experence equals zero. Null and Alternatve Hypotheses H : E( ln W S 6,, E( ln W S 6,, or δ + δ + δ δ + 6δ + δ S H : E( ln W S 6,, E( ln W S 6,, or δ + δ + δ δ + 6δ + δ S Perform a two-tal t-test: The t-statstc s ( ˆ S δˆ ˆ ˆ + δs + δ ( δ + δ S + δ δ ~ t[n K ] (6 sê ( δˆ S t Calculate the estmated varance and standard error of δ ˆ S δˆ + 6δˆ : Vâr ( δˆ S Vâr(ˆ δ + S Côv(ˆ δ, δˆ + Côv(ˆ δ + S Vâr(ˆ δ + Vâr(ˆ δ, δˆ + S Côv(ˆ δ, δˆ. regress lnw s x f fs fx Source SS df MS Number of obs 534 ---------+------------------------------ ( 5, 58 4.68 Model 4.79479 5 8.458958 Prob >. Resdual 6.43396 58.579473 R-squared.83 ---------+------------------------------ Adj R-squared.76 Total 48.449 533.785645 Root MSE.44898 ------------------------------------------------------------------------------ lnw Coef. Std. Err. t P> t [95% Conf. Interval] ---------+-------------------------------------------------------------------- s.936.354 8.8..7936.476 x.63455.3344 7...7597.934 f -.393386.4574 -.65.9 -.874765.888 fs.4448.6689.38.88 -.549.53446 fx -.749.33699 -.3.7 -.4 -.87 _cons.6964687.55538 4.478..39994.8 ------------------------------------------------------------------------------ ECON 35* -- Note 3: Tests for Coeffcent Dfferences: Examples Page 3 of 5 pages
. matrx lst VC symmetrc VC[6,6] s x f fs fx _cons s.476 x 8.54e-6 5.449e-6 f.5678.94.6648 fs -.476-8.54e-6 -.3845.6468 fx -8.54e-6-5.449e-6 -.46734.939.36 _cons -.5678 -.94 -.498.5678.94.498 âr(ˆ δ.6648 âr(ˆ δ.6468 âr(ˆ δ.36 ôv(ˆ δ, ˆ δ.3845 ôv(ˆ δ, ˆ δ.46734 ôv(ˆ δ, ˆ.939 V V V C C C δ Set S 6 and n formula for Vâr( ˆ ˆ S + δ ( ˆ ˆ ˆ Vâr δ + δ S + δ Vâr(ˆ δ + S Vâr(ˆ δ + Vâr(ˆ δ + S Côv(ˆ δ δ :, δˆ + Côv(ˆ δ, δˆ + S Côv(ˆ δ, δˆ Calculate estmated varance of δˆ S δˆ + 6δˆ : Vâr ( δˆ + 6δˆ.6648 + (6 (.6468 + ( (.36 + (6(.3845 + ((.46734 + (6((.939 Results: Vâr δ ˆ + 6δˆ.374538 Vâr(ˆ δ + (6 Vâr(ˆ δ + ( Vâr(ˆ δ + (6Côv(ˆ δ, δˆ + (Côv(ˆ δ, δˆ + (6(Côv(ˆ δ, δˆ ( ( ˆ + 6δˆ Vâr( δˆ + 6δˆ. 374538 sê δ.6995 ECON 35* -- Note 3: Tests for Coeffcent Dfferences: Examples Page 4 of 5 pages
The sample value of the t-statstc under H s calculated from (6 by settng ˆ ˆ ˆ δˆ + 6δˆ.5354 δ + δs + δ δ + δs + δ + 6δ + δ sê t δ ( δ ˆ S sê( ˆ + 6δˆ ( ˆ S δˆ δ.6995 S ( δˆ S δ (6 ( ˆ + 6δˆ sê ( δ + δ S + δ.5354.5354 δ.445.6995.6995 t Null dstrbuton of t : ~ t[n K ] t[534 (3] t[534 6] t[58] t Two-tal crtcal values of t[58]: at the 5% and % sgnfcance levels α.5 α/.5: t.5 [58].964 α. α/.5: t.5 [58].585 Two-tal p-value for t.44. Inference: Snce t.45 <.585 t.5 [58], retan H at the % sgnfcance level. Snce t.45 >.964 t.5 [58], reject H at the 5% sgnfcance level. Snce p-value for t.44 >., retan H at the % sgnfcance level. Snce p-value for t.44 <.5, reject H at the 5% sgnfcance level. How to use Stata to compute ths t-test: Use the followng lncom command.. lncom _b[f] + 6*_b[fs] + *_b[fx] ( f + 6. fs +. fx. ------------------------------------------------------------------------------ lnw Coef. Std. Err. t P> t [95% Conf. Interval] ---------+-------------------------------------------------------------------- ( -.535.6995 -.45.4 -.45348 -.4899 ---------+-------------------------------------------------------------------- ECON 35* -- Note 3: Tests for Coeffcent Dfferences: Examples Page 5 of 5 pages