Tests of Single Linear Coefficient Restrictions: t-tests and F-tests. 1. Basic Rules. 2. Testing Single Linear Coefficient Restrictions

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1 ECONOMICS 35* -- NOTE ECON 35* -- NOTE Tests of Sngle Lnear Coeffcent Restrctons: t-tests and -tests Basc Rules Tests of a sngle lnear coeffcent restrcton can be performed usng ether a two-taled t-test or an -test Tests of two or more lnear coeffcent restrctons can only be performed usng an -test Testng Sngle Lnear Coeffcent Restrctons Consder the followng LOG-LOG (double-log regresson equaton: ln Y + ln X + ln X + u ( The slope coeffcents and are elastcty coeffcents; they are therefore comparable n magntude Common hypothess tests: each nvolves only one lnear coeffcent restrcton or the elastcty of Y wrt X equals the elastcty of Y wrt X the margnal effect on lny of lnx equals the margnal effect on lny of lnx or + the elastcty of Y wrt X s equal n magntude but opposte n sgn to the elastcty of Y wrt X the margnal effect on lny of lnx s equal n magntude but opposte n sgn to the margnal effect on lny of lnx ECON 35* -- Note : Tests of Sngle Lnear Coeffcent Restrctons Page of 3 pages

2 ECONOMICS 35* -- NOTE 3 + (the constant returns-to-scale hypothess the elastctes of Y wrt X and X sum to one; mples that f X and X both change by some proporton λ, then Y changes by the same proporton the margnal effects on lny of lnx and lnx sum to one All three of these hypotheses have a common form: each states that a lnear combnaton of the regresson coeffcents and equals some constant A lnear functon, or lnear combnaton, of the regresson coeffcents and takes the general form c + c where c and c are specfed (known constants Some smple examples: or or ths case, c and c or + or ths case, c and c 3 + or ths case, c and c 4 + or ths case, c and c ECON 35* -- Note : Tests of Sngle Lnear Coeffcent Restrctons Page of 3 pages

3 ECONOMICS 35* -- NOTE 3 General ramework for t-tests and -tests of Lnear Coeffcent Restrctons We want to generalze the t-statstcs and -statstcs for ndvdual coeffcent estmates Recall that the t-statstc for s: ( ~ t[n K] t[n K ] t sê( Recall that the -statstc for s: ( ~ [, N K] [, N K ] ( âr( We now need the t-statstc and the -statstc for lnear combnatons, or lnear functons, of regresson coeffcent estmates such as: c + where c and c are specfed (known constants c ECON 35* -- Note : Tests of Sngle Lnear Coeffcent Restrctons Page 3 of 3 pages

4 ECONOMICS 35* -- NOTE A lnear functon, or lnear combnaton, of the regresson coeffcents and takes the general form c where c and c are specfed (known constants + c A lnear restrcton on the regresson coeffcents and takes the general form c + c c where c s also a specfed (known constant The null and alternatve hypotheses take the general form H : c + c c H : c + c c The t- and -statstcs for testng H aganst H are based on OLS estmates of the unrestrcted model correspondng to the alternatve hypothess H ln Y + ln X + ln X + û The t-statstc for testng H aganst H takes the general form (c ˆ ˆ + c (c + c ~ t[n K] t[n K sê(c + c ( c ˆ + c t where ê(c ˆ + c âr(c + c ˆ s The -statstc for testng H aganst H takes the general form ( [ ˆ ˆ ˆ ˆ (c + c (c + c ] c + c ~ [, N K] [, N K ] âr(c + c ] ECON 35* -- Note : Tests of Sngle Lnear Coeffcent Restrctons Page 4 of 3 pages

5 ECONOMICS 35* -- NOTE General formula for computng the estmated varance of a lnear combnaton of coeffcent estmates The estmated varance of the lnear combnaton of coeffcent estmates c + c s gven by the formula: where ˆ ˆ âr(c c c âr(ˆ c âr(ˆ c c Côv(ˆ, ˆ âr( ˆ the estmated varance of ˆ ; âr( ˆ the estmated varance of $ ; Côv( ˆ, ˆ the estmated covarance of ˆ and $ Note: To compute âr(c ˆ ˆ + c, you need to obtan the values of âr(, and ˆ ˆ âr( ˆ Côv(, These are obtaned from the estmated varance-covarance matrx for the OLS coeffcent estmates ˆ ECON 35* -- Note : Tests of Sngle Lnear Coeffcent Restrctons Page 5 of 3 pages

6 ECONOMICS 35* -- NOTE Examples Evaluate the general formula ˆ ˆ ˆ âr(c c c âr( c âr( ˆ c c Côv( ˆ, ˆ for some specfc lnear combnatons of the two coeffcent estmates ˆ and $ or the lnear combnaton ˆ ˆ, c and c c, c, c c (( âr( âr( âr( ˆ Côv( ˆ, ˆ + or the lnear combnaton ˆ ˆ +, c and c c, c, c c (( ar( ˆ + ar( ˆ + ar( ˆ + Cov( ˆ, 3 or the lnear combnaton ˆ +, c c and c, c 4, c c (( 4 âr( + ˆ âr( 4âr( ˆ 4Côv( ˆ, ˆ or the lnear combnaton ˆ ˆ, c and c c, c ( 4, c c (( 4 âr( ˆ âr( + 4âr( 4Côv(, ˆ ECON 35* -- Note : Tests of Sngle Lnear Coeffcent Restrctons Page 6 of 3 pages

7 ECONOMICS 35* -- NOTE 4 Test of a Sngle Lnear Coeffcent Restrcton: General Example or testng a sngle lnear combnaton of two (or more regresson coeffcents such as c +, use ether a t-test or an -test c The t-statstc for the lnear combnaton of coeffcent estmates c ˆ + c s: ˆ ˆ ( ˆ ˆ (c + c (c + c t c + c ~ t[n K] t[n K] sê(c + c The -statstc for the lnear combnaton of coeffcent estmates c ˆ + c s: ( [ ˆ ˆ ˆ ˆ (c + c (c + c ] c + c ~ [, N K] [, N K ] âr(c + c A Two-Taled t-test of a Sngle Lnear Coeffcent Restrcton Null and alternatve hypotheses H : H : Compute OLS estmates of the unrestrcted model correspondng to the alternatve hypothess H The unrestrcted OLS SRE s: ln Y + ln X + ln X + û (,, N (* Retreve the values of: ˆ, $, âr( ˆ, âr( ˆ and ôv(, ˆ C ECON 35* -- Note : Tests of Sngle Lnear Coeffcent Restrctons Page 7 of 3 pages

8 ECONOMICS 35* -- NOTE Compute sample value of the t-statstc under the null hypothess H The t-statstc for the lnear coeffcent combnaton s ˆ ˆ ( ˆ ˆ ( ( t ~ t[n K] t[n K] t[n sê( Compute âr( âr( + âr( Côv(, ˆ Compute ê( ˆ âr( ˆ s Set, as specfed by the null hypothess H The sample value of the t-statstc under H s therefore t ( ˆ ˆ ˆ ( sê( sê( ( ˆ 3 The null dstrbuton of t ( ˆ dstrbuton: s the t[n K] t[n K] t[n 3] ( ˆ ~ t[n K] t[n K ] t[n 3] t under H 4 Apply the usual decson rule for a two-taled t-test At sgnfcance level α (the α percent sgnfcance level, Reect H f t > t α / [N K] t α / [N 3] or two-tal p-value for t < α ; Retan H f t t α / [N K] t α / [N 3] or two-tal p-value for t α 3] ECON 35* -- Note : Tests of Sngle Lnear Coeffcent Restrctons Page 8 of 3 pages

9 ECONOMICS 35* -- NOTE A Two-Taled -test of a Sngle Lnear Coeffcent Restrcton Null and alternatve hypotheses H : H : Compute OLS estmates of the unrestrcted model correspondng to the alternatve hypothess H The unrestrcted OLS SRE s: ln Y + ln X + ln X + û (,, N (* Retreve the values of: ˆ, $, âr( ˆ, âr( ˆ and ôv(, ˆ C Compute sample value of the -statstc under the null hypothess H The -statstc for the lnear coeffcent combnaton s ˆ ˆ ( ˆ ˆ [( ( ] ~ [, N K] [, N K ] [, N 3] âr( Compute âr(ˆ âr(ˆ + âr(ˆ Côv(, ˆ Set, as specfed by the null hypothess H The sample value of the -statstc under H s therefore ( [( ˆ ˆ ( ] ( ˆ ˆ ˆ ˆ âr( âr( ECON 35* -- Note : Tests of Sngle Lnear Coeffcent Restrctons Page 9 of 3 pages

10 ECONOMICS 35* -- NOTE 3 The null dstrbuton of ( ˆ dstrbuton: s the [, N K] [, N K] [, N 3] ( ˆ ~ [, N K] [, N K ] [, N 3] Note: under H ( [ ( ] ˆ t or t ( ˆ ( 4 Apply the usual decson rule for an -test At sgnfcance level α (the α percent sgnfcance level, Reect H f > α [, N K] α [, N 3] or p-value for < α ; Retan H f α [, N K] α [, N 3] or p-value for α ECON 35* -- Note : Tests of Sngle Lnear Coeffcent Restrctons Page of 3 pages

11 ECONOMICS 35* -- NOTE An Equvalent General -test of a Sngle Lnear Coeffcent Restrcton Null and alternatve hypotheses H : H : Compute OLS estmates of the unrestrcted model correspondng to the alternatve hypothess H The unrestrcted model s gven by the PRE ln Y + ln X + ln X + u ( The unrestrcted OLS SRE obtaned by OLS estmaton of equaton ( s ln Y + ln X + ln X + û (,, N (* Retreve the values of: N RSS RSSU u$ and df N K N K N 3 ormulate the restrcted model correspondng to the null hypothess H Substtute the restrcton nto the unrestrcted regresson equaton (: ln Y ln X ln X (ln X + + ln X ln X + ln X + u + u + u Result: The restrcted model s gven by the PRE ln Y + (ln X + ln X + u ( ECON 35* -- Note : Tests of Sngle Lnear Coeffcent Restrctons Page of 3 pages

12 ECONOMICS 35* -- NOTE 3 Estmate the restrcted model by OLS The restrcted OLS SRE obtaned by OLS estmaton of equaton ( s ~ ~ ln Y + (,, N (* + (ln X ~ + ln X u ~ ~ Note that the restrcted OLS estmate of s smply Retreve the values of N RSS RSSR ~ u and df N K N 4 Compute the sample value of the -statstc under the null hypothess H The requred -statstc s: ( RSS RSS ( df df ~ [ df df df, RSS df ] under H or ths partcular test: df N K N ; df N K N 3 df df N K (N K K K 3 The sample value of the -statstc s therefore ( RSS RSS ( df df ( RSS RSS ( RSS RSS RSS df RSS ( N 3 RSS ( N 3 5 The null dstrbuton of s the [, N 3] dstrbuton 6 Apply the usual decson rule for an -test At sgnfcance level α, Reect H f > α [, N K] α [, N 3] or p-value for < α ; Retan H f α [, N K] α [, N 3] or p-value for α ECON 35* -- Note : Tests of Sngle Lnear Coeffcent Restrctons Page of 3 pages

13 ECONOMICS 35* -- NOTE Equvalence of the t-test and -tests of a Sngle Lnear Coeffcent Restrcton The t-test and -test of a sngle lnear coeffcent restrcton are completely equvalent Ths equvalence follows from two facts: The sample values of the two test statstcs under the null hypothess H are related as follows: ( t or t The square of the sample value of the t-statstc equals the sample value of the - statstc; or the sample value of the t-statstc equals the square root of the sample value of the -statstc At sgnfcance level α, the crtcal values of the null dstrbutons t[n K] and [, N K] are related as follows: ( t [N K] [, N K] α α or t α [N K] α [, N K] The square of the two-taled α/ crtcal value of the t[n K] dstrbuton equals the α-level crtcal value of the [, N K] dstrbuton; or the two-taled α/ crtcal value of the t[n K] dstrbuton equals the square root of the α-level crtcal value of the [, N K] dstrbuton 3 The p-values for the calculated sample values of the test statstcs t and are related as follows: two-taled p-value for t p-value for where two-taled p-value for t Pr( t > t p-value for Pr( > ECON 35* -- Note : Tests of Sngle Lnear Coeffcent Restrctons Page 3 of 3 pages