This document provides guidance on hypothesis testing in linear regression models. It discusses tests of one coefficient restriction and one linear restriction on two or more coefficients. These can be tested using two-tail or one-tail t-tests and F-tests. The test statistics and decision rules for rejecting or retaining the null hypothesis are also outlined.

1. ECON 351* -- A Guide to Hypothesis Testing M.G. Abbott
A Guide to Hypothesis Testing in Linear Regression Models
Tests of One Coefficient Restriction: One Restriction on One Coefficient
H0 specifies only one equality restriction on one coefficient.
Two-Tail Tests of One Restriction on One Coefficient
• Example: H0: βj = bj versus H1: βj ≠ bj where bj is a specified constant.
• Use: either a two-tail t-test or an F-test.
• Test Statistics:
$ ˆ
$ ) = β j − β j ~ t[ N − k ]
t (β j ˆ
⇒ sample value = t 0 (β j ) =
βj − bj
.
$( $
se β j ) ˆ ˆ
se(β ) j
( ) ˆ ) = (β − b )
2
$
βj − βj ˆ 2
$
F(β j ) = ~ F[1, N − k ] ⇒ sample value = F (β j j
.
ˆ ˆ
0 j
$ $
Var (β j ) Var (β j )
Note: [ $ 2
] $ ˆ ˆ
t (β j ) = F(β j ) or t (β j ) = F(β j ) and [t α / 2 [ N − k ]]2 = Fα [1, N − k ] .
• Decision Rules:
Reject H0 if t 0 > t α / 2 [ N − k ] or two-tail p-value for t 0 = Pr ( t > t 0 ) < α ;
F0 > Fα [1, N − k ] or p-value for F0 = Pr( F > F0 ) < α .
Retain H0 if t 0 ≤ t α / 2 [ N − k ] or two-tail p-value for t 0 = Pr ( t > t 0 ) ≥ α ;
F0 ≤ Fα [1, N − k ] or p-value for F0 = Pr( F > F0 ) ≥ α .
ECON 351*: Guide to Hypothesis Testing Page 1 of 5
351guide.doc

2. ECON 351* -- A Guide to Hypothesis Testing M.G. Abbott
One-Tail Tests of One Restriction on One Coefficient
• Examples: H0: βj = bj (or βj ≥ bj) versus H1: βj < bj a left-tail test
H0: βj = bj (or βj ≤ bj) versus H1: βj > bj a right-tail test
• Use: a one-tail t-test.
• Test Statistic:
$ ˆ
$ ) = β j − β j ~ t[ N − k ]
t (β j ⇒ ˆ
sample value = t 0 (β j ) =
βj − bj
.
$( $
se β j ) ˆ ˆ
se(β )
j
• Decision Rules -- left-tail t-test:
Reject H0 if t 0 < − t α [ N − k ] or one-tail p-value for t 0 = Pr( t < t 0 ) < α ;
Retain H0 if t 0 ≥ − t α [ N − k ] or one-tail p-value for t 0 = Pr( t < t 0 ) ≥ α .
• Decision Rules -- right-tail t-test:
Reject H0 if t 0 > t α [ N − k ] or one-tail p-value for t 0 = Pr( t > t 0 ) < α ;
Retain H0 if t 0 ≤ t α [ N − k ] or one-tail p-value for t 0 = Pr( t > t 0 ) ≥ α .
ECON 351*: Guide to Hypothesis Testing Page 2 of 5
351guide.doc

3. ECON 351* -- A Guide to Hypothesis Testing M.G. Abbott
Tests of One Linear Restriction on Two or More Coefficients
H0 specifies only one linear restriction on two or more regression coefficients.
Two-Tail Tests of One Linear Restriction on Two Coefficients
• Example: H0: cjβj + chβh = c0 versus H1: cjβj + chβh ≠ c0.
• Use: either a two-tail t-test or an F-test.
• Test Statistics:
ˆ ˆ
(
t c jβ j h h )
ˆ + c β = (c jβ j + c h β h ) − (c jβ j + c hβ h ) ~ t[ N − k ] = t[ N − k ]
ˆ
ˆ ˆ ˆ
se(c jβ j + c h β h )
1
ˆ ˆ ˆ ˆ ˆ
where: se(c jβ j + c h β h ) = Var (c jβ j + c h β h )
ˆ
ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ
Var (c jβ j + c h β h ) = c 2 Var (β j ) + c 2 Var (β h ) + 2c jc h Cov(β j , β h ) .
ˆ j h
ˆ ˆ
sample value = t 0 c jβ j h h (
ˆ + c β = (c jβ j + c h β h ) − c 0 .
ˆ
ˆ ˆ ˆ
se(c jβ j + c h β h )
)
F(c β + c β ) =
ˆ ˆ [(c β + c β ) − (c β + c β )]
ˆ ˆ
j j h h j j h h
2
~ F[1, N − k ] = F[1, N − k1 ]
j j h h
ˆ ˆ
Var (c β + c β )
ˆ j j h h
ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ
where: Var (c jβ j + c h β h ) = c 2 Var (β j ) + c 2 Var (β h ) + 2c jc h Cov(β j , β h ) .
ˆ j h
sample value = F (c β + c β ) =
ˆ ˆ [(c β + c β ) − c ]
ˆ ˆ
j j h h 0
2
0
ˆ j
ˆ
Var (c β + c β )
ˆ
j h h
j j h h
• Decision Rules:
Reject H0 if t 0 > t α / 2 [ N − k ] or two-tail p-value for t 0 = Pr ( t > t 0 ) < α ;
F0 > Fα [1, N − k ] or p-value for F0 = Pr( F > F0 ) < α .
Retain H0 if t 0 ≤ t α / 2 [ N − k ] or two-tail p-value for t 0 = Pr ( t > t 0 ) ≥ α ;
F0 ≤ Fα [1, N − k ] or p-value for F0 = Pr( F > F0 ) ≥ α .
ECON 351*: Guide to Hypothesis Testing Page 3 of 5
351guide.doc

4. ECON 351* -- A Guide to Hypothesis Testing M.G. Abbott
One-Tail Tests of One Linear Restriction on Two Coefficients
• Examples: H0: cjβj + chβh = c0 vs. H1: cjβj + chβh < c0 a left-tail test
H0: cjβj + chβh = c0 vs. H1: cjβj + chβh > c0 a right-tail test
• Use: a one-tail t-test.
• Test Statistic:
ˆ ˆ
(
t c jβ j h h )
ˆ + c β = (c jβ j + c h β h ) − (c jβ j + c hβ h ) ~ t[ N − k ] = t[ N − k ]
ˆ
ˆ ˆ ˆ
se(c jβ j + c h β h )
1
ˆ ˆ ˆ ˆ ˆ
where se(c jβ j + c h β h ) = Var (c jβ j + c h β h )
ˆ
ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ
Var (c jβ j + c h β h ) = c 2 Var (β j ) + c 2 Var (β h ) + 2c jc h Cov(β j , β h ) .
ˆ j h
ˆ ˆ
(
sample value = t 0 c jβ j h h
ˆ ˆ
)
ˆ + c β = (c jβ j + c h β h ) − c 0 .
ˆ
ˆ
se(c jβ j + c h β h )
• Decision Rules -- left-tail t-test:
Reject H0 if t 0 < − t α [ N − k ] or one-tail p-value for t 0 = Pr( t < t 0 ) < α ;
Retain H0 if t 0 ≥ − t α [ N − k ] or one-tail p-value for t 0 = Pr( t < t 0 ) ≥ α .
• Decision Rules -- right-tail t-test:
Reject H0 if t 0 > t α [ N − k ] or one-tail p-value for t 0 = Pr( t > t 0 ) < α ;
Retain H0 if t 0 ≤ t α [ N − k ] or one-tail p-value for t 0 = Pr( t > t 0 ) ≥ α .
ECON 351*: Guide to Hypothesis Testing Page 4 of 5
351guide.doc

5. ECON 351* -- A Guide to Hypothesis Testing M.G. Abbott
Tests of Two or More Linear Coefficient Restrictions
H0 specifies two or more linear coefficient restrictions.
• Example: H0: β2 = β4 and β3 = β5
H1: β2 ≠ β4 and/or β3 ≠ β5
• Use: a general F-test; only an F-test can be used to test jointly two or more
coefficient restrictions.
• Test Statistics: Either of the following two general F-statistics.
(RSS 0 − RSS1 ) (df 0 − df1 ) (RSS 0 − RSS1 ) (k − k 0 )
F= = .
RSS1 df1 RSS1 ( N − k )
(R 2 − R 2 ) (df 0 − df1 ) (R 2 − R 2 ) (k − k 0 )
F= U R
= U 2R .
(1 − R U ) df1
2
(1 − R U ) ( N − k )
Null distribution: F ~ F[df 0 − df1 , df1 ] = F[k − k 0 , N − k ] .
• Sample value of F-statistic under H0 = F0.
• Decision Rules:
Reject H0 if F0 > Fα [df 0 − df1 , df1 ] = Fα [k − k 0 , N − k ] or
p-value for F0 = Pr( F > F0 ) < α .
Retain H0 if F0 ≤ Fα [df 0 − df 1 , df 1 ] = Fα [k − k 0 , N − k ] or
p-value for F0 = Pr( F > F0 ) ≥ α .
ECON 351*: Guide to Hypothesis Testing Page 5 of 5
351guide.doc