Chapter 6
The Two-Variable Model:
Hypothesis Testing
The Object of Hypothesis Testing
To answer——
? How,good” is the estimated regression line.
? How can we be sure that the estimated regression function
(i.e.,the SRF) is in fact a good estimator of the true PRF?
Yi=B1+B2Xi+μ i Xi———nonstochastic
μ i———stochasti
Yi———stochastic
? Before we tell how good an SRF is as an estimate of the
true PRF,we should assume how the stochastic μ terms
are generated.
6.1 The Classical Linear Regression Model (CLRM)
CLRM assumptions:
? A6.1,The explanatory variable(s) X is uncorrelated with the
disturbance termμ,
? A6.2,Zero mean value assumption:
? ——The expected,or mean,value of the disturbance termμ is zero.
E(μ i)=0 (6.1)
? A6.3,Homoscedasticity assumption:
? ——The variance of each μi is constant,or homoscedastic.
var(μi)=σ2 (6.2)
? A6.4,No autocorrelation assumption:
? ——There is no correlation between two error terms.
? cov(μ i,μ j)=0 i≠ j (6.3)
6.2 Variandes and Standard Errors of Ordinary
Least Squares（ OLS） Estimators
? ——Study the sampling variability of OLS estimators.
? The variances and standard errors of the OLS estimators:
var(b1)= ·σ 2 (6.4)
se(b1) = (6.5)
var(b2)= (6.6)
se(b2) = (6.7)
is an estimator ofσ 2
(6.8)
(6.9)
∑ ei2=RSS(residual sum of squares) =∑ （ Yi-Yi） 2
n-2……,.degrees of freedom
?? 2i
2i
Xn
X
)var(b1
? 2i2Xσ
)var (b2
2σ?
2n
eσ 2i2
??
??
2σσ ?? ?
6.3 The Properties of OLS Estimators
Why Ordinary Least Squares（ OLS）? The OLS method is used
popularly because it has some very strong theoretical properties,
which is known as the Gauss-Markov theorem:
? Gauss-Markov theorem:
——Given the assumptions of the classical linear regression
model,the OLS estimators,in the class of unbiased linear estimators,
have minimum variance; that is,they are BLUE(best linear unbiased
estimators).
That is,the OLS estimators b1 and b2 are:
? 1,Linear,they are linear functions of the random variable Y.
? 2,Unbiased,E( b1 )=B1
E( b2 )=B2
E( )=
? 3,Have minimum variance.
2σ? 2σ
6.4 The Sampling,or Probability,Distributions of OLS
Estimators
? 1.One more assumption of the CLRM.needed:
A6.5,In the PRF Yi=B1+B2Xi+μ i,the error term μ i follows the normal
distribution with mean zero and variance, That is μ i~N(0,) (6.17)
? Central limit theorem:
——If there is a large number of independent and identically
distributed random variables,then,with a few exceptions,the
distribution of their sum tends to be a normal distribution as the number
of such variables increases indefinitely.
? 2,b1 and b2 follow normal distribution
∵ ---μ follows the distribution
---b1 and b2 are linear functions of the normally distributed variable μ,
∴ b1 and b2 are normally distributed.
b1~N(B1,) (6.18)
=var(b1)= (6.4)
b2~N(B2,) (6.19)
=var(b2)= (6.6)
2σ2σ
2b1σ
2b
1σ
?
?
2
i
2
i
χn
X
2b2σ
2b
2σ ?
2
i
2
χ
σ
6.5 Hypothesis Testing
? 1,The confidence interval approach
2,The test of significance approach
? 1,t statistic
? known,~N(0,1)
unknown,we can estimate by using
= ~tn-2 (6.21)
? 2,The Confidence Interval Approach.
（ 1） H0,B2=0
H1,B2≠ 0:
（ 2） Establish a 100(1-α ) confidence interval for B2
P(－ tα /2≤ t≤ tα /2) =1－ α
2σ
?
????
2
i
22
2
22
χσ /
Bb
)s e ( b
BbZ
2σ 2σ 2σ?
)se(b
Bb
2
22 ?
?
?
2
i
22
χ/σ
Bb
?
(6.24)
(6.25)
(6.26)
（ 3） Decision:
· If this interval (i.e.,the acceptance region) includes the null-
hypothesized value of B2,we do not reject the hypothesis.
·If it lies outside the confidence interval (i.e.,it lies in the rejection
region),we reject the null hypothesis.
A Cautionary Note,p166
α1t
χ/σ
BbtP
α /22
i
22
α /2 ??
??
?
?
?
?
??
?
?
?
?
????
??
???
??
?
?
?
?
??
?
?
?
?
????
??
1
χ
σtbB
χ
σtbP
2
i
α /2
222
i
α /2
2
??
?? ? α1)s e (btbB)s e (btbP 2α /2222α /22 ??????
? 3,The Test of Significance Approach
〈 1〉 t test
（ 1） A two-tailed test.
Assume H0:B2=0 and H1:B2 ≠ 0.
Set t statistic,t= (6.29)
Check t table to get the critical t value,if∣ t∣ ＞ critical t value,reject
H0
Check the p value of the t statistic,if p ＜ level of significance,reject
H0
（ 2） A one-tailed test.
H0:B2=0 and H1:B2<0;
T test procedure is the same as the two-tailed test,just:
tα /2 → tα
)se(b
Bb
2
*22 ?
? 〈 2〉 χ 2 test
·H0:
·Get χ 2 statistic,~ (6.31)
·Check χ 2 table,
Find the critical χ 2 value,χ 2＞ critical χ 2 value,reject H0
Find the P value of the χ 2 statistic,p value＞ level of significance,
accept H0
202 σσ ?
2
2
σ
σ2)-(n ? 2 2)(nX ?
6.6 How Good is the Fitted Regression Line,
the Coefficient of Determination,r2
——An overall measure of,goodness of fit” that will tell us how well
the estimated regression line fits the actual Y values.
(5.5)
(6.34)
(6.36)
,the total sum of squares (TSS).
.the explained sum of squares (ESS).
.the residual sum of squares (RSS)
TSS=ESS+RSS (6.38)
(6.39)
(6.40)
r2 the (sample) coefficient of determination.
r2 measures the proportion or percentage of the total variation in Y
explained by the regression model.
iii eYY ?? ?
)Y(Y)YY()Y(Y iiii ?? ?????
iii eyy ?? ?
??? ?? 2i2i2i eyy
?2iy
?2iy?
?2ie
T SSR SST SSE SS1 ??
T SSESSr 2 ?
? Two properties of r2:
① r2>0
② 0≤ r2≤ 1
? Compute r2:
(6.41)
(6.42)
? The coefficient of Correlation,r
r= = （ 6.44）
(6.45)
r=± (6.46)
? △ r has the same sign as the slope coefficient
?
???
??
2
i
2
i2
2
y
e
r
T S S
RS Sr1
?
???
2
i
2
i2
y
e1r
? ?
?
??
??
2i2i
ii
)Y(Y)X(X
)Y) ( YX(X
? ?
?
2
i
2
i
ii
yχ
yχ
2r
6.7 Reporting the Results of
Regression Analysis
6.8 Normality Tests
? 1.Histograms of Residuals
The histogram of the residuals is used to learn something about the
slope of the PDF of a random variable.
If the histogram of the residuals is close to the picture of the normal
distribution,the you can guess μ i is normally distributed.
? 2,Normal Probability Plot
Use normal probability paper:
? the horizontal axis,(X-axis)——values of the variable of interest
(OLS residuals ei)
? the vertical axis,(Y-axis) ——the expected values of this variable
If the variable is in fact from the normal population,the NPP will
approximate a straight line.
? （ 3） Jarque-Bera Test
① Compute the coefficients of skewness,S(a measure of asymmetry
of a PDF),and kurtosis,K of a random variables,(OLS residuals ei)
② Develope the test statistic:
~ (6.49)
③ Get JB from χ 2 table,
if JB=0,the variable is normal distributed
if JB＞,reject H0
6.9 A Word on Computation
6.10 An Illustrative Example:
?????? ??? 4 3)(KS6nJB 22
2(2)χ
2
(2)χ
6.11 Forecasting
? 1,Point Prediction
? 2,Interval Prediction
（ 1） σ 2 known,~N
Mean=E(Y|X0)=B1+B2X0 (6.57)
var= (6.58)
（ 2） σ 2 unknown,~ t(n-2)
Establish a 100（ 1－ α ） confidence interval for the true mean
value of Y
(6.59)
So the forecast error will be between:
（, ）
Note,The width of the confidence band is smallest when X0=,
the width widens sharply as X0 moves away from,
0Y?
?????? ?? ? 2
i
2
02
χ
)X(X
n
1σ
0Y?
? ? )α(1)Ys e (tXbbXBB)Ys e (tXbbP 0α /20210210α /2021 ????????? ??
? ? )α(1)Ys e (tXbb)X|E ( Y)Ys e (tXbbP 0/2α02100/2α021 ???????? ??
)Ys e (tXbb 0α /2021 ??? )Ys e (tXbb 0/2α021 ???
X
X
The Two-Variable Model:
Hypothesis Testing
The Object of Hypothesis Testing
To answer——
? How,good” is the estimated regression line.
? How can we be sure that the estimated regression function
(i.e.,the SRF) is in fact a good estimator of the true PRF?
Yi=B1+B2Xi+μ i Xi———nonstochastic
μ i———stochasti
Yi———stochastic
? Before we tell how good an SRF is as an estimate of the
true PRF,we should assume how the stochastic μ terms
are generated.
6.1 The Classical Linear Regression Model (CLRM)
CLRM assumptions:
? A6.1,The explanatory variable(s) X is uncorrelated with the
disturbance termμ,
? A6.2,Zero mean value assumption:
? ——The expected,or mean,value of the disturbance termμ is zero.
E(μ i)=0 (6.1)
? A6.3,Homoscedasticity assumption:
? ——The variance of each μi is constant,or homoscedastic.
var(μi)=σ2 (6.2)
? A6.4,No autocorrelation assumption:
? ——There is no correlation between two error terms.
? cov(μ i,μ j)=0 i≠ j (6.3)
6.2 Variandes and Standard Errors of Ordinary
Least Squares（ OLS） Estimators
? ——Study the sampling variability of OLS estimators.
? The variances and standard errors of the OLS estimators:
var(b1)= ·σ 2 (6.4)
se(b1) = (6.5)
var(b2)= (6.6)
se(b2) = (6.7)
is an estimator ofσ 2
(6.8)
(6.9)
∑ ei2=RSS(residual sum of squares) =∑ （ Yi-Yi） 2
n-2……,.degrees of freedom
?? 2i
2i
Xn
X
)var(b1
? 2i2Xσ
)var (b2
2σ?
2n
eσ 2i2
??
??
2σσ ?? ?
6.3 The Properties of OLS Estimators
Why Ordinary Least Squares（ OLS）? The OLS method is used
popularly because it has some very strong theoretical properties,
which is known as the Gauss-Markov theorem:
? Gauss-Markov theorem:
——Given the assumptions of the classical linear regression
model,the OLS estimators,in the class of unbiased linear estimators,
have minimum variance; that is,they are BLUE(best linear unbiased
estimators).
That is,the OLS estimators b1 and b2 are:
? 1,Linear,they are linear functions of the random variable Y.
? 2,Unbiased,E( b1 )=B1
E( b2 )=B2
E( )=
? 3,Have minimum variance.
2σ? 2σ
6.4 The Sampling,or Probability,Distributions of OLS
Estimators
? 1.One more assumption of the CLRM.needed:
A6.5,In the PRF Yi=B1+B2Xi+μ i,the error term μ i follows the normal
distribution with mean zero and variance, That is μ i~N(0,) (6.17)
? Central limit theorem:
——If there is a large number of independent and identically
distributed random variables,then,with a few exceptions,the
distribution of their sum tends to be a normal distribution as the number
of such variables increases indefinitely.
? 2,b1 and b2 follow normal distribution
∵ ---μ follows the distribution
---b1 and b2 are linear functions of the normally distributed variable μ,
∴ b1 and b2 are normally distributed.
b1~N(B1,) (6.18)
=var(b1)= (6.4)
b2~N(B2,) (6.19)
=var(b2)= (6.6)
2σ2σ
2b1σ
2b
1σ
?
?
2
i
2
i
χn
X
2b2σ
2b
2σ ?
2
i
2
χ
σ
6.5 Hypothesis Testing
? 1,The confidence interval approach
2,The test of significance approach
? 1,t statistic
? known,~N(0,1)
unknown,we can estimate by using
= ~tn-2 (6.21)
? 2,The Confidence Interval Approach.
（ 1） H0,B2=0
H1,B2≠ 0:
（ 2） Establish a 100(1-α ) confidence interval for B2
P(－ tα /2≤ t≤ tα /2) =1－ α
2σ
?
????
2
i
22
2
22
χσ /
Bb
)s e ( b
BbZ
2σ 2σ 2σ?
)se(b
Bb
2
22 ?
?
?
2
i
22
χ/σ
Bb
?
(6.24)
(6.25)
(6.26)
（ 3） Decision:
· If this interval (i.e.,the acceptance region) includes the null-
hypothesized value of B2,we do not reject the hypothesis.
·If it lies outside the confidence interval (i.e.,it lies in the rejection
region),we reject the null hypothesis.
A Cautionary Note,p166
α1t
χ/σ
BbtP
α /22
i
22
α /2 ??
??
?
?
?
?
??
?
?
?
?
????
??
???
??
?
?
?
?
??
?
?
?
?
????
??
1
χ
σtbB
χ
σtbP
2
i
α /2
222
i
α /2
2
??
?? ? α1)s e (btbB)s e (btbP 2α /2222α /22 ??????
? 3,The Test of Significance Approach
〈 1〉 t test
（ 1） A two-tailed test.
Assume H0:B2=0 and H1:B2 ≠ 0.
Set t statistic,t= (6.29)
Check t table to get the critical t value,if∣ t∣ ＞ critical t value,reject
H0
Check the p value of the t statistic,if p ＜ level of significance,reject
H0
（ 2） A one-tailed test.
H0:B2=0 and H1:B2<0;
T test procedure is the same as the two-tailed test,just:
tα /2 → tα
)se(b
Bb
2
*22 ?
? 〈 2〉 χ 2 test
·H0:
·Get χ 2 statistic,~ (6.31)
·Check χ 2 table,
Find the critical χ 2 value,χ 2＞ critical χ 2 value,reject H0
Find the P value of the χ 2 statistic,p value＞ level of significance,
accept H0
202 σσ ?
2
2
σ
σ2)-(n ? 2 2)(nX ?
6.6 How Good is the Fitted Regression Line,
the Coefficient of Determination,r2
——An overall measure of,goodness of fit” that will tell us how well
the estimated regression line fits the actual Y values.
(5.5)
(6.34)
(6.36)
,the total sum of squares (TSS).
.the explained sum of squares (ESS).
.the residual sum of squares (RSS)
TSS=ESS+RSS (6.38)
(6.39)
(6.40)
r2 the (sample) coefficient of determination.
r2 measures the proportion or percentage of the total variation in Y
explained by the regression model.
iii eYY ?? ?
)Y(Y)YY()Y(Y iiii ?? ?????
iii eyy ?? ?
??? ?? 2i2i2i eyy
?2iy
?2iy?
?2ie
T SSR SST SSE SS1 ??
T SSESSr 2 ?
? Two properties of r2:
① r2>0
② 0≤ r2≤ 1
? Compute r2:
(6.41)
(6.42)
? The coefficient of Correlation,r
r= = （ 6.44）
(6.45)
r=± (6.46)
? △ r has the same sign as the slope coefficient
?
???
??
2
i
2
i2
2
y
e
r
T S S
RS Sr1
?
???
2
i
2
i2
y
e1r
? ?
?
??
??
2i2i
ii
)Y(Y)X(X
)Y) ( YX(X
? ?
?
2
i
2
i
ii
yχ
yχ
2r
6.7 Reporting the Results of
Regression Analysis
6.8 Normality Tests
? 1.Histograms of Residuals
The histogram of the residuals is used to learn something about the
slope of the PDF of a random variable.
If the histogram of the residuals is close to the picture of the normal
distribution,the you can guess μ i is normally distributed.
? 2,Normal Probability Plot
Use normal probability paper:
? the horizontal axis,(X-axis)——values of the variable of interest
(OLS residuals ei)
? the vertical axis,(Y-axis) ——the expected values of this variable
If the variable is in fact from the normal population,the NPP will
approximate a straight line.
? （ 3） Jarque-Bera Test
① Compute the coefficients of skewness,S(a measure of asymmetry
of a PDF),and kurtosis,K of a random variables,(OLS residuals ei)
② Develope the test statistic:
~ (6.49)
③ Get JB from χ 2 table,
if JB=0,the variable is normal distributed
if JB＞,reject H0
6.9 A Word on Computation
6.10 An Illustrative Example:
?????? ??? 4 3)(KS6nJB 22
2(2)χ
2
(2)χ
6.11 Forecasting
? 1,Point Prediction
? 2,Interval Prediction
（ 1） σ 2 known,~N
Mean=E(Y|X0)=B1+B2X0 (6.57)
var= (6.58)
（ 2） σ 2 unknown,~ t(n-2)
Establish a 100（ 1－ α ） confidence interval for the true mean
value of Y
(6.59)
So the forecast error will be between:
（, ）
Note,The width of the confidence band is smallest when X0=,
the width widens sharply as X0 moves away from,
0Y?
?????? ?? ? 2
i
2
02
χ
)X(X
n
1σ
0Y?
? ? )α(1)Ys e (tXbbXBB)Ys e (tXbbP 0α /20210210α /2021 ????????? ??
? ? )α(1)Ys e (tXbb)X|E ( Y)Ys e (tXbbP 0/2α02100/2α021 ???????? ??
)Ys e (tXbb 0α /2021 ??? )Ys e (tXbb 0/2α021 ???
X
X
