Review of Statistical Inference
Prepared by Vera Tabakova, East Carolina University

C.1 A Sample of Data

C.2 An Econometric Model

C.3 Estimating the Mean of a Population

C.4 Estimating the Population Variance and Other Moments

C.5 Interval Estimation
Principles of Econometrics, 3rd Edition
Slide C-2

C.6 Hypothesis Tests About a Population Mean

C.7 Some Other Useful Tests

C.8 Introduction to Maximum Likelihood Estimation

C.9 Algebraic Supplements
Principles of Econometrics, 3rd Edition
Slide C-3
Principles of Econometrics, 3rd Edition
Slide C-4
Figure C.1 Histogram of Hip Sizes
Principles of Econometrics, 3rd Edition
Slide C-5
E[Y ]  
var(Y )  E[Y  E(Y )]2  E[Y ]2  2
Principles of Econometrics, 3rd Edition
(C.1)
(C.2)
Slide C-6
y   yi N
(C.3)
N
Y   Yi / N
(C.4)
i 1
Principles of Econometrics, 3rd Edition
Slide C-7
y   yi N
(C.3)
N
Y   Yi / N
(C.4)
i 1
Principles of Econometrics, 3rd Edition
Slide C-8
Principles of Econometrics, 3rd Edition
Slide C-9
Y
N
1
1
1
1
Y   Yi  Y1  Y2  ...  YN
N
N
N
i 1 N
(C.5)
1 
1 
1 
E[Y ]  E  Y1   E  Y2   ...  E  YN 
N 
N 
N 
1
1
1
 E Y1   E Y2   ...  E YN 
N
N
N
1
1
1
     ...  
N
N
N

Principles of Econometrics, 3rd Edition
Slide C-10
Y
1
1
1

var Y   var  Y1  Y2  ...  YN 
N
N 
N
1
1
1
= 2 var Y1   2 var Y2   ...  2 var YN 
N
N
N
1 2 1 2
1 2
 2   2   ...  2 
N
N
N
2

N
Principles of Econometrics, 3rd Edition
(C.6)
Slide C-11
Y
Figure C.2 Increasing Sample Size and Sampling Distribution of Y
Principles of Econometrics, 3rd Edition
Slide C-12
Y
 1
Y 
1 


P   1  Y    1  P 




/
N

/
N

/
N


1 
 1
 P
Z 

.25
.25


 P  2  Z  2   .9544

If we draw a random sample of size N = 40 from a normal population
with variance 10, the least squares estimator will provide an estimate
within 1 inch of the true value about 95% of the time. If N = 80 the
probability that Y is within 1 inch of μ increases to 0.995.
Principles of Econometrics, 3rd Edition
Slide C-13
Central Limit Theorem: If Y1,…,YN are independent
and identically distributed random variables with mean
μ and variance
σ 2,
Y 
and Y  Yi / N , then Z N 
 N
has a probability distribution that converges to the
standard normal N(0,1) as N  .
Principles of Econometrics, 3rd Edition
Slide C-14
2 y 0  y  1
f  y  
otherwise
0
Y 2/3
ZN 
1/18
N
Principles of Econometrics, 3rd Edition
Slide C-15
Figure C.3 Central Limit Theorem
Principles of Econometrics, 3rd Edition
Slide C-16

A powerful finding about the estimator of the
population mean is that it is the best of all possible
estimators that are both linear and unbiased.

A linear estimator is simply one that is a weighted
average of the Yi’s, such as Y   aiYi , where the ai
are constants.

“Best” means that it is the linear unbiased estimator
with the smallest possible variance.
Principles of Econometrics, 3rd Edition
Slide C-17
r
 r  E  Y    


1

1  E Y      E Y     0


2

 2  E Y       2


3

3  E Y    


4

 4  E Y    


Principles of Econometrics, 3rd Edition
Slide C-18
var Y     E Y   
2
2

ˆ
Principles of Econometrics, 3rd Edition
2
2
Yi  Y 



2
N
Yi  Y 



N 1
2
(C.7)
Slide C-19
var Y   ˆ 2 N
(C.8)
se Y   var Y   ˆ / N
(C.9)
Principles of Econometrics, 3rd Edition
Slide C-20
r
 r  E  Y    


In statistics the Law of Large Numbers says that sample means
converge to population averages (expected values) as the sample size
N → ∞.
 2   Yi  Y  N   2
2
3   Yi  Y  N
3
 4   Yi  Y  N
4
Principles of Econometrics, 3rd Edition
Slide C-21
3
skewness  S  3

4
kurtosis  K  4

Principles of Econometrics, 3rd Edition
Slide C-22
ˆ
2
yi  y 



2
N 1
yi  17.1582 



49
2
159.9995
 3.2653

49
ˆ 2 3.2653
 .0653

var Y  
50
N
se Y   ˆ
Principles of Econometrics, 3rd Edition
N  .2556
Slide C-23

 Yi  Y 
2
N  159.9995 50  1.7889
3   Yi  Y  N  .0791
3
 4   Yi  Y  N  23.8748
4
Principles of Econometrics, 3rd Edition
Slide C-24
 Y   18   
P Y  18   P 






 Y  y 18  17.158 
P Y  18   P 

 P  Z  .4659   .3207

1.8070 
 ˆ
y * 17.1582 
 Y  y y * 17.1582 

P  Y  y *  P 

 P Z 
 .95


1.8070 
1.8070 
 ˆ

y * 17.1582
 1.645  y*  20.1305
1.8070
Principles of Econometrics, 3rd Edition
Slide C-25

C.5.1 Interval Estimation: σ2 Known
N
Y   Yi N
i 1
Y ~ N  ,  2 N 
Y 
Y 
Z

~ N  0,1
2
 N  N
(C.10)
PZ  z    z 
Principles of Econometrics, 3rd Edition
Slide C-26
Figure C.4 Critical Values for the N(0,1) Distribution
Principles of Econometrics, 3rd Edition
Slide C-27
P  Z  1.96  P  Z  1.96  .025
P  1.96  Z  1.96  1  .05  .95
P Y  1.96 
Principles of Econometrics, 3rd Edition
N    Y  1.96 
(C.11)
N   .95
Slide C-28

 

P Y  zc
   Y  zc
 1 

N
N

Y  zc
Principles of Econometrics, 3rd Edition

N
(C.12)
(C.13)
Slide C-29
Principles of Econometrics, 3rd Edition
Slide C-30
Principles of Econometrics, 3rd Edition
Slide C-31

Any one interval estimate may or may not contain the true
population parameter value.

If many samples of size N are obtained, and intervals are
constructed using (C.13) with (1) = .95, then 95% of
them will contain the true parameter value.

A 95% level of “confidence” is the probability that the
interval estimator will provide an interval containing the
true parameter value. Our confidence is in the procedure,
not in any one interval estimate.
Principles of Econometrics, 3rd Edition
Slide C-32

When σ2 is unknown it is natural to replace it with its
estimator ˆ 2 .
 Yi  Y 
N
ˆ 2 
i 1
N 1
Y 
t
ˆ N
Principles of Econometrics, 3rd Edition
2
t( N 1)
(C.14)
Slide C-33


Y 
P  tc 
 tc   1  
ˆ N


ˆ
ˆ 

P Y  tc
   Y  tc
 1 

N
N

Y  tc
Principles of Econometrics, 3rd Edition
ˆ
or Y  tcse Y 
N
(C.15)
Slide C-34
Remark: The confidence interval (C.15) is based upon the
assumption that the population is normally distributed, so that Y is
normally distributed. If the population is not normal, then we
invoke the central limit theorem, and say that Y is approximately
normal in “large” samples, which from Figure C.3 you can see
might be as few as 30 observations. In this case we can use (C.15),
recognizing that there is an approximation error introduced in
smaller samples.
Principles of Econometrics, 3rd Edition
Slide C-35
Principles of Econometrics, 3rd Edition
Slide C-36
Given a random sample of size N = 50 we estimated the
mean U.S. hip width to be = 17.158 inches.
ˆ 2  3.265 therefore ˆ  1.807
ˆ
y  tc
N  1.807
50  .2556
ˆ
1.807
 17.1582  2.01
 16.6447, 17.6717 
N
50
Principles of Econometrics, 3rd Edition
Slide C-37
Components of Hypothesis Tests
A null hypothesis, H0
An alternative hypothesis, H1
A test statistic
A rejection region
A conclusion
Principles of Econometrics, 3rd Edition
Slide C-38

The Null Hypothesis
The “null” hypothesis, which is denoted H0 (H-naught),
specifies a value c for a parameter. We write the null
hypothesis as H 0 :   c. A null hypothesis is the belief we
will maintain until we are convinced by the sample
evidence that it is not true, in which case we reject the null
hypothesis.
Principles of Econometrics, 3rd Edition
Slide C-39

The Alternative Hypothesis
 H1: μ > c If we reject the null hypothesis that μ = c, we
accept the alternative that μ is greater than c.
 H1: μ < c If we reject the null hypothesis that μ = c, we
accept the alternative that μ is less than c.
 H1: μ ≠ c If we reject the null hypothesis that μ = c, we
accept the alternative that μ takes a value other than (not
equal to) c.
Principles of Econometrics, 3rd Edition
Slide C-40

The Test Statistic
A test statistic’s probability distribution is completely
known when the null hypothesis is true, and it has some
other distribution if the null hypothesis is not true.
Y 
t
~ t N 1
ˆ N
If H 0 :   c is true then
Y c
t
~ t N 1
ˆ N
Principles of Econometrics, 3rd Edition
(C.16)
Slide C-41
Remark: The test statistic distribution in (C.16) is
based on an assumption that the population is normally
distributed. If the population is not normal, then we
invoke the central limit theorem, and say that Y is
approximately normal in “large” samples. We can use
(C.16), recognizing that there is an approximation error
introduced if our sample is small.
Principles of Econometrics, 3rd Edition
Slide C-42

The Rejection Region
 If a value of the test statistic is obtained that falls in a region of
low probability, then it is unlikely that the test statistic has the
assumed distribution, and thus it is unlikely that the null hypothesis
is true.
 If the alternative hypothesis is true, then values of the test statistic
will tend to be unusually “large” or unusually “small”, determined
by choosing a probability , called the level of significance of the
test.
 The level of significance of the test  is usually chosen to be .01,
.05 or .10.
Principles of Econometrics, 3rd Edition
Slide C-43

A Conclusion
 When
you have completed a hypothesis test you should
state your conclusion, whether you reject, or do not
reject, the null hypothesis.
 Say what the conclusion means in the economic context
of the problem you are working on, i.e., interpret the
results in a meaningful way.
Principles of Econometrics, 3rd Edition
Slide C-44
Figure C.5 The rejection region for the one-tail test of H1: μ = c against H1: μ > c
Principles of Econometrics, 3rd Edition
Slide C-45
Figure C.6 The rejection region for the one-tail test of H1: μ = c against H1: μ < c
Principles of Econometrics, 3rd Edition
Slide C-46
Figure C.7 The rejection region for a test of H1: μ = c against H1: μ ≠ c
Principles of Econometrics, 3rd Edition
Slide C-47

The null hypothesis is H 0 :   16.5.
The alternative hypothesis is H1 :   16.5.
Y  16.5
~ t( N 1) if the null
 The test statistic t 
ˆ N
hypothesis is true.

The level of significance =.05.
tc  t.95,49  1.6766
Principles of Econometrics, 3rd Edition
Slide C-48

The value of the test statistic is
17.1582  16.5
t
 2.5756.
1.807 50

Conclusion: Since t = 2.5756 > 1.68 we reject the
null hypothesis. The sample information we have is
incompatible with the hypothesis that μ = 16.5. We
accept the alternative that the population mean hip
size is greater than 16.5 inches, at the =.05 level of
significance.
Principles of Econometrics, 3rd Edition
Slide C-49

The null hypothesis is H 0 :   17.
The alternative hypothesis is H1 :   17.
Y  17
~ t( N 1) if the null
 The test statistic t 
hypothesis is true. ˆ N

The level of significance =.05, therefore  2  .025.
tc  t.975,49  2.01
Principles of Econometrics, 3rd Edition
Slide C-50

The value of the test statistic is
17.1582  17
t
 .6191.
1.807 50

Conclusion: Since 2.01  t  .6191  2.01 we do not
reject the null hypothesis. The sample information
we have is compatible with the hypothesis that the
population mean hip size μ = 17.
Principles of Econometrics, 3rd Edition
Slide C-51
Warning: Care must be taken here in interpreting the
outcome of a statistical test. One of the basic precepts
of hypothesis testing is that finding a sample value of
the test statistic in the non-rejection region does not
make the null hypothesis true! The weaker statements
“we do not reject the null hypothesis,” or “we fail to
reject the null hypothesis,” do not send a misleading
message.
Principles of Econometrics, 3rd Edition
Slide C-52
p-value rule: Reject the null hypothesis when the pvalue is less than, or equal to, the level of significance α.
That is, if p ≤ α then reject H0. If p > α then do not reject
H0
Principles of Econometrics, 3rd Edition
Slide C-53

How the p-value is computed depends on the alternative. If
t is the calculated value [not the critical value tc] of the tstatistic with N−1 degrees of freedom, then:
 if H1: μ > c , p = probability to the right of t
 if H1: μ < c , p = probability to the left of t
 if H1: μ ≠ c , p = sum of probabilities to the right of |t|
and to the left of –|t|
Principles of Econometrics, 3rd Edition
Slide C-54
Figure C.8 The p-value for a right-tail test
Principles of Econometrics, 3rd Edition
Slide C-55
Figure C.9 The p-value for a two-tailed test
Principles of Econometrics, 3rd Edition
Slide C-56

A statistical test procedure cannot prove the truth of a null hypothesis.
When we fail to reject a null hypothesis, all the hypothesis test can
establish is that the information in a sample of data is compatible with
the null hypothesis. On the other hand, a statistical test can lead us to
reject the null hypothesis, with only a small probability, , of
rejecting the null hypothesis when it is actually true. Thus rejecting a
null hypothesis is a stronger conclusion than failing to reject it.
Principles of Econometrics, 3rd Edition
Slide C-57
Correct Decisions
The null hypothesis is false and we decide to reject it.
The null hypothesis is true and we decide not to reject it.
Incorrect Decisions
The null hypothesis is true and we decide to reject it (a
Type I error)
The null hypothesis is false and we decide not to reject it
(a Type II error)
Principles of Econometrics, 3rd Edition
Slide C-58

The probability of a Type II error varies inversely with the
level of significance of the test, , which is the probability
of a Type I error. If you choose to make  smaller, the
probability of a Type II error increases.

If the null hypothesis is μ = c, and if the true (unknown)
value of μ is close to c, then the probability of a Type II
error is high.

The larger the sample size N, the lower the probability of a
Type II error, given a level of Type I error .
Principles of Econometrics, 3rd Edition
Slide C-59
H0 :   c
H1 :   c

If we fail to reject the null hypothesis at the  level of significance,
then the value c will fall within a 100(1)% confidence interval
estimate of μ.

If we reject the null hypothesis, then c will fall outside the
100(1)% confidence interval estimate of μ.
Principles of Econometrics, 3rd Edition
Slide C-60

We fail to reject the null hypothesis when tc  t  tc , or
when
Y c
tc 
 tc
ˆ N
Y  tc
Principles of Econometrics, 3rd Edition
ˆ
ˆ
 c  Y  tc
N
N
Slide C-61

C.7.1 Testing the population variance
Y ~ N  ,  2  , Y   Yi N
ˆ   Yi  Y 
2
2
 N  1
H 0 : 2  02
( N  1)ˆ 2
2
V
~

( N 1)
2
0
Principles of Econometrics, 3rd Edition
Slide C-62
If H1 : 2  02 , then the null hypothesis is rejected if
2
V  (.95,
N 1) .
If H1 : 2  02 , then we carry out a two  tail test,
and the null hypothesis is rejected if
2
2
V  (.975,
or
if
V


N 1)
.025, N 1 .
Principles of Econometrics, 3rd Edition
Slide C-63
Case 1: Population variances are equal
12   22   2p
ˆ 2p
N1  1 ˆ 12   N 2  1 ˆ 22


N1  N 2  2
If the null hypothesis H 0 : 1   2  c is true then
t
Principles of Econometrics, 3rd Edition
Y  Y   c
1
2
 1
1 
ˆ 2p 


N
N
2 
 1
~ t( N1  N2 2)
Slide C-64
Case 2: Population variances are unequal
t
df 
Principles of Econometrics, 3rd Edition
*
Y Y   c


1
2
ˆ 12 ˆ 22

N1 N 2
 ˆ
2
1
N1  ˆ N 2 
2
2
2
  ˆ 2 N 2  ˆ 2 N 2 
 1 1  2 2 
 N1  1
N2 1 


Slide C-65
2
2
ˆ
N

1


 1  1 1
N1  1

ˆ 12 12
F
 2 2 ~ F N1 1, N2 1
2
2
 N 2  1 ˆ 2 2 ˆ 2 2
 N 2  1
Principles of Econometrics, 3rd Edition
Slide C-66
The normal distribution is symmetric, and has a bell-shape with a
peakedness and tail-thickness leading to a kurtosis of 3. We can test
for departures from normality by checking the skewness and kurtosis
from a sample of data.
3
skewness  S  3

4
kurtosis  K  4

Principles of Econometrics, 3rd Edition
Slide C-67
The Jarque-Bera test statistic allows a joint test of these two
characteristics,
2

N 2  K  3 
JB   S 


6 
4

If we reject the null hypothesis then we know the data have non-
normal characteristics, but we do not know what distribution the
population might have.
Principles of Econometrics, 3rd Edition
Slide C-68
For the Hip data,
2
2




K

3
2.3315

3



N 2 
50
2
JB   S 
    .0138 
  .9325




6
4
6
4



2
p  P (2)
 .9325  .6273
Principles of Econometrics, 3rd Edition
Slide C-69
Figure C.10 Wheel of Fortune Game
Principles of Econometrics, 3rd Edition
Slide C-70

For wheel A, with p=1/4, the probability of observing WIN,
WIN, LOSS is
1 1 3 3
  
 .0469
4 4 4 64

For wheel B, with p=3/4, the probability of observing WIN,
WIN, LOSS is
Principles of Econometrics, 3rd Edition
3 3 1 9
  
 .1406
4 4 4 64
Slide C-71

If we had to choose wheel A or B based on the available
data, we would choose wheel B because it has a higher
probability of having produced the observed data.

It is more likely that wheel B was spun than wheel A, and
pˆ  3 4 is called the maximum likelihood estimate of p.

The maximum likelihood principle seeks the parameter
values that maximize the probability, or likelihood, of
observing the outcomes actually obtained.
Principles of Econometrics, 3rd Edition
Slide C-72
Suppose p can be any probability between zero and one.
The probability of observing WIN, WIN, LOSS is the
likelihood L, and is
L  p   p  p  1  p   p 2  p3
(C.17)
We would like to find the value of p that maximizes the
likelihood of observing the outcomes actually obtained.
Principles of Econometrics, 3rd Edition
Slide C-73
Figure C.11 A Likelihood Function
Principles of Econometrics, 3rd Edition
Slide C-74
dL  p 
 2 p  3 p2
dp
2 p  3 p2  0  p  2  3 p   0
There are two solutions to this equation, p=0 or p=2/3.
The value that maximizes L(p) is pˆ  2 3, which is the
maximum likelihood estimate.
Principles of Econometrics, 3rd Edition
Slide C-75
Let us define the random variable X that takes the values
x=1 (WIN) and x=0 (LOSS) with probabilities p and 1−p.
P  X  x  f  x | p   p 1  p 
1 x
x
f  x1 ,
, x  0,1
, xN | p   f  x1 | p    f  xN | p 
N  xi
xi

 p 1  p 
 L  p | x1 ,
Principles of Econometrics, 3rd Edition
(C.18)
, xN 
Slide C-76
Figure C.12 A Log-Likelihood Function
Principles of Econometrics, 3rd Edition
Slide C-77
N
ln L  p    ln f  xi | p 
i 1
(C.19)
N
N 


   xi  ln  p    N   xi  ln 1  p 
i 1
 i 1 


d ln L  p   xi N   xi


dp
p
1 p
Principles of Econometrics, 3rd Edition
Slide C-78
 xi  N   xi
pˆ
1  pˆ
0
1  pˆ   xi  pˆ  N   xi   0
xi

pˆ 
N
Principles of Econometrics, 3rd Edition
x
(C.20)
Slide C-79
N
ln L      ln f  xi |  
i 1
ˆ ~a N  ,V 
ˆ  c a
t
~ t N 1
se ˆ

Principles of Econometrics, 3rd Edition
(C.21)
(C.22)
Slide C-80
REMARK: The asymptotic results in (C.21) and (C.22)
hold only in large samples. The distribution of the test
statistic can be approximated by a t-distribution with N−1
degrees of freedom. If N is truly large then the t(N-1)
distribution converges to the standard normal distribution
N(0,1). When the sample size N may not be large, we
prefer using the t-distribution critical values, which are
adjusted for small samples by the degrees of freedom
correction, when obtaining interval estimates and carrying
out hypothesis tests.
Principles of Econometrics, 3rd Edition
Slide C-81
  d ln L     
V  var ˆ    E 

2
  d 
 

Principles of Econometrics, 3rd Edition
2
1
(C.23)
Slide C-82
Figure C.13 Two Log-Likelihood Functions
Principles of Econometrics, 3rd Edition
Slide C-83
d 2 ln L  p 
xi N   xi

 2 
2
2
dp
p
1  p 
(C.24)
E  xi   1 P  xi  1  0  P  xi  0   1 p  0  1  p   p
Principles of Econometrics, 3rd Edition
Slide C-84
 d 2 ln L  p  
E  xi  N   E  xi 

E


2
2
2
dp
p
1  p 


Np N  Np
 2 
2
p
1

p


N

p 1  p 
Principles of Econometrics, 3rd Edition
Slide C-85
1
  d ln L  p   
p 1  p 
V  var  pˆ     E 
 
2
dp
N
 
 
2
 p 1  p  
pˆ ~ N  p,

N


a
Principles of Econometrics, 3rd Edition
Slide C-86
pˆ 1  pˆ 
ˆ
V
N
se  pˆ   Vˆ 
Principles of Econometrics, 3rd Edition
pˆ 1  pˆ 
N
Slide C-87
se  pˆ  
pˆ 1  pˆ 
.375  .625

 .0342
N
200
pˆ  .4 .375  .4
t

 .7303
se  pˆ 
.0342
pˆ  1.96 se  pˆ   .375  1.96 .0342   .3075,.4425
Principles of Econometrics, 3rd Edition
Slide C-88
C.8.4a The likelihood ratio (LR) test
The likelihood ratio statistic which is twice the difference

between ln L ˆ and ln L  c .

LR  2 ln L ˆ  ln L  c  


Principles of Econometrics, 3rd Edition
(C.25)
Slide C-89
Figure C.14 The Likelihood Ratio Test
Principles of Econometrics, 3rd Edition
Slide C-90
Figure C.15 Critical Value for a Chi-Square Distribution
Principles of Econometrics, 3rd Edition
Slide C-91
N
N




ˆ
ˆ
ln L( p)    xi  ln p   N   xi  ln(1  pˆ )
i 1
 i 1 


 Npˆ ln pˆ   N  Npˆ  ln(1  pˆ )
 N  pˆ ln pˆ  1  pˆ  ln(1  pˆ ) 
Principles of Econometrics, 3rd Edition
Slide C-92
For the cereal box problem pˆ  .375 and N = 200.
ln L( pˆ )  200 .375  ln(.375)  (1  .375) ln(1  .375)
 132.3126
Principles of Econometrics, 3rd Edition
Slide C-93
The value of the log-likelihood function assuming H 0 : p  .4
is true is:
N
N 


ln L(.4)    xi  ln(.4)   N   xi  ln(1  .4)
i 1
 i 1 


 75  ln(.4)  (200  75)  ln(.6)
 132.5750
Principles of Econometrics, 3rd Edition
Slide C-94
The problem is to assess whether −132.3126 is significantly
different from −132.5750.
The LR test statistic (C.25) is:
LR  2 [ln L( pˆ )  ln L(.4)]  2   132.3126  (132.575)   .5247
The critical value is 2.95,1  3.84.
Since .5247 < 3.84 we do not reject the null hypothesis.
Principles of Econometrics, 3rd Edition
Slide C-95
Figure C.16 The Wald Statistic
Principles of Econometrics, 3rd Edition
Slide C-96
2

d
ln L    
ˆ
W    c 

2
d




2
(C.26)
If the null hypothesis is true then the Wald statistic (C.26)
has a 
2
1
distribution, and we reject the null hypothesis if
W   (12  ,1) .
Principles of Econometrics, 3rd Edition
Slide C-97
 d 2 ln L    
1
I    E 

V

2
 d



W  ˆ  c I   


2
W  ˆ  c V
Principles of Econometrics, 3rd Edition
1
(C.27)
2

 ˆ  c

(C.28)
2
V
(C.29)
Slide C-98

Vˆ   I ˆ 


1
(C.30)
ˆ  c ˆ  c
W 

t
Vˆ se ˆ

Principles of Econometrics, 3rd Edition
Slide C-99
In the blue box-green box example:
I  pˆ   Vˆ 1 
N
200

 853.3333
pˆ 1  pˆ  .375 1  .375 
W   pˆ  c  I  pˆ   .375  .4   853.3333  .5333
2
Principles of Econometrics, 3rd Edition
2
Slide C-100
Figure C.17 Motivating the Lagrange multiplier test
Principles of Econometrics, 3rd Edition
Slide C-101
d ln L   
s   
d
(C.31)
 s  c 
2
1
LM 
  s  c   I   
I  
(C.32)
2
LM   s  c    I  c 
2

1
 
2
W  ˆ  c I ˆ
Principles of Econometrics, 3rd Edition
Slide C-102
In the blue box-green box example:
xi N   xi

s .4  

c
1 c
75 200  75
 
 20.8333
.4
1  .4
N
200
I .4  

 833.3333
c 1  c  .4 1  .4 
LM   s .4    I .4     20.8333 833.3333  .5208
2
Principles of Econometrics, 3rd Edition
1
2
1
Slide C-103

C.9.1 Derivation of Least Squares Estimator
N
S   ( yi  ) 2
i 1
di  ( yi  ) 2
d i2  ( yi  ) 2
Principles of Econometrics, 3rd Edition
Slide C-104
N
N
S      d   ( yi  )2
i 1
N
2
i
i 1
N
S      y  2 yi  N  2  a0  2a1  a2 2
i 1
2
i
i 1
a0   yi2  14880.1909, a1   yi  857.9100, a2  N  50
Principles of Econometrics, 3rd Edition
Slide C-105
Figure C.18 The Sum of Squares Parabola For the Hip Data
Principles of Econometrics, 3rd Edition
Slide C-106
dS   
 2a1  2a2
d
2a1  2a2ˆ  0
N
y
a1 i 1 i
ˆ  
y
a2
N
N
ˆ 
Principles of Econometrics, 3rd Edition
Y
i 1
N
i
Y
Slide C-107
For the hip data in Table C.1
N
 yi
857.9100
ˆ 

 17.1582
N
50
i 1
Thus we estimate that the average hip size in the population
is 17.1582 inches.
Principles of Econometrics, 3rd Edition
Slide C-108
N
1
1
1
Y   Yi / N  Y1  Y2  ...  YN
N
N
N
i 1
 a1Y1  a2Y2  ...  aN YN
N
  aiYi
i 1
Principles of Econometrics, 3rd Edition
Slide C-109
N
Y   aiYi
i 1
1
a  ai  ci   ci
N

i
Principles of Econometrics, 3rd Edition
Slide C-110
1

Y   a Y     ci  Yi
i 1
i 1  N

N

i i
N
N
N
1
  Yi   ciYi
i 1 N
i 1
N
 Y   ciYi
i 1
Principles of Econometrics, 3rd Edition
Slide C-111
N
N


E Y   E Y   ciYi      ci E Yi 
i 1
i 1


N
    ci
i 1
Principles of Econometrics, 3rd Edition
Slide C-112
N1
  N1


 
var(Y )  var   ai Yi   var     ci  Yi      ci  var(Yi )
 i 1

  i 1  N

 i 1  N
2
N
2
N
N
2
2 N
1

 1

2
2
2 1
     ci      2  ci  ci       ci   ci2 
N
i 1  N
i 1  N
i 1


 N N i 1

2
N
 N 
2
2
N
c
i 1
 var(Y )  
2
2
i
N
(since
 ci  0)
i 1
N
 ci2
i 1
Principles of Econometrics, 3rd Edition
Slide C-113



















alternative hypothesis
asymptotic distribution
BLUE
central limit theorem
central moments
estimate
estimator
experimental design
information measure
interval estimate
Lagrange multiplier test
Law of large numbers
level of significance
likelihood function
likelihood ratio test
linear estimator
log likelihood function
maximum likelihood estimation
null hypothesis
Principles of Econometrics, 3rd Edition



















point estimate
population parameter
p-value
random sample
rejection region
sample mean
sample variance
sampling distribution
sampling variation
standard error
standard error of the mean
standard error of the estimate
statistical inference
test statistic
two-tail tests
Type I error
Type II error
unbiased estimators
Wald test
Slide C-114