This document discusses statistical tests of significance. It explains that statistical tests are used to determine if a result occurred by chance or indicates a real difference. The selection of an appropriate statistical test depends on factors like the scale of measurement, number of groups, and sample size. There are parametric tests that assume a normal distribution and non-parametric tests for qualitative data. Examples of tests mentioned include the t-test, z-test, ANOVA, median test, and chi-square test. The t-test is described in more detail. An example calculation of a t-test is also provided. Finally, uses of statistical tests in fields like medicine are outlined.
Quantitative Analysis for Emperical ResearchAmit Kamble
Overview for Approach Methods for quantitative analysis; which includes
1) Planning of Experiments
2) Data Generation
3) presentation of report
some numerical approach methods; data modeling; hypothesis methods
Quantitative Analysis for Emperical ResearchAmit Kamble
Overview for Approach Methods for quantitative analysis; which includes
1) Planning of Experiments
2) Data Generation
3) presentation of report
some numerical approach methods; data modeling; hypothesis methods
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http://www.youtube.com/onlineteaching
Chapter 9: Inferences from Two Samples
9.2: Two Means, Independent Samples
Please Subscribe to this Channel for more solutions and lectures
http://www.youtube.com/onlineteaching
Chapter 9: Inferences from Two Samples
9.2: Two Means, Independent Samples
ATT00001
ATT00002
ATT00003
ATT00004
ATT00005
CARD.DTA
Card_1995_geo_var_schooling.pdf
Exam2_2014.pdf
ADVANCED ECONOMETRICS
Midterm 2 (Take Home)
Due: Dec.25, 2014
Answer all questions. You should not discuss solutions with your peers but me. Good luck!
Prof. Dr. H. Taştan ,
First Name:...................................................
Last Name:................................................
No:...................................................
1 (20) In class we have shown that when the number of instrumental variables is larger than the number
of endogenous variables the generalized IV estimator (or 2SLS) can be written as
β̂IV =
(
X>PzX
)−1
X>Pzy
where Pz = Z(Z
>Z)−1Z>. In this formulation X is n×k and Z is n× l, l > k.
(a) Show that β̂IV can be obtained as a solution to the following minimization problem
min
β
Q(β) = (y−Xβ)>Pz (y−Xβ)
(b) Show that when k = l the generalized IV estimator reduces to the simple IV estimator:
β̂IV =
(
Z>X
)−1
Z>y
2 (20) Consider the following simple consumption model as a function of permanent income
ci = β1 + β2y
∗
i + ui, ui ∼ iid (0,σ
2
u)
where ci is the logarithm of consumption by household i, and y
∗
i is the permanent income of household
i which is not observed. Instead we observe current income, yi
yi = y
∗
i + vi, vi ∼ iid (0,σ
2
v)
where vi is assumed to be uncorrelated with y
∗
i and ηi. We run the following regression
ci = β1 + β2yi + ηi
(a) Show that yi is negatively correlated with ηi. You can assume β2 > 0.
(b) Evaluate the plim of the OLS estimator β̂2:
β̂2 =
∑n
i=1(yi − ȳ)ci∑n
i=1(yi − ȳ)2
In particular, show that this plim is less than the true β2.
1
3 (30) Use card.dta to answer the following questions. Also read Card (1993), “Using Geographic
Variation in College Proximity to Estimate the Return to Schooling”, NBER Working Paper.
(a) Run the OLS regression of log(wage) on educ, exper, exper2, black, smsa, south, smsa66, reg662
to reg669. Comment on the coefficient estimate of educ.
(b) Estimate the same model by 2SLS using nearc4 as an instrument for educ. Compare the OLS
and IV coefficient estimates on educ. (Note that we partly did this in class). Carry out the
Hausman test.
(c) Use both nearc2 and nearc4 as instruments for educ. Run the reduced form model for educ.
Compare 2SLS estimates to the results obtained in the previous section. Carry out the OID
test.
(d) Discuss the plausibility of Card (1993)’s econometric methodology and empirical findings. Do
you agree with his conclusions?
4 (30) A continuous time model for short term interest rates may be written as a stochastic differential
equation
dr = (α + βr)dt + σrγ�
√
dt
where r is the short term interest rate, � is standard normal random variable, dt is a short time
interval and α,β,γ,σ are parameters. Discrete time approximation is given as
rt+1 − rt = α + βrt + �t+1
with
E(�t+1) = 0, E(�
2
t+1) = σ
2 ...
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1. ch- Sample (Typeset by TYINDEX, Delhi) of May , :
........................................................................................................................
SIGNIFICANT
T E S TS
........................................................................................................................
A fter any experiment we get some results, but we are not sure about this
result whether the result occurred by chance or a real difference. That
time to find truth we will use some statistical tests, these tests are termed as,
‘Tests of Significance’.
S S T:
.................................................................................................................................
The selection of the appropriate statistical test is depends upon:
. The scale of measurement e.g. Ratio, Interval.
. The number of groups e.g. One, Two or More.
. Sample size e.g. If the sample size is less than . Students ‘t’ test is to be
used.
. Measurements e.g. Repeated or Independent measurements.
S T S:
.................................................................................................................................
For application of test sample should be selected randomly. Thus we have in
this case degree of freedom ‘n’ = – = Now, table value of x2 is . t .
2. ch- Sample (Typeset by TYINDEX, Delhi) of May , :
SIGNIFICANT TESTS
Table 13.1 This is Example of Table Sample.
Scale Two groups Three/More groups
Independent Repeated Independent Repeated
Interval Z test Z test ANOVA test ANOVA
and Ratio t test t test (F test) (F test)
Ordinal Median test Wilcox an Median Friedman test
Mann test Kruscal test
Whitney
Nominal X2 test Me Nemar Chi–Square Cochron’s test
test Test
for degree of freedom, which is much less than the obtained value that is
.
T :
.................................................................................................................................
. Parametric Tests
. Non – Parametric Tests
. P T:
.................................................................................................................................
When quantitative data like Weight, Length, Height, and Percentage is given
it is used. These tests were based on the assumption that samples were drawn
from the normally distributed populations.
E.g. Students t test, Z test etc.
. N – P T:
.................................................................................................................................
When qualitative data like Health, Cure rate, Intelligence, Color is given it is
used. Here observations are classified into a particular category or groups.
E.g. Chi square (x2 ) test, Median tests etc.
3. ch- Sample (Typeset by TYINDEX, Delhi) of May , :
SIGNIFICANT TESTS
Table 13.2 This is Example of Table Sample.
Patients Before After
treatment (B) treatment (A)
1 2.4 2.2
2 2.8 2.6
3 3.2 3.0
4 6.4 4.2
5 4.3 2.2
6 2.2 2.0
7 6.2 4.8
8 4.2 2.4
I. T – T:
.................................................................................................................................
W.S. Gosset investigated this test in . It is called Student t – Test because
the pen name of Dr. Gosset was student, hence this test is known as student’s
t – test. It is also called as ‘t- ratio’ because it is a ratio of difference between
two means.
Aylmer Fisher (–) developed students ‘t’ test where samples are
drawn from normal population and are randomly selected.
After comparing the calculated value of ‘t’ with the value given in the ‘t’
table considering degree of freedom we can ascertain its significance.
Is the testing reliable? It is used for comparisons with expectations of the
Normal, Binomial and Poisson distributions and Comparison of a sample
variance with population variance.
S:
Here,
D = 8.3
N=8
D2 = 14.61
8.3
D= = 1.0375
8
4. ch- Sample (Typeset by TYINDEX, Delhi) of May , :
SIGNIFICANT TESTS
∴ Standard deviation of the different between means. Here, the calculated
value for ‘t’ exceeds the tabulated ‘t’ value at p = 0.05 level with df. Therefore
the glucose concentration by the patients after treatment is not significant.
D)2
D2 − ( n
=
N−1
(8.3)2
14.61 − 8
=
7
14.61 − 68.89
8
=
7
14.61 − 8.6112
∴ S.D. =
7
√
= 0.8569
∴ S.D. = 0.9257
Now, standard error of the difference (SED )
SD 0.9257 0.9257
.= √ = √ =
N 8 2.8284
∴ S.E. = 0.3272
D 1.0375
∴t= = = 3.1708
SED 0.3272
Here, the calculated value for ‘t’ exceeds the tabulated ‘t’ value at p = 0.05
level with df. Therefore the glucose concentration by the patients after treat-
ment is not significant.
U:
It is widely used in the field of Medical science, Agriculture and Veterinary as
follows:
r To compare the results of two drugs which is given to same individuals
in the sample at two different situations? E.g. Effect of Bryonia and Ly-
copodium on general symptoms like sleep, appetite etc.
r It is used to study of drug specificity on a particular organ / tissue / cell
level. E.g. Effect of Belberis Vulg. on renal system.
5. ch- Sample (Typeset by TYINDEX, Delhi) of May , :
SIGNIFICANT TESTS
r It is used to compare results of two different methods. E.g. Estimation of
Hb% by Sahlis method and Tallquist method.
r To compare observations made at two different sites of the same body.
E.g. compare blood pressure of arm and thigh.
r To study the accuracy of two different instruments like Thermometer, B.P
apparatus etc.
r To accept the Null Hypothesis that is no difference between the two
means.
r To reject the hypothesis that is the difference between the means of the
two samples is statistically significant.