CS计算机代考程序代写 scheme finance MFIN6201

MFIN6201
Empirical Techniques and Applications in Finance
Week 2
Breastfeeds
University of New South Wales
e-mail: jaehoon.lee@unsw.edu.au
Dr. Jaehoon Lee
School of Banking and Finance
tg§RBM3T*q
Semester 2, 2017
Last update: 2 August 2017

Review of Statistical Theory
• The probability framework for statistical inference • Estimation
• Hypothesis testing
• Confidence intervals
MFIN6201 – Empirical Techniques and Applications in Finance
1

Distribution of a sample of data
Distribution of a sample data drawn randomly from a population: Y1,…,Yn. We will assume simple random sampling
• Choose an individual (district, entity) at random from the population
Randomness and data
• Prior to sample selection, the value of Y is random because the individual selected is random
• Once the individual is selected and the value of Y is observed, then Y is just a number – not random
• The data set is (Y1, Y2, …, Yn), where Yi = value of Y for the i-th individual (district, entity) sampled
MFIN6201 – Empirical Techniques and Applications in Finance
2

Distribution of Y1,…,Yn under simple random sampling Because individual #1 and #2 are selected at random, the value
of Y1 has no information content for Y2. Thus: • Y1 and Y2 are independently distributed
• Y1 and Y2 come from the same distribution, that is, Y1, Y2 are identically distributed
• That is, under simple random sampling, Y1 and Y2 are independently and identically distributed (i.i.d.)
• More generally, under simple random sampling, Yi for i = 1, …, n, are i.i.d.
MFIN6201 – Empirical Techniques and Applications in Finance
3

Estimation
• Suppose independent random draws from an identical distribution, Y1, · · · , Yn.
) this setup is called iid
• Sample average
̄1 1Xn
Y ⌘n(Y1+···+Yn)=n Yi i=1
• Y ̄ is the natural estimator of the mean (E[Y ]), but they are not the same.
• Remember, Y ̄ is the best guess of E[Y ], but not E[Y ] itself!
• Y ̄ is also another random variable.
MFIN6201 – Empirical Techniques and Applications in Finance
4

Sampling distribution of Y ̄
Y ̄ is a random variable, and its properties are determined by the
sampling distribution of Y ̄
• The individuals in the sample are drawn at random.
• Thus the values of (Y1,…,Yn) are random
• Thus functions of (Y1,…,Yn), such as Y ̄, are random: had a di↵erent sample been drawn, they would have taken on a di↵erent value
• The distribution of Y ̄ over di↵erent possible samples of size n is called the sampling distribution of Y ̄.
• The mean and variance of Y ̄ are the mean and variance of its sampling distribution, E(Y ̄) and var(Y ̄).
• The concept of the sampling distribution underpins all of econometrics.
MFIN6201 – Empirical Techniques and Applications in Finance
5

Example:
Then,
8<1 with prob. 0.78 Y = :0 with prob. 0.22 E[Y ] = p = 0.78 var(Y ) = p(1 p) = 0.1716 Sampling distribution of Y ̄: example MFIN6201 - Empirical Techniques and Applications in Finance 6 Sampling distribution of Y ̄: example The sampling distribution of Y ̄ depends on n. Consider n = 2. The sampling distribution of Y ̄ = 12 (Y1 + Y2) is, Pr(Y ̄ = 0) = 0.222 = .0484 P r(Y ̄ = 12) = 2 ⇥ 0.22 ⇥ 0.78 = 0.3432 Pr(Y ̄ = 1) = 0.782 = 0.6084 MFIN6201 - Empirical Techniques and Applications in Finance 7 The sampling distribution of Y ̄ MFIN6201 - Empirical Techniques and Applications in Finance 8 Moments of Y ̄: mean • Mean X E⇥Y ̄⇤=E n1(Y1+···+Yn) 1n = n E [Yi] i=1 =μ – E[Y ̄] = μ, thus Y ̄ is an unbiased estimator of μ MFIN6201 - Empirical Techniques and Applications in Finance 9 Moments of Y ̄: variance • Variance v a r ( Y ̄ ) = v a r X✓ n 1 ( Y 1 + · · · + Y n ) ◆ 1n = n2 var (Yi) ⇣ i=1 ⌘ * cov(Yi,Yj) = 0 for i 6= j because Yi and Yj are iid = 2 n – Thus, var(Y ̄) decreases with n MFIN6201 - Empirical Techniques and Applications in Finance 10 Moments of Y ̄ E ( Y ̄ ) = μ v a r ( Y ̄ ) = 2 n Implications: • Y ̄ is an unbiased estimator of μY (that is, E(Y ̄) = μY ) • var(Y ̄) is inversely proportional to n – The spread of the sampling distribution is proportional to 1/pn – Thus the samplingpuncertainty associated with Y ̄ is proportional to 1/ n (larger samples, less uncertainty, but square-root law) MFIN6201 - Empirical Techniques and Applications in Finance 11 Sampling distribution of Y ̄ when n is large For small sample sizes, the distribution of Y ̄ is complicated, but if n is large, the sampling distribution is simple! • As n increases, the distribution of Y ̄ becomes more tightly centered around μY (Law of Large Numbers) • Moreover, the distribution of pn Y ̄ μY becomes normal (Central Limit Theorem) MFIN6201 - Empirical Techniques and Applications in Finance 12 The Law of Large Numbers An estimator is consistent if the probability that its falls within an interval of the true population value tends to one as the sample size increases. If (Y1, ..., Yn) are i.i.d. and Y2 < 1, then Y ̄ is a consistent estimator of μY , that is, P r [ | Y ̄ μ Y | < " ] ! 1 a s n ! 1 ̄p which can be written, Y ! μY or P r [ | Y ̄ μ Y | > ” ] ! 0 a s n ! 1
̄p ̄p
which can be written, Y ! μY (“Y ! μy” means “converges in
probability to μY ”). The math requires Chebyshev’s inequality: which states that Pr[|X μ| a]  var(X). Thus, as n ! 1,
2 a2
var(Y ̄)= nY !0,whichimpliesthatPr[|Y ̄μY|<"]!1) MFIN6201 - Empirical Techniques and Applications in Finance 13 The Central Limit Theorem (CLT) If (Y1,...,Yn) are i.i.d. and 0 < Y2 < 1 , then when n is large, the distribution of Y ̄ is well approximated by a normal distribution. • Y ̄ is approximately distributed N (μY , Y2 ) (“normal distribution 2 n with mean μY and variance nY ”) Y ̄ E ( Y ̄ ) Y ̄ μ as N(0,1) (standard normal) • The larger is n, the better is the approximation • Standardized Y ̄ ⌘ p = p var(Y ̄) Y/n is approximately distributed Y MFIN6201 - Empirical Techniques and Applications in Finance 14 Sampling distribution of Y ̄ MFIN6201 - Empirical Techniques and Applications in Finance 15 Sampling distribution of pvar(Y ̄) Sampling distribution of Y ̄ Y ̄ E ( Y ̄ ) MFIN6201 - Empirical Techniques and Applications in Finance 16 Summary: Sampling Distribution of Y ̄ For Y1, ..., Yn i.i.d. with 0 < Y2 < 1, • The exact (finite sample) sampling distribution of Y ̄ has mean μY (“Y ̄ is an unbiased estimator of μY ”) and variance Y2 /n • Other than its mean and variance, the exact distribution of Y ̄ is complicated and depends on the distribution of Y (the population distribution) MFIN6201 - Empirical Techniques and Applications in Finance 17 Summary: Sampling Distribution of Y ̄ When n is large, the sampling distribution simplifies • Law of Large Numbers ̄p Y ! μ Y • Central Limit Theorem ! p Y ̄μY d n Y !N(0,1) MFIN6201 - Empirical Techniques and Applications in Finance 18 Y ̄ i s t h e B L U E o f μ Y • Y ̄ is an unbiased estimator of μY E ( Y ̄ ) = μ Y • Y ̄ is a consistent estimator of μY ̄p Y ! μ Y • Y ̄ is the most ecient estimator of μY var(Y ̄) < var(μ ̃Y ) for any other plausible estimator, μ ̃Y • Thus, Y ̄ is the Best Linear Unbiased Estimator (BLUE) of μY MFIN6201 - Empirical Techniques and Applications in Finance 19 How about Y1 instead of Y ̄ as an estimator of μY ? I know it sounds stupid, but why is it stupid? • Unbiasedness : YES • Consistency : NO E[Y1] = μY – Y1 is only Y1. It does not converge to μY even as n increases to infinity. • Eciency : NO – var(Y1) > var(Y ̄), thus Y1 is not ecient.
MFIN6201 – Empirical Techniques and Applications in Finance
20

Y ̄ is the least squares estimator of μY
• E[Yi] = μY means Yi = μY +✏ where ✏ is some random noise.
• SSE (sum of squared errors)
X ✏2 = X (Yi μY )2
• Best estimator would have reduced errors most. Thus, it would be a solution to
minSSE = minX(Yi m)2 m
MFIN6201 – Empirical Techniques and Applications in Finance
21

Y ̄ is the least squares estimator of μY • Optimization (maximization / minimization)
f(x) is maximized / minimized at x⇤ where f0(x⇤) = 0 • To minimize SSE, one need to di↵erentiate it
dX
d mSSE = 2 X(Yi m) = 0
) m = n1 Y i = Y ̄
• Therefore, Y ̄ minimizes SSE. That’s why Y ̄ is also called the
least squares estimator of μY . MFIN6201 – Empirical Techniques and Applications in Finance
22

Sample Selection Bias
What will happen to the estimator Y ̄ when we have non-random sampling?
Textbook example:
Suppose that, to estimate the monthly national unemployment rate, a statistical agency adopts a sampling scheme in which interviewers survey working-age adults sitting in city parks at 10 a.m. on the second Wednesday of the month. Because most employed people are at work at that hour (not sitting in the park!), the unemployed are overly represented in the sample, and an estimate of the unemployment rate based on this sampling plan would be biased. This bias arises because this sampling scheme overrepresents, or oversamples, the unemployed members of the population.
MFIN6201 – Empirical Techniques and Applications in Finance
23

Hypothesis Testing
The hypothesis testing problem (for the mean):
• make a provisional decision based on the evidence at hand whether a null hypothesis is true,
• or instead that some alternative hypothesis is true. • That is, test
– H0 : E(Y) = μY,0 vs. H1 : E(Y) 6= μY,0 (2-sided)
– H0 : E(Y ) = μY,0 vs. H1 : E(Y ) > μY,0 (1-sided, >) – H0 : E(Y ) = μY,0 vs. H1 : E(Y ) < μY,0 (1-sided, <) • μY,0 is the hypothesized value based on your conjecture or null hypothesis MFIN6201 - Empirical Techniques and Applications in Finance 25 Hypothesis Testing Suppose your null hypothesis is H0 : μY = 0 and your actual estimate of Y ̄ is Y ̄act = 1. Can you reject the null hypothesis? • Case 1: H0 is wrong, and the true μY 6=0 • Case 2: H0 is right, but μY 6= Y ̄act because of sampling errors How to decide? You need to compute the probability of each case! MFIN6201 - Empirical Techniques and Applications in Finance 26 Hypothesis testing and p-value P(H0 is right)=P⇣Y ̄actμY,0 is due to sample errors⌘ = P ⇣ Y ̄ μ Y , 0 > Y ̄ a c t μ Y , 0 ⌘ ⌘ p-value
You can reject the null hypothesis if p-value is very small
MFIN6201 – Empirical Techniques and Applications in Finance
27

Some terminology for testing statistical hypotheses
• p-value = probability of drawing a statistic (e.g., Y ̄) at least as adverse to the null as the value actually computed with your data, assuming that the null hypothesis is true.
• The significance level of a test is a pre-specified probability of incorrectly rejecting the null, when the null is true.
• For example, if your p-value is 3%, you can reject the null hypothesis at 5% significance level but not at 1% significance level.
MFIN6201 – Empirical Techniques and Applications in Finance
28

Type I and II errors
• Type I error: incorrect rejection of a true null hypothesis
If significance level is 5%, it would incorrectly reject true null
hypothesis with 5% probability
• Type II error: failure to reject a false null hypothesis
MFIN6201 – Empirical Techniques and Applications in Finance
29

Calculating the p-value p-value = PrH0[| Y ̄ μY,0 |>| Y ̄act μY,0 |]
where Y ̄act is the value of Y ̄ actually observed (nonrandom)
The p-value is the probability of drawing Y ̄ at least as far in the tails of its distribution under the null hypothesis as the sample average you actually computed.
To compute the p-value, you need the to know the sampling distribution of Y ̄, which is complicated if n is small. If n is large, however, you can use the normal approximation (CLT).
MFIN6201 – Empirical Techniques and Applications in Finance
30

Calculating the p-value p-value = Pr h| Y ̄ μY,0 |>| Y ̄act μY,0 |i
H0 “Y ̄ μY,0 Y ̄act μY,0#

=⇠ probability under left+right N(0,1) tails where Y ̄ = Y /pn, the standard deviation of Y ̄
=Pr >
H0 Y ̄ Y ̄
= Pr “|Z| > Y ̄act μY,0# H0 ̄ Y ̄!
= YactμY,0 ⇥2 Y ̄
MFIN6201 – Empirical Techniques and Applications in Finance
31

Calculating the p-value with Y known
For large n, p-value = the probability that a N(0,1) random variable falls outside |(Y ̄act μY,0)/Y ̄|
MFIN6201 – Empirical Techniques and Applications in Finance
32

Calculating the p-value with Y known Null hypothesis can be rejected if
or
because
p-value < 0.05 Y ̄ a c t μ Y , 0 Y /pn > 1.96 P (|Z| > 1.96) = 0.05
MFIN6201 – Empirical Techniques and Applications in Finance
33

Calculating the p-value with Y known From the previous example,
μY,0 = 0, Y ̄act = 1
• If Y = 3 and n = 10,
p-value= Y/pn ⇥2=0.2918
Y ̄ a c t μ Y , 0 ! • If Y = 3 and n = 50,
p-value = 0.0184 p-value = 0.0016
• If Y = 1 and n = 10,
Therefore, it is easy to reject a null hypothesis if n is large or Y is
small
MFIN6201 – Empirical Techniques and Applications in Finance
34

Estimator of Y2
In practice, Y2 is unknown. It must be estimated.
2 1Xn ̄2
sY = n1 (Yi Y) = “sample variance of Y”
i=1
• Fact: If (Y1, …, Yn) are i.i.d. and E(Y 4) < 1 , then 2p2 s Y ! Y • Why does the law of large numbers apply? – Because s2Y is a sample average; see Appendix 3.3 – Technical note: we assume E(Y 4) < 1 because here the average is not of Yi, but of its square; see Appendix 3.3 MFIN6201 - Empirical Techniques and Applications in Finance 35 Estimator of Y2 Why divide not by n but by n1? 2 1Xn ̄2 sY =n1 (YiY) i=1 • PYiY ̄2 P(YiμY)2 because Y ̄ is the least squares estimator. Thus, we need to divide by n 1 to compensate for the di↵erence. • This is also called as degree of freedom. n (number of observations) 1 (compensation for Y ̄) MFIN6201 - Empirical Techniques and Applications in Finance 36 Computing the p-value with Y2 estimated Y ̄ μ Y , 0 Y/pn ⇠N(0,1) Y ̄ μ Y , 0 sY /pn ⇠ t-distribution • The second statistic follows a Student’s t-distribution with n 1 degrees of freedom. • t-distribution converges to standard normal distribution for large n. MFIN6201 - Empirical Techniques and Applications in Finance 37 Student’s t-distribution’s pdf MFIN6201 - Empirical Techniques and Applications in Finance 38 Computing the p-value with Y2 estimated p-value = Pr hY ̄ μY,0 > Y ̄act μY,0i
H0 “Y ̄ μY,0 Y ̄act μY,0# =Pr p> p
H0″Y/n Y/n#
Y ̄ μY,0 Y ̄act μY,0 =Prp> pforlargen
⇠
p-value = PrH0[| t |>| tact |], Y2 estimated
so
H0 sY/ n sY/ n
=⇠ probability under normal tails where t = Y ̄μY,0 (the usual
sY /pn MFIN6201 – Empirical Techniques and Applications in Finance
t-statistics)
39

The p-value and the significance level?
The significance level is prespecified according to one’s tolerance for Type 1 error. For example, if the prespecified significance level is 5%
• You reject if p  0.05
• Approximately, you reject the null hypothesis if | t | 1.96 and
sample size is large enough
• The p-value is sometimes called the marginal significance level. Often, it is better to communicate the p-value than simply whether a test rejects or not
• the p-value contains more information than the “yes/no” statement about whether the test rejects.
MFIN6201 – Empirical Techniques and Applications in Finance
40

t-table and degrees of freedom
At this point, you might be wondering,… What happened to the
t-table and the degrees of freedom? Digression: the Student t distribution
If Yi, i = 1, …, n is i.i.d. N (μY , Y2 ), then the t-statistic has the Student t-distribution with n 1 degrees of freedom. The critical values of the Student t-distribution is tabulated in the back of all statistics books. Remember the recipe?
• Compute the t-statistic
• Compute the degrees of freedom, which is n 1
• Look up the 5% critical value
• If the t-statistic exceeds (in absolute value) this critical value, reject the null hypothesis.
MFIN6201 – Empirical Techniques and Applications in Finance
41

t-table and degrees of freedom
MFIN6201 – Empirical Techniques and Applications in Finance
42

Comments on Student’s t-distribution
The theory of the t-distribution was one of the early triumphs of mathematical statistics. It is astounding, really: if Y is i.i.d. normal, then you can know the exact, finite-sample distribution of the t-statistic – it is the Student t. So, you can construct confidence intervals (using the Student t critical value) that have exactly the right coverage rate, no matter what the sample size. This result was really useful in times when “computer” was a job title, data collection was expensive, and the number of observations was perhaps a dozen. It is also a conceptually beautiful result, and the math is beautiful too which is probably why stats profs love to teach the t-distribution. But….
MFIN6201 – Empirical Techniques and Applications in Finance
43

Comments on Student’s t-distribution
If the sample size is moderate (several dozen) or large (hundreds or more), the di↵erence between the t-distribution and N(0,1) critical values is negligible. Here are some 5% critical values for 2-sided tests:
degrees of freedom(n-1) 10
20
30
60
1
5% t-distribution critical value 2.23
2.09
2.04
2.00
1.96
MFIN6201 – Empirical Techniques and Applications in Finance
44

Hypothesis testing with t-stat: example From the previous example again,
μY,0 = 0, Y ̄act = 1
• If sY = 3 and n = 10,
t = sY /pn = 1.054
Y ̄ a c t μ Y , 0
p-value = P (|tdf=9| >= 1.054) = 0.319
• If sY = 3 and n = 50,
t = 2.357, p-value = 0.022
• If sY = 1 and n = 10,
t = 3.162, p-value = 0.012
MFIN6201 – Empirical Techniques and Applications in Finance
45

Summary: Student’s t-distribution
• The assumption that Y is distributed N (μY , Y2 ) is rarely
plausible in practice (returns? income?)
• For n > 30, the t-distribution and N(0,1) are very close (as n
grows large, the tn1 distribution converges to N(0,1))
• The t-distribution is an artifact from days when sample sizes
were small and “computers” were people
• For historical reasons, statistical software typically uses the t-distribution to compute p-values – but this is irrelevant when the sample size is moderate or large.
• For these reasons, in this class we will focus on the large-n approximation given by the CLT
MFIN6201 – Empirical Techniques and Applications in Finance
46

Confidence intervals
• A 95% confidence interval for μY is an interval that contains the true value of μY in 95% of repeated samples
• Digression: What is random here? The values of Y1,…,Yn and thus any functions of them, including the confidence interval
• The population parameter, μY , is not random; we just don’t know it
MFIN6201 – Empirical Techniques and Applications in Finance
48

Confidence intervals
A 95% confidence interval can always be constructed as the set of values of μY not rejected by a hypothesis test with a 5% significance level.
Y ̄ μ Y Y ̄ μ Y
{μY 😐 sY/pn |1.96}={μY :1.96 sY/pn 1.96}
̄ sY ̄ sY = {μY : Y 1.96pn  μY  Y +1.96pn}
̄ sY ̄ sY
= {μY 2 (Y 1.96pn, Y + 1.96pn)}
This confidence interval relies on the large-n results that Y ̄ is 2p2
approximately normally distributed and sY ! Y . MFIN6201 – Empirical Techniques and Applications in Finance
49

Standard deviation vs. standard error
• The standard error (SE) of a statistic is the standard deviation of its sampling distribution, or an estimate of that standard deviation.
• For example, the standard error of Y ̄ is an estimator of the standard deviation of Y ̄.
standard deviation of Y = Y p standard deviation of Y ̄ = Y / n
standard error of Y ̄ = sY /pn • Thus, 95% confidence interval would be
Y ̄ ± 1.96 ⇥ SE(Y ̄) MFIN6201 – Empirical Techniques and Applications in Finance
50

Confidence intervals: example
From the previous example again,
Y ̄ a c t = 1
•IfsY =3andn=10,
95% C.I. = Y 2.262pn, Y + 2.262pn
!
̄ sY ̄ sY = (1.146, 3.146)
• If sY = 3 and n = 50,
95% C.I. = (0.147, 1.853)
• If sY = 1 and n = 10,
95% C.I. = (0.285, 1.715)
MFIN6201 – Empirical Techniques and Applications in Finance
51

Confidence intervals: example
• Confidence intervals would become narrower if n is large or Y is small.
• H0 :μY,0 =0 is not rejected in case 1 because μY,0 =0 is within its confidence interval.
• However, the null is rejected in case 2 and 3 because it is out of the intervals.
MFIN6201 – Empirical Techniques and Applications in Finance
52

Let’s go back to the class-size:test-score example
Compare districts with “small” (STR < 20) and “large” (STR 20) class sizes: • Estimation of = di↵erence between group means • Test the hypothesis that = 0 • Construct a confidence interval for MFIN6201 - Empirical Techniques and Applications in Finance 53 Estimation 1 nlarge ⌘Y ̄Y ̄= XY XY 1 nsmall small large nsmall i nlarge i i=1 i=1 = 657.4 650.0 = 7.4 • Is this a large di↵erence in a real-world sense? • Standard deviation across districts = 19.1 • Di↵erence between 60th and 75th percentiles of test score distribution is 666.7 659.4 = 7.3 • This is a big enough di↵erence to be important for school reform discussions, for parents, or for a school committee? MFIN6201 - Empirical Techniques and Applications in Finance 54 Hypothesis testing Di↵erence-in-means test: compute the t-statistic t= where var() = var ⇣Y ̄ Y ̄ ⌘ small large SE() = v a r Y ̄ + v a r ⇣ Y ̄ ⌘ small large ⇡ s2small + s2large nsmall nlarge This derivation is based on two assumptions. • Y ̄ and Y ̄ are independent. small large • Y,small 6= Y,large MFIN6201 - Empirical Techniques and Applications in Finance 55 Hypothesis testing Therefore, Y ̄ small large Y ̄ t = S E ( Y ̄ Y ̄ Y ̄ Y ̄ small large small large ) = s 2 2 ssmall + slarge nsmall nlarge MFIN6201 - Empirical Techniques and Applications in Finance 56 Hypothesis testing Y ̄ Y ̄ small large 657.4 650.0 7.4 = r19.42 + 17.92 = 1.83 = 4.05 238 182 hypothesis that the two means are the same. t = ss2small s2 + large nsmall nlarge |t| > 1.96, so reject (at the 5% significance level) the null
MFIN6201 – Empirical Techniques and Applications in Finance
57

Confidence interval
A 95% confidence interval for the di↵erence between the means is, (Y ̄ Y ̄ ) ± 1.96 ⇥ SE(Y ̄ Y ̄ )
small large small large = 7.4 ± 1.96 ⇥ 1.83
= (3.8, 11.0) Two equivalent statements:
• The 95% confidence interval for does not include 0; • The hypothesis that = 0 is rejected at the 5% level.
MFIN6201 – Empirical Techniques and Applications in Finance
58

Hypothesis testing with a pooled variance estimator
If we can assume Y,small = Y,large, then s 2 = 1 24 X Y Y ̄
2 + X ⇣ Y Y ̄
i large
⌘ 2 35
pooled nsmall + nlarge 2 i small small
large Plugging the pooled variance estimator into t-statistic,
t= SE()
Y ̄ Y ̄ small large
= ss2small + s2large nsmall nlarge
Y ̄ Y ̄ small large
=r11 spooled nsmall + nlarge
MFIN6201 – Empirical Techniques and Applications in Finance
59

Practice questions
• Try question 3.1, 3.2, 3.6, 3.13, 3.18. • Answers will be provided next week.
• This is not an assessment.
• They are just for practice.
MFIN6201 – Empirical Techniques and Applications in Finance
60

Related Posts