MFIN6201
Empirical Techniques and Applications in Finance
Week 4
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Dr. Jaehoon Lee
School of Banking and Finance University of New South Wales
e-mail: jaehoon.lee@unsw.edu.au
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Semester 2, 2017
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Last update: 15 August 2017
Summary of the last week
• Population regression model
Yi = 0 + 1Xi + ui, i = 1, · · · , n
• Least Squares Assumptions
– Assumption #1. E (u | X = x) = 0
– Assumption #2. (Xi,Yi), i = 1,··· ,n are i.i.d.
– Assumption #3. Large outliers in X and/or Y are rare.
MFIN6201 – Empirical Techniques and Applications in Finance
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Summary of the last week
• Population parameter
1 = cov(X, Y )
var(X)
• Sample OLS estimator
ˆ = s X Y
1 s2X • Goodness of fit: R2, SER, RMSE
MFIN6201 – Empirical Techniques and Applications in Finance
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• Mean
• Variance
Summary of the last week
E ( ˆ ) = 11
v a r ⇣ ˆ ⌘ = 1 v a r [ ( X i μ X ) u i ] 1 n [var(Xi)]2
• Asymptotic distribution ! ˆ⇠N ,1var[(Xi μX)ui]
1 1 n [var(Xi)]2 MFIN6201 – Empirical Techniques and Applications in Finance
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Regression with a Single Regressor
Hypothesis Tests and Confidence Intervals
• The standard error of ˆ 1
• Hypothesis tests concerning 1
• Confidence intervals for 1
• Regression when X is binary
• Heteroskedasticity and homoskedasticity
• E ciency of OLS and the Student t distribution
MFIN6201 – Empirical Techniques and Applications in Finance
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A big picture review of where we are going
We want to learn about the slope of the population regression line. We have data from a sample, so there is sampling uncertainty. There are five steps towards this goal.
1. State the population object of interest
2. Provide an estimator of this population object
3. Derive the sampling distribution of the estimator (this requires certain assumptions). In large samples this sampling distribution will be normal by the CLT.
4. The square root of the estimated variance of the sampling distribution is the standard error (SE) of the estimator
5. Use the SE to construct t-statistics (for hypothesis tests) and confidence intervals.
MFIN6201 – Empirical Techniques and Applications in Finance
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Step 1–3 are learned from the previous week
1. State the population object of interest
Yi = 0 + 1Xi +ui, i = 1,2,··· ,n
We are interested in 1 = Y/ X, the response of Y for an autonomous change in X (causal e↵ect)
2. Provide an estimator of this population object ˆ = s X Y
1 s2X MFIN6201 – Empirical Techniques and Applications in Finance
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Step 1–3 are learned from the previous week
3. Derive the sampling distribution of the estimator
To derive the large-sample distribution of ˆ , we make the 1
following assumptions.
• E(u|X = x) = 0
• (Xi,Yi), i = 1,2,··· ,n are i.i.d • Large outliers are rare
Under the assumptions, for n large, ˆ is approximately 1
distributed as
ˆ⇠N ,1var[(Xi μX)ui]!
1 1 n [var(Xi)]2 MFIN6201 – Empirical Techniques and Applications in Finance
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We have also seen step 4–5 already
4. The square root of the estimated variance of the sampling distribution is the standard error (SE) of the estimator
5. Use the SE to construct t-statistics (for hypothesis tests) and confidence intervals.
MFIN6201 – Empirical Techniques and Applications in Finance
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Hypothesis Testing and the Standard Error of ˆ 1
The objective is to test a hypothesis, like 1 = 0, using data – to reach a tentative conclusion whether the (null) hypothesis is correct or incorrect.
General setup
Null hypothesis and two-sided alternative:
H0 : 1 = 1,0 vs. H1 : 1 6= 1,0 where 1,0 is the hypothesized value under the null.
Null hypothesis and one-sided alternative:
H0 : 1 = 1,0 vs. H1 : 1 <