Course 323063 - Empirical Methods in Finance Summary
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Course
Empirical Methods in Finance (323063)
Institution
Tilburg University (UVT)
Summary for the first part of Course - Empirical Methods in Finance given in the first block of academic year . It's a summary including the most important parts of the slides, it does not include specific examples.
Other Important Stuff
Ordinary Least Squares (OLS)
1. Take vertical distances, defines as 𝑢̂𝑖 = 𝑦𝑖 − 𝑦̂𝑖 between each point in the graph and each potential candidate
fitted line.
2. Take the square of each distance and sums them: ∑𝑁 2
𝑖=1 û𝑖
3. Find the estimated coefficients 𝛼̂ and 𝛽̂ that minimize the sum of the squared residuals ∑𝑁 𝑖=1 û𝑖
2
a. We know that the fitted value of the dependent variable is 𝑦̂𝑖 = 𝛼̂ + 𝛽̂ 𝑥𝑖
b. We know that the true value of the dependent variable is 𝑦𝑖 = 𝛼 + 𝛽𝑥𝑖 + 𝑢𝑖
2
c. Minimize the following L function: 𝐿 = ∑𝑁 2 𝑁
𝑖=1 û𝑖 = ∑𝑖=1(𝑦𝑖 − 𝑦 ̂𝑖 )2 = ∑𝑁 ̂ − 𝛽̂ 𝑥𝑖 )
𝑖=1(𝑦𝑖 − 𝛼
𝐸[(𝑦−𝑦̅)(𝑥−𝑥̅ )] 𝐶𝑜𝑣(𝑥,𝑦)
i. This gives: 𝛼̂ = 𝑦̅ − 𝛽̂𝑥̅ where 𝛽̂ = =
𝐸[(𝑥−𝑥̅ )2] 𝑉𝑎𝑟(𝑥)
Interpretations of β Under Log
Model DV IV Interpretation of β
Level-level Y X ∆𝑦 = 𝛽∆𝑥
Level-log Y Log(x) ∆𝑦 = (𝛽/100)%∆𝑥
Log-level Log(y) X %∆𝑦 = 100𝛽∆𝑥
Log-log Log(y) Log(x) %∆𝑦 = 𝛽%∆𝑥
OLS Properties
1. Estimator: 𝛼̂ and 𝛽̂ are estimators of the true values α and β.
2. Linear: 𝛼̂ and 𝛽̂ are linear estimators, linear combinations of y.
3. Unbiased: OLS estimators 𝛼̂ and 𝛽̂ are unbiased if on average they are equal to the true values α and β.
a. This implies that if we take the distribution of 𝛼̂ and 𝛽̂, derived estimating our model across many
samples, the mean of each estimator will be equal to the true values of α and β.
b. 𝐸(𝛼̂ ) = 𝛼, 𝐸(𝛽̂) = 𝛽
4. Best: OLS estimators 𝛼̂ and 𝛽̂ have the minimum variance among the class of linear unbiased estimators.
a. This implies that if we take the distribution of 𝛼̂ and 𝛽̂, derived estimating our model across many
samples, the variance of each estimator will be the minimum across all linear unbiased estimators (also
known as efficiency).
Large sample properties of OLS
1. Consistency: the estimates 𝛼̂ and 𝛽̂ will converge to the true values α and β as the sample size N increases to
infinity.
2. Asymptotic normality: the estimates 𝛼̂ and 𝛽̂ are approximately normally distributed in large enough samples.
OLS Assumptions (Bivariate Model)
1. The population model is linear in parameters: 𝑦𝑖 = 𝛼 + 𝛽𝑥𝑖 + 𝑢𝑖
2. We have a random sample from the population
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