Summary Empirical Finance

Summary of all the lectures of Empirical Finance, including additional notes of dr Opschoor, and the Stata clips

Preview samenvatting (3 van de 134 pagina's)

Week 1
Clip 1
Part I – Classical linear regression model
After these clips on the classical linear regression model, you should
 Understand what the classical linear regression model (CLRM) is and how it can be
used in empirical finance
 Know key concepts: estimation, inference, estimator, estimate, parameters, dummy
variables, outliers
 Understand what assumptions are needed for valid inference in the CLRM and why
 Understand the t and F test, and model adequacy measures such as the R2 and 𝑅̅
The following video clips are involved with Pack 1:
 Clip I: Motivation/Notation/Transforming variables  Stata Clip I
 Clip II: Dummies (Level/Slope dummies, notation, perfect multi-collinearity)  Stata
Clip II
 Clip III: Model Adequacy and outliers  Stata Clip III
 Clip IV: Parameter estimation and inference I (two parts)
 Clip V: Inference II (t and F test)  Stata Clip IV

Motivation: explaining income
 Suppose you would like to explain wage
 Which factors could possible affect wage?
o Education? Experience?
o Other factors?  include 𝜖 (or 𝜖 for measurement error)
 How do these factors affect wage? (linear – going up from 1 to 2 years is the same as
going from 20 to 21 years, non-linear – going up from 1 to 2 years is more
meaningful than going from 20 to 21 years)
 Most simple relationship:
o 𝑊𝑎𝑔𝑒𝑖 = 𝛽0 + 𝛽1𝐸𝑑𝑢𝑐𝑖 + 𝛽2𝐸𝑥𝑝𝑖 + 𝜖𝑖

What is random?
 What is random and non-random in the linear regression model?
 Our example: 𝑊𝑎𝑔𝑒𝑖 = 𝛽0 + 𝛽1𝐸𝑑𝑢𝑐𝑖 + 𝛽2𝐸𝑥𝑝𝑖 + 𝜖𝑖
o 𝜖𝑖 = unobserved and random: error term
o 𝑊𝑎𝑔𝑒𝑖 = observed, random: dependent variable
o 𝛽0, 𝛽1, 𝛽2 = fixed non-random, but not known: parameters
o 𝐸𝑑𝑢𝑐1, 𝐸𝑥𝑝𝑖 = observed, possibly random/non-random, regressors
 Therefore, anything that depends on data will be (a) random (variable) and will have
distributional properties (as we took a sample)
Vector notation (Sec. 4.1 – 4.2)
Consider again our example 𝑊𝑎𝑔𝑒𝑖 = 𝛽0 + 𝛽1𝐸𝑑𝑢𝑐𝑖 + 𝛽2𝐸𝑥𝑝𝑖 + 𝜖𝑖
Gather parameters into a parameter vector:
𝛽0
𝛽 = ( 𝛽1)
𝛽2

Vectors in my class are always column vectors, unless transposed, e.g.
𝛽′ = ( 𝛽0 𝛽1 𝛽2 )

Gather the firm characteristics into a vector
𝑥𝑖 = ( 𝐸𝑑𝑢𝑐𝑖)
𝐸𝑥𝑝𝑖
𝛽0
Note: 𝑥′
𝛽
𝑖 𝛽 = (1 𝐸𝑑𝑢𝑐𝑖 𝐸𝑥𝑝𝑖) ( 1) = 𝛽0 + 𝛽1𝐸𝑑𝑢𝑐𝑖 + 𝛽2𝐸𝑥𝑝𝑖
𝛽2
Therefore, 𝑊𝑎𝑔𝑒
′
𝑖 = 𝑥𝑖 𝛽 + 𝜖𝑖

Defining 𝑦𝑖 = 𝑊𝑎𝑔𝑒𝑖, we obtain our first standard notation for a linear regression model
𝑦
′
𝑖 = 𝑥𝑖 𝛽 + 𝜖𝑖 with 𝑦𝑖 ∈ ℝ1, 𝛽 ∈ ℝ𝑘𝑥1, 𝑥𝑖 ∈ ℝ𝑘𝑥1, 𝜖𝑖 ∈ ℝ1 with in our example k = 3
regressors (also the constant 𝛽0) (check formats matrix multiplication) Further notation: we stack all observation i=1, …, 3010
𝑦1
𝐸𝑑𝑢𝑐1
𝐸𝑥𝑝1
𝜖1
𝑦
𝐸𝑑𝑢𝑐
𝐸𝑥𝑝
𝜖
𝑦 = (
) , 𝑋 = (
) , 𝜖 = (
),
𝑦3010
1 𝐸𝑑𝑢𝑐
𝜖
3010
𝐸𝑥𝑝3010
3010

We obtain our second standard notation for the linear regression model, the matrix
notation: 𝑦 = 𝑋𝛽 + 𝜖
(never use this for thesis, you want to show what x and y are)
Thus, X = 3010 x 3, and 𝛽 is 3x1, so the product X𝛽 is defined and is 3010x1, just like y and 𝜖 Check the following:
𝑦 = 𝑋𝛽 + 𝜖
𝑦1
𝐸𝑑𝑢𝑐1
𝐸𝑥𝑝1
𝜖
𝛽
𝑦
𝐸𝑑𝑢𝑐
𝐸𝑥𝑝
𝜖
) = 𝑋 = (
) ( 𝛽 ) + (
𝑦
𝛽
3010
𝐸𝑑𝑢𝑐
𝜖
3010
𝐸𝑥𝑝3010
3010
𝑊𝑎𝑔𝑒1
= 𝛽0 + 𝐸𝑑𝑢𝑐1𝛽1
𝐸𝑥𝑝1𝛽2
+𝜖1
𝑊𝑎𝑔𝑒
𝛽
𝛽
+𝜖
= (
= 𝛽0 + 𝐸𝑑𝑢𝑐2. 1
𝐸𝑥𝑝2 2
𝑊𝑎𝑔𝑒
+𝜖
3010
= 𝛽0 + 𝐸𝑑𝑢𝑐3010𝛽1 𝐸𝑥𝑝3010𝛽2
3010

As an aside: the constant term is a column of ones in the X matrix
So, it is all the same, but written differently. Why? In some case, some notations are more
convenient than others! The most sophisticated is 𝑊𝑎𝑔𝑒𝑖 = 𝛽0 + 𝛽1𝐸𝑑𝑢𝑐𝑖 + 𝛽2𝐸𝑥𝑝𝑖 + 𝜖𝑖
When can we use linear regression?
 If the model is linear in the parameters
(𝛽1 𝑒𝑡𝑐. )
 It might be non-linear in the variables
(𝐸𝑑𝑢𝑐𝑖 𝑒𝑡𝑐. )
 Examples:
o Linear in parameters, linear in variables:
linear regression
𝑊𝑎𝑔𝑒𝑖 = 𝛽0 + 𝛽1𝐸𝑑𝑢𝑐𝑖 + 𝜖𝑖
o Linear in parameters, non-linear in variables:
linear regression
𝑊𝑎𝑔𝑒
𝑖 = 𝛽0 + 𝛽1𝐸𝑑𝑢𝑐𝑖 + 𝛽2𝐸𝑥𝑝𝑖 + 𝜖𝑖
o Non-linear in parameters, linear in variables:
non-linear
𝑊𝑎𝑔𝑒
𝑖 = 𝛽0 + 𝛽1𝐸𝑑𝑢𝑐𝑖 + 𝛽2 𝐸𝑥𝑝𝑖 + 𝜖𝑖
o Non-linear in parameters, non-linear in variables:
non-linear
𝑊𝑎𝑔𝑒
−1
𝑖 = 𝛽0 + 𝛽1𝛽2 (𝐸𝑑𝑢𝑐𝑖 − 1)𝛽2 + 𝜖𝑖

Non-linear in parameters  can’t use the linear regression model.
𝑊𝑎𝑔𝑒

𝑖 = 𝛽0 + 𝛽1𝐸𝑑𝑢𝑐𝑖 + 𝜖𝑖
if Education goes up by one year, my wage goes up by 𝛽1,
given the amount of Experience stays the same. Transforming variables
 Sometimes, you can transform a non-linear model to make it linear: this changes the
interpretation of the coefficients.
 Example:
o Transformation of a model: Cobb-Douglas production function
𝑌
𝛽1 𝛽2
𝑖 = 𝛽0𝐾
𝐿 , take logs to obtain the linear (in the parameters) specification
𝑖
𝑖
𝑦
𝑖 = 𝛽𝑜 + 𝛽1𝑘𝑖 + 𝛽2𝑙𝑖, with 𝛽𝑜 a constant (equal to ln𝛽0), and 𝑘𝑖 = 𝑙𝑛𝐾𝑖 etc.
𝜕𝑙𝑛𝑦
The coefficient 𝛽
𝑖
2 is now an elasticity
(relative change %) rather than
𝜕𝑙𝑛𝑥𝑖
𝜕𝑦
𝜕𝑊𝑎𝑔𝑒
the usual marginal effect 𝑖 (=
𝑖
, 𝑐𝑒𝑡𝑒𝑟𝑖𝑠 𝑝𝑎𝑟𝑖𝑏𝑢𝑠!)
𝜕𝑥𝑖
𝜕𝐸𝑑𝑢𝑐𝑎𝑡𝑖𝑜𝑛𝑖

Transforming variables (logs)
1. Taking a logarithm can often help to rescale the data so that their variance is more
constant, which overcomes a common statistical problem known as
heteroskedasticity (variance will be much more constant, week 3)
2. Logarithmic transfers can help to make a positively (right) skewed distribution closer
to a normal distribution (such as firm size)
3. Taking logarithms can also be a way to make a non-linear, multiplicative relationship
between variables into a linear, additive one (see example above)

Transforming variables – when to take the log?
 It depends on the economic settings: variables may need to be transformed before
being put into a regression (make plots to see!)
 Some variables are non-intuitive when untransformed, e.g. FX rates (all exchange
rates have their own scale), it is necessary to transform them. (e.g. Increase of 1
Japanese Yen is completely different than increase in Dollar against a Euro)
 In finance: pay attention on the application (e.g. modelling log-returns, but portfolios
𝑝
𝑝
need simple returns! (ln ( 𝑡 ) versus 𝑡−𝑝𝑡−1 ))
𝑝𝑡−1
𝑝𝑡−1