Summary: Linear Regression

Read the summary and the most important questions on Linear Regression

• 1 Linear Regression

• 1.1 introduction, motivation and notation

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• After the clips on Lin regression you can

  - Understand the CLRM model and how it can be used in empirical finance
  - know key concepts: inference, estimation, estimator, estimate, parameters, dummy variables and 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, R ̄2

• The epsilon i is the error term for

  - other factors driving the dependent forgotten to include in the model
  - functional misspecification if its not a linear relationship,
  that will all be mopped up in E,i 
  - measurement error; maybe we could not measure perfectly, e.g with GDP; can't really measure it, how to approximate it, what's wrong will be in e,i. 

• Notation is important: i - subscript for instance

  An i: cross-sectional data; I ask many persons about all variables at one period. 
  time-series, t: I ask one person all variables through all times.
  panel; i,t: ask all persons all variables through time 

• The equation from a statistical point of view is as follows:

  - The y is observed, random (because of e,i) 
  - The beta is fixed, not random but not known 
  - The parameter (X,i) is observed, possibly random/nonrandom 
  - The error term is unobserved and random.

• 1.2 transformations of variables (log-y instead of y, log-x instead of x)

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• 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 heteroscedasticity.

• 2  Logarithmic transforms can help to make

  a positively skewed distribution closer to a normal distribution (such as firm size)

• Depending on the economic settings, variables may need to be 

  transformed before being put into a regression (make plots!!)

• Some variables are nonintuitive when untransformed,

  
  
  
  
  e.g., FX rates

• In finance: pay attention on the application

  
  
  
  
  (e.g., modeling log-returns, but portfolios need simple returns!!)

• linear Yi = β0 + β1Xi + ui

  
  
  
  
  slope: Xi increases by 1, y increases by β1