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MF-852 Financial Econometrics

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The Two-Variable Linear Model. Y: dependent variable of interest ... Linear model is always an approximation. What are and ? Why is there an error term ei? ... – PowerPoint PPT presentation

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Title: MF-852 Financial Econometrics


1
MF-852 Financial Econometrics
  • Lecture 6
  • Linear Regression I
  • Roy J. Epstein
  • Fall 2003

2
The Two-Variable Linear Model
  • Y dependent variable of interest
  • May be complicated, e.g., health care expenditure
  • X an explanatory variable that causes Y.
  • E.g., income.
  • How to quantify effect of X on Y?

3
A Linear Regression
  • Most simple model is linear relationship
  • Yi ? ?Xi ei
  • Linear model is always an approximation.
  • What are ? and ??
  • Why is there an error term ei?
  • ei is also called the residual

4
Linear Regression Purpose
  • Two goals
  • Make useful prediction of Y, given X.
  • Get estimate of ?.

5
The Error Term
  • The error term should be pure noise that is not
    useful for predicting Y.
  • What are desirable properties for ei?

6
Error Term First Property
  • E(ei) 0, on average error does not predict a
    value for Y.
  • Means that E(Y) ? ?Xi , so the regression
    prediction is unbiased.

7
Error Term Second Property
  • var(ei) ?2
  • Each Y observation has same variance.
  • Means that all observations equally informative.
  • Needed for accurate calculation of standard
    errors of ? and ?.

8
Error Term Third Property
  • Cov(X, ei) 0
  • Needed for accurate estimate of ?.
  • Otherwise effect of e on Y would be attributed to
    X.

9
Error Term Fourth Property
  • Covar(ei, ej) 0
  • Error for one observation independent of error
    for another observation.
  • Needed for accurate calculation of standard
    errors of ? and ?.

10
Ordinary Least Squares (OLS) Regression
  • Rule minimize variance of the prediction error
    ei Yi (? ?Xi).
  • If error has zero variance, then prediction would
    be perfect!
  • OLS estimation find ? and ? to make
  • as small as possible.

11
Ordinary Least Squares (OLS)
  • Treat model as a prediction of Y.
  • Rule minimize variance of the prediction error
    ei Yi (? ?Xi).
  • If error has zero variance, then prediction would
    be perfect!
  • OLS estimation find ? and ? to make
  • as small as possible.

12
Data and the Regression Line
13
Actual and Fitted Relationships
  • What are the data points?
  • What is the regression line?
  • What are the error terms?

14
The Estimated Residuals
15
Regressions in Excel
  • Make sure you have installed the Data Analysis
    Tool Pack!
  • You need this for Excel to do regressions
    automatically.

16
OLS Regression Coefficients
  • The estimated coefficients are random variables.
  • In this example,
  • ? 0.173, standard error 1.32
  • ? 0.144, standard error 0.0094

17
Statistical Significance
  • Suppose H0 ? 0
  • Is the estimated ? statistically significant?
  • Suppose H0 ? 0
  • Is the estimated ? statistically significant?
  • Suppose H0 ? 0 AND ? 0
  • Is the joint hypothesis accepted or rejected?

18
More Hypothesis Tests
  • Suppose H0 ? 0.16
  • Do you accept or reject H0?
  • Suppose H0 ? 2
  • Do you accept or reject H0?
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