MF-852 Financial Econometrics

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MF-852 Financial Econometrics

• Lecture 6
• Linear Regression I
• Roy J. Epstein
• Fall 2003

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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?

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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

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Linear Regression Purpose

• Two goals
• Make useful prediction of Y, given X.
• Get estimate of ?.

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The Error Term

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

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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.

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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 ?.

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Error Term Third Property

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

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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 ?.

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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.

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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.

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Data and the Regression Line
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Actual and Fitted Relationships

• What are the data points?
• What is the regression line?
• What are the error terms?

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The Estimated Residuals
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Regressions in Excel

• Make sure you have installed the Data Analysis
  Tool Pack!
• You need this for Excel to do regressions
  automatically.

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OLS Regression Coefficients

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

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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?

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More Hypothesis Tests

• Suppose H0 ? 0.16
• Do you accept or reject H0?
• Suppose H0 ? 2
• Do you accept or reject H0?