MF-852 Financial Econometrics

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MF-852 Financial Econometrics

• Lecture 7
• Hypothesis Testing in Bivariate Regression
• Roy J. Epstein
• Fall 2003

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Topics

• Two-Sided vs. One-Sided Hypothesis Tests
• Confidence Intervals and P-Values
• R2 and F in Linear Model
• Regression Example Beta Coefficients
• Modeling Strategy Cocaine and Sentencing

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Two-Sided Confidence Interval

• The 95 confidence interval (C.I.) for
  (normally distributed) xbar is
• This is a two-sided test
• H0 ? ?0 vs.
• H1 ? ? ?0 (i.e., ? gt ?0 or ? lt ?0)

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One-Sided Confidence Interval

• One-sided confidence interval used to find upper
  or lower limit for ?.
• 95 upper limit
• 95 C.I. is

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Example

• Children with lead poisoning have lower blood
  hemoglobin than normal children. Want to find
  95 upper limit for ? for lead poisoned children.
• 25 hemoglobin samples yield xbar 10.6 with
  standard deviation 2.
• 95 C.I. is (?, 10.6 2/5)

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Other Confidence Intervals

• Customary to use a 95 C.I.
• What is 90 C.I.? 99 C.I.?

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P-Value

• Assuming H0, what is the probability that the
  sample value would be as extreme as the value
  actually observed?
• Alternative to pre-determined confidence
  interval.
• Lets the data tell you the confidence level.

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P-Value Example

• Sample yields xbar 7 with standard error of 4.
  Assume normality.
• H0 ? 0
• (xbar0)/4 has standard normal dist.
• Critical value is (70)/4 1.75
• P(z ? 1.75) 0.04

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P-Value Example

• If H0 was true, then 4 chance of observing z as
  large as 1.75.
• Two-tailed test
• Significant at 8 level
• C.I. would be
• One-tailed test significant at 4 level.

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Linear Model OLS Estimation

• Regression model
• Yi ? ?Xi ei
• Estimated coefficients are
• Predicted Yi
• Predicted ei
• Note

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R2

• It can be shown that
• Total variance of Y equals predicted variance
  error variance
• R2
• fraction of variance explained by model.

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F

• Used for hypothesis tests with variances.
• Test of significance of R2 (goodness of fit)

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Regression From Last Time
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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
• R2 0.90
• F(1,26) 234.26

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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?

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Regression Intuition

• Suppose you run a regression of Y just on an
  intercept (no X variables).
• What will be the value of alphahat?
• What is the R2 in this regression?
• Suppose the model is Y a bX.
• What is yhat when Xxbar?

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Example Beta Coefficient

• We will estimate the CAPM.

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Example Cocaine Sentencing

• You will propose a model and hypotheses!