Violation of Assumptions

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Violation of Assumptions
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Omitted Variables

• This problem occurs if a variable is omitted from
  the specification either due to an error by the
  researcher or lack of data.
• If the variable is uncorrelated with the included
  variables
• The estimated slopes are inefficient (their
  variance is too large).
• The estimated slopes are unbiased.

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Omitted Variables

• If the variable is correlated with the included
  variables
• The t-tests are biased (the estimated variance of
  the slopes is too small).
• The estimated slopes are biased.
• This is a serious problem because it leads us to
  reject true null hypotheses too often.

4
Omitted Variables

• This suggests that great care be taken in model
  building. It is generally not good procedure to
  allow the sample to dictate the model. It is
  better to include a variable that should not be
  there than exclude a variable that should.

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EXAMPLE
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EXAMPLE
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Effect of Omitted Variable
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Measurement Error

• In the dependent variable
• Slopes are biased toward zero - null hypotheses
  that are false are more difficult to reject.
  Measurement error makes it more difficult to
  reject null hypotheses.
• In an independent variable
• Slope is biased toward zero. Slopes of other
  variables that are correlated with this variable
  can also be biased. Measurement error can lead
  to rejecting true nulls.

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Measurement Error

• Implications
• Your dependent variable is hard to measure
  product satisfaction or quality of work. If you
  do find results they would be even stronger if
  you could measure the variable accurately. A
  significant result with a variable that is
  difficult to measure should not be dismissed!

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Measurement Error

• Implications
• Your independent variable is hard to measure
  product satisfaction or quality of work. Same as
  dependent variable (a significant result would be
  even more significant). HOWEVER, poor
  measurement can lead you to give MORE credit than
  is due to another variable.

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Measurement Error

• Conclusions
• Measure your variables as accurately as possible
  to improve the power of your tests
• If your independent variable is difficult to
  measure, you must worry about the results for
  other variables in the model.

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Heteroscedasticity

• Typically a problem in cross sectional data
• Slopes are unbiased, but inefficient.
• However, this is often an indication of an
  omitted variable problem, in which case the
  slopes are potentially biased.

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Heteroscedasticity

• Usually occurs due to a few outliers. Possible
  cures
• Drop the outliers.
• Use a transformation like a log transformation
  that eliminates the problem.
• Use advanced procedures to correct the problem
  (Weighted Least Squares Generalized Least
  Squares)

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Heteroscedasticity

• Examples
• Data on firms of different sizes - there is
  likely to be more heterogeneity in management for
  small firms
• Small firms -gt big errors
• Large firms -gt small errors
• Data on proportions gathered from groups of
  different sizes
• Large groups likely to give better estimates
• Example College graduation rates

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Autocorrelation

• This occurs when the error in one observation is
  correlated with the error in another observation.
• This is generally a time series problem.
• This correlation can be quite simple, or very
  complicated.
• If the correlation is with the previous
  observation error, this is called 1st order
  autocorrelation.

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Example Plots of Residuals
Positive Autocorrelation Negative
Autocorrelation None
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The Durbin-Watson Statistic

• Used when data is collected over time to detect
  autocorrelation (Residuals in one time period
  are related to residuals in another period)
• Measures Violation of independence assumption

• Approximately 0positive autocorrelation
• Approximately 2none
• Approximately 4negative autocorrelation.

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The Durbin-Watson Statistic
Durbin-Watson table (one-tailed critical values)
Durbin-Watson table (one-tailed critical
values) ?.05
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The Durbin-Watson Statistic
Durbin-Watson table (one-tailed critical values)
Durbin-Watson table (one-tailed critical
values) ?.05
dgtDH indicates ACCEPT NULL
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The Durbin-Watson Statistic
Durbin-Watson table (one-tailed critical values)
Durbin-Watson table (one-tailed critical
values) ?.05
dltDL indicates REJECT NULL
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The Durbin-Watson Statistic
Durbin-Watson table (one-tailed critical values)
Durbin-Watson table (one-tailed critical
values) ?.05
dgtDL and dltDH is inconclusive
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The Durbin-Watson Statistic
Durbin-Watson table (one-tailed critical
values) ?.05
Test for NEGATIVE autocorrelation USE
4-d Example d3.5 n15, p2 use d.5 reject
null
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Durbin-Watson Example
Relationship between sales and customers
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Durbin-Watson Example
DW.88 p1 ( of variables) n15 ( of
observations) dl1.08 dh1.36
Conclusion Reject null of no positive
autocorrelation (DWlt dl)
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Problem and Cure

• Autocorrelation present
• t-tests are biased - estimated standard error too
  small
• Degree of autocorrelation known (or estimated)
• Remove by differencing the data.
• Special case correlation 1 -gt first
  difference the data

We then run the regression using Y and X
instead of Y and X
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EXAMPLE
How is birth rate related to wars and women in
the labor force? WLFlabor force participation of
women Divorcedivorce rate returnyr3 years
following a war
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Results
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Residuals
DW.55 reject Null of no autocorrelation
Estimate rho as r1-d/2 1-.55/2.725
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Differencing Data
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Results
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Multicollinearity

• Does not violate assumptions of least squares
  (unless it is perfectly collinear)
• Estimates have low ability to reject false null
  hypotheses (low power).
• A post hoc problem.
• Little that can be done - eliminating a variable
  could cause omitted variable bias.

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Multicollinearity

• May require testing groups of variables instead
  of individual slopes.
• Use F-test for a group of variables that are
  measuring a similar idea rather than testing the
  idea by looking at individual t-tests

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Example of Collinearity

• How is MPG influenced by car characteristics?

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Regression Results
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Correlations of Independent Variables
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Qualitative Dependent Variables

• The dependent variable is a category or
  membership in a group
• Are you a union member?
• Which type of transportation do you use?
• Did you graduate from college?
• Type of worker (Good Average Poor)

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Bivariate Dependent Variable

• Only two values (union-nonunion graduate or not)
• Values can be coded as zero or 1
• What is the probability that the person is a
  graduate?
• It is likely that this probability changes in a
  nonlinear way

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Probability changing with independent variable
values
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Transforming the dependent variable

• Two most popular transformations are LOGIT and
  PROBIT
• LOGIT is log of odds
• log
• or where e is the
• natural log

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LOGIT

• Around .5, estimates from linear and nonlinear
  model yield similar results
• LOGIT or PROBIT constrain forecasts to the zero
  to 1 range, making them more efficient
• LOGIT or PROBIT yield coefficients like
  regression that are interpreted as changes in
  probabilities

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LOGIT

• Coefficients are distributed as Z (normal)
• Since this is a multiplicative model, the actual
  change in probability depends on the level of the
  other independent variables

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Example Logit Application Measuring Bank
Discrimination
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Example Logit Application Measuring Bank
Discrimination
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Example Predicting Promotion
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More than two categories

• If the categories are ORDERED (1 to 3, three
  being best) can use ORDERED LOGIT or PROBIT
• If the categories are random, need a
  multinomial LOGIT OR PROBIT (type of
  transportation used)

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Chapter Summary

• Presented The Multiple Regression Model
• Considered Contribution of Individual Independent
  Variables
• Discussed Coefficient of Determination
• Addressed Categorical Explanatory Variables
• Considered Transformation of Variables