Ch5 Relaxing the Assumptions of the Classical Model

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Ch5 Relaxing the Assumptions of the Classical
Model

• 1. Multicollinearity What Happens if the
  Regressors Are Correlated?
• 2.Heteroscedasticity What Happens if the Error
  Variance Is Nonconstant?
• 3. Autocorrelation What Happens if the Error
  Terms Are correlated?

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1.Multicollinearity

• Perfect Collinearity
• Multicollinearity two or more variables are
  highly(but not perfectly) correlated with each
  other.
• The easiest way to test multicollinearity is to
  examine the standard errors of the coefficients.
• Reasonable method to relieve multicollinearity is
  to drop some highly correlated variables.

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1.Test of Multicollinearity

• A relatively high in an equation with few
  significant t statistics
• Relatively high simple correlations between one
  or more pairs of explanatory variables
• But the above criterion is not very applicable
  for time series data. And it can not test
  multicollinearity that arises because three or
  four variables are related to each other, either.

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2.Heteroscedasticity

• Impact of heteroscedasticity on parameter
  estimators
• Corrections for heteroscedasticity
• Tests for Corrections for heteroscedasticity.

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2.Impact of Heteroscedasticity

• Existence of heteroscedasticity would make OLS
  parameter estimators not efficient, although
  these estimators are still unbiased and
  consistent
• Often occurs when dealing with cross-sectional
  data.

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2.Correction of Heteroscedasticity

• Known Variance
• Unknown Variance (Error Variance Varies Directly
  with an Independent Variable)

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Known Variance
Two-variable regression model
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Known Variance
Multiple linear regression model
let Because
Therefore WLS is BLUE
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Unknown Variance
Let Because
Therefore WLS is BLUE
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2.Tests of Heteroscedasticity

• Informal Test Method
• Observe the residuals to see whether
  estimated variances
• differ from observation to observation.
• Formal Test Methods
• Goldfeld-Quandt Test
• Breusch-Pagan Test and The White Test

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Goldfeld-Quandt Test

• Steps
• Order the data by the magnitude of Independent
  Variable
• Omit the middle d observations
• Fit two separate regression, the first for the
  smaller X and the second for the larger X
• Calculate the residual sum of squares of each
  regression RSS1 and RSS2
• Assuming the error process is normally
  distributed, then RSS2/RSS1F((N-d-2k)/2,
  (N-d-2k)/2) .

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Breusch-Pagan Test

• Steps
• First calculate the least-squares residuals
  and use these residuals to estimate
  
• Run the followingregression
• If the error term is normally distributed and the
  null hypothesis is valid, then

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White Test

• Steps
• First calculate the least-squares residuals
  and use these residuals to estimate
  
• Run the following regression
• When the null hypothesis is valid, then

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3.Serial Correlation

• Impact of serial correlation on OLS estimators
• Corrections for serial correlation
• Tests for serial correlation.

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Serial Correlation

• Serial Correlation often occurs in time-series
  studies
• Fist-order serial correlation
• Positive serial correlation

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Impact of Serial Correlation

• Serial correlation will not affect the
  unbiasedness or consistency of the OLS
  estimators, but it does not affect their
  efficiency.

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Correction of Serial Correlation

• The model with serial correlated error terms
  usually is described as
• Formula for first-order serial-correlation
  coefficient

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Correction of Serial Correlation
When is knownGeneralized Differencing
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Methods for estimating

• The Cochrane-Orcutt Procedure
• The Hildreth-Lu Procedure.

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Cochrane-Orcutt Procedure

• Steps
• Using OLS to estimate the original model
• Using the residuals from the above equation to
  perform the regression
• Using the estimated value of to perform the
  generalized differencing transformation process
  and yield new parameters.
• Substituting these revised parameters into the
  original equation and obtaining the new estimated
  residuals

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Cochrane-Orcutt Procedure

• Steps
• Using these second-round residuals to run the
  regression and obtain new estimate of
• The above iterative process can be carried on
  many times until the new estimates of differ
  from the old ones by less than 0.01 or 0.005.

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The Hildreth-Lu Procedure

• Steps
• Specifying a set of grid values for
• For each value of , estimating the transformed
  equation
• Selecting the equation with the lowest
  sum-of-squared residuals as the best equation
• The above procedure can be continued with new
  grid values chosen in the neighborhood of the
  value that is first selected until the desired
  accuracy is attained.

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Test of Serial Correlation

• Durbin-Watson Test
• Test statistic is
• DW0,4, value near 2 indicating no
  first-order serial correlation. Positive serial
  correlation is associated with DW values below 2,
  and negative serial correlation is associated
  with DW values above 2.

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Range of the DW Statistic
Value of DW
Result 4-dlltDWlt4 Reject the null
hypothesis negative serial 4-dultDWlt4-dl
Result indeterminate 2ltDWlt4-du
Accept null hypothesis dultDWlt2
Accept null hypothesis dlltDWltdu
Result indeterminate 0ltDWltdl
Reject null hypothesis positive serial