Chapter 9: Serial Correlation

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Chapter 9Serial Correlation
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• 1. Introduction

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Introduction

• An assumption of the CLRM is that there is no
  correlation between the error terms
• cov (or E) (ui,uj) 0 for i not equal to j
• One error term is not affected by another error
  term
• Autocorrelation is usually associated with time
  series data

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Residuals

• As with heteroskedasticity, we can look at the
  residuals of our model, but this time plot them
  against time.
• Autocorrelation can be positive or negative, but
  usually positive for economic variables.

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• 2. Why Does Autocorrelation Occur?

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Inertia

• Business cycles occur in most economic variables
• A certain momentum is built into these variables,
  making them interdependent.
• A a high positive disturbance in one period is
  likely followed by a high positive disturbance in
  the next.

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Model Specification

• If the model is underspecified or has the wrong
  functional form
• The residuals will have a pattern.
• Suppose examine the demand for beef and include
  price of beef and income, but not price of pork.
• If price of pork does affect the demand for beef,
  then the error term will capture this systematic
  effect.

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Cobwebs

• The supply of many agricultural commodities
  reacts to price but with a lag, because it takes
  time to produce crops.
• Produce to a high price at the start of year, but
  price falls by harvest
• There is oversupply, and farmers react by
  producing less.
• Overproduce in one year and underproduce the
  next.

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Spatial Autocorrelation

• In regional cross-section data a random shock
  affecting activity in one region may influence
  economic activity in an adjacent region because
  of close economic ties.

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Data Manipulation

• Smoothing the data by averaging over months or
  quarters can sometimes lead to a systematic
  pattern in the disturbances
• This averages out the true disturbances over time

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• 3. Consequences

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Consequences

• OLS estimators are linear and unbiased
• Not efficient
• They do not have minimum variance, so are not
  BLUE
• t tests not reliable
• Often the standard errors too small--ts too high
• - R2 not a good measure

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• 4. How to Detect Autocorrelation

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Examine Residuals

• Plot residuals against time
• Check for up and down pattern, or a quadratic
  pattern, or a positive or negative slope.
• Plot the residuals against lagged residuals-
• Check for bunching in one or two quadrants
• Residuals at time t have high values as do
  residuals at time t-1

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Runs Test

• This is a more formal method
• If we expect random residuals we would have
  negatives and positives all mixed up.
• If residuals are correlated then we may get a run
  of negatives followed by a run of positives.

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Runs Test

• Count the number of runs in data,and positive and
  negative residuals.
• Then look up in runs tables.
• N1 is number of positives
• N2 is number of negatives.
• If number of runs is equal or less than the
  critical value in part (a), reject the null
  hypothesis positive autocorrelation exists.

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Runs Test

• If number of runs is equal or more than the
  critical value in part (b), reject the null
  hypothesis the sequence is random--negative
  autocorrelation.

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Runs Test Example

• Suppose we have 19 positives and 11 negatives and
  7 runs
• Critical value of runs is 9 for part (a)
• Positive autocorrelation.

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Durbin-Watson Test

• This is the most frequently used test.

So just the sum of squared differences in lagged
residuals divided by the ESS
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Durbin-Watson Test

• SAS will provide d test statistic
• Put / DW at end of model statement

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Durbin-Watson Test

• Durbin-Watson statistic assumes that the error
  terms are generated by the following scheme
• ut ??ut-1 vt
• where -1lt ?? lt 1
• Value of the error term at time t depends on its
  value in period t-1 plus some random term

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Durbin-Watson Test

• Dependency on the past value is measured by ?,
  the coefficient of autocorrelation.
• This is a first-order autoregressive scheme or
  AR(1)
• First order because it is only lagged one period.
• Assume an AR1 scheme in most econometric work,
  because this a tractable solution.

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Durbin-Watson Test

• Computed d value lies between 0 and 4.
• The closer it is to 0 then we get positive
  autocorrelation.
• The closer it is to 4 we get negative
  autocorrelation.
• The closer it is to 2 we get no autocorrelation.

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Durbin-Watson Test

• We can look at DW tables for a more definite
  assessment.
• N 25, 6 explanatory variable, including
  intercept, gives us a range dl - du, 0.953-1.886.

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Durbin-Watson Test

• Where does our own DW fall?
• 0 positive a/c
• dl .953 indecision
• du 1.886 indecision
• 2 no a/c
• 4-du 2.124 indecision
• 4-dl 3.147 indecision
• 4 negative a/c

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• 5. Remedial
• Measures

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Transform Model

• Assume that the error term follows an AR1 scheme.
• Then transform our original model
• Yt b1 b2Xt ut
• Write above with a one-period lag
• Yt-1 b1 b2Xt-1 ut-1
• Multiply on both sides by ?
• ?Yt-1 ?b1 ?b2Xt-1 ?ut-1

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Transform Model

• Subtract this from first
• Yt- ?Yt-1 b1(1- ?) b2(Xt-?b2Xt-1) ut-
  ?ut-1
• Run OLS on this model
• This is called generalized difference equation
• It gives less weight to large residuals that
  follow large residuals
• We can also apply this to higher order schemes.

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• 6. How to Estimate ?

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First Difference Method

• Assume ? 1
• True for many econ. time series
• Error terms are perfectly positively correlated.
• The generalized difference equation reduces to
  the first difference equation
• Yt- Yt-1 b2 (Xt - Xt-1) vt
• No intercept in this model

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Estimating ?

• Estimate from the DW statistic.
• d 2(1-?)
• ? 1-(d/2)
• Estimate from OLS residuals
• et ?et-1 vt

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Cochrane-Orcutt

• Regress OLS residuals on residuals lagged one
  period to get an estimate of ?
• Substitute ? in the generalized difference
  process and run another regression.
• Get parameters put back in original model get
  new residuals
• Continue until changes in ? are very small.

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Durbin

• Dependent variable is regressed on itself lagged,
  and all the independent variables and lagged
  independent variables.
• The estimated coefficient of the lagged dependent
  variable is used as an estimate of ?
• Substitute in generalized difference process.

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Hildreth-Lu

• Specify some ? values and estimate the
  generalized difference process assuming these
  values
• Pick the one with the lowest sum of squared
  residuals.

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• 7. Example

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Example

• Suppose estimate the following model for US
  1970-87
• Yt b1 b2Xt ut
• Yt is the stock price index NYSE
• Xt is GNP in billions of dollars
• Results
• Y 10.78 0.025 X t (1.17)
  (7.47)
• R2 0.77 DW 0.4618

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Example

• Look up critical DW
• n18, k1
• 5 critical values are 1.158 and 1.391.
• Since the computed d value is below this, then
  positive autocorrelation is likely.

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Remedial Measures

• Calculate ? from the d statistic
• ? 1-(d/2) 1-(0.4618/2) 0.7691
• Estimate the generalized difference equation
  using ?
• Yt-.77Yt-1b1(1-.77)b2(Xt-.77b2Xt-1) vt
• Create these variables - lags etc. (intercept
  included)

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Remedial Measures

• Run model (dropping the first observation because
  of lag)
• Results
• Y -26.701 0.038X
• t (-0.89) (4.48)
• R2 0.911 d 1.3645
• d is much better
• Looks like got rid of autocorrelation

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Alternatives

• Could also estimate ? from residuals
• Estimate et et-1 vt
• Results
• et 0.8916et-1 and use in generalized difference
  equation.
• Could also use first difference method.