Autocorrelation

1
Autocorrelation

• Lecture 20

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Todays plan

• Definition and implications
• How to test for first order autocorrelation
• Note well only be taking a detailed look at 1st
  order autocorrelation, but higher orders exist
• e.g. quarterly data is likely to have 4th order
  autocorrelation
• How to correct for first-order autocorrelation
  and how to estimate allowing for autocorrelation
• Again well use the Phillips curve as an example

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Definitions and implications

• Autocorrelation is a time-series phenomenon
• 1st-order autocorrelation implies that
  neighboring observations are correlated
• the observations arent independent draws from
  the sample

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Definitions and implications (2)

• In terms of the Gauss-Markov (or BLUE) theorem
• The model is still linear and unbiased if
  autocorrelation exists

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Definitions and implications (3)

• Autocorrelation will affect the variance
• if s t, then we would have
• but if s ? t, and Y observations are not
  independent, we have nonzero covariance terms

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Definitions and implications (4)

• Think of a numerical example to demonstrate this
• assuming t 3 if we wanted to estimate
• We want to consider the efficiency, or the
  variance of
• If BLUE all covariance terms are zero.
• If covariance terms are nonzero we no longer
  have minimum variance
• minimum variance is defined as

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Summary of implications

• 1) Estimates are linear and unbiased
• 2) Estimates are not efficient. We no longer
  have minimum variance
• 3) Estimated variances are biased either
  positively or negatively
• 4) Unreliable t and F test results
• 5) Computed variances and standard errors for
  predictions are biased
• Main idea autocorrelation affects the efficiency
  of estimators

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How does autocorrelation occur?

• Autocorrelation occurs through one of the
  following avenues
• 1) Intertia in economic series through
  construction
• With regards to unemployment this was called
  hysteresis this means that certain sections of
  society who are prone to unemployment
• 2) Incorrect model specification
• There might be missing variables or we might have
  transformed the model to create correlation
  across variables

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How does autocorrelation occur?

• 3) Cobweb phenomenon
• agents respond to information with lags to
• this is usually related to agricultural markets
• 4) Data manipulation
• example constructing annual information based on
  quarterly data

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Graphical results

• With no autocorrelation in the error term we
  would expect all errors to be randomly dispersed
  around zero within reasonable boundaries
• Simply graphing the estimated errors against time
  indicates the possibility of autocorrelation
• we look for patterns of errors over time
• patterns can be positive, negative, or zero
• Graph error vs. time, we have positive
  correlation in the error term
• errors from one time period to the next tend to
  move in 1 direction, with a positive slope

11
Phillips Curve

• L_20.xls Phillips Curve data
• Can calculate predicted wage inflation using the
  observed unemployment rate and the estimated
  regression coefficients
• Can then calculate the estimated error of the
  regression equation
• Can also calculate the error lagged one time
  period

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Durbin-Watson statistic

• We will use the Durbin-Watson statistic to test
  for autocorrelation
• This is computed by looking over T-1 observations
  where t 1, T

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Durbin-Watson statistic (2)

• The assumptions behind the Durbin-Watson
  statistic are
• 1) You must include an intercept in the
  regression
• 2) Values of X are independent and fixed
• 3) Disturbances, or errors, are generated by
• this says that errors in this time period are
  correlated with errors in the last time period
  and some random error vt
• ? is the coefficient of autocorrelation and is
  bounded -1 ? ? ? 1
• ? can be calculated as

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How to estimate ?

• This estimation matters because it will be used
  in the model correction
• ? can be estimated by this equation
• Once we have the Durbin-Watson statistic d, you
  can obtain an estimate for ?

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How to estimate ? (2)

• How the test works
• The values for d range between 0 and 4 with 2 as
  the midpoint
• using

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How to estimate ? (3)

• We can represent this in the following figure

• dL represents the D-W upper bound
• dU represents the D-W lower bound
• The mirror image of dL and dU are 4- dL and 4-dU

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Procedure

• Table on the second handout for today is the
  Durbin-Watson statistical table and an additional
  table for this analysis
• 1) Run model
• 2) Compute
• 3) Compute d statistic
• 4) Find dL and dU from the tables K is the
  number of parameters minus the constant and T is
  the number of observations
• 5) Test to see if autocorrelation is present

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Example (2)

• Returning to L_20.xls

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Generalized least squares

• What can we do about autocorrelation?
• Recall that our model is Yt a bXt et
  (1)
• We also know et ?et-1 vt
• We will have to estimate the model using
  generalized least squares (GLS)

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Generalized least squares (2)

• Lets take our model and lag it by one time
  period
• Yt-1 a bXt-1 et-1
• Multiplying by ?
• ?Yt-1 ?a ?bXt-1 ?et-1 (2)
• Subtracting our (2) from (1), we get
• Yt - ?Yt-1 a(1-?) b(Xt-1 - ? Xt-1) vt
• where
• vt et - ?et-1

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Generalized least squares (3)

• Now we need an estimate of ? we can transform
  the variables such that
• where
• Estimating equation (3) allows us to estimate
  without first-order autocorrelation

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Estimating ?

• There are several approaches
• One way is by using a short cut
• thinking back to the Durbin-Watson statistic,
• we can rewrite the expression for d as

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Estimating ? (2)

• Collecting like terms, we have
• Solving for ?, we can get an estimate in terms of
  d
• Since earlier we defined ? as
• we can use this to get a more precise estimate
• There are three or four other methods in the text