Class Outline

1
Class Outline

• Autcorrelation
• Consequences for OLS estimation
• Patterns of Autocorrelation
• Estationarity
• AR(1) process
• AR(p) process
• Testing for Autocorrelation
• Durbin-Watson
• Breusch-Pagan
• Estimation under Autocorrelation
• Reading Chapter 14 Textbook

2
Autocorrelation

• Consider the K variable linear model

• The Assumptions of this model are
• E(ut)0, i1,,n
• Var(ut)E(ut-E(ut))2E(ut2)?2 i1,,n
  (The variance of the error
  term is constant for all observations
  (homoscedasticity).
• Cov(ut,us)0 ? i?j (no serial correlation).
• The explanatory variables are non-stochastic and
  no exact linear relationship exists between them.

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Autocorrelation

• One of the assumptions is the no serial
  correlation

When each observation corresponds to a different
period we say that we have time series data Time
provides a natural way to sort observations In
time series the no-serial correlation assumption
is labeled the no autocorrelation assumption It
means that the error term corresponding a
particular period is not linearly related to the
error term of any other observation in the past
or future
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Autocorrelation

• When we have autocorrelation,

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Autocorrelation
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Why Autocorrelation Occur?

• Inertia
• Specification Bias
• Excluded variables
• Incorrect functional form
• Cobweb Phenomenon
• Lags
• Manipulation of Data
• Data Transformation
• Nonstationarity

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Consequences for OLS estimation

• What impact we have on our model if we ignore
  autocorrelation?
• The least squares estimator is still unbiased,
  but it is no longer best
• The formulas for the standard errors are no
  longer correct

8
Consequences for OLS estimation

• Least Square Estimator,
• Then,
• ?OLS is still a linear estimator
• ?OLS is still unbiased (when we proved
  unbiasedness we did not use the assumption
  constant variance
• ?OLS is no longer the BLUE of ?
• Now, S2(XX)-1 is no longer an unbiased estimator
  of V(?)(XX)-1(X?X)(XX)-1

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Patterns of Autocorrelation

• How is the structure of the errors?
• Consider a time series process with T ordered
  observations
• An important characteristics of this process is
  stationarity. The process ut is stationary if and
  only if the following holds
• E(ut)?lt? ?t
• Cov(ut,ut-j)?jlt ? ?t, j

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Patterns of Autocorrelation

• Property one requires the mean to exists and to
  be constant across time
• The second property requires the covariance to
  exist and to depend only on the distance between
  observations and not on the period where they are
  measured.
• (for example, stationarity requires the
  covariance between consumption in 1981 and 1983
  to be the same as the covariance between 1996 and
  1998).
• The second requisite implies that the variances
  are constant (V(ut)Cov(ut,ut) which is a
  constant)

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Patterns of Autocorrelation

• First Order Autorregressive process AR(1)
• We say that ut follows a zero mean AR(1) if,
• ut?ut-1?t
• Where ?t is a white noise process. It can be show
  that the AR(1) process is stationary if and only
  if ?lt1. We will assume that this stationarity
  condition holds

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Patterns of Autocorrelation

• AR(p) process
• A natural generalization is the AR(p) process,
• ut?1ut-1?2ut-2?put-p?t

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Testing for Autocorrelation

• There are two main tests for Autocorrelation
• Durbin-Watson test
• Breusch-Pagan LM test

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Durbin Watson Test

• Consider the simple linear model with AR(1)
  autocorrelation,
• where ? is a white noise process normally
  distributed

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Durbin Watson Test

• The Test is
• H0?0
• H1??0
• The test is easy if we know the error term ut.
• We would simply regress ut on ut-1 and test the
  null hypothesis. But in practice we do not know
  the values of the uts.

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Durbin Watson Test

• Instead we will use the residuals from the OLS
  model ignoring the problem of autocorrelation.
• However, things are not easy, since the
  Durbin-Watson test is based in the following
  statistic,

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Durbin Watson Test

• In order to get some intuition from the DW test,
  lets solve the square in the numerator,

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Durbin Watson Test

• Assume that the number of observations is large,
  then
• The first term should be close to one
• The second term should be close to one also
• In the case of the third term we have that
• then the third term is an estimate of ? having
  replaced ut by

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Durbin Watson Test

• Then, the following approximation for the Durbin
  Watson holds,
• DW11-2 2(1- )
• Then, for the null hypothesis H0?0, the DW
  should be close to 2.
• On the alternative hypothesis DW should be lower
  than 2, approximating to zero as ? increases.

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Durbin Watson Test

• If we follow the procedure of other tests, we
  should find a critical value dc, compute DW and
  evaluate if we accept or reject the Null
  hypothesis.
• However, because the distribution of the DW
  depends on the values of X, and since the X vary
  case by case, it is impossible to tabulate the
  distribution of DW for every possible problem.

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Durbin Watson Test

• Luckily, Durbin and Watson find that the critical
  value, dc is bounded by two values dl and du, and
  these values do not depend on the data.
• dlltdcltdu
• Then, once we calculated DW, we should follow the
  following
• DWltdl then DWltdc Reject H0 and Accept H1?gt0
• DWgtdu then DWgtdc Accept H0
• dlltDWltdu Test is inconclusive

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Durbin Watson Test
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Durbin Watson Test

• Comments about the DW test
• This is a finite sample test since the
  distribution of the DW test can be derived from
  the normality assumption of ? for every sample
  size
• The model must include an intercept
• It is crucial that the X is non-stochastic
• This test is for AR(1) process, but it is not
  informative about general processes like AR(p)

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The Breusch-Pagan LM Test

• Consider the following model,
• H0?1?2?p0
• H1 ?1?0 or ?2?0 oror ?p?0

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The Breusch-Pagan LM Test

• Breusch and Pagan propose the following
  procedure
• 1. Estimate the OLS model and save the residuals
• 2. Regress
• 3. Compute the test statistic (T-p)R2
• Under the null hypothesis this statistic has a ?2
  distribution with p degrees of freedom.

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The Breusch-Pagan LM Test

• Characteristics of this test
• Is an asymptotic test (large number of
  observations)
• We do not need non-stochastic X
• It explores different AR(p) processes including
  AR(1)

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Estimation under Autocorrelation

• Consider the simple model,
• Since the model is valid for any period, the
  following statement holds

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Estimation under Autocorrelation

• Now, multiply both sides of the last equation by
  ?,
• subtract one equation from the other and we will
  have that
• where

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Estimation under Autocorrelation

• If ? is known this procedure is straightforward
• If we do not know ?, we can estimate it by some
  technique. One approximation is DW?2(1- )
• First estimate ?
• Second, transform the variables and obtain y and
  x
• Third, run the OLS model of the transformed
  variables, knowing that the estimators are now
  BLUE
• Estimator ?1 can be easily recovered since