Autocorrelation

1
Chapter 6

• Autocorrelation

2
What is in this Chapter?

• How do we detect this problem?
• What are the consequences?
• What are the solutions?

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What is in this Chapter?

• Regarding the problem of detection, we start with
  the Durbin-Watson (DW) statistic, and discuss its
  several limitations and extensions. We discuss
  Durbin's h-test for models with lagged dependent
  variables and tests for higher-order serial
  correlation.
• We discuss (in Section 6.5) the consequences of
  serially correlated errors and OLS estimators.

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What is in this Chapter?

• The solutions to the problem of serial
  correlation are discussed in Section 6.3
  (estimation in levels versus first differences),
  Section 6.9 (strategies when the DW test
  statistic is significant), and Section 6.10
  (trends and random walks).
• This chapter is very important and the several
  ideas have to be understood thoroughly.

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6.1 Introduction

• The order of autocorrelation
• In the following sections we discuss how to
• 1. Test for the presence of serial correlation.
• 2. Estimate the regression equation when the
  errors are serially correlated.

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6.2 Durbin-Watson Test
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6.2 Durbin-Watson Test
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6.2 Durbin-Watson Test
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6.2 Durbin-Watson Test
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6.2 Durbin-Watson Test
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6.3 Estimation in Levels Versus First Differences

• Simple solutions to the serial correlation
  problem First Difference
• If the DW test rejects the hypothesis of zero
  serial correlation, what is the next step?
• In such cases one estimates a regression by
  transforming all the variables by ?-differencing
  (quasi-first difference) or first-difference

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6.3 Estimation in Levels Versus First Differences
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6.3 Estimation in Levels Versus First Differences
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6.3 Estimation in Levels Versus First Differences

• When comparing equations in levels and first
  differences, one cannot compare the R2 because
  the explained variables are different.
• One can compare the residual sum of squares but
  only after making a rough adjustment. (Please
  refer to P.231)

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6.3 Estimation in Levels Versus First Differences
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6.3 Estimation in Levels Versus First Differences
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6.3 Estimation in Levels Versus First Differences

• Since we have comparable residual sum of squares
  (RSS), we can get the comparable R2 as well,
  using the relationship RSS Syy(l R2)

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6.3 Estimation in Levels Versus First Differences
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6.3 Estimation in Levels Versus First Differences

• Illustrative Examples

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6.3 Estimation in Levels Versus First Differences
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6.3 Estimation in Levels Versus First Differences
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6.3 Estimation in Levels Versus First Differences
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6.3 Estimation in Levels Versus First Differences
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6.3 Estimation in Levels Versus First Differences

• Usually, with time-series data, one gets high R2
  values if the regressions are estimated with the
  levels yt and Xt but one gets low R2 values if
  the regressions are estimated in first
  differences (yt yt-1) and (xt xt-1)
• Since a high R2 is usually considered as proof of
  a strong relationship between the variables under
  investigation, there is a strong tendency to
  estimate the equations in levels rather than in
  first differences.
• This is sometimes called the R2 syndrome."

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6.3 Estimation in Levels Versus First Differences

• However, if the DW statistic is very low, it
  often implies a misspecified equation, no matter
  what the value of the R2 is
• In such cases one should estimate the regression
  equation in first differences and if the R2 is
  low, this merely indicates that the variables y
  and x are not related to each other.

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6.3 Estimation in Levels Versus First Differences

• Granger and Newbold present some examples with
  artificially generated data where y, x, and the
  error u are each generated independently so that
  there is no relationship between y and x
• But the correlations between yt and yt-1,.Xt and
  Xt-1, and ut and ut-1 are very high
• Although there is no relationship between y and x
  the regression of y on x gives a high R2 but a
  low DW statistic

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6.3 Estimation in Levels Versus First Differences

• When the regression is run in first differences,
  the R2 is close to zero and the DW statistic is
  close to 2
• Thus demonstrating that there is indeed no
  relationship between y and x and that the R2
  obtained earlier is spurious
• Thus regressions in first differences might often
  reveal the true nature of the relationship
  between y and x.
• Further discussion of this problem is in Sections
  6.10 and 14.7

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Homework

• Find the data
• Y is the Taiwan stock index
• X is the U.S. stock index
• Run two equations
• The equation in levels (log-based price)
• The equation in the first differences
• A comparison between the two equations
• The beta estimate and its significance
• The R square
• The value of DW statistic
• Q Adopt the equation in levels or the first
  differences?

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6.3 Estimation in Levels Versus First Differences

• For instance, suppose that we have quarterly
  data then it is possible that the errors in any
  quarter this year are most highly correlated with
  the errors in the corresponding quarter last year
  rather than the errors in the preceding quarter
• That is, ut could be uncorrelated with ut-1 but
  it could be highly correlated with ut-4.
• If this is the case, the DW statistic will fail
  to detect it
• What we should be using is a modified statistic
  defined as

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6.3 Estimation in Levels Versus First Differences
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6.4 Estimation Procedures with Autocorrelated
Errors
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6.4 Estimation Procedures with Autocorrelated
Errors
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6.4 Estimation Procedures with Autocorrelated
Errors
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6.4 Estimation Procedures with Autocorrelated
Errors
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6.4 Estimation Procedures with Autocorrelated
Errors

• GLS (Generalized least squares)

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6.4 Estimation Procedures with Autocorrelated
Errors
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6.4 Estimation Procedures with Autocorrelated
Errors

• In actual practice ? is not known
• There are two types of procedures for estimating
• 1. Iterative procedures
• 2. Grid-search procedures.

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6.4 Estimation Procedures with Autocorrelated
Errors
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6.4 Estimation Procedures with Autocorrelated
Errors
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6.4 Estimation Procedures with Autocorrelated
Errors
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6.4 Estimation Procedures with Autocorrelated
Errors
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6.4 Estimation Procedures with Autocorrelated
Errors
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6.4 Estimation Procedures with Autocorrelated
Errors
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Homework

• Redo the example (see Table 3.11 for the data) in
  the Textbook
• OLS
• C-O procedure
• H-L procedure with the interval of 0.01
• Compare the R2 (Note please calculate the
  comparable R2 form the levels equation)

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6.5 Effect of AR(1) Errors on OLS Estimates

• In Section 6.4 we described different procedures
  for the estimation of regression models with
  AR(1) errors
• We will now answer two questions that might arise
  with the use of these procedures
• 1. What do we gain from using these procedures?
• 2. When should we not use these procedures?

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6.5 Effect of AR(1) Errors on OLS Estimates

• First, in the case we are considering (i.e., the
  case where the explanatory variable Xt is
  independent of the error ut), the OLS estimates
  are unbiased
• However, they will not be efficient
• Further, the tests of significance we apply,
  which will be based on the wrong covariance
  matrix, will be wrong.

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6.5 Effect of AR(1) Errors on OLS Estimates

• In the case where the explanatory variables
  include lagged dependent variables, we will have
  some further problems, which we discuss in
  Section 6.7
• For the present, let us consider the simple
  regression model

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6.5 Effect of AR(1) Errors on OLS Estimates
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6.5 Effect of AR(1) Errors on OLS Estimates
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6.5 Effect of AR(1) Errors on OLS Estimates
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6.5 Effect of AR(1) Errors on OLS Estimates
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6.5 Effect of AR(1) Errors on OLS Estimates
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6.5 Effect of AR(1) Errors on OLS Estimates
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6.5 Effect of AR(1) Errors on OLS Estimates
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6.5 Effect of AR(1) Errors on OLS Estimates
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An Alternative Method to Prove the Above
Characteristics???

• Use simulation method as shown at Chapter 5
• Write your program by the Gauss program
• Take the program at Chapter 5 and make some
  modifications on it

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6.5 Effect of AR(1) Errors on OLS Estimates

• Thus the consequences of autocorrelated errors
  are
• 1. The least squares estimators are unbiased but
  are not efficient. Sometimes they are
  considerably less efficient than the procedures
  that take account of the autocorrelation
• 2. The sampling variances are biased and
  sometimes likely to be seriously understated.
  Thus R2 as well as t and F statistics tend to be
  exaggerated.

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6.5 Effect of AR(1) Errors on OLS Estimates
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6.5 Effect of AR(1) Errors on OLS Estimates

• 2. The discussion above assumes that the true
  errors are first-order autoregressive. If they
  have a more complicated structure (e.g.,
  second-order autoregressive), it might be thought
  that it would still be better to proceed on the
  assumption that the errors are first-order
  autoregressive rather than ignore the problem
  completely and use the OLS method???
• Engle shows that this is not necessarily true
  (i.e., sometimes one can be worse off making the
  assumption of first-order autocorrelation than
  ignoring the problem completely).

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6.5 Effect of AR(1) Errors on OLS Estimates
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6.5 Effect of AR(1) Errors on OLS Estimates
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6.5 Effect of AR(1) Errors on OLS Estimates
63
6.7 Tests for Serial Correlation in Models with
Lagged Dependent Variables

• In previous sections we considered explanatory
  variables that were uncorrelated with the error
  term
• This will not be the case if we have lagged
  dependent variables among the explanatory
  variables and we have serially correlated errors
• There are several situations under which we would
  be considering lagged dependent variables as
  explanatory variables
• These could arise through expectations,
  adjustment lags, and so on.

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6.7 Tests for Serial Correlation in Models with
Lagged Dependent Variables

• The various situations and models are explained
  in Chapter 10. For the present we will not be
  concerned with how the models arise. We will
  merely study the problem of testing for
  autocorrelation in these models
• Let us consider a simple model

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6.7 Tests for Serial Correlation in Models with
Lagged Dependent Variables
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6.7 Tests for Serial Correlation in Models with
Lagged Dependent Variables
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• new
• format /m1 /rd 9,3
• beta2
• T30 _at_ sample number _at_
• uRndn(T,1)
• xRndn(T,1)0u
• ybetaxu
• _at_ OLS _at_
• Beta_OLSolsqr(y,x)
• print " OLS beta estimate "
• Beta_OLS

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• new
• format /m1 /rd 9,3
• beta2
• T50000 _at_ sample number _at_
• uRndn(T,1)
• xRndn(T,1)0u
• ybetaxu
• _at_ OLS _at_
• Beta_OLSolsqr(y,x)
• print " OLS beta estimate "
• Beta_OLS

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• new
• format /m1 /rd 9,3
• beta2
• T50000 _at_ sample number _at_
• uRndn(T,1)
• xRndn(T,1)0.5u
• ybetaxu
• _at_ OLS _at_
• Beta_OLSolsqr(y,x)
• print " OLS beta estimate "
• Beta_OLS

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6.7 Tests for Serial Correlation in Models with
Lagged Dependent Variables
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6.7 Tests for Serial Correlation in Models with
Lagged Dependent Variables
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6.7 Tests for Serial Correlation in Models with
Lagged Dependent Variables
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6.7 Tests for Serial Correlation in Models with
Lagged Dependent Variables
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6.7 Tests for Serial Correlation in Models with
Lagged Dependent Variables
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6.8 A General Test for Higher-Order Serial
Correlation The LM Test

• The h-test we have discussed is, like the
  Durbin-Watson test, a test for first-order
  autoregression.
• Breusch and Godfrey discuss some general tests
  that are easy to apply and are valid for very
  general hypotheses about the serial correlation
  in the errors
• These tests are derived from a general principle
  called the Lagrange multiplier (LM) principle
• A discussion of this principle is beyond the
  scope of this book. For the present we will
  explain what the test is
• The test is similar to Durbin's second test that
  we have discussed

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6.8 A General Test for Higher-Order Serial
Correlation The LM Test
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6.8 A General Test for Higher-Order Serial
Correlation The LM Test
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6.8 A General Test for Higher-Order Serial
Correlation The LM Test
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6.8 A General Test for Higher-Order Serial
Correlation The LM Test
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6.8 A General Test for Higher-Order Serial
Correlation The LM Test
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6.9 Strategies When the DW Test Statistic is
Significant

• The DW test is designed as a test for the
  hypothesis ? 0 if the errors follow a
  first-order autoregressive process
• However, the test has been found to be robust
  against other alternatives such as AR(2), MA(1),
  ARMA(1, 1), and so on.
• Further, and more disturbingly, it catches
  specification errors like omitted variables that
  are themselves autocorrelated, and misspecified
  dynamics (a term that we will explain). Thus the
  strategy to adopt, if the DW test statistic is
  significant, is not clear. We discuss three
  different strategies

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6.9 Strategies When the DW Test Statistic is
Significant

• 1. Assume that the significant DW statistic is an
  indication of serial correlation but may not be
  due to AR(1) errors
• 2. Test whether serial correlation is due to
  omitted variables.
• 3. Test whether serial correlation is due to
  misspecified dynamics.

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6.9 Strategies When the DW Test Statistic is
Significant
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6.9 Strategies When the DW Test Statistic is
Significant
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6.9 Strategies When the DW Test Statistic is
Significant
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6.9 Strategies When the DW Test Statistic is
Significant
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6.9 Strategies When the DW Test Statistic is
Significant
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6.9 Strategies When the DW Test Statistic is
Significant

• Serial correlation due to misspecification
  dynamics

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6.9 Strategies When the DW Test Statistic is
Significant
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6.9 Strategies When the DW Test Statistic is
Significant
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6.9 Strategies When the DW Test Statistic is
Significant
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6.9 Strategies When the DW Test Statistic is
Significant
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6.10 Trends and Random Walks
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6.10 Trends and Random Walks
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6.10 Trends and Random Walks
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6.10 Trends and Random Walks
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6.10 Trends and Random Walks

• Both the models exhibit a linear trend. But the
  appropriate method of eliminating the trend
  differs
• To test the hypothesis that a time series belongs
  to the TSP class against the alternative that it
  belongs to the DSP class, Nelson and Plosser use
  a test developed by Dickey and Fuller

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6.10 Trends and Random Walks
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6.10 Trends and Random Walks
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Three Types of RW

• RW without drift Yt1Yt-1ut
• RW with drift Ytalpha1Yt-1ut
• RW with drift and time trend Ytalphabetat1Yt
  -1ut
• utiid(0,sigma)

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RW or Unit Root tests by E-view

• Additional Slides
• Augmented D-F tests
• Yta1Yt-1ut
• Yt-Yt-1(a1-1)Yt-1ut
• ?Yt(a1-1)Yt-1ut
• ?Yt?Yt-1ut
• H0a11 H0 ?0
• ?Yt?Yt-1S?Yt-iut

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6.10 Trends and Random Walks
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6.10 Trends and Random Walks

• As an illustration consider the example given by
  Dickey and Fuller.36 For the logarithm of the
  quarterly Federal Reserve Board Production Index
  1950-1 through 1977-4 they assume that the time
  series is adequately represented by the model

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6.10 Trends and Random Walks
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6.10 Trends and Random Walks
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6.10 Trends and Random Walks

• 6. Regression of one random walk on another, with
  time included for trend, is strongly subject to
  the spurious regression phenomenon. That is, the
  conventional t-test will tend to indicate a
  relationship between the variables when none is
  present.

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6.10 Trends and Random Walks

• The main conclusion is that using a regression on
  time has serious consequences when, in fact, the
  time series is of the DSP type and, hence,
  differencing is the appropriate procedure for
  trend elimination
• Plosser and Schwert also argue that with most
  economic time series it is always best to work
  with differenced data rather than data in levels
• The reason is that if indeed the data series are
  of the DSP type, the errors in the levels
  equation will have variances increasing over time

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6.10 Trends and Random Walks

• Under these circumstances many of the properties
  of least squares estimators as well as tests of
  significance are invalid
• On the other hand, suppose that the levels
  equation is correctly specified. Then all
  differencing will do is produce a moving average
  error and at worst ignoring it will give
  inefficient estimates
• For instance, suppose that we have the model

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6.10 Trends and Random Walks
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6.10 Trends and Random Walks

• Differencing and Long-Run EffectsThe Concept of
  Cointegration
• One drawback of the procedure of differencing is
  that it results in a loss of valuable "long-run
  information" in the data
• Recently, the concept of cointegrated series has
  been suggested as one solution to this problem.39
  First, we need to define the term
  "cointegration.
• Although we do not need the assumption of
  normality and independence, we will define the
  terms under this assumption.

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6.10 Trends and Random Walks
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6.10 Trends and Random Walks

• YtI(1)
• Yt is a random walk
• ?Yt is a white noise, or iid
• No one could predict the future price change
• The market is efficient
• The impact of previous shock on the price will
  remain and not approach to zero

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6.10 Trends and Random Walks
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6.10 Trends and Random Walks
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6.10 Trends and Random Walks
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Cointegration
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Cointegration
125
Cointegration

• Run the VECM (vector error correction model) by
  E-view
• Additional slides

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Cointegration
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Lead-lag relation obtained with VECM model

• If beta_A is significant and beta_U is
  insignificant,
• the price adjustment mainly depends on ADR
  markets
• ADR prices converge to UND prices
• UND prices lead ADR prices in price discovery
  process
• UND prices provide an information advantage

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• If beta_U is significant and beta_A is
  insignificant,
• the price adjustment mainly depends on UND
  markets
• UND prices converge to ADR prices
• ADR prices lead UND prices in price discovery
  process
• ADR prices provide an information advantage

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• If both of beta_U and beta_A are significant
• suggesting a bidirectional error correction
• The equilibrium prices line within ADR and UND
  prices
• Both ADR and UND prices converge to the
  equilibrium prices

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• If both of beta_U and beta_A are significant, but
  the beta_U is greater than beta_A in absolute
  velue
• The finding denotes that it is the UND price that
  makes greater adjustment in order to reestablish
  the equilibrium
• That is, most of the price discovery takes place
  at the ADR market.

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Homework

• Find the spot and futures prices
• Daily and 5-year data at least
• Run the cointegration test
• Run the VECM
• Lead-lag relationship

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6.11 ARCH Models and Serial Correlation

• We saw in Section 6.9 that a significant DW
  statistic can arise through a number of
  misspecifications.
• We will now discuss one other source. This is the
  ARCH model suggested by Engle which has, in
  recent years, been found useful in the analysis
  of speculative prices.
• ARCH stands for "autoregressive conditional
  heteroskedasticity."

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6.11 ARCH Models and Serial Correlation

• GARCH (p,q) Model

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6.11 ARCH Models and Serial Correlation

• The high level of persistence in GARCH models
• the sum of the two GARCH parameter estimates
  approximates unity in most cases
• Li and Lin (2003) This finding provides some
  support for the notion that GARCH models are
  handicapped by the inability to account for
  structural changes during the estimation period
  and thus suffers from a high persistence problem
  in variance settings.

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6.11 ARCH Models and Serial Correlation

• Find the stock returns
• Daily and 5-year data at least
• Run the GARCH(1,1) model
• Check the sum of the two GARCH parameter
  estimates
• Parameter estimates
• Graph the time-varying variance estimates

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Could we identify RW? Low test power of the DF
test

• The Power of the test?
• The H0 is not true, but we accept the H0
• The data series is I(0), but we conclude it is
  I(1)

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Several Key Problems for Unit Root Tests

• Low test power
• Structural change problem
• Size distortion
• RW or non-stationary or I(1)
• Yt1Yt-1ut
• Stationary Process or I(0)
• Yt0.99Yt-1ut-1, T1,000
• Yt0.98Yt-1ut-1, T50 or 1000

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Spurious Regression

• RW 1 Yt0.051Yt-1ut
• RW 2 Xt0.031Xt-1vt

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Spurious Regression

• new
• format /m1 /rd 9,3
• _at_ Data Gerneration Process _at_
• Yzeros(1000,1) u2Rndn(1000,1)
• Xzeros(1000,1) v1Rndn(1000,1)
• i2
• do until igt1000
• Yi,10.051Yi-1,1ui,1
• Xi,10.031Xi-1,1vi,1
• ii1
• endo

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