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?

3
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.
• Durbin's h-test for models with lagged dependent
  variables
• Tests for higher-order serial correlation.
• We discuss (in Section 6.5) the consequences of
  serially correlated errors and OLS estimators.

4
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
• Section 6.10 trends and random walks
• This chapter is very important and the several
  ideas have to be understood thoroughly.

5
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.

6
6.2 Durbin-Watson Test
7
6.2 Durbin-Watson Test

• The sampling distribution of d depends on values
  of the explanatory variables and hence Durbin and
  Watson derived upper limits and lower
  limits for the significance level for d.
• There are tables to test the hypothesis of zero
  autocorrelation against the hypothesis of
  first-order positive autocorrelation. ( For
  negative autocorrelation we interchange
  .)

8
6.2 Durbin-Watson Test

• If , we reject the null hypothesis
  of no autocorrelation.
• If , we do not reject the null
  hypothesis.
• If , the test is
  inconclusive.

9
6.2 Durbin-Watson Test

• Although we have said that
  this approximation is valid only in large
  samples.
• The mean of d when has been shown to
  be given approximately by (the proof is rather
  complicated for our purpose)
• where k is the number of regression
  parameters estimated (including the constant
  term) and n is the sample size.

10
6.2 Durbin-Watson Test

• Thus, even for zero serial correlation, the
  statistic is biased upward form 2.
• If k5 and n15, the bias is as large as 0.8.
• We illustrate the use of the DW test with an
  example.

11
6.2 Durbin-Watson Test

• Illustrative Example
• Consider the data in Table 3.11. The estimated
  production function is
• Referring to the DW table with k2 and n39 for
  5 significance level, we see that
  .
• Since the observed , we
  reject the hypothesis at the 5 level.

12
6.2 Limitations of D-W Test

• It test for only first-order serial correlation.
• The test is inconclusive if the computed value
  lies between .
• The test cannot be applied in models with lagged
  dependent variables.

13
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)
• First-difference

14
6.3 Estimation in Levels Versus First Differences
15
6.3 Estimation in Levels Versus First Differences
16
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)

17
6.3 Estimation in Levels Versus First Differences
18
6.3 Estimation in Levels Versus First Differences

• For instance, if the residual sum of squares is ,
  say, 1.2 by the level equation, and 0.8 by the
  difference equation and n 11, k1, DW0.9, then
  the adjusted residual sum of squares with the
  levels equation is (9/10)(0.9)(1.2)0.97 which is
  the number to be compared with 0.8.

19
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 S y y (l - R2)

20
6.3 Estimation in Levels Versus First Differences

• Let
• from the first difference equation
• residual sum of squares from the
  levels equation
• residual sum of squares from the
  first difference
• equation
• comparable from the levels
  equation
• Then

21
6.3 Estimation in Levels Versus First Differences

• Illustrative Examples
• Consider the simple Keynesian model discussed by
  Friedman and Meiselman. The equation estimated in
  levels is
• where Ct personal consumption expenditure
• (current dollars)
• At autonomous expenditure
• (current dollars)

22
6.3 Estimation in Levels Versus First Differences

• The model fitted for the 1929-1030 period gave
  (figures in parentheses are standard)

23
6.3 Estimation in Levels Versus First Differences

• This is to be compared with
  from the equation in first differences.

24
6.3 Estimation in Levels Versus First Differences

• For the production function data in Table 3.11
  the first difference equation is
• The comparable figures the levels equation
  reported earlier in Chapter 4, equation (4.24) are

25
6.3 Estimation in Levels Versus First Differences

• This is to be compared with
  from the equation in first differences.

26
6.3 Estimation in Levels Versus First Differences

• Harvey gives a different definition of .He
  defines it as
• This does not adjust for the face that the error
  variances in the levels equations and the first
  difference equation are not the same.
• The arguments for his suggestion are given in his
  paper.

27
6.3 Estimation in Levels Versus First Differences

• In the example with the Friedman-Meiselman data
  his measure of is given by
• Although cannot be greater than 1, it can
  be negative.
• This would be the case when
  ,that is, when the level model is giving a
  poorer explanation than the naïve model, which
  says that is a constant.

28
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).

29
6.3 Estimation in Levels Versus First Differences

• 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."
• An example

30
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.

31
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.

32
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.
• An example

33
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?

34
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.

35
6.3 Estimation in Levels Versus First Differences

• What we should be using is a modified statistic
  defined as
• The quarterly data (e.g. GDP)
• The monthly data (e.g. Industrial product index)

36
6.4 Estimation Procedures with Autocorrelated
Errors

• Now we will derive var(ut) and the correlations
  between ut and lagged values of ut. ..
• From equation(6.1) note that ut depends on et and
  ut-1 ,ut-1 depends on et-1 and ut-2 ,and so on.

37
6.4 Estimation Procedures with Autocorrelated
Errors

• This ut depends on et ,et-1 ,et-2 ,.Since et
  are serially independent, and ut-1 depends on
  et-1 ,et-2 and so on, but not et ,we have
• Since , we have
  for all t.

38
6.4 Estimation Procedures with Autocorrelated
Errors

• If we denote var(ut) by , we have
• Thus we have
• This gives the variance of ut in terms of the
  variance of et and the parameter .

39
6.4 Estimation Procedures with Autocorrelated
Errors

• Let us now derive the correlations. Denoting the
  correlation between ut and ut-1 (which is called
  the correlation of lag s) by , we get
• But
• Hence
• or

40
6.4 Estimation Procedures with Autocorrelated
Errors

• Since we get by successive substitution
• Thus the lag correlations are all powers of
  and decline geometrically.

41
6.4 Estimation Procedures with Autocorrelated
Errors

• GLS (Generalized least squares)

42
6.4 Estimation Procedures with Autocorrelated
Errors
43
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.

44
6.4 Estimation Procedures with Autocorrelated
Errors

• Iterative Procedures
• Among the iterative procedures, the earliest was
  the Cochrane-Orcutt (C-O) procedure.
• In the Cochrane-Otcutt procedure we estimate
  equation(6.2) by OLS, get the estimated residuals
  , and estimate
  .

45
6.4 Estimation Procedures with Autocorrelated
Errors

• Durbin suggested an alternative method of
  estimating .
• In this procedure, we write equation (6.5) as
• We regress
  and take the estimated coefficient of as
  an estimate of .

46
6.4 Estimation Procedures with Autocorrelated
Errors

• Use equation (6.6) and (6.6) and estimate a
  regression of y on x.
• The only thing to note is that the slop
  coefficient in this equation is , but the
  intercept is .
• Thus after estimating the regression of y on x,
  we have to adjust the constant term
  appropriately to get estimates of the parameters
  of the original equation (6.2).

47
6.4 Estimation Procedures with Autocorrelated
Errors

• Further, the standard errors we compute from the
  regression of y on x are now asymptotic
  standard errors because of the fact that has
  been estimated.
• If there are lagged values of y as explanatory
  variables, these standard errors are not correct
  even asymptotically.
• The adjustment needed in this case is discussed
  in Section 6.7.

48
6.4 Estimation Procedures with Autocorrelated
Errors

• Grid-Search Procedures
• One of the first grid-search procedures is the
  Hildreth and Lu procedure suggested in 1960.
• The procedure is as follows. Calculate
  in equation(6.6) for different values of
  at intervals of 0.1 in the rang
  .
• Estimate the regression of and
  calculate the residual sum of squares RSS in each
  case.

49
6.4 Estimation Procedures with Autocorrelated
Errors

• Choose the value of for which the RSS is
  minimum.
• Again repeat this procedure for smaller intervals
  of around this value.
• For instance, if the value of for which RSS
  is minimum is -0.4, repeat this search procedure
  for values of at intervals of 0.01 in the
  range .

50
6.4 Estimation Procedures with Autocorrelated
Errors

• This procedure is not the same as the ML
  procedure. If the errors et are normally
  distributed, we can write the log-likelihood
  function as (derivation is omitted)
• where
• Thus minimizing Q is not the same as maximizing
  log L.
• We can use the grid-search procedure to get the
  ML estimate.

51
6.4 Estimation Procedures with Autocorrelated
Errors

• Consider the data in Table 3.11 and the
  estimation of the production function
• The OLS estimation gave a DW statistic of 0.86,
  suggesting significant positive autocorrelation.
• Assuming that the errors were AR(1), two
  estimation procedures were used the Hildreth-Lu
  grid search and the iterative Cochrane-Orcutt
  (C-O).

52
6.4 Estimation Procedures with Autocorrelated
Errors

• The other procedures we have described can also
  be tried, but this is left as an exercise.
• The Hildreth-Lu procedure gave .
• The iterative C-O procedure gave .
• The DW test statistic implied that
  .

53
6.4 Estimation Procedures with Autocorrelated
Errors

• The estimates of the parameters (with standard
  errors in parentheses) were as follows

54
6.4 Estimation Procedures with Autocorrelated
Errors

• In this example the parameter estimates given by
  Hildreth-Lu and the iterative C-O procedures are
  pretty close to each other.
• Correcting for the autocorrelation in the errors
  has resulted in a significant change in the
  estimates of .

55
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?

56
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.

57
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

58
6.5 Effect of AR(1) Errors on OLS Estimates

• For the present, let us consider the simple
  regression model
• Let
• If ut are AR(1), we have .
  

59
6.5 Effect of AR(1) Errors on OLS Estimates
60
6.5 Effect of AR(1) Errors on OLS Estimates

• If we ignore the autocorrelation problem, we
  would be computing
  .Thus we would be ignoring the expression in the
  parentheses of equation (6.10).
• To get an idea of the magnitude of this
  expression, let us assume that the xt series also
  follow an AR(1) process with
  

61
6.5 Effect of AR(1) Errors on OLS Estimates

• Since we are now assuming xt to be stochastic,
  we will consider the asymptotic variance of
  .
• The expression in parentheses in equation (6.10)
  is now

62
6.5 Effect of AR(1) Errors on OLS Estimates

• Thus
• where T is the number of observations.
• If ,then
• Thus ignoring the expression in the parenthesis
  of equation (6.10) results in an underestimation
  by close to 78 for the variance of .

63
6.5 Effect of AR(1) Errors on OLS Estimates

• If ,this is an unbiased estimate .
• If ,then under the assumptions we are
  making, we have approximately
• Again if ,
  we have

64
6.5 Effect of AR(1) Errors on OLS Estimates

• We can also derive the asymptotic variance of the
  ML estimator when both x and u are
  first-order autoregressive as follow. Note that
  the ML estimator of is asymptotically
  equivalent to the estimator obtained from a
  regression of
  .

65
6.5 Effect of AR(1) Errors on OLS Estimates

• Hence
• where

66
6.5 Effect of AR(1) Errors on OLS Estimates

• When xt is autoregressive we have
• Hence by substitution we get the asymptotic
  variance of as

67
6.5 Effect of AR(1) Errors on OLS Estimates

• Thus the efficiency of the OLS estimator is
• One can compute this for different values of
  .
• For this efficiency is 0.21.

68
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.

69
6.5 Effect of AR(1) Errors on OLS Estimates

• The solution to these problems is to use the
  maximum likelihood procedure (one-step procedure)
  or some other procedure mentioned earlier
  (two-step procedure) that takes account of the
  autocorrelation.
• However, there are four important points to note

70
6.5 Effect of AR(1) Errors on OLS Estimates

• If is known, it is true that one can get
  estimators better than OLS that take account of
  autocorrelation. However, in practice is
  known and has to be estimated. In small samples
  it is not necessarily true that one gains (in
  terms of mean-square error for ) by
  estimating .
• This problem has been investigated by Rao
  and Griliches, who suggest the rule of thumb (for
  sample of size 20) that one can use the methods
  that take account of autocorrelation if
  ,where is the estimated first-order
  serial correlation from an OLS regression. In
  samples of larger sizes it would be worthwhile
  using these methods for smaller than 0.3.

71
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).

72
6.5 Effect of AR(1) Errors on OLS Estimates

• In regressions with quarterly (or monthly) data,
  one might find that the errors exhibit fourth (or
  twelfth)-order autocorrelation because of not
  making adequate allowance for seasonal effects.
  In such case if one looks for only first-order
  autocorrelation, one might not find any. This
  does not mean that autocorrelation is not a
  problem. In this case the appropriate
  specification for the error term may be
  for quarterly data and
  monthly data.

73
6.5 Effect of AR(1) Errors on OLS Estimates

• Finally, and most important, it is often possible
  to confuse misspecified dynamics with serial
  correlation in the errors. For instance, a static
  regression model with first-order autocorrelation
  in the errors, that is,
  ,can written as

74
6.5 Effect of AR(1) Errors on OLS Estimates

• 4. The model is the same as
• with the restriction .
• We can estimate the model (6.11) and test
  this restriction. If it is rejected, clearly it
  is not valid to estimate (6.11).(the test
  procedure is described in Section 6.8.)

75
6.5 Effect of AR(1) Errors on OLS Estimates

• The errors would be serially correlated but not
  because the errors follow a first-order
  autoregressive process but because the term xt-1
  and yt-1 have been omitted.
• Thus is what is meant by misspecified dynamics.
  Thus significant serial correlation in the
  estimated residuals does not necessarily imply
  that we should estimate a serial correlation
  model.

76
6.5 Effect of AR(1) Errors on OLS Estimates

• Some further tests are necessary (like the
  restriction in the
  above-mentioned case).
• In face, it is always best to start with an
  equation like (6.11) and test this restriction
  before applying any test for serial correlation.

77
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.
• Let us consider a simple model

78
6.7 Tests for Serial Correlation in Models with
Lagged Dependent Variables

• Let us consider a simple model
• et are independent with mean 0 and variance s2
  and .
• Because ut depends on ut-1 and yt-1 depends on
  ut-1, the two variables yt-1 and ut will be
  correlated.

79
6.7 Tests for Serial Correlation in Models with
Lagged Dependent Variables
An example
80
6.7 Tests for Serial Correlation in Models with
Lagged Dependent Variables

• Durbins h-Test
• Since the DW test is not applicable in these
  models, Durbin suggests an alternative test,
  called the h-test.
• This test uses
• as a standard normal variable.

81
6.7 Tests for Serial Correlation in Models with
Lagged Dependent Variables

• Hence is the estimated first-order serial
  correlation from the OLS residual, is
  the estimated variance of the OLS estimate of a,
  and n is the sample size.
• If , the test is not applicable.
  In this case Durbin suggests the following test.

82
6.7 Tests for Serial Correlation in Models with
Lagged Dependent Variables

• Durbins Alternative Test
• From the OLS estimation of equation(6.12) compute
  the residuals .
• Then regress
• The test for ?0 is carried out by testing the
  significance of the coefficient of in the
  latter regression.

83
6.7 Tests for Serial Correlation in Models with
Lagged Dependent Variables

• An equation of demand for food estimated from 50
  observations gave the following results (figures
  in parentheses are standard errors)
• where qtfood consumption per capita
• ptfood price (retail price deflated
  by the consumer
• price index)
• ytper capita disposable income
  deflated by the
• consumer price index

84
6.7 Tests for Serial Correlation in Models with
Lagged Dependent Variables

• We have
• Hence Duribins h-statistic is
• This is significant at the1level.
• Thus we reject the hypothesis ?0, even though
  the DW statistic is close to 2 and the estimate
  from the OLS residuals is

85
6.7 Tests for Serial Correlation in Models with
Lagged Dependent Variables

• Let us keep all the numbers the same and just
  change the standard error of .
• The following are the results
• Thus, other things equal, the precision with
  which is estimated has estimated has
  significant effect on the outcome of the h-test.

86
6.7 Tests for Serial Correlation in Models with
Lagged Dependent Variables

• In the case where the h-test cannot be used, we
  can use the alternative test suggested by Durbin,
  
• However, the Monte Carlo study by Maddala and Rao
  suggests that this test does not have good power
  in those cases where the h-test cannot be used.

87
6.7 Tests for Serial Correlation in Models with
Lagged Dependent Variables

• On the other hand, in cases where the h-test can
  be used, Durbins second test is almost as
  powerful.
• It is not often used because it involves more
  computations.
• However, we will show that Durbins second test
  can be generalized to higher-order
  autoregressions, whereas the h-test cannot.

88
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.

89
6.8 A General Test for Higher-Order Serial
Correlation The LM Test

• 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

90
6.8 A General Test for Higher-Order Serial
Correlation The LM Test

• Consider the regression model
• We are interested in testing H0?1?2?p0.
• The xs in equation(6.14) include lagged
  dependent variables as well.
• The LM test is as follows.

91
6.8 A General Test for Higher-Order Serial
Correlation The LM Test

• First, estimate (6.14) by OLS and obtain the
  least squares residuals .
• Next, estimate the regression equation
• And test whether the coefficient of are
  all zero.
• We take the conventional F-statistic and use p.F
  as ?2 with d.f. of p.
• We use the ?2-test rather than the F-test because
  the LM test is a large sample test.

92
6.8 A General Test for Higher-Order Serial
Correlation The LM Test

• The test can be used for different specifications
  of the error process.
• For instance, for the problem of testing for
  fourth-order autocorrelation
• we just estimate
• Instead of (6.16) and test ?40

93
6.8 A General Test for Higher-Order Serial
Correlation The LM Test

• The test procedure is the same for autoregressive
  or moving average errors.
• For instance if we have a moving average (MA)
  error
• instead of (6.17), the test procedure is still
  to estimate (6.18) and test .

94
6.8 A General Test for Higher-Order Serial
Correlation The LM Test

• In all these case, we just test H0 by estimating
  equation (6.16) with p2 and test ?1?20.
• What is of importance is the degree of
  autoregression, not its nature.

95
6.8 A General Test for Higher-Order Serial
Correlation The LM Test

• Finally, an alternative to the estimation of
  (6.16) is to estimate the equation

96
6.8 A General Test for Higher-Order Serial
Correlation The LM Test

• The LM test for serial correlation is
• Estimate equation (6.14) by OLS get the residual
  .
• Estimate equation (6.16) or (6.19) by OLS and
  compute the F-statistic for testing the
  hypothesis
  .
• Use with p degrees of freedom.

97
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

98
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.

99
6.9 Strategies When the DW Test Statistic is
Significant

• Errors Not AR(1)
• In case 1, if the DW statistic is significant,
  since it does not necessarily mean that the
  errors are AR(1), we should check for
  higher-order autoregressions by estimating
  equations of the form
• Once the order been determined, we can estimate
  the model with appropriate assumptions about the
  error stricture by the methods described in
  Section 6.4.

100
6.9 Strategies When the DW Test Statistic is
Significant

• Moving average (MA) errors and ARMA errors?
• Estimation with MA errors and ARMA errors is more
  complicated than with AR errors.
• However, researchers suggest that it is the order
  of the error process that is more important than
  the particular form the practical point of view,
  for most economic data, it is just sufficient to
  determine the order of the AR process.
• Thus if a significant DW statistic is observed,
  the appropriate strategy would be to try to see
  whether the errors are generated by a
  higher-order AR process than AR(1) and then
  undertake estimation.

101
6.9 Strategies When the DW Test Statistic is
Significant

• Autocorrelation Caused by Omitted Variables
• Suppose that the true regression equation is
• and instead we estimate

102
6.9 Strategies When the DW Test Statistic is
Significant

• Then since , if xt is
  autocorrelated, this will produce autocorrelation
  in vt.
• However vt is no longer independent of xt.
• This not only are the OLS estimators of ß0 and ß1
  from (6.20) inefficient, they are inconsistent as
  well.

103
6.9 Strategies When the DW Test Statistic is
Significant

• Serial correlation due to misspecification
  dynamics.
• In a seminal paper published in 1964, Sargan
  pointed out that a significant DW statistic does
  not necessarily imply that we have a serial
  correlation problem.
• This point was also emphasized by Henry and
  Mizon.
• The argument goes as follows.
• Consider
• and et are independent with a common variable
  s2.
• We can write this model as

104
6.9 Strategies When the DW Test Statistic is
Significant

• Consider an alternative stable dynamic model
• Equation (6.25) is the same as equation(6.26)
  with the restriction

105
6.9 Strategies When the DW Test Statistic is
Significant

• A test for ?0 is a test for ß10 (and ß30).
• But before we test this, what Sargan says is that
  we should first test the restriction (6.27) and
  test for ?0 only if the hypothesis
  is not rejected.
• If this hypothesis is rejected, we do not have a
  serial correlation model and the serial
  correlation in the errors in (6.24) is due to
  misspecified dynamics, that is the omission of
  the variable yt-1 and xt-1 from the equation.

106
6.9 Strategies When the DW Test Statistic is
Significant

• If the DW test statistic is significant, a proper
  approach is to test the restriction(6.27) to make
  sure that what we have is a serial correlation
  model before we undertake any autoregressive
  transformation of the variables.
• In fact, Sargan suggests starting with the
  general model (6.26) and testing the restriction
  (6.27) first, before attempting any test for
  serial correlation.

107
6.9 Strategies When the DW Test Statistic is
Significant

• Illustrative Example
• Consider the data in Table 3.11 and the
  estimation of the production function (4.24).
• In Section 6.4 we presented estimates of the
  equation assuming that the errors are AR(1).
• This was based on a DW test statistic of 0.86.
• Suppose that we estimate an equation of the
  form(6.26).
• The results are as follows (all variables in
  logs figures in parentheses are standard errors)

108
6.9 Strategies When the DW Test Statistic is
Significant

• Illustrative Example
• Under the assumption that the errors are AR(1),
  the residual sum of squares, obtained from the
  Hildreth-Lu procedure we used in Section 6.4, is
  RSS10.02635

109
6.9 Strategies When the DW Test Statistic is
Significant

• Since we have two slope coefficients, we have two
  restrictions of the form (6.27).
• Note that for the general dynamic model we
  estimating six parameters (a and five ßs).
• For the serial correlation model we are
  estimating four parameters (a, two ßs, and ?).
• We will use the likelihood ratio test (LR)

110
6.9 Strategies When the DW Test Statistic is
Significant

• and -2loge?has a ?2 -distribution with d.f.
  2(number of restrictions).
• In our example
• which is significant at the 1 level.
• Thus the hypothesis if a first-order
  autocorrelation is rejected.
• Although the DW statistic is significant, this
  does not mean that the errors are AR(1).

111
6.10 Trends and Random Walks

• Throughout our discussion we have assumed that
  for all t, and
  
• for all t and
  k, where is serial correlation of lag k
  (this is simply a function of the lag k and does
  not depend on t).
• If these assumptions are satisfied, the series ut
  is called covariance stationary (covariances are
  constant over time) or just stationary.

112
6.10 Trends and Random Walks

• Many economic time series are clearly
  nonstationary in the sense that the mean and
  variance depend on time, and they tend to depart
  ever further from any given value as time goes
  on.
• If this movement is predominantly in one
  direction (up or down) we say that series
  exhibits a trend.
• More detailed discussion of the topics covered
  briefly here can be found in Chapter 14.

113
6.10 Trends and Random Walks

• Nonstationary time series are frequently
  de-trended before further analysis is done.
• There are two procedures used for de-trending
• Estimating regressions on time.
• Successive differencing.

114
6.10 Trends and Random Walks

• In the regression approach it is assumed that the
  series yt is generated by the mechanism
• where f(t) is the trend and ut is a
  stationary series with mean zero and variance
  .
• Let us suppose that f(t) is linear so that we have

115
6.10 Trends and Random Walks

• Note that the trend-eliminated series is ,
  the least squares residuals that satisfy the
  relationship .
  
• If differencing is used to eliminate the trend we
  get .
• We have to take a first difference again to
  eliminate ß and we get
  as the de-trended series.

116
6.10 Trends and Random Walks

• On the other hand, suppose we assume that yt is
  generated by the model
• Where et is a stationary series with mean zero
  and variance .
• In this case the first difference of yt is
  stationary with mean ß.
• This model is also known as the random-walk
  model.
• Accumulating yt starting with an initial value y0
  we get from eauation(6.30).

117
6.10 Trends and Random Walks

• which has the same form as (6.29) except for
  the fact that the disturbance is not stationary,
  it has variance ts2 that increase over time.
• Nelson and Plosser call the model (6.29)
  tren-stationary processes (TSP) and model (6.30)
  difference-stationary processes (DSP)

118
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
• which belong to the DSP class if ?1,ß0 and
  the TSP class if .

119
6.10 Trends and Random Walks

• Thus we have to test the hypothesis ?1,ß0
  against .
• The problem here is that we cannot use the usual
  least squares distribution theory when ?1.
• Dickey and Fuller show that the least squares
  estimate of ? is not distributed around unity
  under the DSP hypothesis (that is, the true value
  ?1) but rather around a value less than one.
• However, the negative bias diminishes as the
  number of observations increases.

120
6.10 Trends and Random Walks

• They tabulate the significance points for testing
  the hypothesis ?1 against .
• Nelson and Plosser applied the Dickey-Fuller test
  to a wide range of historical time series for the
  U.S. economy and found that the DSP hypothesis
  was accepted in all cases, with the exception of
  the unemployment rate.
• They conclude that for most economic time series
  the DSP model is more appropriate

121
6.10 Trends and Random Walks

• The problem of testing the hypothesis ?1 in the
  first-order autoregressive equation of the form
• is called testing for unit roots.
• There is an enormous literature on this problem
  but one of the most commonly used tests is the
  Dickey-Fuller test.
• The standard expression for the large sample
  variance of the least squares estimator
  which would be zero under the null
  hypothesis.
• Hence, one needs to derive the limiting
  distribution of under H0, ?1 to apply the
  test.

122
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)
• An example

123
Augmented D-F (ADF) 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
• Unit root test an example
• Limitations of ADF Tests

124
6.10 Trends and Random Walks

• Spurious Trends
• If ß0 in equation (6.30) the model is called a
  trendless random walk or a random walk with zero
  drift.
• However, from equation(6.31) note that even
  though there is no trend in the mean, there is a
  trend in the variance.
• Suppose that the true model is of the DSP type
  with ß0.
• What happens if we estimated a TSP type model?

125
6.10 Trends and Random Walks

• That is, the true model is one with no trend in
  the mean but only a trend in the variance, and we
  estimate a model with a trend in the mean but no
  trend in the variance.
• It is intuitively clear that the trend in the
  variance will be transmitted to the mean and we
  will find a significant coefficient for t even
  though in reality there is no trend in the mean.
• How serious is this problem?

126
6.10 Trends and Random Walks

• Nelson and Kang analyze this and conclude that
• 1. Regression of a random walk on time by least
  squares will produce R2 values of around 0.44
  regardless of sample size when, in fact, the mean
  of the variable has no relationship with time
  whatever.
• 2. In the case of random walks with drift, that
  is ß?0, the R2 will be higher and will increase
  with the sample size, reaching one in the limit
  regardless of the value of ß.

127
6.10 Trends and Random Walks

• 3. The residual from the regression on time which
  we take as the de-trended series, has on the
  average only about 14 of the true stochastic
  variance of the original series.
• 4. The residuals from the regression on time are
  also autocorrelated being roughly(1-10/N) at lag
  one, where N is the sample size.

128
6.10 Trends and Random Walks

• 5. Conventional t-tests to test the significance
  of some of the regressors are not valid. They
  tend to reject the null hypothesis of no
  dependence on time, with very high frequency.

129
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.
• An spurious regression example

130
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

131
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

132
6.10 Trends and Random Walks

• where ut are independent with mean zero and
  common variance .
• If we difference this equation, we get
• where the error is a moving average, and,
  hence, not serially independent.
• But estimating the first difference equation by
  least squares still gives us unbiased/consistent
  estimates.

133
6.10 Trends and Random Walks

• Thus, the consequences of differencing when it is
  not needed are much less serious than those of
  failing to difference when it is appropriate
  (when the true model is of the DSP type).
• In practice, it is best to use the Dickey-Fuller
  test to check whether the data are of DSP or TSP
  type.
• Otherwise, it is better to use differencing and
  regressions in first differences, rather than
  regressions in levels with time as an extra
  explanatory variable.

134
6.10 Trends and Random Walks

• The Concept of Cointegration Differencing vs.
  Long-Run Effects
• 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.
  First, we need to define the term
  "cointegration.

135
6.10 Trends and Random Walks

• If yt follow a random walk model, that is
• then we get by successive substitution,
• Thus, yt is a summation of ej , and

136
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

137
6.10 Trends and Random Walks

• We say in this case yt is I(1)intergrated to
  order one. If yt is I(1) and we add to this zt
  which is I(0), then will be I(1).
• When we specify regression models in time series,
  we have to make sure that the different variables
  are integrated to the same degree. Otherwise, the
  equation does not make sense.
• For instance, if we specify the regression model
• and we say that , that is
  ut is I(0), we have to make sure that yt and xt
  are integrated to the same order.

138
6.10 Trends and Random Walks

• For instance, if yt is I(1) and xt is I(0) there
  will not be any ß that will satisfy the
  relationship(6.34).
• Suppose yt is I(1) and xt is I(1) then if there
  is a nonzero ß such that is I(0),
  then yt and xt are said to be cointegrated.

139
6.10 Trends and Random Walks

• Suppose that yt and xt are both random walks, so
  that they are both I(1).Then an equation in first
  differences of the form
• is a valid equation, since
  , and vt are all I(0).
• Equation(6.34) is considered a long-run
  relationship between yt and xt and
  equation(6.35)describes short-run dynamics.
• Engle and Granger suggest estimating(6.34) by
  ordinary least squares, obtaining the estimator
  and substituting it in equation (6.35) to
  estimate the parameters aand ?.

140
6.10 Trends and Random Walks

• This two-step estimation procedure, however,
  rests on the assumption that yt and xt are
  cointegrated.
• It is, therefore, important to test for
  cointegration.
• Engle and Granger suggest estimaing(6.34) by
  ordinary least squares, getting the residual

141
6.10 Trends and Random Walks

• test amounts to is testing the hypothesis ?1 in
• that is, testing the hypothesis
• H0 ut is I(1)
• In essence, we testing the null hypothesis that
  yt and xt are not cointegrated.
• Note that yt is I(1) and xt is I(1), so we are
  trying to see that ut is not I(1).

142
Cointegration

• Co-integration test an example
• Run the VECM (vector error correction model) by
  E-view

143
Homework

• Find the spot and futures prices
• 5-year daily data at least
• Run the cointegration test
• Run the VECM
• Check the lead-lag relationship using the EC
  parameter estimate

144
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 (1982) which has,
  in recent years, been found useful in the
  analysis of speculative prices.
• ARCH stands for "autoregressive conditional
  heteroskedasticity.
• Robert Engle and Clive Granger
• NOBEL PRIZE WINNERS FOR ECONOMICS, 2003

145
6.11 ARCH Models and Serial Correlation

• GARCH (p,q) Model (by Bollerslev, 1986)

146
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
• An example GARCH (1,1)

147
6.11 ARCH Models and Serial Correlation

• Find the stock returns
• 5-year daily 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