III' Bivariate model nonstationary time series

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III. Bivariate model nonstationary time series

• A. Spurious regression
• What is spurious regression?
• Monte Carlo simulations
• Examples
• B. Cointegration
• Basic concepts
• Engle-Granger test
• An alternative test of cointegration

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III. A. Spurious regression1. What is spurious
regression?

• Xt and Yt are I(1) but unrelated variables
  researchers may think they are related and run a
  spurious regression
• Yt b1 b2 Xtet
• Granger and Newbold (1974) drew attention to the
  issue in econometrics
• Phillips (1986) theoretical explanations

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2. Monte Carlo results a). X and Y are
cointegrated

• OLS estimators are unbiased and the t statistic
  close to the theoretical t distribution.
• There is a further increase in the concentration
  of the distribution of b in case (ii). As T
  increases the s.d. of b drops.

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2. Monte Carlo results b). no cointegration ?
spurious regression

• Patterson, Table 8.5 X and Y are not
  cointegrated
• Large Var(bols) its highly possible to get a
  nonzero estimate
• (i) Misleading implication about whether X and Y
  are related or not

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Spurious Regression -- Unsafe to use the
traditional t distribution

• (ii) Misleading critical values of the
  theoretical t distribution
• Empirical distribution T50
• Lower 5 critical value -8.38
• Higher 5 critical value 8.37
• Theoretical distribution
• 10, two-sided test critical values, ?1.67,
• Size problem low power of the test
• too often reject the null of no relationship
• (71 for T50 80 for T100 90 for T500)

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2. Monte Carlo results b). spurious regression

• (iii). Misleading R2
• Patterson, Table 8.6 For unrelated I(1)
  processes, conventional R2 is misleading since
• Mean(R2)0.24
• Prob(R2 gt0.6)0.1
• R2 is a random variable (high R2 for unrelated
  I(1) variables does not disappear as sample size
  increases)

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From spurious to cointegration

• The tests of significance for the spurious
  regression should be interpreted with great care.
  
• The series are typically highly correlated even
  when there is no underlying relationship between
  them.

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3. Example 1 C us and Y uk

• The coeff b is significant since both series have
  a time trend. To be non-spurious, we require that
  the regression removes the stochastic trend from
  the dependent variables, leaving stationary
  residuals.

• Two signals about spurious regression the visual
  impression of the residuals and the small
  reported DW statistic 2(1-r) r is the OLS
  coeff. from the regression of on

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Example 2

• Yt b Xt ut
• Yt number of unemployed people in Taiwan
• Xt accumulated number of sun spots in the same
  period
• It is very possible to find the OLS estimate of b
  significant, even though there is no relationship
  between the two time series

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Example 3 H1t , H2t I(1) uncorrelated
H1t
H2t
But if we regress H1t on H2t we have the
following results
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Example 3 Significant t-value and high R2 but
Poor fit residuals are far from WN
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Example 3 Unit root in residuals

• The following test results suggest that there is
  a unit root in the residual series

The OLS estimate of the coefficient does not
converge to zero. The t-stat tT also diverges
and appears significant.
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III. B. Cointegration 1. basic concept 1

• Y and X share the same stochastic trend so that
  they are tied together in the long run that is
  even if they deviate from each other in the short
  run, they tend to return to the trend in the long
  run

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Example Time series of consumption and income
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Example Time series of consumption and income

• Construct a new series Yt its statistical
  properties Yt 180 Con 1.18 Inc

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Example Time series of consumption and income

• Yts ACF
• The ADF test on yt

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Spurious or cointegrated?

• Yt is I(0), thus Con and Inc are cointegrated.
• Two I(1) series yt and xt
• Spurious regression if the estimation residuals
  ut are unit root process
• The same regression produces a cointegration
  relationship if ut are stationary process.

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1. Basic concept 2
An (n1) vector time series Zt is said to be
cointegrated if each of the series taken
individually is I(1) with some linear combination
of the series is stationary or I(0) for some
nonzero (n 1) vector a
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Examples

• Davidson, Hendry, Srba, and Yeo (1978)
• C (t) Yp(t)
• lnC (t) and lnY (t) I(1), but
• lnC (t) - lnY (t) I(0)
• Long run consumption tends to be a roughly
  constant portion of income
• PPP P(t)E(t)P (t) p(t)e(t)p(t)
• weak version of PPP z(t)p(t)-e(t)-p(t)I(0)
• p(t), e(t), and p(t) are I(1)

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2. Test for cointegrationEngle-Granger approach

• I. If cointegrating vector is known
• Step one
• unit root tests for z1t, z2t I(1)
• if cannot reject null then,
• Step two
• unit root tests for etI(0) or I(1)

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2. Test for cointegration Engle-Granger approach

• II. If cointegrating vector is unknown
• EG regression in the levels of the I(1)
  variables
• z1t agz2t et
• Step one estimate a and g by OLS
• unit root tests for z1t, z2t I(1)
• OLS estimate is superconsistent (converging to
  true mean at a rate T, rather than T½)
• Step two OLS residuals
• unit root tests for I(0) or I(1)

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Null and alternative hypotheses

• et bet-1 ht
• H0 b1
• nonrejection of the null et I(1)
• z1t and z2t are not cointegrated
• HA blt1
• if this is true, et I(0)
• z1t and z2t are cointegrated

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Critical values for EG test

• If regression coeffs are known rather than
  estimated, for example a0 and g1, than we can
  use the ordinary DF critical values
• Otherwise, the critical values are larger
  negative values than for the corresponding DF
  test statistics (Patterson, Table 8.7)

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Illustrating of the EG two-step test

• Taiwan real income, consumption, and savings
  ratio
• lnC lnY ln(C/Y) ln(1-s)-s
• SY-C sS/Y
• Visual impressions
• 1. C and Y have a clear trend
• 2. Savings ratio shows deviations from the mean
  long memory or RW?
• Unit root tests Eviews output

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Taiwan GNP, Consumption Expenditure, (Y-C)/Y
1951-2001
Y
C
s
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Step 1 Results of unit root tests

• lnc ADF(1) w/trend Test Statistic -3.526769
• lny ADF(1) w/trend Test Statistic -2.780085
• 1 Critical Value -4.1584
• 5 Critical Value -3.5045
• 10 Critical Value -3.1816
• lnc PP Test Statistic -1.951612
• lny PP Test Statistic -2.056268
• 1 Critical Value -4.1540
• 5 Critical Value -3.5025
• 10 Critical Value -3.1804

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Step 2 EG Cointegrating regression

• Eqcy and eqyc
• lnC0.016769200160.9514837016 lnY
• lnY0.021558312471.047411126 lnC
• R2 is impressive but spurious if lnC and lnY are
  not cointegrated
• Estimated income elasticity is almost 1,
  residuals savings ratio
• t statistic is misleading if stationary
  variables are excluded, disturbance is serially
  correlated, the OLS s.e. are incorrect

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Step 3 unit root tests for estimated residuals

• Residuals lnC - lnY
• DF Test Statistic -6.256393
• 1 Critical Value -2.6120
• 5 Critical Value -1.9478
• 10 Critical Value -1.6195
• Residuals lnY - lnC
• DF Test Statistic -5.390573

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Second stage of the EG approach

• Variables in the short run may adjust in order to
  return to their long run paths
• Cointegration implies short run dynamics in
  variables
• Links between cointegration and error correction
  models
• When C(t-1)ltY(t-1) C(t) will increase
• C(t) also changes when Y(t) changes according the
  long-run relationship

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Error Correction Model

• When actual consumption exceeds target last
  period, i.e., errors show up, current consumption
  will decrease
• Consumers also response to the stimulus of a
  change in the variables determining equilibrium

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Estimation method

• Stage I
• OLS estimation of f1 and f2
• Use the OLS residuals to test for cointegration
• Stage II
• if reject the null of non-cointegration,
  replacing the xt-1 by OLS estimates of it and
  estimate

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Problems of standard inference

• Stock (1987)
• 1. first stage estimators of the long-run coeffs
  are superconsistent but not normally distributed
  even asymptotically. Inference on the coeffs
  using standard tables is invalid.
• 2. second stage estimator of the remaining I(0)
  coeff is consistent and asymptotically normally
  distributed.

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Review We have learned

• EGs two-step method of testing cointegration
• Step 1 test for the order of integration of the
  variables involved in the postulated LR
  relationship
• Step 2a known CI vector ctyt
• DF cointegration test

Critical values are the same as standard DF and
ADF unit root tests
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EGs two-step method of testing cointegration

• Step 2b unknown CI vector

Regressing an I(0) variable on an I(1) variable
does not have the standard t distribution if the
null is true. It depends on the number of coeffs
estimated.
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More variables in LR relationship

• mgt2
• See tables in handout for critical values.
• Table 2 m2, n30, 5

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Review spurious regression

• Static regression
• Spurious regression disappears current and
  lagged UKs disposable income are insignificant
  in explaining USs current consumption
• Dynamic multivariate time-series models are the
  right foundation for macroeconometrics, in case
  of solving related non-stationary problems.

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Dynamic models no spurious results?

• Simple version of the dynamic specification of
  consumption
• b ? a2

• Error correction mechanism
• if ggt0 or a1 lt1 ?
• ct ?,? iff ct-1 lt,gt ct-1byt-1

LR xy
SR xy
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ECM ? CI

• With an ECM representation, consumption in the LR
  converges to its equilibrium value c

ct-ct I(0) or ct-byt I(0) when ct , yt I(1)
Thus, ct , yt CI(1,1)
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Granger representation- link between
cointegration and ECM

• Engle and Granger (1987)
• If ct, yt I(1), CI(1,1), then there always
  exists an EC representation

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Granger representation

• Is GR balance?
• Are time-series properties (I(0) or I(1)) of LHS
  and RHS consistent ?
• ct, yt I(1) ? ?ct, ?yt I(0)
• If ct, yt CI(1,1) ? I(0)
• If ct and yt are I(1) and cointegrated then
  knowledge of one variable helps forecast of the
  other at least in one direction
• The GR does not exist if ct and yt are not
  cointegrated.

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3. An alternative test of cointegration

• If ct and yt I(1) and cointegrated then
  at least one of them is nonzero.
• Test of cointegration can be based on the
  significance of these coeffs
• H0 q2c0
• Ha q2clt0
• t stat non-standard distribution

Rejecting null is in favor of cointegration
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KED (1992) known cointegration coeffs

• Monte Carlo simulation
• using tecm is more powerful than using DF-t with
  the residuals from the 1st stage EG regression of
  Yt on Xt. ( if q1 1, tecm t, Why? Check this in
  Patterson, pp343-5)
• Its possible to improve upon the DF test by not
  imposing the restriction that the SR and LR
  elasticities are both equal to one

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Monte Carlo simulation of KED

• T25, q2 -0.05
• X, Y cointegrated but slow adjustment
• Using 5 DF critical value, -1.95
• the power of t 9
• the power of tecm 10 for q0 92
  for q8
• (qthe var of (?1-1)DX/var(e))
• Its possible to improve upon the DF test by not
  imposing the restriction that the SR and LR
  elasticities are both equal to one
• In practice, the statistical model of ECM may not
  be so simple, misspecification may lead to mixed
  results of cointegration

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KED (1992) unknown cointegration coeffs

• The modified tecm test

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KED (1992) unknown cointegration coeffs

• 2. The modified DF test allow for the
  possibility that the restrictions are incorrect
• Case 1 if y1

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Modified DF test i. j1
xt Yt-jXt

• DF t test stat.
• H0 q20,
• Non-rejection of null, i.e., non-cointegration
• But ut(q1-1)?Xtet serial correlation
• DF- t is invalid if q1?1
• We should regress ()

Including DXt as a regressor can correct for the
possible invalid restriction of q11
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Modified DF test ii.  j unknown

• xt Yt-jXt
• the modified DF regression
• ?xtgxt-1 ut
• ?(Yt-jXt) g (Yt-1-jXt-1) ut
• () ?Yt j ?Xtg (Yt-1-jXt-1)ut
• SR elast LR elast
• The restriction implied by this regression is
  that coeff on DXt is the same as the LR coeff j.
• We also can say that () is a restricted ECM

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Unrestricted ECM

• ?Ytj?Xtq2(Yt-1-jXt-1)et(q1-j)?Xt

• ?(Yt-jXt) q2(Yt-1-jXt-1)et(q1-j)?Xt
• ?xtq2xt-1 ut
• where ut et(q1-j)DXt
• if the restriction is valid, q1j, the last term
  vanishes

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Unrestricted ECM

• To correct the possible misspecification of the
  standard DF regression. add ?Xt to the RHS of ()
• i.e. Dxt dDXtgxt-1et
• Where dq1-j , gq2
• In practice,
• H0 g 0 modified DF test (Table 8.11)
• t-statistic has a large negative value
• rejection of null, xt I(0),
  cointegration

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References for distribution tables

• Banerjee, Dolado, Gralbraith, and Hendry,
  Cointegration, Error-correction and the
  Econometrics, 1993, Oxford Univ Press.
• Kremers, Ericsson and Dolado, 1992, The power of
  cointegration tests, Oxford Bulletin of
  Economics and Statistics, Vol. 54, pp. 325-48.