UNIT ROOTS

1
LECTURE-7

• UNIT ROOTS
• COINTEGRATION
• ENGLE-GRANGER METHOD
• 1st Assignment is due on 5th October
• (less than 10 pages)

2
So far we used GETS

• ECM formulation
• General ARDL(p,q) formulation
• Search for final parsimonious equation
• does not mean LSE-Hendry's method is better than
  alternatives
• GETS is simple, gives good results helps to
  understand alternative methods

3
In terms of our car driving analogy

• With GETS we were on a highway
• alternative techniques like driving on a
  country road
• ? pay utmost attention to how you drive from now
  on
• some knowledge of car mechanics maintenance
  (econometric theory) is useful

4
WE NEED UNIT ROOT TESTS FOR

• Engle-Granger (EG)
• Phillips-Hansens FMOLS
• Johansens VECM
• all need pre-testing
• can be applied only if variables are unit root
  variables

5
UNIT ROOT TESTS

• If Y X have unit roots, they are
  non-stationary (NS) variables
• OLS (NLLS etc) regressions with NS variables
  give spurious results
• no unit roots in Y X means they are stationary
• standard methods (OLS, NLLS etc) are valid

6
Early symptoms to notice for Unit Roots
Spurious Results

• High R-bar squares (e.g.0.9) low DW (e.g.0.3)
• In general DW lt R-bar square
• High t-ratios
• Example US consumption

7

• Ordinary Least Squares Estimation
  
• 
  
• Dependent variable is CON
  
• 166 observations used for estimation from 1954Q1
  to 1995Q2
• 
  
• Regressor Coefficient Standard Error
  T-RatioProb
• C -21.6063
  3.7600 -5.7464.000
• YD .92689
  .0016478 562.5010.000
• 
  (very high t-ratio)
• 
  
• R-Squared .99948 R-Bar-Squared (very
  high) .99948
• S.E. of Regression 31.6845 F-stat.
  F( 1, 164) 316407.3.000
• Mean of Dependent Variable 1578.3 S.D. of
  Dependent Variable 1387.8
• Residual Sum of Squares 164641.3 Equation
  Log-likelihood -808.2053
• Akaike Info. Criterion -810.2053 Schwarz
  Bayesian Criterion -813.3173
• DW-statistic .35034 (Low DW
  lt R bar sq)
• 
  

All the symptoms indicate CON YD have unit roots
8
Alternative Unit Root Tests

• About half-a-dozen unit root tests
• standard software can be used
• popular tests are
• Dicky-Fuller (DF) Augmented D-F (ADF) tests.
  ADF is frequently used
• The Phillips-Perron (PP) test

9
Alternative Unit Root Tests

• Essentially, Y is a unit root (NS) variable if in
  the following regression (gamma) ? 1

Subtract lagged Y from both sides
10
Needs a different critical value

• This equation can be estimated with OLS, but the
  test statistic based on t-distribution is not
  valid. We need the DF test statistic

11
Furthermore. Since we are driving on a country
road

• If there is serial correlation in the residuals,
  we need correction
• previous DF test statistic is not valid
• The new (corrected) test equation is known as the
  augmented Dicky-Fuller (ADF) equation
• And the test is known as ADF test
• Sounds complicated, but easy

12
2 Common ADF Equations

• Note there are (augmented) additional lagged
  differences to get rid-off serial correlation
• In (A) no trend. (often used for testing if ?Y is
  NS)
• T is often included because variables like Y, M,
  P etc are strongly trended
• ADF test critical values for (A no trend) (B
  trend) are different

13
In practice it is easy

• All that you do is just insert the ADF command in
  Mfit ADF Y(4)
• (4) means include 4 lagged values of ?Y
• Microfit gives the results for both (A) (B)
• Here is a sample for US consumption

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Without Trend

• Unit root tests for variable CON
  
• The Dickey-Fuller regressions include an
  intercept but not a trend
• 
  
• 157 observations used in the estimation of all
  ADF regressions.
• Sample period from 1956Q2 to 1995Q2
  
• 
  
• Test Statistic LL
  AIC SBC HQC
• DF 18.4252 -627.2557 -629.2557
  -632.3119 -630.4969
• ADF(1) 8.3508 -624.2492 -627.2492
  -631.8335 -629.1110
• ADF(2) 5.1213 -619.1382 -623.1382
  -629.2507 -625.6207
• ADF(3) 3.4017 -613.0077 -618.0077
  -625.6483 -621.1108
• ADF(4) 3.4603 -612.7288 -618.7288
  -627.8976 -622.4526
• ADF(5) 3.0321 -612.4042 -619.4042
  -630.1011 -623.7486
• ADF(6) 2.2389 -609.3998 -617.3998
  -629.6248 -622.3648
• ADF(7) 1.8751 -608.5430 -617.5430
  -631.2961 -623.1286
• ADF(8) 1.4443 -606.8570 -616.8570
  -632.1383 -623.0633
• 
  
• 95 critical value for the augmented
  Dickey-Fuller statistic -2.8799
• LL Maximized log-likelihood AIC Akaike
  Information Criterion

15
With Trend

• Unit root tests for variable CON
  
• The Dickey-Fuller regressions include an
  intercept and a linear trend
• 
  
• 157 observations used in the estimation of all
  ADF regressions.
• Sample period from 1956Q2 to 1995Q2
  
• 
  
• Test Statistic LL
  AIC SBC HQC
• DF 1.0626 -610.1342 -613.1342
  -617.7186 -614.9961
• ADF(1) 1.0394 -610.1275 -614.1275
  -620.2400 -616.6100
• ADF(2) .82217 -608.9939 -613.9939
  -621.6345 -617.0970
• ADF(3) .53158 -606.1306 -612.1306
  -621.2993 -615.8543
• ADF(4) .71663 -604.6974 -611.6974
  -622.3943 -616.0418
• ADF(5) .72953 -604.6839 -612.6839
  -624.9089 -617.6489
• ADF(6) .48482 -603.2903 -612.2903
  -626.0434 -617.8759
• ADF(7) .38798 -603.0423 -613.0423
  -628.3235 -619.2486
• ADF(8) .20144 -602.1125 -613.1125
  -629.9218 -619.9393
• 
  
• 95 critical value for the augmented
  Dickey-Fuller statistic -3.4391
• LL Maximized log-likelihood AIC Akaike
  Information Criterion

Look for the highest values of AIC SBC
H0 variable is non-stationary
Critical value
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conclusion

• CON (consumption) is NS variable
• how do we decide if it is a unit root variable,
  i.e. it is I(1)?
• If CON becomes stationary, after differencing
  once, then it is a I(1) or unit root variable in
  its levels
• If differencing is required twice, then CON is
  I(2)
• If differencing is required 3 times, then CON is
  I(3) and so on

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Some special changes for this example

• Here is the ADF test result for ?CON
• Note there is a mild trend in ?CON
• ? the table with TREND should be used
• Also, reduced sample size to 1972Q4, to avoid the
  effects of oil shock see lecture notes

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Trend in ?CON
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ADF test result for ?CON

• Unit root tests for variable DCON
  
• The Dickey-Fuller regressions include an
  intercept and a linear trend
• 
  
• 70 observations used in the estimation of all
  ADF regressions.
• Sample period from 1955Q3 to 1972Q4
  
• 
  
• Test Statistic LL
  AIC SBC HQC
• DF -5.9606 -176.1326
  -179.1326 -182.5054 -180.4723
• ADF(1) -3.3632 -173.5618 -177.5618
  -182.0588 -179.3480
• ADF(2) -2.6267 -173.2065 -178.2065
  -183.8277 -180.4393
• ADF(3) -2.6374 -172.9911 -178.9911
  -185.7366 -181.6705
• ADF(4) -2.7944 -172.4714 -179.4714
  -187.3411 -182.5973
• 
  
• 95 critical value for the augmented
  Dickey-Fuller statistic -3.4739
• LL Maximized log-likelihood AIC Akaike
  Information Criterion
• SBC Schwarz Bayesian Criterion HQC
  Hannan-Quinn Criterion

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Some confusion

• test result is ambiguous (use absolute vales)
• 95 level (computed) ADF of 3.3632 lt CV 3.4739 by
  a small amount
• Looks that computed value will exceed CV at 90
• ?, as a check, we conduct another test, usually
  the PP test (ask Rup in lab)

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Some confusion

• Critical Values (CVs) for PP test are same as in
  ADF (-3.4739)
• Computed PP stat is -5.1733
• We use absolute values reject unit root null
  for ?CON
• ? ?CON is stationary I(0) CON is I(1)

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Try these tests in the lab

• Test if YD is I(1) ?YD is I(0)
• Use both ADF PP
• Note Sample period ends in 1972Q4
• As I said earlier, we are now on a country road!
• I will assume YD is I(1) in levels I(0) in
  first differences

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• 1st Assignment is due on September 5 Monday
• Break for 10 minutes

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COINTEGRATIONEngle-Granger procedure (EG)

• In all alternatives (to GETS), there are 4 steps
• First, get the ECM through cointegration
  technique (e.g. with EG)
• 1.A. Test if this is actually a CR (only for EG)
• identify whether this CR is for consumption
  and/or income
• specify the ARDL
• search for parsimonious ARDL

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COINTEGRATIONEngle-Granger procedure (EG)

• First, estimate with OLS the ECM (simple)
• (Other estimation methods are used in FMOLS
  Johansen VECM etc)
• a test is performed in only in EG (not in
  others) if in fact that ECM (a combination of
  level I(1) variables) now become I(0)
• This test is CRADF test

26
COINTEGRATIONEngle-Granger

• Let us then estimate the first stage EG OLS
  equation for US consumption
• Instead of CON YD, I will use their logs
• Note Sample period ends at 1972Q4
• (You will get strange results if you use the
  entire sample with post Oil shock data)

27
EG Equation known asEG cointegrating equation

• Ordinary Least Squares
  Estimation
• 
  
• Dependent variable is LC
  
• 76 observations used for estimation from 1954Q1
  to 1972Q4
• 
  
• Regressor Coefficient
  Standard Error T-RatioProb
• C .25753
  .12932 1.9915.050
• T .6565E-3
  .3794E-3 1.7302.088
• LY .93814
  .023562 39.8167.000
• 
  
• R-Squared .99948
  R-Bar-Squared .99947
• S.E. of Regression .0080350 F-stat.
  F( 2, 73) 70348.2.000
• Mean of Dependent Variable 6.0092 S.D. of
  Dependent Variable .34810
• Residual Sum of Squares .0047129 Equation
  Log-likelihood 260.3115
• Akaike Info. Criterion 257.3115 Schwarz
  Bayesian Criterion 253.8154
• DW-statistic .65106
  
• 
  

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How do we test?CRADF test

• LC .25753 .6565E-3 T .93814 LY
• Note the implied ECM below
• LC (.25753 .6565E-3 T .93814 LY)
• We perform an ADF type test on the residuals
• i.e. we are testing if this ECM is I(0)
• We dont include intercept trend for this test

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CRADF Test

• In practice this test is easy
• We simply ask Mfit to do this
• Let us see this
• GO TO POST REGRESSION MENU
• SELECT HYPOTHESES TESTS
• SELECT UNIT ROOT TEST FOR RESIDUALS

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CRADF Test Results

• Unit root tests for residuals
  
• 
  
• Based on OLS regression of LC on
  
• C T LY
  
• 76 observations used for estimation from 1954Q1
  to 1972Q4
• 
  
• Test Statistic LL
  AIC SBC HQC
• DF -3.9012 265.8025 264.8025
  263.6712 264.3527
• ADF(1) -3.6168 265.8223 263.8223
  261.5597 262.9225
• ADF(2) -4.2910 268.2729 265.2729
  261.8789 263.9232
• ADF(3) -4.9711 270.9146 266.9146
  262.3892 265.1150
• ADF(4) -4.5661 271.1245 266.1245
  260.4678 263.8751
• 
  
• 95 critical value for the Dickey-Fuller
  statistic -3.9166
• LL Maximized log-likelihood AIC Akaike
  Information Criterion
• SBC Schwarz Bayesian Criterion HQC
  Hannan-Quinn Criterion

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Note AIC SBC differ in this particular case

• CV 95 -3.9166
• AIC ADF(3) -4.9711
• SBC DF -3.9012
• SBC is only marginally less in absolute value
  (SBC generally has more penalty)
• We may conclude that CREG is I(0)

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How do we know which are the dependent
independent variables in a CR?

• Difficult, controversial often misused
• In LSE GETS, economic theory is accepted (it is
  not really a solution, however)
• In all others, CRs tell which combination of
  levels of I(1) variables will be I(0)
• Nothing more..
• This is the famous identification problem about
  which Sims his VAR are critical (see lecture-1)

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Identification Granger causality tests

• Please read the lecture notes to understand what
  this is
• Here, I want to say that Granger Causality tests
  are not cause effect tests, although this test
  is misused by many
• It only tells, whether we should estimate just
  one ECMARDL equation or more

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Identification Granger Causality Test (GCT)

• US consumption2 variables, LC LY
• GCT helps to find if LY is or not affected by
  (i.e. depends on LCON) disequilibrium in LC
• Disequilibrium in LC ECMt-1
• So, is this ECMt-1 significant in the ARDL for
  LY?

35
GCT Application(simple but adequate for our
purpose)

• Test if ECMt-1 is significant or not in the
  following 2 equations. Use OLS
• ?LC -?1 LC t-1 (.25753 .6565E-3 T .93814
  LY t-1)
• ?LY -?2 LC t-1 (.25753 .6565E-3 T .93814
  LY t-1)
• Here are the results

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GCT

• Ordinary Least Squares Estimation
  
• 
  
• Dependent variable is DLC
  
• 71 observations used for estimation from 1955Q2
  to 1972Q4
• 
  
• Regressor Coefficient
  Standard Error T-RatioProb
• C .016214
  .8334E-3 19.4550.000
• RESEG72(-1) -.19615 .10419
  -1.8827.064
• 
  
• 
  Just
  significant
• Ordinary Least Squares
  Estimation
• 
  
• Dependent variable is DLY
  
• 71 observations used for estimation from 1955Q2
  to 1972Q4
• 
  
• Regressor Coefficient
  Standard Error T-RatioProb
• C .016847
  .9009E-3 18.6999.000
• RESEG72(-1) .14754 .11262
  1.3101.195
• 
  

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GCT

• This does not mean LY causes LC LC does not
  cause LY (recall YCIG)
• It only means that we can estimate ARDL for ?LC,
  disregarding the ARDL for ?LY
• If lagged ECM were significant in both then we
  should estimate 2 equations

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• If lagged ECM were significant in both then we
  should estimate 2 equations
• ?LC -?1 LC t-1 (.25753 .6565E-3 T .93814
  LY t-1)
• lagged (?LC and ?LY )
• ?LY -?2 LC t-1 (.25753 .6565E-3 T .93814
  LY t-1)
• lagged (?LC and ?LY )
• This is necessary to get good forecasts of LC
  LY
• What causes what depends on economic theory not
  on GCT
• Otherwise weathermans forecasts Granger cause
  rain!

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So, what is the final Consumption equation based
on EG?

• Ordinary Least Squares Estimation Note
  ARDL(3,5)
• 
  
• Dependent variable is DLC
  
• 68 observations used for estimation from 1956Q1
  to 1972Q4
• 
  
• Regressor Coefficient
  Standard Error T-RatioProb
• C .0021636
  .0022556 .95920.341
• RESEG72(-1) -.40719 .078835
  -5.1651.000 (lagged ECM)
• DLC(-2) .39363
  .094813 4.1516.000
• DLC(-3) .33820
  .11051 3.0602.003
• DLY .59317
  .074919 7.9174.000
• (DLY should be dropped if LY is not exo)
• DLY(-3) -.28375
  .11745 -2.4160.019
• DLY(-5) -.19081
  .088380 -2.1589.035
• 
  
• R-Squared .63304
  R-Bar-Squared .59694
• S.E. of Regression .0046299 F-stat.
  F( 6, 61) 17.5383.000
• Mean of Dependent Variable .016209 S.D. of
  Dependent Variable .0072928
• Residual Sum of Squares .0013076 Equation
  Log-likelihood 272.7202

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Just out of curiosityWhat do we get with GETS?

• Non-Linear Least Squares Estimation
  
• The estimation method converged
  after 4 iterations
• 
  
• Non-linear regression formula
  ECM in GETS
• DLCA0 CLAMBDA( LC(-1)-B1 LY(-1))
• G1DLC(-2)G2DLC(-3)
  G3DLYG4DLY(-3)G5DLY(-5)
  
• 68 observations used for estimation from 1956Q1
  to 1972Q4
• 
  
• Parameter Estimate
  Standard Error T-RatioProb
• A0 -.014988
  .013579 -1.1038.274
• LAMBDA -.36900 .073305
  -5.0337.000
• B1 .99456
  .0075566 131.6152.000
• G1 .29573
  .099850 2.9617.004
• G2 .26035
  .11049 2.3563.022
• G3 .49559
  .083760 5.9168.000
• G4 -.28950
  .11473 -2.5233.014
• G5 -.26368
  .089092 -2.9596.004
• 
  
• R-Squared .65726
  R-Bar-Squared .61728

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Not much difference

• GETS gives a higher income elasticity (0.99) EG
  gives lower elasticity (0.93)
• GETS has a marginally smaller SEE higher R bar
  sq. Implies better forecasts
• The point is they give close results

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• END OF LECTURE-7