Spurious regression and cointegration

upani · Unread post by **upani** » Thu Jul 09, 2015 2:57 pm

Hi Estima,

I have two non stationary series but they are cointegrated. I want to know is there any spurious regression is there between the series.I have used the following commands from RATS programming manual.

Code: Select all

linreg lfp / resids
# constant lsp
corr(num=8,results=cors,partial=partial,picture="##.###",qstats) resids
graph(nodates,number=0,style=bar,key=below,footer="ACF and PACF") 2
# cors
# partial
*
* Do E-G test with fixed lags
*
diff resids / dresids
linreg dresids
# resids{1} dresids{1 to 8}
*
* Do E-G test with different lag lengths
*
compute egstart=%regstart()
do i = 0,8
linreg(noprint) dresids egstart *
# resids{1} dresids{1 to i}
com aic = -2.0*%logl + %nreg*2
com sbc = -2.0*%logl + %nreg*log(%nobs)
dis "Lags: " i "T-stat" %tstats(1) "The aic = " aic " and sbc = " sbc
end do i
linreg dresids
# resids{1} dresids{1 to 6}
@regcrits
@regcorrs(number=24,qstats,report)
*
@egtest(lags=8,method=aic)
#lfp lsp

I got the following output and i am struggling to interpret the results.

Code: Select all

Linear Regression - Estimation by Least Squares
Dependent Variable LFP
Usable Observations                      2037
Degrees of Freedom                       2035
Centered R^2                        0.9860833
R-Bar^2                             0.9860765
Uncentered R^2                      0.9999863
Mean of Dependent Variable       4.6606554921
Std Error of Dependent Variable  0.1465366463
Standard Error of Estimate       0.0172910238
Sum of Squared Residuals         0.6084232898
Regression F(1,2035)              144192.4320
Significance Level of F             0.0000000
Log Likelihood                      5375.8882
Durbin-Watson Statistic                0.9217

    Variable                        Coeff      Std Error      T-Stat      Signif
************************************************************************************
1.  Constant                     0.0007201326 0.0122777897      0.05865  0.95323403
2.  LSP                          1.0026875783 0.0026405500    379.72679  0.00000000


Correlations of Series RESIDS

Autocorrelations
  1        2          3        4      5      6      7      8
 0.539      0.464      0.419  0.377  0.364  0.317  0.295  0.295

Partial Autocorrelations
  1        2          3        4      5      6      7      8
 0.539      0.244      0.148  0.086  0.091  0.021  0.030  0.056

Ljung-Box Q-Statistics
  Lags  Statistic Signif Lvl
     8   2512.031   0.000000


Linear Regression - Estimation by Least Squares
Dependent Variable DRESIDS
Usable Observations                      2028
Degrees of Freedom                       2019
Centered R^2                        0.3161544
R-Bar^2                             0.3134448
Uncentered R^2                      0.3161551
Mean of Dependent Variable       0.0000162406
Std Error of Dependent Variable  0.0166013162
Standard Error of Estimate       0.0137556225
Sum of Squared Residuals         0.3820294261
Log Likelihood                      5819.5345
Durbin-Watson Statistic                2.0124

    Variable                        Coeff      Std Error      T-Stat      Signif
************************************************************************************
1.  RESIDS{1}                    -0.197362821  0.025440640     -7.75778  0.00000000
2.  DRESIDS{1}                   -0.463708441  0.030332225    -15.28765  0.00000000
3.  DRESIDS{2}                   -0.312951271  0.031215554    -10.02549  0.00000000
4.  DRESIDS{3}                   -0.221299671  0.031172760     -7.09914  0.00000000
5.  DRESIDS{4}                   -0.184567409  0.030778411     -5.99665  0.00000000
6.  DRESIDS{5}                   -0.113856669  0.029871796     -3.81151  0.00014226
7.  DRESIDS{6}                   -0.122255608  0.028529749     -4.28520  0.00001912
8.  DRESIDS{7}                   -0.127090596  0.026338473     -4.82528  0.00000150
9.  DRESIDS{8}                   -0.105577160  0.022062884     -4.78528  0.00000183

Lags:  0 T-stat     -24.73586 The aic =   -11401.34869  and sbc =   -11395.73389
Lags:  1 T-stat     -16.93379 The aic =   -11523.85599  and sbc =   -11512.62638
Lags:  2 T-stat     -13.67489 The aic =   -11567.01703  and sbc =   -11550.17261
Lags:  3 T-stat     -12.00269 The aic =   -11580.52348  and sbc =   -11558.06426
Lags:  4 T-stat     -10.57125 The aic =   -11596.35109  and sbc =   -11568.27706
Lags:  5 T-stat     -10.07312 The aic =   -11595.28429  and sbc =   -11561.59546
Lags:  6 T-stat      -9.52526 The aic =   -11595.31075  and sbc =   -11556.00711
Lags:  7 T-stat      -8.79200 The aic =   -11600.19753  and sbc =   -11555.27909
Lags:  8 T-stat      -7.75778 The aic =   -11621.06909  and sbc =   -11570.53584

Linear Regression - Estimation by Least Squares
Dependent Variable DRESIDS
Usable Observations                      2030
Degrees of Freedom                       2023
Centered R^2                        0.3044601
R-Bar^2                             0.3023972
Uncentered R^2                      0.3044601
Mean of Dependent Variable       0.0000039702
Std Error of Dependent Variable  0.0166042761
Standard Error of Estimate       0.0138683264
Sum of Squared Residuals         0.3890845568
Log Likelihood                      5807.7007
Durbin-Watson Statistic                2.0040

    Variable                        Coeff      Std Error      T-Stat      Signif
************************************************************************************
1.  RESIDS{1}                    -0.233421370  0.024628068     -9.47786  0.00000000
2.  DRESIDS{1}                   -0.418570535  0.029361642    -14.25569  0.00000000
3.  DRESIDS{2}                   -0.264492340  0.030178347     -8.76431  0.00000000
4.  DRESIDS{3}                   -0.167779800  0.029710846     -5.64709  0.00000002
5.  DRESIDS{4}                   -0.120938772  0.028590449     -4.23004  0.00002441
6.  DRESIDS{5}                   -0.041701889  0.026494492     -1.57398  0.11564767
7.  DRESIDS{6}                   -0.030381884  0.022195837     -1.36881  0.17121064


Information Criteria
AIC          -5.715
SBC          -5.696
Hannan-Quinn -5.708
(log) FPE    -5.715

Lag  Corr  Partial   LB Q    Q Signif
  1 -0.002  -0.002  0.009645    0.9218
  2 -0.006  -0.006  0.094194    0.9540
  3 -0.011  -0.011  0.328140    0.9547
  4 -0.016  -0.016  0.871864    0.9286
  5 -0.022  -0.022  1.843944    0.8703
  6 -0.035  -0.035  4.286888    0.6379
  7 -0.059  -0.060 11.315774    0.1254
  8 -0.019  -0.021 12.069212    0.1481
  9  0.064   0.061 20.339604    0.0159
 10  0.049   0.047 25.236209    0.0049
 11  0.002  -0.000 25.246271    0.0084
 12  0.031   0.029 27.263532    0.0071
 13  0.032   0.031 29.414685    0.0057
 14  0.014   0.014 29.790178    0.0082
 15  0.027   0.032 31.260000    0.0081
 16  0.018   0.031 31.895087    0.0103
 17 -0.053  -0.042 37.710896    0.0027
 18  0.016   0.018 38.241475    0.0036
 19  0.015   0.016 38.684984    0.0049
 20  0.009   0.014 38.868113    0.0069
 21  0.036   0.038 41.545383    0.0048
 22  0.068   0.068 51.059406    0.0004
 23  0.040   0.040 54.286408    0.0002
 24  0.020   0.015 55.114828    0.0003


Linear Regression - Estimation by Least Squares
Dependent Variable LFP
Usable Observations                      2037
Degrees of Freedom                       2035
Centered R^2                        0.9860833
R-Bar^2                             0.9860765
Uncentered R^2                      0.9999863
Mean of Dependent Variable       4.6606554921
Std Error of Dependent Variable  0.1465366463
Standard Error of Estimate       0.0172910238
Sum of Squared Residuals         0.6084232898
Regression F(1,2035)              144192.4320
Significance Level of F             0.0000000
Log Likelihood                      5375.8882
Durbin-Watson Statistic                0.9217

    Variable                        Coeff      Std Error      T-Stat      Signif
************************************************************************************
1.  LSP                          1.0026875783 0.0026405500    379.72679  0.00000000
2.  Constant                     0.0007201326 0.0122777897      0.05865  0.95323403


Engle-Granger Cointegration Test
Null is no cointegration (residual has unit root)
Regression Run From 10 to 2037
Observations         2029
With 8 lags chosen from 8 by AIC
Constant in cointegrating vector
Critical Values from MacKinnon for 2 Variables

Test Statistic -7.75778**
1%(**)         -3.90530
5%(*)          -3.34064
10%            -3.04821

I want to know two issues.

1. R square values is high compared to DW value-What does it mean?
2. Autocorrelation problem is high for the residuals at level (lag 1)-what is the conclusion?

please guide me how to interpret the results?

With sincere regards,
Upananda

TomDoan · Unread post by **TomDoan** » Thu Jul 09, 2015 3:41 pm

If the series are cointegrated (and your EG test rather clearly indicates that they are), then the Y1 on Y2 regression isn't spurious. It's just that you can't tell the difference between a spurious regression (where the beta doesn't converge to zero even with no relationship between the series) and a cointegrating regression (where the beta is "superconsistent") without digging deeper into the dynamics. In both cases, the static regression is dynamically misspecified (it doesn't try to deal with the serially correlated residuals). In the spurious regression situation, the residuals maintain a unit root, so they will have a really low DW (.2, .1 and even lower are common). However, in your case, the residuals from the static regression have a fair bit of correlation, hence a fairly low DW---a bit below 1.

Everything else goes to having quite a bit of short-run noise in the process. It looks like you may need to boost the maximum number of lags in the EGTEST---with 2000+ data points, 8 isn't very much.

upani · Unread post by **upani** » Fri Jul 10, 2015 10:03 pm

Dear Sir,

I am grateful to you for clearing doubt. As you have mentioned in case of cointegrating regression beta should be super consistent. Is this means beta value will be close to 1. Ideally what should be beta value? whether i have to proceed for ECM for further drawing inference regarding the regarding the variables?

With regards,
Upananda

TomDoan · Unread post by **TomDoan** » Sat Jul 11, 2015 10:27 am

upani wrote:Dear Sir,

I am grateful to you for clearing doubt. As you have mentioned in case of cointegrating regression beta should be super consistent. Is this means beta value will be close to 1. Ideally what should be beta value? whether i have to proceed for ECM for further drawing inference regarding the regarding the variables?

With regards,
Upananda

Cointegration is covered in virtually all graduate level textbooks in econometrics now. If you're unfamiliar with it, you probably would benefit from reviewing that.

No. Superconsistent means that the estimate converges to the "truth" faster than would be typical from a stationary regressor (so the standard error in your static regression isn't asympotically valid). In some models, the true value is 1; in others it isn't---whether the theoretical value is 1 in your case will depend upon the combination of variables. Have you reviewed the literature on the variable-pair that you're studying?

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