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  • Originally posted by Jonathan Hillgren View Post
    Sebastian Kripfganz thanks for all your quick and kind replys and I am really grateful for your responses. Would it in theory be possible for a variable to show a negative long term coefficient but in the short run be positive? best regards Jonathan Hillgren
    Yes, this can happen but only if that variable enters the ARDL specification with some lags (not just contemporaneously).
    https://twitter.com/Kripfganz

    Comment



    • Hi again,

      Thanks a lot for your previous replies! We have some more questions about the interpretation of the model from someone who is more experienced with the ardl process. If our independent variable Vol (volatility) is based on/estimated from ln-values, we have decided not to use as an ln-variable so it is not written in the ln-form in our regression, how should we interpret its effect on the regressand if that one is in ln-form, when doing the ardl ec1 regression? The same goes for the dummy variable, which is not in ln-form either.

      We are also wondering how to interpret the coefficient of the constant? How does this affect our ln-dependent variable? If it’s significant, should it be included in the model and if not significant, removed? And does the ADJ-term affect all of the independent variables? Best regards Jonathan & Emma

      Comment


      • The coefficient of a variable without a log transformation in an ARDL regression with a log-transformed dependent variable is interpreted as in standard linear regression models as a semi-elasticity: how much does the dependent variable change in percent given a one unit change of the independent variable.

        Please also see: How to interpret coefficients when with ln transformed dependent variable?

        The constant term is usually not interpretable. It captures the means (or the drift in non-stationary models) of all the variables in the model, both from the left- and right-hand side of the regression equation. There is rarely a situation where I would recommend to not include the constant in the model unless you are sure that all the variables are stationary with mean zero (because you may have demeaned the variables first). There is not much that you can lose by including the constant, but your results can easily become rubbish if you exclude it because you would be forcing the regression line to pass through the origin.
        Last edited by Sebastian Kripfganz; 02 May 2017, 08:26.
        https://twitter.com/Kripfganz

        Comment


        • Hello,

          Can one in some way measure the first-order autocorrelation coefficient for the ARDL-model?

          Thanks for the help!

          Comment


          • Assuming that you are interested in the residual autocorrelation, you could do something like the following:
            Code:
            webuse lutkepohl2
            ardl ln_inv ln_inc ln_consump
            predict res, resid
            corrgram res
            Please see help corrgram for details.
            https://twitter.com/Kripfganz

            Comment


            • Hi Sebastian Kripfganz and thanks alot for all your help. We have one final question, How does it work if we have autocorrelation in the residuals, we regressed the model given to us by ARDL, AIC that determined our lags and then did the bgodfrey test with one more lag "estat bgodfrey, lags(5) and encountered autocorrelation. Should we use Newey West standard errors? or is the modell based on AIC giving correct estimates?

              Comment


              • It depends on the purpose of your analysis.

                If you want to test for the existence of a long-run relationship with the bounds test (estat btest), it is crucial to avoid autocorrelation. Newey-West standard errors are not of help for this test. To be on the safe side, you could overwrite the "optimal" lag order by choosing higher lag orders with the lags() option. You could also increase the maximum lag order with maxlags() and see if the AIC picks a higher lag order.

                On the other side, for prediction purposes you would usually prefer a more parsimonious model as this gives you a smaller root mean squared error (at the potential cost of a slightly larger bias).
                https://twitter.com/Kripfganz

                Comment


                • Dear Sebastian Kripfganz ,

                  I am testing for the presence of a long-run relationship between a set of macroeconomic variables, after reading the slides of Stata meeting held in Chicago.

                  I used the following command:

                  Code:
                  ardl mgsv rmp nc itv xgsv LVix LGlobalEPUPPP if ifscode==946, maxlags(7) aic maxcombs(2000000) dots fast
                  matrix list e(lags)
                  which provides the a set of lags I employ in the following specification:
                  Code:
                  ardl mgsv rmp nc itv xgsv LVix LGlobalEPUPPP if ifscode==946, ec1 lags(6 7 6 0 7 7 7) regstore(ardl5)
                  The result looks like the following:
                  Code:
                  . ardl mgsv rmp nc itv xgsv LVix LGlobalEPUPPP if ifscode==946, ec1 lags(6 7 6 0 7 7 7) regstore(ardl5)
                  
                  ARDL regression
                  Model: ec
                  
                  Sample: 1998q4 - 2015q4 
                  Number of obs  = 69
                  Log likelihood = 207.7786
                  R-squared      = .95588501
                  Adj R-squared  = .86364457
                  Root MSE       = .02109561
                  
                  -------------------------------------------------------------------------------
                         D.mgsv |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
                  --------------+----------------------------------------------------------------
                  ADJ           |
                           mgsv |
                            L1. |   .0011824   .3684211     0.00   0.997    -.7628762     .765241
                  --------------+----------------------------------------------------------------
                  LR            |
                            rmp |
                            L1. |  -117.3742   36536.76    -0.00   0.997    -75889.97    75655.22
                                |
                             nc |
                            L1. |    452.768   140809.5     0.00   0.997    -291568.3    292473.8
                                |
                            itv |
                            L1. |  -309.3532   96365.31    -0.00   0.997    -200158.8    199540.1
                                |
                           xgsv |
                            L1. |  -92.96919    29150.9    -0.00   0.997    -60548.23    60362.29
                                |
                           LVix |
                            L1. |  -111.0917   34574.54    -0.00   0.997     -71814.3    71592.11
                                |
                  LGlobalEPUPPP |
                            L1. |   196.7787   61272.01     0.00   0.997    -126873.6    127267.1
                  --------------+----------------------------------------------------------------
                  SR            |
                           mgsv |
                            LD. |  -.7698193   .4133583    -1.86   0.076    -1.627072    .0874333
                           L2D. |  -.6011546   .3336204    -1.80   0.085    -1.293041    .0907316
                           L3D. |  -.4666905   .2788486    -1.67   0.108    -1.044987    .1116061
                           L4D. |  -.1707356   .2243865    -0.76   0.455    -.6360848    .2946135
                           L5D. |   .2738664   .1452609     1.89   0.073    -.0273864    .5751192
                                |
                            rmp |
                            D1. |  -.4977635   .1471148    -3.38   0.003    -.8028609    -.192666
                            LD. |  -.3390906    .237754    -1.43   0.168    -.8321622     .153981
                           L2D. |  -.6463966   .2260095    -2.86   0.009    -1.115112   -.1776816
                           L3D. |  -.5194799   .2229064    -2.33   0.029    -.9817594   -.0572003
                           L4D. |  -.6262338   .2044137    -3.06   0.006    -1.050162   -.2023057
                           L5D. |  -.5097497   .1873629    -2.72   0.012    -.8983166   -.1211828
                           L6D. |  -.2131168   .1856316    -1.15   0.263    -.5980932    .1718595
                                |
                             nc |
                            D1. |  -.1878331   .3366906    -0.56   0.583    -.8860866    .5104204
                            LD. |   .2164175   .3163145     0.68   0.501    -.4395786    .8724135
                           L2D. |   .2843947   .3660218     0.78   0.445    -.4746881    1.043477
                           L3D. |   .6038347   .3562524     1.69   0.104    -.1349876    1.342657
                           L4D. |   .2999871    .338704     0.89   0.385    -.4024421    1.002416
                           L5D. |  -.5897248    .278342    -2.12   0.046    -1.166971   -.0124789
                                |
                            itv |
                            D1. |   .3657799   .1022349     3.58   0.002     .1537577    .5778022
                                |
                           xgsv |
                            D1. |   .8381486   .1627303     5.15   0.000     .5006665    1.175631
                            LD. |   .5862168   .4027664     1.46   0.160    -.2490695    1.421503
                           L2D. |   .8408149   .3208903     2.62   0.016     .1753292    1.506301
                           L3D. |   .4102869   .2401392     1.71   0.102    -.0877314    .9083051
                           L4D. |  -.1290902   .2492349    -0.52   0.610    -.6459717    .3877914
                           L5D. |    -.28598   .1775715    -1.61   0.122    -.6542407    .0822807
                           L6D. |   .3315378   .1827353     1.81   0.083     -.047432    .7105076
                                |
                           LVix |
                            D1. |  -.1122472    .043141    -2.60   0.016    -.2017162   -.0227782
                            LD. |  -.2618214   .0884512    -2.96   0.007    -.4452579   -.0783849
                           L2D. |   -.288866   .0914218    -3.16   0.005    -.4784632   -.0992689
                           L3D. |  -.2192862   .0805578    -2.72   0.012    -.3863529   -.0522195
                           L4D. |  -.1940336   .0660847    -2.94   0.008    -.3310849   -.0569823
                           L5D. |  -.1066013   .0429121    -2.48   0.021    -.1955955   -.0176071
                           L6D. |  -.0592069   .0313052    -1.89   0.072      -.12413    .0057161
                                |
                  LGlobalEPUPPP |
                            D1. |   .0342402    .035667     0.96   0.347    -.0397286    .1082089
                            LD. |   .2170323   .0686799     3.16   0.005      .074599    .3594655
                           L2D. |   .2440361    .067782     3.60   0.002     .1034649    .3846073
                           L3D. |   .1735577   .0548971     3.16   0.005      .059708    .2874074
                           L4D. |    .162349   .0461701     3.52   0.002     .0665981    .2580999
                           L5D. |   .0919748   .0319221     2.88   0.009     .0257723    .1581772
                           L6D. |    .057422   .0292392     1.96   0.062    -.0032164    .1180603
                                |
                          _cons |    .776483   .8081197     0.96   0.347    -.8994546    2.452421
                  -------------------------------------------------------------------------------
                  What puzzles me are the coefficients related to the long-run behaviour. They are extremely large and not significant.

                  I tried to run the same regression reducing randomly the number of lags, just to check if the number of lags could be an issue. Here is the example:

                  Code:
                  ardl mgsv rmp nc itv xgsv LVix LGlobalEPUPPP if ifscode==946, ec1 lags(3 2 2 1 2 2 2) regstore(ardl5)
                  The results is as follows:

                  Code:
                  . ardl mgsv rmp nc itv xgsv LVix LGlobalEPUPPP if ifscode==946, ec1 lags(3 2 2 1 2 2 2) regstore(ardl5)
                  
                  ARDL regression
                  Model: ec
                  
                  Sample: 1997q3 - 2015q4 
                  Number of obs  = 74
                  Log likelihood = 174.94337
                  R-squared      = .83380359
                  Adj R-squared  = .77108797
                  Root MSE       = .02688595
                  
                  -------------------------------------------------------------------------------
                         D.mgsv |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
                  --------------+----------------------------------------------------------------
                  ADJ           |
                           mgsv |
                            L1. |  -.6339296   .1788977    -3.54   0.001     -.992753   -.2751062
                  --------------+----------------------------------------------------------------
                  LR            |
                            rmp |
                            L1. |   .0134759   .0908202     0.15   0.883    -.1686863    .1956381
                                |
                             nc |
                            L1. |   .3847702   .2116265     1.82   0.075    -.0396989    .8092393
                                |
                            itv |
                            L1. |   .2217854   .0904437     2.45   0.018     .0403784    .4031925
                                |
                           xgsv |
                            L1. |   .7106944   .0744039     9.55   0.000     .5614592    .8599297
                                |
                           LVix |
                            L1. |   .0537966    .047911     1.12   0.267    -.0423007    .1498939
                                |
                  LGlobalEPUPPP |
                            L1. |  -.0659986   .0510523    -1.29   0.202    -.1683965    .0363994
                  --------------+----------------------------------------------------------------
                  SR            |
                           mgsv |
                            LD. |   .1521405   .1463569     1.04   0.303    -.1414143    .4456953
                           L2D. |   .0566878   .0886861     0.64   0.525    -.1211941    .2345696
                                |
                            rmp |
                            D1. |  -.1423269   .1090739    -1.30   0.198    -.3611013    .0764476
                            LD. |   .1216298   .1126371     1.08   0.285    -.1042916    .3475512
                                |
                             nc |
                            D1. |   .6104873   .2715063     2.25   0.029     .0659147     1.15506
                            LD. |  -.0225563   .2432914    -0.09   0.926    -.5105371    .4654245
                                |
                            itv |
                            D1. |    .306998    .070086     4.38   0.000     .1664233    .4475726
                                |
                           xgsv |
                            D1. |   .6816911   .1130562     6.03   0.000     .4549291    .9084532
                            LD. |  -.1829763   .1634703    -1.12   0.268    -.5108563    .1449037
                                |
                           LVix |
                            D1. |  -.0070046    .028669    -0.24   0.808    -.0645074    .0504982
                            LD. |  -.0211439   .0269239    -0.79   0.436    -.0751465    .0328586
                                |
                  LGlobalEPUPPP |
                            D1. |  -.0101336   .0305751    -0.33   0.742    -.0714595    .0511924
                            LD. |  -.0088528   .0298187    -0.30   0.768    -.0686616     .050956
                                |
                          _cons |  -.8602113   .3521431    -2.44   0.018    -1.566521   -.1539016
                  -------------------------------------------------------------------------------
                  In this case the problem is not present and the adjustment coeffcient correctly lies between 0 and -1.

                  I tried to run a similar regression, i.e. with the same explanatory variables, for other two countries and the issue does not appear.

                  I am wondering what could be the cause leading to the first result.

                  Many many thanks.

                  Marco

                  Comment


                  • The long-run coefficients are a function of the speed-of-adjustment coefficient. The latter appears in the denominator of the long-run coefficients. Because it is very small (almost zero), the reported long-run coefficients necessarily become very large. Note that a speed-of-adjustment coefficient of zero technically implies that there is no cointegrating relationship among the dependent variable and the independent variables, and that the dependent variable is non-stationary. Hence, long-run coefficients are meaningless.

                    That said, you are estimating 41 parameters with 69 observations. This cannot yield reliable results. A reduction of the number of lags is required, as you have done in your second example. Instead of randomly reducing the number of lags, I would suggest to use the maxlags() option of ardl.
                    https://twitter.com/Kripfganz

                    Comment


                    • Dear Sebastian Kripfganz,

                      I do thank you very much for you kind and prompt reply.

                      I am concerned with the number of maxlags be included. Is there a criterion that I can rely on to pick the optimum maxlags?

                      Many thanks again.

                      Marco

                      Comment


                      • Out of the top of my head, I am not aware of a suitable rule to determine the maximum lag order to be considered in an ARDL model.
                        https://twitter.com/Kripfganz

                        Comment


                        • Dear Sebastian Kripfganz,

                          Thank you very much for your reply.

                          I was also wondering if there might be a trade off between the maximum number of lags and the presence of serial correlation. The higher the number of lags the lower the risk of serial correlation, right?

                          Many thanks

                          Marco

                          Comment


                          • That's generally true, yes. On the other side, as you have experienced, including too many lags results in serious overfitting problems. That is why a reasonable lag order selection is both important and difficult at the same time.
                            https://twitter.com/Kripfganz

                            Comment


                            • Dear sebastian,

                              Thank you very much.

                              My apologies for the numerous questions and if this one is silly. What would be the steps to find a reasonable lag, or a set of reasonable lags?

                              Many thanks.

                              Marco

                              Comment


                              • As I said, I am not aware of any general rule about how to determine an initial maximum lag order. Say, you stick with the default of ardl, that is maxlags(4). You could subsequently run a standard time-series test for autocorrelation of the residuals; see help regress postestimation time series, for example:
                                Code:
                                . webuse lutkepohl2
                                
                                . ardl ln_inv ln_inc ln_consump, regstore(ardl_bic)
                                
                                ARDL regression
                                Model: level
                                
                                Sample: 1961q1 - 1982q4
                                Number of obs  = 88
                                Log likelihood = 158.83176
                                R-squared      = .9918295
                                Adj R-squared  = .9913313
                                Root MSE       = .04123219
                                
                                ------------------------------------------------------------------------------
                                      ln_inv |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
                                -------------+----------------------------------------------------------------
                                      ln_inv |
                                         L1. |   .8432219   .0588646    14.32   0.000     .7261214    .9603224
                                             |
                                      ln_inc |  -.4477328   .3143463    -1.42   0.158    -1.073068    .1776022
                                             |
                                  ln_consump |
                                         --. |     1.9247   .5487929     3.51   0.001     .8329761    3.016424
                                         L1. |  -.3682414   .5622263    -0.65   0.514    -1.486689    .7502058
                                         L2. |  -.9598887   .4300221    -2.23   0.028     -1.81534   -.1044377
                                             |
                                       _cons |  -.0460065   .0706528    -0.65   0.517    -.1865575    .0945445
                                ------------------------------------------------------------------------------
                                
                                . estimates restore ardl_bic
                                (results ardl_bic are active now)
                                
                                . estat durbinalt
                                
                                Durbin's alternative test for autocorrelation
                                ---------------------------------------------------------------------------
                                    lags(p)  |          chi2               df                 Prob > chi2
                                -------------+-------------------------------------------------------------
                                       1     |          3.730               1                   0.0534
                                ---------------------------------------------------------------------------
                                                        H0: no serial correlation
                                
                                . estat bgodfrey
                                
                                Breusch-Godfrey LM test for autocorrelation
                                ---------------------------------------------------------------------------
                                    lags(p)  |          chi2               df                 Prob > chi2
                                -------------+-------------------------------------------------------------
                                       1     |          3.874               1                   0.0490
                                ---------------------------------------------------------------------------
                                                        H0: no serial correlation
                                (Notice that I have not used estat dwatson because that would require strictly exogenous regressors but the lagged dependent variable in an ARDL model is not strictly exogenous by construction.)

                                Both the alternative Durbin test and the Breusch-Godfrey-LM test have a p-value close to 5% which leaves considerable doubt about potential serial correlation. By default, the lag selection above was done with the Schwarz-Bayesian information criterion (BIC) that tends to choose more parsimonious models than the Akaike information criterion (AIC). Let us redo the analysis with the AIC instead:
                                Code:
                                . ardl ln_inv ln_inc ln_consump, regstore(ardl_aic) aic
                                
                                ARDL regression
                                Model: level
                                
                                Sample: 1961q1 - 1982q4
                                Number of obs  = 88
                                Log likelihood = 163.42433
                                R-squared      = .99263931
                                Adj R-squared  = .99189393
                                Root MSE       = .03987169
                                
                                ------------------------------------------------------------------------------
                                      ln_inv |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
                                -------------+----------------------------------------------------------------
                                      ln_inv |
                                         L1. |   .6361522   .1066167     5.97   0.000     .4239369    .8483674
                                         L2. |   .0544957   .1292958     0.42   0.675    -.2028612    .3118526
                                         L3. |   .1947748   .1074963     1.81   0.074    -.0191911    .4087407
                                             |
                                      ln_inc |  -.7132999   .3172674    -2.25   0.027    -1.344805    -.081795
                                             |
                                  ln_consump |
                                         --. |   2.200332   .5534362     3.98   0.000     1.098745    3.301919
                                         L1. |  -.0080402   .5827281    -0.01   0.989    -1.167931    1.151851
                                         L2. |  -.4564319   .5622027    -0.81   0.419    -1.575468    .6626045
                                         L3. |  -.9016915   .4374505    -2.06   0.043    -1.772415    -.030968
                                             |
                                       _cons |  -.0867693   .0709145    -1.22   0.225    -.2279212    .0543825
                                ------------------------------------------------------------------------------
                                
                                . estimates restore ardl_aic
                                (results ardl_aic are active now)
                                
                                . estat durbinalt
                                
                                Durbin's alternative test for autocorrelation
                                ---------------------------------------------------------------------------
                                    lags(p)  |          chi2               df                 Prob > chi2
                                -------------+-------------------------------------------------------------
                                       1     |          0.021               1                   0.8840
                                ---------------------------------------------------------------------------
                                                        H0: no serial correlation
                                
                                . estat bgodfrey
                                
                                Breusch-Godfrey LM test for autocorrelation
                                ---------------------------------------------------------------------------
                                    lags(p)  |          chi2               df                 Prob > chi2
                                -------------+-------------------------------------------------------------
                                       1     |          0.024               1                   0.8769
                                ---------------------------------------------------------------------------
                                                        H0: no serial correlation
                                We notice that the optimal lag order based on the AIC is higher for the lagged dependent variable and ln_consump than based on the BIC. The serial-correlation tests then both clearly do not reject the null hypothesis of no serial correlation any more. Thus, this is probably a reasonable lag selection and we do not have to increase the maximum lag order any further. Another indication is that all variables' lag orders are smaller than the maximum lag order of 4.

                                If the latter was not the case and/or we would still find evidence of remaining serial correlation, we could increase the maximum lag order to, say, 5, and redo the analysis. This stepwise procedure has different problems (keyword: pretesting) but I would not worry too much about it, provided your initial maximum lag order is neither too small nor too large.

                                Another way to think about the initial maximum lag order is motivated from the frequency of your data. With quarterly data, a maximum lag order of 4 seems very reasonable to capture seasonal fluctuations. With monthly data, you might want to start with a maximum lag order of 12, provided your time series is long enough such that the number of estimated parameters is still reasonably small relative to the sample size.

                                Once you have found a reasonable lag order, you can then reestimate the ARDL model in the error-correction representation and subsequently run the bounds test:
                                Code:
                                . ardl ln_inv ln_inc ln_consump, lags(3 0 3) ec1
                                
                                ARDL regression
                                Model: ec
                                
                                Sample: 1960q4 - 1982q4
                                Number of obs  = 89
                                Log likelihood = 165.33303
                                R-squared      = .29433631
                                Adj R-squared  = .22376994
                                Root MSE       = .0398231
                                
                                ------------------------------------------------------------------------------
                                    D.ln_inv |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
                                -------------+----------------------------------------------------------------
                                ADJ          |
                                      ln_inv |
                                         L1. |  -.1104725   .0610698    -1.81   0.074    -.2320052    .0110601
                                -------------+----------------------------------------------------------------
                                LR           |
                                      ln_inc |
                                         L1. |  -6.032112   5.026647    -1.20   0.234    -16.03546    3.971234
                                             |
                                  ln_consump |
                                         L1. |   7.090765   5.183768     1.37   0.175    -3.225261    17.40679
                                -------------+----------------------------------------------------------------
                                SR           |
                                      ln_inv |
                                         LD. |  -.2491467   .1081415    -2.30   0.024    -.4643551   -.0339383
                                        L2D. |  -.1870067   .1070165    -1.75   0.084    -.3999763    .0259629
                                             |
                                      ln_inc |
                                         D1. |  -.6663827   .3125476    -2.13   0.036    -1.288372   -.0443931
                                             |
                                  ln_consump |
                                         D1. |   2.093388   .5397916     3.88   0.000     1.019169    3.167608
                                         LD. |   1.300412   .4360514     2.98   0.004     .4326424    2.168182
                                        L2D. |   .9061069   .4368897     2.07   0.041     .0366686    1.775545
                                             |
                                       _cons |  -.0896716   .0707544    -1.27   0.209    -.2304773    .0511341
                                ------------------------------------------------------------------------------
                                
                                . estat btest
                                
                                Pesaran/Shin/Smith (2001) ARDL Bounds Test
                                H0: no levels relationship             F =  3.873
                                                                       t = -1.809
                                
                                Critical Values (0.1-0.01), F-statistic, Case 3
                                
                                      | [I_0]   [I_1]  | [I_0]   [I_1]  | [I_0]   [I_1]  | [I_0]   [I_1]
                                      |    L_1     L_1 |   L_05    L_05 |  L_025   L_025 |   L_01    L_01
                                ------+----------------+----------------+----------------+---------------
                                  k_2 |   3.17    4.14 |   3.79    4.85 |   4.41    5.52 |   5.15    6.36
                                accept if F < critical value for I(0) regressors
                                reject if F > critical value for I(1) regressors
                                
                                Critical Values (0.1-0.01), t-statistic, Case 3
                                
                                      | [I_0]   [I_1]  | [I_0]   [I_1]  | [I_0]   [I_1]  | [I_0]   [I_1]
                                      |    L_1     L_1 |   L_05    L_05 |  L_025   L_025 |   L_01    L_01
                                ------+----------------+----------------+----------------+---------------
                                  k_2 |  -2.57   -3.21 |  -2.86   -3.53 |  -3.13   -3.80 |  -3.43   -4.10
                                accept if t > critical value for I(0) regressors
                                reject if t < critical value for I(1) regressors
                                
                                k: # of non-deterministic regressors in long-run relationship
                                Critical values from Pesaran/Shin/Smith (2001)
                                
                                . estat btest, n
                                
                                Pesaran/Shin/Smith (2001) ARDL Bounds Test
                                H0: no levels relationship             F =  3.873
                                
                                Critical Values (0.1-0.01), F-statistic, Case 3
                                
                                      | [I_0]   [I_1]  | [I_0]   [I_1]  | [I_0]   [I_1]
                                      |    L_1     L_1 |   L_05    L_05 |   L_01    L_01
                                ------+----------------+----------------+---------------
                                  k_2 |   3.26    4.25 |   3.94    5.04 |   5.41    6.78
                                accept if F < critical value for I(0) regressors
                                reject if F > critical value for I(1) regressors
                                
                                k: # of non-deterministic regressors in long-run relationship
                                Critical values from Narayan (2005), N=80
                                Based on the asymptotic critical values, the F-test is inconclusive at the 5% level (but close to the lower bound) and does not reject the null hypothesis at smaller levels. The t-test is clearly not rejected at any standard significance level. If we look at the small-sample critical values for the F-test, the lower bound critical value exceeds the test statistic even at the 5% level. Taken together, we found clear evidence that there is no long-run relationship based on this example.

                                I hope that helps.
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