Originally posted by Jonathan Hillgren
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Yes, this can happen but only if that variable enters the ARDL specification with some lags (not just contemporaneously). -
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 & EmmaComment
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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, 09:26.Comment
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Hello,
Can one in some way measure the first-order autocorrelation coefficient for the ARDL-model?
Thanks for the help!Comment
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Assuming that you are interested in the residual autocorrelation, you could do something like the following:
Please see help corrgram for details.Code:webuse lutkepohl2 ardl ln_inv ln_inc ln_consump predict res, resid corrgram res
Comment
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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
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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).Comment
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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:
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, maxlags(7) aic maxcombs(2000000) dots fast matrix list e(lags)
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)
What puzzles me are the coefficients related to the long-run behaviour. They are extremely large and not significant.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 -------------------------------------------------------------------------------
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:
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)
In this case the problem is not present and the adjustment coeffcient correctly lies between 0 and -1.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 -------------------------------------------------------------------------------
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.
MarcoComment
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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.Comment
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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.
MarcoComment
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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.Comment
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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
MarcoComment
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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.Comment
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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.
MarcoComment
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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:
(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.)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
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:
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.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
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:
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.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
I hope that helps.Comment
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