I have non-stationary time-series data for variables such as Energy Consumption, Trade, Oil Prices, etc and I want to study the impact of these variables on the growth in electricity generation from renewable sources (I have taken the natural logarithms for all the variables).
I performed a linear regression which gave me spurious results (r-squared >0.9)
After testing these time series for unit roots using Augmented Dickey- Fuller test all of them were found to be non-stationary and hence the spurious regression. However their first differences for some of them, and second differences for the others, were found to be stationary.
Now when I test the new linear regressions with the proper order of integration for each variables (in order to have a stationary model) the statistical results are not good (high p-value for some variables and low r-squared (0.25))
My question is how should I proceed now? Should i change my variables?
your analysis is wrong , you can't use a classical regression .
you must use dynamic regression , it is a suitable approach for time series.
why you are use a natural logarithms for all the variables? You can use it if the data not stationary in a variance .
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Are you using regression analysis on single individual (time series data) or on multiple individuals (panel data)?
Dear Gonçalo Maria ,
Please note that transforming variable(s) does NOT make the series stationary, but rather makes the distribution(s) symmetrical. Application of logarithmic transformation needs to be exercised with extreme caution regarding properties of the series, underlying theory and the implied logical/correct interpretation of the relationships between the dep variable and associated selected regressors.
Reverting to your question, the proposed solution would be to use the Autoregressive Distributed Lag (ARDL) model approach, which is suitable for datasets containing a mixture of variables with different orders of integration. Kindly read the manuscripts attached for your info.
All the best!
Gonçalo Maria ,
In addition to the suggestions/inputs I made earlier, please carefully consider the units of measurement of your variables, how you will correctly interpret your results after transforming them into logarithmic forms and conduct estimation using the suitable method, as well as the theoretical framework (relevance quite key) underpinning your empirical study/analysis.
Best!
Maybe you can consider the recursive least squares algorithm (RLS) with forgetting factor (RLS-FF). RLS is the recursive application of the well-known least squares (LS) regression algorithm, so that each new data point is taken in account to modify (correct) a previous estimate of the parameters from some linear (or linearized) correlation thought to model the observed system. The method allows for the dynamical application of LS to time series acquired in real-time. As with LS, there may be several correlation equations with the corresponding set of dependent (observed) variables. For the RLS-FF algorithm, acquired data is weighted according to its age, with increased weight given to the most recent data. The correlation parameters are updated gradually.
Application example ― I have applied the RLS-FF algorithm to estimate the parameters from the KLa correlation, used to predict the O2 gas-liquid mass-transfer, hence giving increased weight to most recent data:
Thesis Controlo do Oxigénio Dissolvido em Fermentadores para Minimi...
Thank you all for the feedback. I started the analysis with single individuals. However when I did the ADF test the order of integration was different for the variables (some I(1) and the others I(2)). So now I'm testing with a panel data João Bento , and the results are better since now all the variables are I(1). They're no cointegrated and for that reason, I applied the Vector Autoregression Model (VAR).
The situation that I have now is that the optimal lag according to AIC (Akaike information criterion) and LR is 8. Is that a problem? If yes what can I do to reduce this lag?
Elvis Munyaradzi Ganyaupfu the ARDL is the most suitable for my case?
1. The feasible transformation must depend on the specification. If one assumes a linear relation between an explaining and the dependent variable, a ln-transformation is not allowed.
2. A ln-estimation is comfortable, because the coefficients can be read as elasticities, but ln-variables are not a good picture of the reality. As, in general, regressions are for behavioural equations (sum of the behaviour of agents), one should take into account the information of the agents. I do not know anybody, who is thinking in logs. Normally, people calculate with absolute values or differences, relations or percentage changes.
3. I do not know how you measure renewable electricity. If it is the capacity, than it is likely that the quantity of a period is near the preceding periods quantity plus the investment measured in electric capacity. Therefore, I recommend taking differences - in (not only) this case - for the estimation. Regressions with absolute variables (or logs of them) mostly lead to spurious results.
4. Although VAR and ARDL are rather modern nowadays, I see no autoregressivity in the relations you want to analyse. Why and how should an increase of renewable electricity in one period influence the change of renewable electricity in coming periods?
5. A lag of 8 seems too long (even for quarterly data). I see no realistic explanation for such a long lag.
Dear Gonçalo Maria ,
Please forgive me for getting entirely confused about what you are exactly doing based on what you mentioned in your previous update, and I quote:
"However when I did the ADF test the order of integration was different for the variables (some I(1) and the others I(2)). So now I'm testing with a panel data João Bento , and the results are better since now all the variables are I(1). They're no cointegrated and for that reason, I applied the Vector Autoregression Model (VAR)".
The first statement in your response as quoted above states that you have a mixture of variables in terms of integration - where some are I(1) while others are I(2). In such a scenario, an ARDL model remains as the suitable approach yes.
The second statement in your response as quoted above states that you then tested a panel data ==== that brings absolute confusion to me! In your first enquiry you indicated your dataset is timeseries in nature, but in your update above you indicated you then tested a panel data and the results got better with all variables being I(1) as you wrote to João Bento. Timeseries and panel data sets are purely different data structures and as such even suitable tests for stationarity are distinct. Is your dataset timeseries and longitudinal (panel)?
If your dataset is panel or longitudinal in nature, then a suitable stationarity test technique should carefully be selected from a set of different options/methods such as the Levin-Lin–Chu (2002), Harris-Tzavalis (1999), Breitung (2000), Breitung and Das 2005), Im-Pesaran-Shin (2003) and Fisher-type (Choi, 2001) tests, which all hold the null hypothesis that all panels contain a unit root; while the Hadri (2000) Lagrange multiplier (LM) test holds the null hypothesis that all the panels are (trend) stationary.
Please see attached a snapshot alternative unit root tests for a panel dataset. You can read the following papers:
[1]. Breitung, J. 2000.The local power of some unit root tests for panel data. Advances in Econometrics, Volume 15: Nonstationary Panels, Panel Cointegration, and Dynamic Panels, ed. B. H. Baltagi, 161–178. Amsterdam: JAY Press.
[2]. Hadri, K. 2000.Testing for stationarity in heterogeneous panel data. Econometrics Journal, 3: 148–161.
[3]. Levin, A., C.-F. Lin, and C.-S. J. Chu. 2002.Unit root tests in panel data: Asymptotic and finite-sample properties. Journal of Econometrics, 108: 1–24.
[4]. Im, K. S., M. H. Pesaran, and Y. Shin. 2003.Testing for unit roots in heterogeneous panels. Journal of Econometrics, 115: 53–74.
Reverting to the initial scenario in which you indicated that your dataset variables are timeseries in nature, if variables are all stationary then application of VAR is indeed appropriate.
Regarding your question whether or not there is a problem with the optimal lag equal to 8 selected based on the AIC and LR statistic, it would be advisable for you to first indicate the frequency (eg. monthly, quarterly or annual) of your series. Such information would guide in providing informed inputs on whether or not certain aspects would need to be taken into account in conducting the VAR lag optimal order selection, given the critical importance of optimal lag length in empirical research that involves use of VAR models in modelling time series data. Inferences made from estimated VAR models depend on correct specification of models, hence determination of an ideal lag order plays a critical role towards ensuring precision of estimates, as well as impulse response functions (IRFs) and variance decompositions produced from estimates of VAR models.
From an abstract standpoint, if the frequency of your data is either quarterly or annual, I feel a lag order equal to 8 is too high. In standard practice, use of a lag length higher than the optimal order leads to higher mean square forecast errors (MSFE) of the VAR, while use of lag length lower than the correct order regularly leads to computation of autocorrelation errors (Lutkepohl, 1993). Therefore, use of incorrect lag lengths leads to inaccurate estimation of IRFs and variance decompositions, and further adversely affects examination or estimation of cointegration properties of data.
Looking forward to your response and update - quite interesting!
Best,
Elvis
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