Enhancing Model Selection by Obtaining Optimal Tuning Parameters in ElasticNet Quantile Regression, Application to Crude Oil Prices
Abstract
:1. Introduction
2. Methodology
2.1. Quantile Regression
2.2. Elastic Net Regression
2.3. DFold CrossValidation
Algorithm 1: Dfold CrossValidation 

2.4. Proposed Penalized Quantile Regression Method
 Apply the QR method at τ = (0.25, 0.50, 0.75) using all the variables:$${\widehat{\beta}}_{\tau}^{QR}=\underset{\beta}{min}{\sum}_{i=1}^{n}{\rho}_{\tau}({y}_{i}{x}_{i}^{T}\beta )$$
 Using the training set only, select the optimal parameters via the DCV method at D = 10 as follows:
 The regularization parameter ${\alpha}_{opt}$ value of the sequence 0 < α < 1, where ${\alpha}_{opt}$ represents the relative contribution of the L_{1} penalty versus L_{2} penalty.$${\alpha}_{opt}={argmin}_{k=\overline{1:K}}\left\{{CV}_{{\alpha}_{k}}\right\};$$$${CV}_{{\alpha}_{k}}={\displaystyle \frac{1}{10}}\sum _{d=1}^{10}{PE}_{{\alpha}_{k}};{\alpha}_{k}\in (0,1)$$
 The tuning parameter ${\lambda}_{opt}$ value is at ${\alpha}_{opt}$$${\lambda}_{opt}={argmin}_{s=\overline{1:S}}\left\{{CV}_{{\alpha}_{opt},{\lambda}_{s}}\right\};$$$${CV}_{{\alpha}_{opt},{\lambda}_{s}}={\displaystyle \frac{1}{10}}{\sum}_{d=1}^{10}{MSE}_{{\alpha}_{opt},{\lambda}_{s}}$$
 Based on the Equations (7) and (10) at ${\alpha}_{opt}$ and ${\lambda}_{opt}$, the ELNET penalized regression is used as the following formula:$${\widehat{\beta}}_{\tau}^{ELNET.QR}=\underset{\beta}{min}{\sum}_{i=1}^{n}{\rho}_{\tau}\left({y}_{i}{x}_{i}^{T}\beta \right)+{\lambda}_{opt}\left({\alpha}_{opt}{\Vert \widehat{\beta}\Vert}_{1}+{\displaystyle \frac{(1{\alpha}_{opt})}{2}}{\Vert \widehat{\beta}\Vert}_{2}^{2}\right);$$$${\rho}_{\tau}\left(v\right)=v(\tau {I}_{\{v<0\}})$$
3. Application
3.1. Simulation Study
3.2. Application Datasets
4. Results and Discussion
4.1. Simulation Results
4.2. Application Results and Discussion
5. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
References
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Method  $\mathit{\lambda}$  RSS  RMSE  MAE  MAPE  MASE 

τ = 0.25  
RR.QR  ${\lambda}_{min}$  35.5827  0.88624  0.71977  3.657265  0.794615 
${\lambda}_{1se}$  39.4718  0.93395  0.76183  4.024955  0.841051  
LASSO.QR  ${\lambda}_{min}$  14.1830  0.56093  0.40053  1.746325  0.442176 
${\lambda}_{1se}$  15.3082  0.58278  0.43724  1.999109  0.482706  
AdLASSO.QR (RR.W. λ_{min})  ${\lambda}_{min}$  14.5157  0.56762  0.41579  2.082011  0.459031 
${\lambda}_{1se}$  16.0379  0.59633  0.45133  2.030926  0.498263  
AdLASSO.QR (RR.W. λ_{1se})  ${\lambda}_{min}$  14.7190  0.57034  0.41830  2.092005  0.461806 
${\lambda}_{1se}$  16.2224  0.59878  0.45343  2.044566  0.500584  
$\mathrm{ELNET}.\mathrm{QR}\alpha =0.25$  ${\lambda}_{min}$  19.0338  0.64701  0.50269  2.272781  0.554958 
${\lambda}_{1se}$  22.9179  0.71031  0.56536  2.504381  0.624143  
$\mathrm{ELNET}.\mathrm{QR}\alpha =0.5$  ${\lambda}_{min}$  14.7433  0.57177  0.41665  1.89485  0.459972 
${\lambda}_{1se}$  17.4372  0.62082  0.47837  2.131715  0.528107  
$\mathrm{ELNET}.\mathrm{QR}\alpha =0.75$  ${\lambda}_{min}$  14.2895  0.56306  0.40377  1.780631  0.445752 
${\lambda}_{1se}$  15.7379  0.59067  0.44563  2.044958  0.491972  
$\mathrm{ELNET}.\mathrm{QR}{\alpha}_{opt}$  ${\lambda}_{min}$  13.9615  0.55671  0.39654  1.776101  0.437778 
${\lambda}_{1se}$  15.7195  0.59049  0.44696  2.020386  0.493431  
τ = 0.5  
RR.QR  ${\lambda}_{min}$  30.8563  0.82567  0.69494  2.127862  0.76720 
${\lambda}_{1se}$  33.5667  0.86207  0.72893  2.019126  0.80472  
LASSO.QR  ${\lambda}_{min}$  11.7178  0.50823  0.39718  2.592074  0.438483 
${\lambda}_{1se}$  13.6771  0.54948  0.45072  2.456135  0.497588  
AdLASSO.QR (RR.W. λ_{min})  ${\lambda}_{min}$  13.385  0.53733  0.44051  2.604391  0.486318 
${\lambda}_{1se}$  14.9137  0.56951  0.46914  2.488478  0.517925  
AdLASSO.QR (RR.W. λ_{1se})  ${\lambda}_{min}$  13.4163  0.53805  0.44168  2.602395  0.487607 
${\lambda}_{1se}$  14.9282  0.56976  0.46944  2.48853  0.518257  
$\mathrm{ELNET}.\mathrm{QR}\alpha =0.25$  ${\lambda}_{min}$  16.1670  0.59738  0.48819  2.668777  0.538961 
${\lambda}_{1se}$  19.3541  0.65371  0.53790  2.602162  0.593840  
$\mathrm{ELNET}.\mathrm{QR}\alpha =0.5$  ${\lambda}_{min}$  13.1976  0.53957  0.44388  2.665215  0.490041 
${\lambda}_{1se}$  15.5554  0.58596  0.47915  2.659132  0.528980  
$\mathrm{ELNET}.\mathrm{QR}\alpha =0.75$  ${\lambda}_{min}$  12.3296  0.52141  0.42639  2.639871  0.470737 
${\lambda}_{1se}$  14.3324  0.56247  0.46268  2.544796  0.510792  
$\mathrm{ELNET}.\mathrm{QR}{\alpha}_{opt}$  ${\lambda}_{min}$  10.3090  0.47689  0.37236  2.564913  0.411081 
${\lambda}_{1se}$  13.6800  0.54953  0.45087  2.515077  0.497758  
τ = 0.75  
RR.QR  ${\lambda}_{min}$  42.7458  0.96993  0.79680  6.288212  0.879659 
${\lambda}_{1se}$  46.8397  1.01652  0.83617  6.47382  0.923126  
LASSO.QR  ${\lambda}_{min}$  12.9671  0.53623  0.45390  4.716719  0.501104 
${\lambda}_{1se}$  14.3718  0.5644  0.46955  4.696969  0.518382  
AdLASSO.QR (RR.W. λ_{min})  ${\lambda}_{min}$  14.4291  0.55778  0.47079  4.889332  0.519747 
${\lambda}_{1se}$  15.8043  0.58392  0.48687  4.910642  0.537499  
AdLASSO.QR (RR.W. λ_{1se})  ${\lambda}_{min}$  14.5222  0.55917  0.47169  4.892964  0.520746 
${\lambda}_{1se}$  15.8926  0.58537  0.48803  4.916813  0.538782  
$\mathrm{ELNET}.\mathrm{QR}\alpha =0.25$  ${\lambda}_{min}$  20.6517  0.67191  0.55133  5.073862  0.608665 
${\lambda}_{1se}$  23.6645  0.71873  0.58854  5.305833  0.649741  
$\mathrm{ELNET}.\mathrm{QR}\alpha =0.5$  ${\lambda}_{min}$  13.7992  0.55295  0.46196  4.620298  0.510000 
${\lambda}_{1se}$  15.1615  0.57941  0.47896  4.653668  0.528763  
$\mathrm{ELNET}.\mathrm{QR}\alpha =0.75$  ${\lambda}_{min}$  13.0885  0.53868  0.45467  4.68214  0.501955 
${\lambda}_{1se}$  14.5195  0.56725  0.47102  4.675978  0.520007  
$\mathrm{ELNET}.\mathrm{QR}{\alpha}_{opt}$  ${\lambda}_{min}$  12.8468  0.53376  0.45244  4.714046  0.499487 
${\lambda}_{1se}$  14.3309  0.56360  0.46864  4.683324  0.517377 
Method  $\mathit{\lambda}$  RSS  Num. of V.S.  V.S. 

RR  ${\lambda}_{min}=$ 0.041839  109.177  10  ${x}_{1},\dots ,{x}_{10}$ 
${\lambda}_{1se}=$ 0.74835  77.0141  10  ${x}_{1},\dots ,{x}_{10}$  
LASSO  ${\lambda}_{min}=$ 0.003994  120.091  9  ${x}_{1},\dots ,{x}_{4},{x}_{6},{\dots ,x}_{10}$ 
${\lambda}_{1se}=$ 0.078399  82.3651  7  ${x}_{1},\dots ,{x}_{4},{x}_{6},{{x}_{8},x}_{10}$  
$\mathrm{ELNET}\alpha =0.25$  ${\lambda}_{min}=$ 0.013263  116.129  10  ${x}_{1},\dots ,{x}_{10}$ 
${\lambda}_{1se}=$ 0.260352  76.9288  7  ${x}_{1},\dots ,{x}_{4},{x}_{6},{{x}_{8},x}_{10}$  
$\mathrm{ELNET}\alpha =0.5$  ${\lambda}_{min}=$ 0.007987  117.951  9  ${x}_{1},\dots ,{x}_{4},{x}_{6},{\dots ,x}_{10}$ 
${\lambda}_{1se}=$ 0.142868  79.5524  7  ${x}_{1},\dots ,{x}_{4},{x}_{6},{{x}_{8},x}_{10}$  
$\mathrm{ELNET}\alpha =0.75$  ${\lambda}_{min}=$ 0.005844  118.623  9  ${x}_{1},\dots ,{x}_{4},{x}_{6},{\dots ,x}_{10}$ 
${\lambda}_{1se}=0.104532$  80.8656  7  ${x}_{1},\dots ,{x}_{4},{x}_{6},{{x}_{8},x}_{10}$  
τ = 0.25  
RR.QR  ${\lambda}_{min}=0.1322$  103.907  10  ${x}_{1},\dots ,{x}_{10}$ 
${\lambda}_{1se}=0.5480$  92.046  10  ${x}_{1},\dots ,{x}_{10}$  
LASSO.QR  ${\lambda}_{min}=0.0471$  87.492  4  ${x}_{1},\dots ,{x}_{4}$ 
${\lambda}_{1se}=0.0720$  93.659  3  ${x}_{1},{x}_{2},{x}_{4}$  
AdLASSO.QR (RR.W. λ_{min})  ${\lambda}_{min}=0.0008$  100.139  4  ${x}_{1},{x}_{3},{x}_{4},{x}_{10}$ 
${\lambda}_{1se}=0.0028$  100.797  3  ${x}_{1},{x}_{3},{x}_{4}$  
AdLASSO.QR (RR.W. λ_{1se})  ${\lambda}_{min}=0.0003$  96.542  4  ${x}_{1},\dots ,{x}_{4}$ 
${\lambda}_{1se}=0.0012$  99.369  3  ${x}_{1},{x}_{3},{x}_{4}$  
$\mathrm{ELNET}.\mathrm{QR}\alpha =0.25$  ${\lambda}_{min}=0.0911$  86.377  8  ${x}_{1},\dots ,{x}_{6},{x}_{8},{x}_{10}$ 
${\lambda}_{1se}=$ 0.2002  87.614  5  ${x}_{1},\dots ,{x}_{4},{x}_{10}$  
$\mathrm{ELNET}.\mathrm{QR}\alpha =0.5$  ${\lambda}_{min}=0.0317$  100.762  9  ${x}_{1},\dots ,{x}_{6},{x}_{8},{{x}_{9},x}_{10}$ 
${\lambda}_{1se}=$ 0.1275  90.6738  4  ${x}_{1},\dots ,{x}_{4}$  
$\mathrm{ELNET}.\mathrm{QR}\alpha =0.75$  ${\lambda}_{min}=0.054$  87.045  5  ${x}_{1},\dots ,{x}_{4},{x}_{6}$ 
${\lambda}_{1se}=0.0931$  92.967  4  ${x}_{1},\dots ,{x}_{4}$  
$\mathrm{ELNET}.\mathrm{QR}{\alpha}_{opt}=0.38$  ${\lambda}_{min}=0.0787$  84.426  8  ${x}_{1},\dots ,{x}_{6},{x}_{8},{x}_{10}$ 
${\lambda}_{1se}=0.1532$  89.458  5  ${x}_{1},\dots ,{x}_{4},{x}_{10}$  
τ = 0.5  
RR.QR  ${\lambda}_{min}=0.0678$  79.0412  10  ${x}_{1},\dots ,{x}_{10}$ 
${\lambda}_{1se}=0.6563$  74.8137  10  ${x}_{1},\dots ,{x}_{10}$  
LASSO.QR  ${\lambda}_{min}=0.0228$  77.2748  8  ${x}_{1},\dots ,{x}_{4},{x}_{6},{x}_{8}\dots ,{x}_{10}$ 
${\lambda}_{1se}=0.0531$  75.0064  5  ${x}_{1},\dots ,{x}_{4},{x}_{8}$  
AdLASSO.QR (RR.W. λ_{min})  ${\lambda}_{min}=0.0006$  79.3498  3  ${x}_{1},{x}_{3},{x}_{4}$ 
${\lambda}_{1se}=0.0035$  77.0606  2  ${x}_{1},{x}_{3}$  
AdLASSO.QR (RR.W. λ_{1se})  ${\lambda}_{min}=0.0001$  73.2189  5  ${x}_{1},\dots ,{x}_{4},{x}_{10}$ 
${\lambda}_{1se}=0.0009$  80.3982  3  ${x}_{1},{x}_{3},{x}_{4}$  
$\mathrm{ELNET}.\mathrm{QR}\alpha =0.25$  ${\lambda}_{min}=$ 0.0390  76.283  9  ${x}_{1},\dots ,{x}_{6},{x}_{8},{x}_{9},{x}_{10}$ 
${\lambda}_{1se}=$ 0.1719  74.4287  8  ${x}_{1},\dots ,{x}_{4},{x}_{6},{x}_{8}\dots ,{x}_{10}$  
$\mathrm{ELNET}.\mathrm{QR}\alpha =0.5$  ${\lambda}_{min}=$ 0.0357  77.3792  9  ${x}_{1},\dots ,{x}_{6},{x}_{8},{x}_{9},{x}_{10}$ 
${\lambda}_{1se}=$ 0.1000  74.9740  6  ${x}_{1},\dots ,{x}_{4},{x}_{8},{x}_{10}$  
$\mathrm{ELNET}.\mathrm{QR}\alpha =0.75$  ${\lambda}_{min}=$ 0.0277  77.5181  8  ${x}_{1},\dots ,{x}_{4},{x}_{6},{x}_{8},{x}_{9},{x}_{10}$ 
${\lambda}_{1se}=$ 0.0687  74.9286  6  ${x}_{1},\dots ,{x}_{4},{x}_{8},{x}_{10}$  
$\mathrm{ELNET}.\mathrm{QR}{\alpha}_{opt}=0.02$  ${\lambda}_{min}=0.3392$  72.9008  10  ${x}_{1},\dots ,{x}_{10}$ 
${\lambda}_{1se}=0.7450$  74.9729  9  ${x}_{1},\dots ,{x}_{4},{x}_{6},{x}_{7}\dots ,{x}_{10}$  
τ = 0.75  
RR.QR  ${\lambda}_{min}=0.0511$  99.8799  10  ${x}_{1},\dots ,{x}_{10}$ 
${\lambda}_{1se}=0.6298$  99.3733  10  ${x}_{1},\dots ,{x}_{10}$  
LASSO.QR  ${\lambda}_{min}=0.0053$  108.9124  8  ${x}_{1},\dots ,{x}_{4},{x}_{6},{x}_{8},\dots ,{x}_{10}$ 
${\lambda}_{1se}=0.0354$  101.6289  5  ${x}_{1},{x}_{3},{x}_{4},{x}_{6},{x}_{10}$  
AdLASSO.QR (RR.W. λ_{min})  ${\lambda}_{min}=0.0004$  113.2957  5  ${x}_{1},{x}_{3},{x}_{4},{x}_{6},{x}_{10}$ 
${\lambda}_{1se}=0.0011$  118.7315  4  ${x}_{1},{x}_{3},{x}_{6},{x}_{10}$  
AdLASSO.QR (RR.W. λ_{1se})  ${\lambda}_{min}=0.0001$  100.7581  5  ${x}_{1},{x}_{3},{{x}_{4},x}_{6},{x}_{10}$ 
${\lambda}_{1se}=0.0003$  105.5192  3  ${x}_{1},{x}_{3},{x}_{4}$  
$\mathrm{ELNET}.\mathrm{QR}\alpha =0.25$  ${\lambda}_{min}=$ 0.0204  103.2303  8  ${x}_{1},\dots {x}_{4},{x}_{6},{x}_{8}\dots ,{x}_{10}$ 
${\lambda}_{1se}=$ 0.1334  102.9762  6  ${x}_{1}{x}_{3},{x}_{4}{,x}_{6},{x}_{8},{x}_{10}$  
$\mathrm{ELNET}.\mathrm{QR}\alpha =0.5$  ${\lambda}_{min}=$ 0.0102  106.1962  8  ${x}_{1},\dots {x}_{4},{x}_{6},{x}_{8}\dots ,{x}_{10}$ 
${\lambda}_{1se}=0.0688$  101.7987  5  ${x}_{1}{x}_{3},{x}_{4}{,x}_{6},{x}_{10}$  
$\mathrm{ELNET}.\mathrm{QR}\alpha =0.75$  ${\lambda}_{min}=$ 0.0068  107.9645  8  ${x}_{1},\dots {x}_{4},{x}_{6},{x}_{8}\dots ,{x}_{10}$ 
${\lambda}_{1se}=$ 0.0458  101.4287  5  ${x}_{1}{x}_{3},{x}_{4}{,x}_{6},{x}_{10}$  
$\mathrm{ELNET}.\mathrm{QR}{\alpha}_{opt}=0.02$  ${\lambda}_{min}=0.2555$  97.6474  9  ${x}_{1},\dots ,{x}_{6},{x}_{8}\dots ,{x}_{10}$ 
${\lambda}_{1se}=0.8571$  105.4114  8  ${x}_{1},\dots {x}_{4},{x}_{6},{x}_{8}\dots ,{x}_{10}$ 
Method  $\mathit{\lambda}$  RMSE  MAE  MAPE  MASE 

RR  ${\lambda}_{min}=$ 0.041839  1.1469  0.7806  1.7260  0.8729 
${\lambda}_{1se}=$ 0.74835  0.9633  0.7105  1.2235  0.7945  
LASSO  ${\lambda}_{min}=$ 0.003994  1.2029  0.8069  1.8265  0.9022 
${\lambda}_{1se}=$ 0.078399  0.9962  0.7161  1.3324  0.8008  
ELNET $\alpha =0.25$  ${\lambda}_{min}=$ 0.013263  1.1829  0.7974  1.7923  0.8916 
${\lambda}_{1se}=$ 0.260352  0.9627  0.7046  1.2343  0.7879  
ELNET $\alpha =0.5$  ${\lambda}_{min}=$ 0.007987  1.1921  0.8018  1.8080  0.8966 
${\lambda}_{1se}=$ 0.142868  0.9790  0.7096  1.2885  0.7935  
ELNET $\alpha =0.75$  ${\lambda}_{min}=$ 0.005844  1.1955  0.8035  1.8136  0.8985 
${\lambda}_{1se}=0.104532$  0.9871  0.7129  1.3112  0.7972  
τ = 0.25  
RR.QR  ${\lambda}_{min}=0.1322$  1.1189  0.8876  2.537  0.9924 
${\lambda}_{1se}=0.5480$  1.0531  0.8610  2.727  0.9628  
LASSO.QR  ${\lambda}_{min}=0.0471$  1.0267  0.8217  2.514  0.9187 
${\lambda}_{1se}=0.0720$  1.0623  0.8520  2.571  0.9527  
AdLASSO.QR (RR.W. λ_{min})  ${\lambda}_{min}=0.0008$  1.0984  0.8644  2.618  0.9667 
${\lambda}_{1se}=0.0028$  1.1020  0.8742  2.639  0.9776  
AdLASSO.QR (RR.W. λ_{1se})  ${\lambda}_{min}=0.0003$  1.0785  0.8527  2.704  0.9534 
${\lambda}_{1se}=0.0012$  1.0942  0.8661  2.671  0.9685  
$\mathrm{ELNET}.\mathrm{QR}\alpha =0.25$  ${\lambda}_{min}=0.0911$  1.0201  0.8259  2.4273  0.9236 
${\lambda}_{1se}=$ 0.2002  1.0274  0.8387  2.5879  0.9378  
$\mathrm{ELNET}.\mathrm{QR}\alpha =0.5$  ${\lambda}_{min}=0.0317$  1.1018  0.8779  2.5650  0.9817 
${\lambda}_{1se}=$ 0.1275  1.0452  0.8432  2.5592  0.9429  
$\mathrm{ELNET}.\mathrm{QR}\alpha =0.75$  ${\lambda}_{min}=0.054$  1.0241  0.8195  2.5257  0.9164 
${\lambda}_{1se}=0.0931$  1.0583  0.8500  2.5636  0.9505  
$\mathrm{ELNET}.\mathrm{QR}{\alpha}_{opt}=0.38$  ${\lambda}_{min}=0.0787$  1.0086  0.8180  2.444  0.9146 
${\lambda}_{1se}=0.1532$  1.0382  0.8404  2.565  0.9398  
τ = 0.5  
RR.QR  ${\lambda}_{min}=0.0678$  0.9759  0.7058  1.6486  0.7893 
${\lambda}_{1se}=0.6563$  0.9494  0.6876  1.2275  0.7689  
LASSO.QR  ${\lambda}_{min}=0.0228$  0.9649  0.7037  1.6693  0.7869 
${\lambda}_{1se}=0.0531$  0.9506  0.6907  1.4299  0.7724  
AdLASSO.QR (RR.W. λ_{min})  ${\lambda}_{min}=0.0006$  0.9778  0.7109  1.7108  0.7949 
${\lambda}_{1se}=0.0035$  0.9636  0.7151  1.6833  0.7996  
AdLASSO.QR (RR.W. λ_{1se})  ${\lambda}_{min}=0.0001$  0.9392  0.6872  1.5984  0.7685 
${\lambda}_{1se}=0.0009$  0.9842  0.7099  1.3913  0.7938  
$\mathrm{ELNET}.\mathrm{QR}\alpha =0.25$  ${\lambda}_{min}=$ 0.0390  0.9590  0.6993  1.6403  0.7820 
${\lambda}_{1se}=$ 0.1719  0.9470  0.6876  1.3001  0.7689  
$\mathrm{ELNET}.\mathrm{QR}\alpha =0.5$  ${\lambda}_{min}=$ 0.0357  0.9655  0.7031  1.6494  0.7863 
${\lambda}_{1se}=$ 0.1000  0.9504  0.6895  1.3276  0.7710  
$\mathrm{ELNET}.\mathrm{QR}\alpha =0.75$  ${\lambda}_{min}=$ 0.0277  0.9664  0.7040  1.6612  0.7872 
${\lambda}_{1se}=$ 0.0687  0.9501  0.6901  1.3927  0.7717  
$\mathrm{ELNET}.\mathrm{QR}{\alpha}_{opt}=0.02$  ${\lambda}_{min}=0.3392$  0.9372  0.6827  1.3390  0.7634 
${\lambda}_{1se}=0.7450$  0.9504  0.6914  1.1955  0.7731  
τ = 0.75  
RR.QR  ${\lambda}_{min}=0.0511$  1.0970  0.7913  2.9823  0.8848 
${\lambda}_{1se}=0.6298$  1.0942  0.7899  2.9192  0.8833  
LASSO.QR  ${\lambda}_{min}=0.0053$  1.1455  0.8323  3.1377  0.9307 
${\lambda}_{1se}=0.0354$  1.1065  0.7785  2.9182  0.8705  
AdLASSO.QR (RR.W. λ_{min})  ${\lambda}_{min}=0.0004$  1.1683  0.8644  3.1510  0.9665 
${\lambda}_{1se}=0.0011$  1.1960  0.9016  3.4228  1.0082  
AdLASSO.QR (RR.W. λ_{1se})  ${\lambda}_{min}=0.0001$  1.1018  0.7837  2.909  0.8764 
${\lambda}_{1se}=0.0003$  1.1275  0.8137  3.1028  0.9099  
$\mathrm{ELNET}.\mathrm{QR}\alpha =0.25$  ${\lambda}_{min}=$ 0.0204  1.1152  0.8057  3.0121  0.9010 
${\lambda}_{1se}=$ 0.1334  1.1139  0.7947  2.9758  0.8886  
$\mathrm{ELNET}.\mathrm{QR}\alpha =0.5$  ${\lambda}_{min}=$ 0.0102  1.1311  0.8191  3.0654  0.9159 
${\lambda}_{1se}=0.0688$  1.1075  0.7847  2.9390  0.8775  
$\mathrm{ELNET}.\mathrm{QR}\alpha =0.75$  ${\lambda}_{min}=$ 0.0068  1.1405  0.8278  3.1141  0.9256 
${\lambda}_{1se}=$ 0.0458  1.1055  0.7801  2.9212  0.8723  
$\mathrm{ELNET}.\mathrm{QR}{\alpha}_{opt}=$ 0.02  ${\lambda}_{min}=0.2555$  1.0847  0.7812  2.8845  0.8736 
${\lambda}_{1se}=0.8571$  1.1270  0.8156  3.0404  0.9120 
τ $=0.25,{\mathit{\alpha}}_{\mathit{o}\mathit{p}\mathit{t}}=$ 0.38  τ $=0.5,{\mathit{\alpha}}_{\mathit{o}\mathit{p}\mathit{t}}=$ 0.02  τ $=0.75,{\mathit{\alpha}}_{\mathit{o}\mathit{p}\mathit{t}}=$ 0.02  

${\mathit{\lambda}}_{\mathit{m}\mathit{i}\mathit{n}}$  ${\mathit{\lambda}}_{1\mathit{s}\mathit{e}}$  ${\mathit{\lambda}}_{\mathit{m}\mathit{i}\mathit{n}}$  ${\mathit{\lambda}}_{1\mathit{s}\mathit{e}}$  ${\mathit{\lambda}}_{\mathit{m}\mathit{i}\mathit{n}}$  ${\mathit{\lambda}}_{1\mathit{s}\mathit{e}}$  
${\widehat{\beta}}_{1}$  0.1686  0.1313  0.1545  0.0904  0.1054  0.0548 
${\widehat{\beta}}_{2}$  0.1225  0.0415  0.1066  0.0614  0.0306  0.0184 
${\widehat{\beta}}_{3}$  0.3038  0.2110  0.2072  0.1061  0.1421  0.0703 
${\widehat{\beta}}_{4}$  0.1151  0.0948  0.1360  0.0879  0.0761  0.0487 
${\widehat{\beta}}_{5}$  0.0150  0  −0.0172  0  −0.0277  0 
${\widehat{\beta}}_{6}$  0.0529  0  0.0413  0.0123  0.0648  0.0158 
${\widehat{\beta}}_{7}$  0  0  0.0033  0.0026  0  0 
${\widehat{\beta}}_{8}$  0.0037  0  0.0448  0.0128  0.0489  0.0093 
${\widehat{\beta}}_{9}$  0  0  0.0133  0.0013  −0.0357  −0.0048 
${\widehat{\beta}}_{10}$  0.0802  0.0062  0.0862  0.0531  0.1022  0.0425 
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© 2024 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).
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AlJawarneh, A.S.; Alsayed, A.R.M.; Ayyoub, H.N.; Ismail, M.T.; Sek, S.K.; Ariç, K.H.; Manzi, G. Enhancing Model Selection by Obtaining Optimal Tuning Parameters in ElasticNet Quantile Regression, Application to Crude Oil Prices. J. Risk Financial Manag. 2024, 17, 323. https://doi.org/10.3390/jrfm17080323
AlJawarneh AS, Alsayed ARM, Ayyoub HN, Ismail MT, Sek SK, Ariç KH, Manzi G. Enhancing Model Selection by Obtaining Optimal Tuning Parameters in ElasticNet Quantile Regression, Application to Crude Oil Prices. Journal of Risk and Financial Management. 2024; 17(8):323. https://doi.org/10.3390/jrfm17080323
Chicago/Turabian StyleAlJawarneh, Abdullah S., Ahmed R. M. Alsayed, Heba N. Ayyoub, Mohd Tahir Ismail, Siok Kun Sek, Kivanç Halil Ariç, and Giancarlo Manzi. 2024. "Enhancing Model Selection by Obtaining Optimal Tuning Parameters in ElasticNet Quantile Regression, Application to Crude Oil Prices" Journal of Risk and Financial Management 17, no. 8: 323. https://doi.org/10.3390/jrfm17080323