Dominance analysis in linear regressions with ARIMA errors

Question

I have a question regarding dominance analysis in linear regressions with ARIMA errors. I am currently working with stress models for the banking industry. In certain cases, we are using dynamic regressions for these purposes using macroeconomic data, such as exogenous variables. We want to assess the relative importance of the macroeconomic variable within the fitted equation. In any case, the macroeconomic variable should have a higher relative importance than the ARIMA errors.

I have been doing some research online, but I could not find anything related to dynamic regressions. On the contrary, for linear regressions I have found a variety of methods: standardized coefficients, Shapley regressions and dispersion importance (Johnson and Lebreton - 2004). I have found a R package "relaimpo" that implements some of these methods.

Are you aware of any methodology developed so far regarding this matter?

Answer 1

@Anderson Arroyo

A model like this is possible to estimate but, to my understanding, is not implemented in R/on CRAN as a package currently.

For instance, you could implement a model similar to the one you are interested in here using Stata and the SSC module domme (see Luchman, Xue, and Kaplan, 2020 for a discussion of the approach as well as this github page).

Below there is an example from an ARMA model with a first order moving average, first order autoregression and an exogenous variable. The approach is a dominance analysis adapted to focus on parameter estimation as opposed to independent variables and estimates a McFadden's pseudo-R^2.

. webuse friedman2, clear

. arima consump m2 if tin(, 1981q4), ar(1) ma(1)

(setting optimization to BHHH)
Iteration 0:   log likelihood = -344.67575  
Iteration 1:   log likelihood = -341.57248  
Iteration 2:   log likelihood = -340.67391  
Iteration 3:   log likelihood = -340.57229  
Iteration 4:   log likelihood =  -340.5608  
(switching optimization to BFGS)
Iteration 5:   log likelihood =  -340.5515  
Iteration 6:   log likelihood = -340.51272  
Iteration 7:   log likelihood = -340.50949  
Iteration 8:   log likelihood =  -340.5079  
Iteration 9:   log likelihood = -340.50775  
Iteration 10:  log likelihood = -340.50774  

ARIMA regression

Sample:  1959q1 - 1981q4                        Number of obs     =         92
                                                Wald chi2(3)      =    4394.80
Log likelihood = -340.5077                      Prob > chi2       =     0.0000

------------------------------------------------------------------------------
             |                 OPG
     consump |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
consump      |
          m2 |   1.122029   .0363563    30.86   0.000     1.050772    1.193286
       _cons |  -36.09872   56.56703    -0.64   0.523    -146.9681    74.77062
-------------+----------------------------------------------------------------
ARMA         |
          ar |
         L1. |   .9348486   .0411323    22.73   0.000     .8542308    1.015467
             |
          ma |
         L1. |   .3090592   .0885883     3.49   0.000     .1354293    .4826891
-------------+----------------------------------------------------------------
      /sigma |   9.655308   .5635157    17.13   0.000     8.550837    10.75978
------------------------------------------------------------------------------
Note: The test of the variance against zero is one sided, and the two-sided
      confidence interval is truncated at zero.

. matrix list e(b)

e(b)[1,5]
       consump:    consump:       ARMA:       ARMA:      sigma:
                                     L.          L.            
            m2       _cons          ar          ma       _cons
y1   1.1220286  -36.098721   .93484865   .30905921   9.6553076

. domme (consump = m2) (ARMA = L.ar L.ma) if tin(, 1981q4), reg(arima consump m2) ropt(ar(1) ma(1)) fitstat(e(),
>  mcf)

Total of 7 models/regressions

General dominance statistics: ARIMA regression
Number of obs             =                      92
Overall Fit Statistic     =                  0.5125

            |      Dominance      Standardized      Ranking
            |      Stat.          Domin. Stat.
------------+------------------------------------------------------------------------
consump     |
 m2         |         0.2181      0.4256            2 
ARMA        |
 L.ar       |         0.2449      0.4778            1 
 L.ma       |         0.0495      0.0965            3 
-------------------------------------------------------------------------------------
Conditional dominance statistics
-------------------------------------------------------------------------------------

             #param_ests:  #param_ests:  #param_ests:
                       1             2             3
consump:m2        0.3482        0.2265        0.0797
 ARMA:L.ar        0.3905        0.2533        0.0910
 ARMA:L.ma        0.0856        0.0578        0.0050
-------------------------------------------------------------------------------------
Complete dominance designation
-------------------------------------------------------------------------------------

                  dominated?:  dominated?:  dominated?:
                          m2         L_ar         L_ma
  dominates?:m2            0           -1            1
dominates?:L_ar            1            0            1
dominates?:L_ma           -1           -1            0
-------------------------------------------------------------------------------------

Strongest dominance designations

ARMA:L.ar completely dominates consump:m2
consump:m2 completely dominates ARMA:L.ma
ARMA:L.ar completely dominates ARMA:L.ma

I have been working on a port of this module to R but it is not yet ready to accommodate an AR(I)MA model. Hope to add such capabilities in the future and, eventually, release on CRAN.