## Abstract

Knowing the factors that influence the secondary refined copper supply behavior has been fundamental in generating a copper market model and developing public and corporate policies. When analyzing the explanatory variables used in the existing models in the literature, it is possible to observe high variability in estimating the parameters when modifying the availability of information or changing the observation period. Based on this, we argue that only some explanatory variables will have robust estimated parameters, which means that they are unbiased, stable (i.e., they do not vary significantly when the specification of the equation or the number of observations changes), and with asymptotic convergence over time. This work defines and validates a method to select robust explanatory variables capable of quantifying the refined secondary supply of copper (or any other variable) in a given period. Using a database with 23 explanatory variables in the period 1960–2017, we characterize the estimated parameters with high and low robustness, thus supporting the proposed hypothesis. The results obtained allowed identifying those variables with low uncertainty in estimating their parameters, with a high statistical significance, and with a low standard deviation. This allows to obtain a robust function for the secondary refined copper supply in the long term, capturing essential elements of reality.

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## Notes

This scrap has excellent quality, and in most cases, it only needs to be melted down to re-enter the economy (Tilton and Guzmán 2016).

The stability of the estimated parameter could also be influenced by the expert judgment of different econometricians.

Another way that a mathematical function could be variable over time is precisely by using models with variable coefficients (see for example Durán Santomil et al. 2011).

Different estimation techniques such as auto-regressive distributed delay models (ARDL), generalized linear models (GLM), two-stage least squares (TSLS), and author regressive moving average models (ARMA) suffer from the same problem suggested (Green 2018).

The standard deviation \((\sigma )\) is defined as \(\sigma =\sqrt{\frac{1}{N}\sum_{i=1}^{N}{({x}_{i}-\mu )}^{2}}\), where the sum of each data \({x}_{i}\) minus the sample average μ is squared and divided by the sample number

*N*.Fisher et al. (1972), Slade (1980), and Vial (1988) calculated the explanatory variables stock and flow of scrap, assuming that 100% of the copper that entered the market later reemerged as old scrap, without considering a fraction dissipated and lost in the process. However, it is essential to mention that calculating the flow and stock of scrap metal is much more complex (see Rivera et al. 2021), but in terms of exercise, what was existent in the literature was used.

When considering working with the dollar/euro index, a more global analysis of the market is being included, with the euro being a measure of weight that counteracts the dollar, particularly in the period 1975–2017.

White noise is a series such that its mean is zero, the variance is constant, and it is unrelated across time (Green 2018).

Null hypothesis: H

_{0}: δ = 0 (that is, there is a unit root, the time series is non-stationary or has a stochastic tendency). Alternative hypothesis: H_{1}: δ < 0 (that is, the time series is stationary, possibly around a deterministic trend) (Greene 2018).When the dispersion range between the variables is between 97 and 100%, it is considered a perfect linear relationship (Walpole et al. 2012).

This iteration period was considered to obtain a gradual representation; an iteration greater than 4 years probably does not show significant changes in the estimated parameter \(\left(\widehat{\beta }\right)\).

Discarding those 23 pairs of variables with a correlation coefficient greater than and equal to 97%, using the 23 explanatory variables, it is possible to build a total universe of 7,322,143 different models (\(23 \sum_{n=2}^{23}\left\{\left(\frac{23}{n}\right)-23\left(\frac{25-n}{n-1}\right)\right\}=\mathrm{7.322.143})\).

The results of all the models generated are found as an annex to the paper.

Autocorrelation occurs when the independent variables have a temporal structure that is repeated on certain periods over time. Then, today’s residuals

*(t)*will depend on past residuals*(t-1)*and the assumption of independence of the classical linear model will not be fulfilled (Green 2018).Durbin and Watson (1951) tabulated these limits for observations from 6 to 200 and up to 20 explanatory variables.

The results of all the models evaluated throughout this section are found as an annex to the paper.

Simultaneous equations contain more than one dependent variable, or endogenous, which require a number of equations equal to the number of endogenous variables, a condition that violates the strict exogeneity assumption (Greene 2018).

To verify the stability of the parameters of function 4, the CUSUM and CUSUM

^{2}tests were performed. It was obtained as a result that there is structural permanence with 95% confidence, and there are no structural changes with 95% confidence. These results indicate that the variables selected to compose the robust model of function 4 show stability in estimating the coefficients.

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## Acknowledgements

All persons who have made substantial contributions to the work reported in the manuscript (e.g., technical help, writing and editing assistance, general support), but who do not meet the criteria for authorship, are named in the Acknowledgements and have given us their written permission to be named. If we have not included an Acknowledgements, then that indicates that we have not received substantial contributions from non-authors.

## Author information

### Authors and Affiliations

### Contributions

*Category 1*

Conception and design of study:

Nilza Rivera, Juan Ignacio Guzmán, Gustavo Lagos

Acquisition of data:

Nilza Rivera, Juan Ignacio Guzmán, Gustavo Lagos

Analysis and/or interpretation of data:

Nilza Rivera, Juan Ignacio Guzmán, Gustavo Lagos

*Category 2*

Drafting the manuscript:

Nilza Rivera, Juan Ignacio Guzmán, Gustavo Lagos

Revising the manuscript critically for important intellectual content:

Nilza Rivera, Juan Ignacio Guzmán, Gustavo Lagos

*Category 3*

Approval of the version of the manuscript to be published (the names of all authors must be listed):

1. Nilza Rivera

2. Juan Ignacio Guzmán

3. Gustavo Lagos

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### Cite this article

Rivera, N., Guzmán, J.I. & Lagos, G. A method to estimate the robustness of the secondary refined copper supply function.
*Miner Econ* (2022). https://doi.org/10.1007/s13563-022-00308-4

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DOI: https://doi.org/10.1007/s13563-022-00308-4

### Keywords

- Parameter stability
- Robustness
- Estimated parameters
- Copper scrap
- Econometric model
- Secondary production

### JEL Classification

- Q24;
- Q31
- L72