Abstract
This paper compares ordinary least squares (OLS), weighted least squares (WLS), and adaptive least squares (ALS) by means of a Monte Carlo study and an application to two empirical data sets. Overall, ALS emerges as the winner: It achieves most or even all of the efficiency gains of WLS over OLS when WLS outperforms OLS, but it only has very limited downside risk compared to OLS when OLS outperforms WLS.
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Notes
- 1.
The reason for introducing a small constant \(\delta > 0\) on the left-hand side of (9) is that, because one is taking logs, one needs to avoid a residual of zero, or even very near zero. The choice \(\delta = 0.1\) seems to work well in practice.
- 2.
See Sect. 4.1 of [13] for a detailed description of the Max standard error. In a nutshell, the Max standard error is the maximum of the HC standard error and the ‘textbook’ standard error from an OLS regression, which assumes conditional homoskedasticity.
- 3.
The second performance measure does not depend on the nominal confidence level, since by definition (19), it is equivalent to the ratio of the average standard error of a given method to the average OLS-HC standard error.
- 4.
It can be shown [7, e.g.] that \(p{\slash }n\) corresponds to the average element of the hat matrix.
- 5.
The two data sets are available under the names CEOSAL2 and HPRICE2, respectively at http://fmwww.bc.edu/ec-p/data/wooldridge/datasets.list.html.
- 6.
The log always corresponds to the natural logarithm.
- 7.
This regression results in taking the log of log(sales) and log(mktval) on the right-hand side; taking absolute values is not necessary, since log(sales) and log(mktval) are always positive. Furthermore, some observations have a value of zero for ceoten; we replace those values by 0.01 before taking logs.
- 8.
[13] only use univariate regressions in their Monte Carlo study and do not provide any applications to empirical data sets.
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A Figures and Tables
A Figures and Tables
Density plots for the estimators of \(\beta _1\) for Specification S.1 and its four parameter values. The sample size is 100, the regressors are U[1, 4]-distributed and the error terms follow a standard normal distribution
Boxplots of the ratios of the eMSE of WLS (left) and ALS (right) to the eMSE of OLS. For a given sample size \(n =20,50,100\), the boxplots are over all 27 combinations of specification of the skedastic function, parameter value, distribution of the regressors, and distribution of the error terms
Boxplots of the ratios of the average length of WLS confidence intervals for \(\beta _1\) (left) and ALS confidence intervals for \(\beta _1\) (right) to the average length of OLS confidence intervals for \(\beta _1\). For the given sample size \(n = 100\), the boxplots are over all 27 combinations of specification of the skedastic function, parameter value, distribution of the regressors, and distribution of the error terms
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Sterchi, M., Wolf, M. (2017). Weighted Least Squares and Adaptive Least Squares: Further Empirical Evidence. In: Kreinovich, V., Sriboonchitta, S., Huynh, VN. (eds) Robustness in Econometrics. Studies in Computational Intelligence, vol 692. Springer, Cham. https://doi.org/10.1007/978-3-319-50742-2_9
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DOI: https://doi.org/10.1007/978-3-319-50742-2_9
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