Robust monitoring machine: a machine learning solution for out-of-sample R $$^2$$ -hacking in return predictability monitoring

Financial Innovation

Table 3 Robust Monitoring Performance Measures

Forecast		Monitoring	Monitoring	$E[d_i]$	$Var[d_i]$	Diebold-Mariano
Forecast		Risk Premium	Alpha	$\times 10^{-6}$	$\times 10^{-8}$	Statistic
Robust	Estimate	1.15	0.61	9.81	0.99	2.87
Monitoring (m)	(p-value)	(0.017)	(0.021)			(0.004)
Proposed (a)	Estimate	1	0	8.52	1.86	1.82
		(by definition)	(by definition)			(0.068)
Benchmark (b)	By definition	0	0	0	0	N/A

This table reports monitoring performance measures: monitoring risk premium and monitoring alpha. $E[d_{i}]$ and $Var[d_{i}]$ for $i \in \{a, m, b\}$ denote the expected value and variance of loss difference, which represent average forecasting performance and its relative risk, respectively. Diebold-Mariano tests are to show if a forecast has significantly different predictability from a benchmark. The (two-sided) p-values in parentheses are calculated via bootstrapping for Monitoring Alpha and Monitoring Risk Premium. All forecasts are free from look-ahead bias