General remarks

National stock indices have a much greater explanatory power of miners’ share prices than the TSI index
The S&P TSX and Baltic Dry Index indices have little use in all cases
The logic for bringing the BDI is that TSI 62% is for delivery in Tianjin, China and that producers have to get the iron ore there (BDI is considered the best proxy
for sea transport prices).
The logic for bringing the S&P TSX was to have a ‘global mining’ index instead of a national multi-industry one. As it happens national indices
are more meaningful in all cases

FMG

The correlation between FMG returns and the TSI AUD returns is 44%, higher than for any of the other miners considered
The correlation between FMG returns and the AS25 returns is 51.5%

40% of the variance in FMG returns can be explained by TSI AUD and AS25
The regression equation is as folows:
FMG return = -0.00044975-0.16599 * TSI AUD return + 0.3493372 * AS25 return

The S7P TSX Global is not useful as an explanatory factor, nor is the Baltic Dry Index

Results being encouraging for this stock, I tested whether lagged stock index data was better at explaining FMG returns: it is not, explanatory power goes to near 0
This is consistent with markets being efficient, in the sense that they reflect all available info without delay

Atlas

Share price behaves erratically, cannot be explained by either TSI (3.8% explanatory power) or Stock index prices (4.4% explanatory power), or any other factor considered.
It is difficult/impossible to imagine factors that would be more relevant that what the company sells (TSI AUD) or the stock market it is in (AS25)
=> the logical conclusion is that trying to obtain a regression formula for Atlas is not possible, share price behaves in a firm-specific way
independently of any significant outside factor
I went through their last published quarterly report, which confirms the above conclusion (if any 2 factors could explain prices it would be those 2, but they don’t)

Arrium

I sourced some daily data on top of the monthly one you supplied
The correlation between Arrium returns and TSI returns is again very low (2.4%) and regressing the returns on TSI returns gives meaningless results
The correlation between Arrium returns and AS25 returns is 47%, and the AS25 returns explain 21% of the variance in Arrium returns
The S7P TSX has again little explanatory power