Why Mean Reversion Conflicts with Index Replication

# Mean-reverting backtest between index and components


# Mean-reverting backtest between index and components












I am a beginner with ETF replication: I have to make a code to make the value of my assets go back to the average of the index Eurostoxx 50 with a subset of components. I am not sure how to implement it.

I have understood that typically, a cross-sectional mean reversion strategy is fed a universe of stocks, where each stock has its relative returns compared to the mean returns of the universe.

- a stock with a positive relative return is shorted

- a stock with a negative relative return is bought

But is it what I have to adapte to replicate an ETF?

I know how to code a mean-reverting strategy:

```
from backtesting import Backtest, Strategy
from backtesting.lib import crossover

from backtesting.test import SMA, GOOG

class SmaCross(Strategy):
    def init(self):
        Close = self.data.Close
        self.ma1 = self.I(SMA, Close, 10)
        self.ma2 = self.I(SMA, Close, 20)
        #self.boll = bt.indicators.BollingerBands(period=self.p.period, devfactor=self.p.devfactor)

    def next(self):
        if crossover(self.ma1, self.ma2):
            self.buy()
        elif crossover(self.ma2, self.ma1):
            self.sell()

bt = Backtest(GOOG, SmaCross,
              cash=10000, commission=.002)
bt.run()
bt.plot()
```

But, first I don't know how to do mean reverting to go back to the index (SX5E) and, second, I don't know how to adapt my dataset to the one `Backtest` is expecting `('data' must be a pandas.DataFrame with columns 'Open', 'High', 'Low', 'Close', and (optionally) 'Volume`).

My dataset looks is fully accessible here and looks like the following:

```
    ABI BB Equity   AD NA Equity    ADS GY Equity   AI FP Equity    AIR FP Equity   AMS SQ Equity   ASML NA Equity  BAS GY Equity   BAYN GY Equity  BBVA SQ Equity  BMW GY Equity   BN FP Equity    BNP FP Equity   CRH ID Equity   CS FP Equity    DAI GY Equity   DG FP Equity    DPW GY Equity   DTE GY Equity   EL FP Equity    ENEL IM Equity  ENGI FP Equity  ENI IM Equity   FP FP Equity    FRE GY Equity   GLE FP Equity   IBE SQ Equity   INGA NA Equity  ISP IM Equity   ITX SQ Equity   KER FP Equity   LIN GY Equity   MC FP Equity    MUV2 GY Equity  NOKIA FH Equity     OR FP Equity    ORA FP Equity   PHIA NA Equity  SAF FP Equity   SAN FP Equity   SAN SQ Equity   SAP GY Equity   SIE GY Equity   SU FP Equity    TEF SQ Equity   URW NA Equity   VIV FP Equity   VOW3 GY Equity  SX5E Index
Dates                                                                                                                                                                                                   
2019-02-27 09:00:00     65.65   22.870  212.40  109.95  113.32  69.24   162.14  67.25   68.79   5.2590  73.67   67.18   43.275  27.60   21.830  53.00   82.42   26.680  14.455  106.60  5.314   14.065  15.220  49.910  48.500  25.805  7.372   11.338  2.0830  26.22   484.20  152.80  303.55  207.1   5.3660  222.0   13.320  35.105  118.40  72.93   4.1550  94.00   97.30   68.64   7.573   143.46  24.23   150.30  3279.78
2019-02-27 09:10:00     65.69   22.490  212.20  109.90  113.20  69.16   162.32  67.21   67.96   5.2310  73.59   67.16   43.045  27.53   21.840  52.82   82.50   26.700  14.485  107.05  5.320   14.065  15.212  49.925  48.530  25.700  7.364   11.300  2.0790  26.09   483.90  152.95  303.20  206.8   5.3540  221.6   13.360  35.095  118.75  72.95   4.1265  94.27   97.20   68.76   7.558   143.04  24.25   149.80  3275.70
...   ...
```

The last column is the index.

## Answer by Pepino (score 0)

https://quant.stackexchange.com/a/50832

You cannot have a strategy which will at the same time perform mean reversion and track an index.

Firstly, index replication is a long-only strategy. No names are shorted in an index. Secondly, every time your mean reversion strategy trades, it will deviate from the index, increasing the tracking error and eventually diverging from it.

What you could have is a portfolio constructed with the biggest weights of a index (your subset) and trade a mean reversion strategy on each of them, appropriately matching the size on each of the names with their weighting in the index, and rebalancing to mirror the index whenever necessary/appropriate.

Yet, this strategy will replicate (part of) the index only when all names are long.