State-dependent stock selection in index tracking: a machine learning approach

For example, at the end of August 2019, passive US stock funds managed $4.27 trillion—slightly more than the $4.25 trillion in actively managed US stock funds (www.​reuters.​com). This means index funds now control half the US stock mutual fund market.

For example, see Malkiel (1995), Moskowitz (2000), Kosowski (2011) and Kacperczyk et al. (2014), among others.

For example, Alexander and Dimitriu (2004) propose a model to compose an optimal tracking portfolio using principal component analysis in the stock returns. Focardi and Fabozzi (2004) choose a clustering-based methodology to compose an optimal tracking portfolio, in which Euclidean distances among stock price series are employed for clustering. Corielli and Marcellino (2006) use several factors by employing a heuristic selection to compose a tracking portfolio that discards long-term factors that could result in tracking error. Wu et al. (2007) use a goal programming approach by defining appropriate values for both the tracking error and its volatility. Canakgoz and Beasley (2009) and Li et al. (2011) use multi-objective optimization to maximize return and minimize tracking error. One of the most common approaches is the cointegration technique, in which the index is developed based on historical data on the long-term relationships between the subsets of stocks constituting the portfolio (e.g., Alexander and Dimitriu (2002, 2005a).

Other studies that use market capitalization include Meade and Salkin (1990), Bamberg and Wagner (2000), among others.

Some of the economic variables for mean predictability include term spread (Campbell 1987; Fama and French 1988, 1989; Ferson and Harvey (1991)), default spread (Fama and French 1988, 1989; Keim and Stambaugh 1986) and Treasury bill yield (Fama and Schwert 1977; Ferson and Harvey 1991). The variables for predicting variances include lagged squared return and/or lagged variance (Bollerslev 1986; Engle 1982; French et al. 1987; Harvey 2001; Schwert 1989; Whitelaw 1994), default spread (Harvey 2001; Whitelaw 1994), dividend yield (Harvey 2001) and debt-to-equity ratio (Schwert 1989). Finally, the predictability of covariances is attributed to variables such as lagged covariances, lagged cross-products of returns (Bollerslev et al. 1988), term spread (Campbell 1987; Harvey 2001), default spread and dividend yield (Harvey 2001).

Correlation can be very sensitive to the presence of outliers and is only a short-term statistic and lacks stability. However, since both cointegration and correlation approaches stand out for the index tracking problem, Alexander and Dimitriu (2005a) present a study focused on comparing the two methods, and the results are similar for both. Sant’Anna et al. (2016) attempt to compare correlation and cointegration using IT strategy. This comparison is similar to that carried out by Alexander and Dimitriu (2005a) for IT and long-short strategies.

A sigmoid function that generates output values between 0 and 1.

Some of these state variables include Term Spread (Campbell 1987; Fama and French 1988, 1989; Ferson and Harvey 1991), Default Spread (Fama and French 1988, 1989; Keim and Stambaugh 1986) and Treasury Bill Yield (Fama and Schwert 1977; Ferson and Harvey 1991). Default Spread (Harvey 2001; Whitelaw 1994), Dividend Yield (Harvey 2001) and Debt-to-Equity ratio (Schwert 1989).

We de-seasonalize Unemployment Rate, Industrial Production Index and Consumer Price Index by applying the first difference method.

The DAX, as the Germany's primary stock index, includes the 30 largest companies that trade on the Frankfurt Stock Exchange.

Our findings are not sensitive to the length of in- or out-of-sample periods. Results based on various choices are available upon request.

We follow previous studies and use different numbers of stocks in TPs to show how our proposed approach works in different size scenarios. The exact number of stocks used in tracking portfolios varies in these studies, even for the same index. For example, Alexander and Dimitriu (2005a) use 20, 25 and 30 stocks to track DJIA index which includes 30 stocks. Filbeck and Visscher (1997) select ten stocks to track FTSE 100, whereas Rafaely and Bennell (2006) use between five to 30 stocks to track the same index. Meade and Salkin (1990) use 25 stocks to track FT500 index, and Zheng et al. (2020) use 30, 40 and 50 stocks to construct their tracking portfolios to track S&P 500 index.