## 3 Methodology
In total, there are eleven portfolios in our experiment with different covariance estimators, including:  
* the sample covariance matrix (denoted as sample);
* the shrinkage to the identity matrix (denoted as STIM): this is the linear shrinkage estimator which combines the sample covariance matrix and the identity matrix by an optimal shrinkage intensity [17];
* the shrinkage to single-index model (denoted as SSIM): this is the linear shrinkage estimator which uses the single-index model of [22] as the target matrix and then combines with the sample covariance by an optimal shrinkage intensity [18];
* The Shrinkage to constant correlation matrix (denoted as SCCM): this is the linear shrinkage estimator, a target matrix assumes the correlations of each pair of stocks are the same to estimate a constant-correlation covariance matrix and then combine with the sample covariance matrix by an optimal shrinkage intensity [19];
* the combination of the sample matrix, the diagonal matrix and the single-index model (denoted as sTS): this is a portfolio with covariance matrix as a combination of basic elements, including the sample matrix, the diagonal matrix and the single-index model, then simply take an equally average instead of optimal weighting [5];the combination of the sample matrix, the single-index model and the constant correlation matrix (denoted as sSC): this is a portfolio with covariance matrix as a combination of basic elements, including the sample matrix, the single-index model and the constant correlation matrix, then simply take an equally average instead of optimal weighting [6];
* the combination of the sample matrix, the diagonal matrix and the constant correlation matrix (denoted as sTC): this is a portfolio with covariance matrix as a combination of basic elements, including the sample matrix, the diagonal matrix and the constant correlation matrix, then simply take an equally average instead of optimal weighting;
* given one of the following estimators: the sample covariance estimator, the linear shrinkage to the identity matrix estimator, the linear shrinkage to the single-index model estimator and the linear shrinkage to constant correlation estimator; we apply our method on a given estimator to build our GMVPs, they are denoted as \(k\)-sample, \(k\)-STIM, \(k\)-SSIM and \(k\)-SCCM respectively. A backtesting process for the portfolios above is as follows: given one of the above covariance estimators, at the end of each week a GMVP is built on the latest data then at the first trading day of next week, it will execute and match by the closing prices of that day. We hold it until the weekend then re-compute and rebalance the portfolio: in other words, the out-of-sample period is one week. Particularly, the training data for the covariance estimator is two-year weekly data: in other words, the in-sample period is two years. Considering the data quality, at each date, we use only the assets which have at least a half of non-null data: in other words, one-year after their IPOs. In our backtesting process, to evaluate a portfolio precisely, we follow real regulations and restrictions of the market. Firstly, instead of assuming no impact of commission, we apply a real commission fee on every transaction. Secondly, we use a long-term interest rate of government bonds in the portfolio performance calculations. Thirdly, we limit the maximum number of shares that could be traded at each date but to simplify the process, we assume that we could long/short up to the real traded volume in the dataset and also there is no slippage in our transactions. Last but not least, to avoid a lookahead bias, at the end of each date the latest data we can use is the historical data up to that date and then the orders will fit with the next date's prices.