shall combine optimisation and statistical techniques, where for the latter we employ the bootstrapping re-sampling technique.
The remainder of the chapter is organised as follows. In Section 2 we give further insight into the bootstrapping technique. In Section 3 we introduce our formulation for the proposed method. Computational results are presented in Section 4 and finally in Section 5 we summarise and give concluding remarks.
5.2
Bootstrappping Procedure
Bootstrapping is a common statistical tool for generating an approximate sampling distribution of a statistic from one sample, in order to estimate a parameter. The idea was first introduced by the seminal work of Efron(1979) and has since become very popular due to its intuitiveness and the fact that no stringent conditions are attached to its application. More recently the popularity of this computationally intensive approach has increased due to technological advancements and availability of relatively cheap, powerful and efficient computers.
In this section we present the idea of the bootstrap approach from an application perspective by way of an example. Theoretical treatment of the subject can be found in Efron and Tibshirani (1993);Shao and Tu (1995) and Davison and Hinkley (1997).
Suppose that one is interested in the average height of some population of interest. However, due to some constraints, such as financial and time, only a fraction (n) of the entire population (m) is sampled with realisations x = {x1, x2, ..., xn} and where n < m. From this realisation one can calculate
a sample mean x = (Pni=1xi)/n. In order to make inference for the whole
5.2. Bootstrappping Procedure 93
distribution of likely values for the unobserved population mean, µ, and given this information a probabilistic statement can be formulated on the likely range of µ with a given degree of confidence. For instance,
Pr[ˆµL< µ < ˆµU] = 0.95 (5.1)
to indicate a 95% confidence that the true unknown mean µ would lie between ˆ
µL and ˆµU, where the subscript L and U denote lower and upper bounds
respectively and in the content of the example they represent 2.5 and 97.5 percentiles.
From the theory of statistics, the calculation of the lower and upper bounds depends on the assumption attached to the data. The most common of these assumptions are the data are identically distributed from a normal distribution with unknown mean µ and standard deviation σ, N(µ, σ). If the sample size n is large (n ≥ 30) then ˆµL and ˆµU forming a 100(1 − θ)%
confidence interval (θ ∈ (0, 1)) can be respectively calculated as x − z1−θ/2σ/ˆ
√ n and x + z1−θ/2σ/ˆ
√
n where zθ denotes the inverse cumulative distribution of a
Normal distribution at level θ and ˆσ2 = 1
n−1
Pn
i=1(xi− x)2 is an estimate of the
true but unknown σ2. Where the sample is small (n < 30) z
1−θ/2 is replaced by
student-t distribution with n − 1 degrees of freedom, t1−θ/2,n−1.
In order to obtain the limits ˆµL and ˆµU using the re-sampling technique,
the idea of a non-parametric bootstrap involves sampling with replacement a large number of times from the original sample, x. More specifically, one samples directly from x and obtains x∗ = {x∗
1, x∗2, ..., x∗n} followed by a
recalculation of x∗. By repeating this process a large number of times, say
1000, one is able to map out the distribution of x and from this the lower and upper bounds for a given confidence level can be calculated. That is, once
5.2. Bootstrappping Procedure 94
the bootstrapped samples {x1∗, x2∗, ..., x1000∗} are obtained they are sorted in
ascending order and the lower and upper limits ˆµLand ˆµU are respectively given
by the [n(θ/2)] and [n(1 − θ/2)] data points, where [a] denotes an integer part of a.
This idea forms the foundation of the methodology presented in the following discussion as it extends to constructing confidence intervals for unknown regression parameters as shall become apparent.
5.2.1
Portfolio selection using the bootstrapping technique
Liang et al. (1996) offered bootstrap simulation as a tool for quantifying the uncertainty in the composition of portfolios. They used this bootstrap simulation in an attempt to estimate the amount of real estate investors should hold to achieve optimum portfolio performance. The bootstrap method has shown itself to be useful in situations where the number of available data points is relatively small and the assumptions of parametric techniques do not hold. However, the confidence intervals produced were large.
Hatemi and Roca (2006) examined the simple case of international portfolio diversification involving the three largest stock markets in the world US, UK and Japan. Based on standard portfolio analysis, they first examined whether or not using diversification by US investors into the UK and Japanese markets would have been beneficial. They compared the risk-adjusted returns in the US market and that of the tangency portfolio consisting of the US, UK and Japanese markets. Then they undertook an analysis of the three markets based on a bootstrapping approach. They considered that the results of this study can be expanded to accommodate more markets and can also be done from the point of view of investors from the other markets. It can also be replicated on disaggregated scales.
5.2. Bootstrappping Procedure 95
Bartlmae (2009) introduced a framework for constructing portfolios, addressing two of the major problems of classical mean-variance optimization in practice: low diversification and sensitivity to information ambiguity. In order to address these issues, he used a bootstrapping method to incorporate the effects of input parameter variation. He investigated these methods by the use of Monte Carlo sampling. Firstly in order to overcome the problem of non-intuitive and undiversified portfolios, he introduced a method to construct portfolios that show a higher degree of diversification. He did this by introducing a diversification on the portfolio weights. In a second step, he applied bootstrapping to assess the input parameter ambiguity. By this method, more robust portfolios can be found. He incorporated these methods into a portfolio construction procedure.
Chen et al. (2012) applied the bootstrapping technique proposed by Kosowski et al. (2006) to examine whether the performance of enhanced-return index funds are based on luck or superior enhancing skills. They showed the advantages of using the bootstrap to rank fund performance. Their results show evidence of enhanced-return index funds with positive and significant alphas after controlling for luck and sampling variability.
Kopa (2012) considered robustness and bootstrap techniques in portfolio efficiency testing with respect to second-order stochastic dominance (SSD). He applied a computational method to test whether a US market portfolio, is (SSD) efficient with respect to 48 US industry representative portfolios. Moreover, he presented a robust version of a (SSD) portfolio efficiency test that allows for small errors in data and he analysed their impact on the market portfolio (SSD) efficiency. He improved his results, by applying the bootstrap technique to estimate the p-value of market portfolio (SSD) efficiency.