Repeat the process b times and obtain a bootstrapped

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- Repeat the process B times and obtain a bootstrapped sample of B parameter estimates; - We use this sample to estimate a confidence interval. 2. Dealing with data dependency: the main limitation of the bootstrap is that standard boostrap procedures presuppose that observations are independent, and they can be unreliable if this assumption does not hold. - If we are prepared to make parametric assumptions, we can model the dependence parametrically (e.g., using a GARCH procedure). We can then bootstrap from the residuals. However, the drawback of this solution is that is requires us to make parametric assumptions and of course presupposes that those assumptions are valid. - An alternative is to use a block approach: we divide sample data into non-overlapping blocks of equal length, and select a block at random. However, this approach can be tricky to implement and can lead to problems because it tends to ‘whiten’ the data. - A third solution is to modify the probabilities with which individual observations are chosen. Instead of assuming that each observation is chosen with the same probability, we can make the probabilities of selection dependent on the time indices of recently selected observations. T2 Principal Component Analysis 1. The theory: - Assume x is an mx1 random vector, with covariance matrix Σ , and let Λ be a diagonal matrix whose diagonal elements are eigenvalues of Σ , let A is the matrix of eigenvectors of Σ . Then T A A Σ = Λ - The principal component of x are the linear combinations of the individual x-variables produced by pre-multiplying x by A: p= Ax Æ the variance-covariance matrix of p, VC(p), is then: ( ) ( ) T VC p VC Ax A A = = Σ = Λ - since Λ is a diagonal matrix. Æ the different principal components are uncorrelated with each other. And the variances of principal components are given - 30 -
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Study Notes: Risk Management and Financial Institutions By Zhipeng Yan by the diagonal elements of Λ , the eigenvalues. - In addition, we can choose the order of our principal components so that the eigenvalues are in declining order. The first principal component therefore ‘explains’ more of the variability of our original data than the second principal component, and so on. - In short, the principal components of our m original variables are m artificial variables constructed so that the first principal component ‘explains’ as much as it can of the variance of these variables; the second principal component ‘explains’ as much as it can of the remaining variance, but it uncorrelated with the first component; and so forth. 2. Estimates of principal components based on historical data can be quite unstable. Some simple rules of thumb – such as taking moving averages of our principal components – can help to mitigate this instability. We also should be careful about using too many principal components. We only want to add principal components that represent stable relationships that are good for forecasting, and there will often come a point where additional principal components merely lead to the model tracking noise – and so undermine the forecasting ability of our model.
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