Question

I. Outline step-by-step the Johansen’s methodology for testing for cointegration between a set of variables in the context of a VAR.

II. Suppose you are interested in portfolio allocation across ten of the ghana stock exchange sectors. Using a step-by-step

procedure demonstrate how you would use PCA or factor analysis to allocate the portfolio.

Answer 1

I.

II.

Principle component analysis (PCA) is a widely used dimension reduction technique. Through finding structure in the covariance or correlation matrix we use this structure to locate low-dimensional subspaces containing most of the variation in the data.

PCA starts with a sample Y i = (Yi,1,...,Yi,d), i = 1,...,n, of d-dimensional random vectors with mean vector μ and covariance matrix Σ. One goal of PCA is finding “structure” in Σ. (Ruppert 443)

According to Haugh, in the context of risk management, we take this vector to represent the (normalized) changes, over some appropriately chosen time horizon, of an n-dimensional vector of risk factors. These risk factors could represent security price returns, returns on futures contracts of varying maturities, or changes in spot interest rates, again of varying maturities.

Let Y = (Y1 , . . . , Yn ) T denote an n-dimensional random vector with variancecovariance matrix, Σ. The goal of PCA is to construct linear combinations in such a way that:

Pi= �!"Yj ! !!! , for i=1,...,n

(1) The Pi’s are orthogonal so that E[Pi Pj] = 0 for i ≠ j, and

(2) The Pi’s are ordered so that: (i )P1 explains the largest percentage of the total variability in the system and (ii) each Pi explains the largest percentage of the total variability in the system that has not already been explained by P1, . . . , Pi−1. (Haugh 7)

According to Hauge, if the normalized random variables satisfy E[Yi] = 0 and Var(Yi) = 1 it is very common to apply PCA in practice. This is achieved by subtracting the means from the original random variables and dividing by their respective standard deviations. We do this to ensure that no single component of Y can influence the analysis by virtue of that component’s measurement units. We will therefore assume that the Yi’s have already been normalized.

The key tool of PCA is the spectral decomposition from linear algebra which states that any symmetric matrix A∈ ℝ!×! can be written A=Γ∆Γ!,

where (i) ∆ is a diagonal matrix, diag(λ1, ···,λn), of the eigenvalues of A

which, without loss of generality, are ordered so that

λ1 ≥λ2 ≥···≥λn, and (ii) Γ is an orthogonal matrix with the ith column of Γ containing the ith standardized eigen-vector, �!of A. The orthogonality of Γ implies Γ ΓT = ΓT Γ = In.

Since Σ is symmetric we can take A = Σ in (1), and the positive semidefiniteness of Σ implies λi ≥ 0 for all i = 1,...,n. The principal components of Y are then given by P = (P1,...,Pn) satisfying P = ΓT Y. (Haugh 8)

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