GRS.test: A Test of the Efficiency of a Given Portfolio (Gibbons, Ross,...

In PengWu12245/EcoPleio: Testing Factor Models for Asset Returns: Based on Pleiotropy Model

Description

The test is for the following multivariate linear regression model:

y_j = α_j + ∑_{k=1}^q β_{jk}x_k + \varepsilon_j, j=1,...,p.

where y_j denotes the excess return on asset j; (x_1,...,x_q) is the excess return on the porfolio whose efficiency is being tested; and \varepsilon_j is the disturbance term for asset j. The disturbances are assumed to be jointly normally distributed with mean zero and nonsingular covariance matrix Σ, conditional on the excess returns for portfolios (x_1,...,x_q).

Usage

GRS.test(x, y)

Arguments

x	samples of predictor which is a n*q matrix.
y	samples of response which is a n*p vector.

Details

The test of the efficiency of a given portfolio is equivalent to the following hypothesis test.

• H0: all the Intercepts α are zero, i.e. α_j=0, \forall j=1,...,p.

• H1: otherwise.

Value

A list with the following elements:

• p.value: the p-value of the GRS test.

• grs.stat: the test statistic of the GRS test.

References

Gibbons, M. R., Ross, S. A. and Shanken, J. (1989). A test of the efficiency of a given portfolio. Econometrica, 57(5), 1121-1152.

Examples

## Quick example for the GRS test


set.seed(1)
n = 200; q = 2; p = 3
x <- matrix( rnorm(n*q), nrow = n)

# Generate data under H0
y <- matrix(NA, nrow = n, ncol = p)
y[,1] <- x[,1] + x[,2] + rnorm(n, sd = 0.5)
y[,2] <- x[,1] + 2 * x[,2] + rnorm(n, sd = 0.5)
y[,3] <- x[,1] + 3 * x[,2] + rnorm(n, sd = 0.5)
GRS.test(x, y)$p.value

# Generate data under H1
y <- matrix(NA, nrow = n, ncol = p)
y[,1] <- 1 + x[,1] + x[,2] + rnorm(n, sd = 0.5)
y[,2] <- 1 + x[,1] + 2 * x[,2] + rnorm(n, sd = 0.5)
y[,3] <- x[,1] + 3 * x[,2] + rnorm(n, sd = 0.5)
GRS.test(x, y)$p.value

PengWu12245/EcoPleio documentation built on Jan. 7, 2020, 12:23 a.m.