Background

Engle and Sheppard (2001) proposed a test for constant correlation:

H₀ : R_t = R̄  ∀ t ∈ T

against

H_a : vech^u(R_t) = vech^u(R̄) + β₁vech^u(R_t − 1) + β₂vech^u(R_t − 2) + ⋯ + β_pvech^u(R_t − p)

where vech^u is a modified vech that only selects elements above the diagonal. The testing procedure first estimates a constant correlation model, with first stage GARCH processes whose standardized residuals are whitened using the symmetric square root decomposition of the matrix R̄ (estimated in the second stage).

Under the null of constant correlation, these residuals should be i.i.d. with a variance-covariance matrix given by I_k, the identity matrix. Consider the artificial regressions of the outer products of the residuals on a constant and their lags, in stacked form:

Y_t = vech^u[(R̄^−1/2D_t⁻¹r_t)(R̄^−1/2D_t⁻¹r_t)^′ − I_k]

where R̄^−1/2D_t⁻¹r_t is a k × 1 vector of residuals jointly standardized under the null, with D_t the diagonal matrix of time varying standard deviations from the first stage GARCH process. The (vector) autoregression with s lags is given by:

Y_t = α + β₁Y_t − 1 + ⋯ + β_sY_t − s + η_t

Under the null, the intercept and all of the lag parameters in the model should be zero. In order to estimate the test statistic, all that is necessary is to make the T × k vector of outer-products for each univariate regressor and the T × s + 1 matrix of regressors for each set of regressors. Then the parameters can be estimated by stacking the k(k − 1)/2 vector of regressands and regressors and performing a seemingly unrelated regression.

The test can then be conducted as:

$$ \frac{\hat{\delta}^\prime X^\prime X \hat{\delta}}{\hat{\sigma}^2} $$

which is asymptotically χ_(s + 1)², with δ̂ the estimated regression parameters, X the matrix of regressors (including the constant term) and σ̂² the variance of the regression residuals.¹

Demo

We use 10 global indices from the globalindices dataset to illustrate the test and output which has both a print and as_flextable method for nice printing.

suppressMessages(library(tsmarch))
suppressMessages(library(xts))
data(globalindices)
Sys.setenv(TZ = "UTC")
train_set <- 1:1600
series <- 1:10
y <- as.xts(globalindices[, series])
train <- y[train_set,]
test <- escc_test(y, lags = 2, constant = TRUE)

as_flextable(test, include.decision = TRUE, use.symbols = TRUE, footnote.reference = TRUE)

Not surprisingly, the test rejects the null of constant correlation, though some caution should be exercised in the presence of structural breaks which may lead to invalid inferences.

Conclusion

The constant correlation test is a simple test, but also quite restrictive in that it ignores structural breaks and is really only optimal within a specific class of time varying conditional correlations. More robust tests do exist², and may be included in the future.