---
title: "Constant correlation test"
author: "Alexios Galanos"
date: "`r Sys.Date()`"
output: 
    rmarkdown::html_vignette:
        css: custom.css
        code_folding: show
        toc: true
link-citations: yes
bibliography-style: apalike
natbiboptions: round
bibliography: tsmarch.bib
vignette: >
  %\VignetteIndexEntry{Constant correlation test}
  %\VignetteEngine{knitr::rmarkdown}
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---

```{r setup, include = FALSE}
knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>"
)
```

# Background

@Engle2001 proposed a test for constant correlation:

$$
H_0: R_t = \bar{R} \quad \forall \, t \in T
$$

against

$$
H_a: \text{vech}^u(R_t) = \text{vech}^u(\bar{R}) + \beta_1\text{vech}^u(R_{t-1}) + \beta_2\text{vech}^u(R_{t-2}) + \cdots + \beta_p\text{vech}^u(R_{t-p})
$$

where $\text{vech}^u$ is a modified $\text{vech}$ 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 $\bar{R}$ (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:

$$
Y_t = \text{vech}^u \left[ (\bar{R}^{-1/2} D_t^{-1} r_t)(\bar{R}^{-1/2} D_t^{-1} r_t)^\prime - I_k \right]
$$

where $\bar{R}^{-1/2} D_t^{-1} r_t$ is a $k \times 1$ 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:

$$
Y_t = \alpha + \beta_1 Y_{t-1} + \cdots + \beta_s Y_{t-s} + \eta_t
$$

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 $T \times k$ 
vector of outer-products for each univariate regressor and the $T \times s + 1$ 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 $\chi^2_{(s+1)}$, with $\hat{\delta}$ the estimated regression 
parameters, $X$ the matrix of regressors (including the constant term) and $\hat\sigma^2$
the variance of the regression residuals.^[adjusted for the number of lags $s$ and
the constant term.]

# 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.

```{r mod_calc,highlight=TRUE}
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)
```

```{r print_test,warning=FALSE,message=FALSE}
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 exist^[see for example
@Mccloud2011 who also provide an excellent review.], and may be included in the future.

# References