hyptest: Test for Time-Varying Covariance via Local PCA and Bootstrap

In TVMVP: Time-Varying Minimum Variance Portfolio

Test for Time-Varying Covariance via Local PCA and Bootstrap

Description

This function performs a hypothesis test for time-varying covariance in asset returns based on Su and Wang (2017). It first standardizes the input returns and then computes a time-varying covariance estimator using a local principal component analysis (Local PCA) approach. The test statistic J_{pT} is computed and its significance is assessed using a bootstrap procedure. The procedure is available either as a stand-alone function or as a method in the 'TVMVP' R6 class.

Usage

hyptest(returns, m, B = 200, kernel_func = epanechnikov_kernel)

Arguments

returns	A numeric matrix of asset returns with dimensions T × p (time periods by assets).
m	Integer. The number of factors to extract in the local PCA. See determine_factors.
B	Integer. The number of bootstrap replications to perform. Default is 200
kernel_func	Function. A kernel function for weighting observations in the local PCA. Default is epanechnikov_kernel.

Details

Two usage styles:

# Function interface
hyptest(returns, m=2)

# R6 method interface
tv <- TVMVP$new()
tv$set_data(returns)
tv$determine_factors(max_m=5)
tv$hyptest()
tv
tv$get_bootstrap()         # prints bootstrap test statistics

When using the method form, if 'm' are omitted, they default to values stored in the object. Results are cached and retrievable via class methods.

The function follows the steps below:

1. Standardizes the returns.

2. Computes the optimal bandwidth using the Silverman rule.

3. Performs a local PCA on the standardized returns to extract local factors and loadings.

4. Computes a global factor model via singular value decomposition (SVD) to obtain global factors.

5. Calculates residuals by comparing the local PCA reconstruction to the standardized returns.

6. Computes a test statistic J_{pT} based on a function of the residuals and covariance estimates as:

   \hat{J}_{pT} = \frac{T p^{1/2} h^{1/2} \hat{M} - \hat{\mathbb{B}}_{pT}}{\sqrt{\hat{\mathbb{V}}_{pT}}},

   where:

   \hat{M} = \frac{1}{pT} \sum_{i=1}^p \sum_{t=1}^T \left(\hat{\lambda}_{it}' \hat{F}_t - \tilde{\lambda}_{i0}' \tilde{F}_t\right),

   \hat{\mathbb{B}}_{pT} = \frac{h^{1/2}}{T^2 p^{1/2}} \sum_{i=1}^p \sum_{t=1}^T \sum_{s=1}^T \left(k_{h,st} \hat{F}_s' \hat{F}_t - \tilde{F}_s' \tilde{F}_t\right)^2 \hat{e}_{is}^2,

   and

   \hat{\mathbb{V}}_{pT} = \frac{2}{p h T^2} \sum_{1\leq s \neq r \leq T} \bar{k}_{sr}^2 \left(\hat{F}_s' \hat{\Sigma}_F \hat{F}_r \right)^2 \left(\hat{e}_r' \hat{e}_s \right)^2.

7. A bootstrap procedure is then used to compute the distribution of J_{pT} and derive a p-value.

The function prints a message indicating the strength of evidence for time-varying covariance based on the p-value.

Value

A list containing:

J_NT	The test statistic J_{pT} computed on the original data.
p_value	The bootstrap p-value, indicating the significance of time variation in covariance.
J_pT_bootstrap	A numeric vector of bootstrap test statistics from each replication.

References

Su, L., & Wang, X. (2017). On time-varying factor models: Estimation and testing. Journal of Econometrics, 198(1), 84–101

Examples

# Simulate some random returns (e.g., 100 periods, 30 assets)
set.seed(123)
returns <- matrix(rnorm(100*30, mean = 0, sd = 0.02), nrow = 100, ncol = 30)

# Test for time-varying covariance using 3 factors and 10 bootstrap replications
test_result <- hyptest(returns, m = 3, B = 10, kernel_func = epanechnikov_kernel)

# Print test statistic and p-value
print(test_result$J_NT)
print(test_result$p_value)

# Or use R6 method interface
tv <- TVMVP$new()
tv$set_data(returns)
tv$determine_factors(max_m=5)
tv$hyptest(iB = 10, kernel_func = epanechnikov_kernel)
tv
tv$get_bootstrap()         # prints bootstrap test statistics

TVMVP documentation built on June 28, 2025, 1:08 a.m.