Changepoint Detection in Heteroscedastic Random Coefficient Autoregressive Models

Abstract

We propose a family of CUSUM-based statistics to detect the presence of changepoints in the deterministic part of the autoregressive parameter in a Random Coefficient Autoregressive (RCA) sequence. Our tests can be applied irrespective of whether the sequence is stationary or not, and no prior knowledge of stationarity or lack thereof is required. Similarly, our tests can be applied even when the error term and the stochastic part of the autoregressive coefficient are non iid, covering the cases of conditional volatility and shifts in the variance, again without requiring any prior knowledge as to the presence or type thereof. In order to ensure the ability to detect breaks at sample endpoints, we propose weighted CUSUM statistics, deriving the asymptotics for virtually all possible weighing schemes, including the standardized CUSUM process (for which we derive a Darling-Erdős theorem) and even heavier weights (so-called Rényi statistics). Simulations show that our procedures work very well in finite samples. We complement our theory with an application to several financial time series.

Keywords:

• Changepoint problem
• Heteroscedasticity
• Nonstationarity
• Random coefficient autoRegression
• Weighted CUSUM process

Supplementary Materials

All technical lemmas and proofs, further results, Monte Carlo experiments and further data analysis are relegated to the supplementary material.

Notes

1 See also, in the context of sequential monitoring, the related contributions by Na, Lee, and Lee (2010), Li, Tian, and Qi (2015), and Li et al. (2015).

2 We use ${(1+y_{i-1}^2)}^{-1}$ as weights. The rationale for this choice is based on considering the “error term” ${ϵ}_{i,1}{y}_{i-1}+{ϵ}_{i,2}$, whose (conditional) variance is $\text{var}({ϵ}_{0,1})y_{i-1}^2+\text{var}({ϵ}_{0,2})$, whence the weights containing the $y_{i-1}^2$ term. Koul and Schick (1996) show that the WLS estimator is first order equivalent to MLE. The use of the weight function ${(1+y_{i-1}^2)}^{-1}$ is proposed in Schick (1996), where it is also shown that efficiency is attained when using $\text{var}({ϵ}_{0,1})$ and $\text{var}({ϵ}_{0,2})$, or consistent estimators thereof. However, in our context we also allow for nonstationarity, and in this case $\text{var}({ϵ}_{0,2})$ cannot be estimated consistently (see e.g., Lemma A10 in Horváth and Trapani 2019).

3 In Section B in the supplementary materials, we report further experiments, including using demeaning as described in Section 3.3.

4 When using (3.19), we use L = 200. Results are however not particulary sensitive to this specification.

5 As far as housing prices are concerned, we use the daily data constructed by Bollerslev, Patton, and Wang (2016), and we refer to that paper for a description of the datasets. We have focused on the housing markets in Boston and Los Angeles, following a similar analysis (albeit carried out with monthly data and a different time span) in Horváth, Liu, and Lu (2021). For completeness, we consider eight further cities in Section C in the supplementary materials.