# Journal of Time Series Analysis

## Detecting at‐Most‐m Changes in Linear Regression Models

### Journal Article

In this article, we provide a new procedure to test for at‐most‐ $\mathfrak{m}$ changes in the time‐dependent regression model ${y}_{t}={\mathbf{x}}_{t}^{\top }{\mathbit{\beta }}_{t}+{e}_{t},1⩽t⩽T$, that is, β1 = β2 = ⋯ = βT under the no‐change null hypothesis against the alternative ${y}_{t}={\mathbit{x}}_{t}^{\top }{\mathbit{\beta }}^{\left(i\right)}+{e}_{t},$ if ${k}_{i-1}^{\ast } and β(j) ≠ β() for some $1⩽j,\ell ⩽\mathfrak{m}+1$ with ${k}_{0}^{\ast }=0,1<{k}_{1}^{\ast }<{k}_{2}^{\ast }<\cdots <{k}_{\mathfrak{m}}^{\ast }. Our procedure is based on weighted sums of the residuals, incorporating the possibility of $\mathfrak{m}$ changes. The weak limit of the proposed test statistic is the sum of two double‐exponential random variables. A small Monte Carlo simulation illustrates the applicability of the limit results in case of small and moderate sample sizes. We compare the new method to the cumulative sum control chart (CUSUM) and standardized (weighted) CUSUM procedures and obtain the power curves of the test statistics under the alternative. We apply our method to find changes in the unconditional four‐factor capital asset pricing model.

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