We consider an infinite-order long-memory heterogeneous autoregressive (HAR) model, which is motivated by a long-memory property of realized volatilities (RVs), as an extension of the finite order HAR-RV model. We develop bootstrap tests for structural mean or variance changes in the infinite-order HAR model via stationary bootstrapping. A functional central limit theorem is proved for stationary bootstrap sample, which enables us to develop stationary bootstrap cumulative sum (CUSUM) tests: a bootstrap test for mean break and a bootstrap test for variance break. Consistencies of the bootstrap null distributions of the CUSUM tests are proved. Consistencies of the bootstrap CUSUM tests are also proved under alternative hypotheses of mean or variance changes. A Monte-Carlo simulation shows that stationary bootstrapping improves the sizes of existing tests.
Corsi (2009) and Hwang and Shin (2014) recently proposed autoregressive models called heterogeneous autoregressions (HAR) of realized volatility (RV) to address the long-memory properties of financial market volatilities. Corsi (2004, 2009) proposed an additive cascade model having three volatility components defined over three different time periods, called the HAR(3) model. The HAR(3) model has been shown to successively achieve the purpose of reproducing the main empirical features of financial return volatilities such as long memory, fat tails and self-similarity. However, as noted by Corsi (2009), the HAR(3) model has a short memory with a exponentially decreasing autocorrelation function (ACF) because it can be expressed as a stationary AR(22) model. Hwang and Shin (2014) proposed a genuine long-memory HAR model with algebraically decreasing ACF, an infinite-order HAR(∞) model, as an extension of the Corsi (2009)’s HAR(3) model. They characterized stationary conditions for the model, provided some probability theories and statistical methods in terms of consistency and limited the normality of the ordinary least squares estimator (OLSE) and forecasting.
Long memory of realized volatility is occasionally accompanied by structural changes. The problem of testing for structural changes has been a most important issue in time series regression or dynamic economic models. For this purpose, cumulative sum (CUSUM) tests have been widely used because the change-points are not known. See Brown
All the break tests for the HAR model except the CUSUMSQ test of Hwang and Shin (2015) have undesirable size distortions. The aim of this paper is to develop bootstrap tests for mean or variance changes in the HAR(∞) model, which remedy the size distortion problem. Based on the result of Lee (2014), we establish a bootstrap FCLT. Block bootstrapping methods are more well-suited than identically distributed (iid) bootstrapping because realized volatilities have long memories. Among the various block bootstrapping methods used, we consider the stationary bootstrapping (SB) of Politis and Romano (1994). The SB is one of the most widely adopted block bootstrapping methods for the dependent samples and is characterized by geometrically distributed random block lengths.
The partial sum process of the SB sample is shown to converge to the standard Brownian motion that enables us to construct SB CUSUM tests: a bootstrap mean break test and a bootstrap variance break test. Asymptotic critical values can be obtained from the stationary bootstrap distribution of the CUSUM tests. Consistencies of the null bootstrapping distributions of the CUSUM tests are proved. Consistencies of the bootstrapping CUSUM tests are also proved under alternative hypotheses of mean or variance breaks.
A Monte-Carlo experiment is conducted to show that SB significantly improves the sizes of the CUSUM tests of Lee (2014) for mean break and for variance break, which are badly sized in a finite sample. It also shows some improvement of the CUSUM test of Hwang and Shin (2013) for the mean break. The size improvement is achieved without power loss.
The remaining of the paper is organized as follows. The HAR models are described and Section 2 presents the existing results Section 3 discusses the main results, including the SB functional central theorem and the bootstrap CUSUM tests. Section 4 deals with the Monte-Carlo study and Section 5 gives the concluding remarks. The
First, we describe the 3
where
As an extension of Corsi’s model (
where
Here, we review the discussion of Hwang and Shin (2014) for the HAR(∞) process
with
The coefficients
with
We refer to Remarks 1 and 2 of Hwang and Shin (2014) for a necessary and sufficient condition of (A1) and for the absolutely summability of φ
For the long memory property of
For generic constants
According to Hwang and Shin (2014), the long-memory property of the HAR(∞) model has been investigated, which are stated in Propositions 2 and 3. In particular, Proposition 3 tells us that, given
Estimation theories were provided by Hwang and Shin (2014) in which the infinite-order HAR model in (
Recently, Lee (2014) has established a FCLT for the HAR(∞) model by showing that
We say that {
where
The FCLT in Proposition 4 requires conditions on
Let a data set {
The first strategy was considered by Lee (2014). Applying the FCLT in Proposition 4, Lee (2014) considered the CUSUM test for mean break and the CUSUM test for the variance break as well as derived the limiting distributions of the CUSUM tests. The mean break test and variance break test are
respectively, where
is a consistent estimator of the long-run variance lim
The second strategy was considered by Hwang and Shin (2013, 2015). The HAR(
as given by
where
The tests
Large values of
whose distribution function is given by
where
We construct SB tests and prove their asymptotic validity. Assume that sample {
SB versions (
Consistencies of the null bootstrapping distributions of (
We briefly describe how to construct SB sample from the original sample. Let {
The stationary bootstrap versions of
where
In order to prove consistencies of the null distributions of
for 0 ≤
and
Let
The SB versions
Let
for 0 ≤
Thanks to the consistencies in Theorems 2 and 4, asymptotic critical values of mean break tests
Let
Consistencies of the stationary bootstrap tests will be proved under alternative hypotheses of a single mean break at time
or of a variance break at time
In the followings, Pr^{*} denotes the conditional probability given on sample {
A simulation experiment is conducted to investigate finite sample sizes and powers of the proposed tests for breaks in the memory parameters
where
The normal errors
and the variance break test statistics
For computing the stationary bootstrap tests
Tables 2 and 3 report sizes and powers of the mean break tests and volatility break tests, respectively, which are based on 1,000 independent replications with
SB improves sizes of
Among the four mean break tests
Among the four variance break tests
We next investigate the performance of the
For mean break test, we see, for
For the mean break test, we see the power of the test with
For break test, there is no significance difference in size and power performances of tests with
From this experiment, we can say that SB considerably improve the sizes of the existing tests, especially mean break tests, without power loss. No substantial improvement is achieved for the variance break test
We established the stationary bootstrap functional central limit theorem (FCLT) for the HAR(∞) model, which is a genuine long-memory model for realized volatility in financial economics. The bootstrap version of the cumulative sum of the HAR(∞) process is shown to converge to the standard Brownian motion. Applying the FCLT, under the null hypothesis of no break, we have established consistencies of the bootstrap null distributions of the SB CUSUM tests for structural mean change and for structural variance change. Consistencies of the stationary bootstrap tests are also established under alternative hypotheses of mean break and of variance break. Monte-Carlo simulation shows that SB improves size performance of existing tests especially for mean break tests.
This study was supported by grants from the National Research Foundation of Korea (2016R1A2B400 8780, NRF-2015-1006133).
Proof is given in a similar way to that of Theorem 1 of Parker
where
For
where ∑ = ((
Verifications of (
It is obvious from the result in Theorem 1.
Let
Following the same arguments as in Step 1 of the proof of Theorem 2.4 by Hwang and Shin (2013), together with the asymptotic normality of the stationary sequence {
where
The model (
where
where
where
To prove the first convergence in distribution for
and we observe, by (
It is clear that
To prove the second convergence in distribution for
and letting
where
and
It is obvious from the results in Theorem 3.
Let
First, for
We show the second limiting in (
where
Noting that
For
By the assumption, 0
Also in case that
Thus
Note that since 1 −
for some positive
Parameters for data generating process (DGP)
DGP | ||||||||
---|---|---|---|---|---|---|---|---|
0.370 | 0.222 | 0.133 | 0.080 | 0.048 | 0.029 | 0.017 | ||
0.173 | 0.155 | 0.140 | 0.126 | 0.113 | 0.102 | 0.092 | ||
0.372 | 0.343 | 0.224 | ||||||
0.039 | 0.412 | 0.361 |
Rejection rates (%) of the level 5% mean break tests
DGP | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Size | Power | Size | Power | |||||||||||||||
HAR(7) | 1,000 | 0 | 92.5 | 4.6 | 3.2 | 3.9 | 99 | 29 | 39 | 25 | 95.7 | 4.7 | 3.1 | 3.7 | 100 | 29 | 39 | 25 |
20 | 7.8 | 3.6 | - | - | 50 | 26 | - | - | 8.4 | 3.6 | - | - | 52 | 26 | - | - | ||
80 | 2.4 | 4.5 | - | - | 26 | 28 | - | - | 2.4 | 4.4 | - | - | 26 | 28 | - | - | ||
160 | 0.6 | 4.8 | - | - | 2 | 18 | - | - | 0.5 | 4.9 | - | - | 2 | 18 | - | - | ||
2,000 | 0 | 94.5 | 3.9 | 5.9 | 4.4 | 100 | 59 | 71 | 56 | 96.5 | 4.2 | 5.9 | 4.2 | 100 | 59 | 70 | 56 | |
20 | 8.7 | 3.5 | - | - | 76 | 49 | - | - | 10.0 | 3.2 | - | - | 77 | 49 | - | - | ||
80 | 2.8 | 3.2 | - | - | 58 | 50 | - | - | 2.8 | 3.3 | - | - | 59 | 50 | - | - | ||
160 | 2.8 | 5.4 | - | - | 39 | 45 | - | - | 2.7 | 5.5 | - | - | 39 | 45 | - | - | ||
4,000 | 0 | 94.7 | 4.2 | 5.1 | 4.2 | 100 | 89 | 95 | 89 | 98.1 | 4.4 | 5.1 | 4.2 | 100 | 89 | 95 | 89 | |
20 | 6.9 | 2.5 | - | - | 97 | 85 | - | - | 8.5 | 2.2 | - | - | 97 | 85 | - | - | ||
80 | 3.6 | 3.9 | - | - | 93 | 87 | - | - | 3.6 | 3.9 | - | - | 94 | 87 | - | - | ||
160 | 4.3 | 5.5 | - | - | 88 | 84 | - | - | 4.2 | 5.7 | - | - | 88 | 84 | - | - | ||
DGP | ||||||||||||||||||
Size | Power | Size | Power | |||||||||||||||
Historic HAR(3) | 1,000 | 0 | 100.0 | 13.8 | 1.1 | 4.2 | 100 | 55 | 3 | 7 | 98.8 | 8.0 | 2.1 | 3.3 | 100 | 41 | 11 | 11 |
20 | 67.7 | 9.7 | - | - | 93 | 50 | - | - | 44.2 | 6.3 | - | - | 85 | 40 | - | - | ||
80 | 13.7 | 7.6 | - | - | 57 | 34 | - | - | 6.6 | 5.2 | - | - | 42 | 30 | - | - | ||
160 | 0.1 | 3.4 | - | - | 0 | 11 | - | - | 0.1 | 4.0 | - | 0.0 | 1 | 14 | - | - | ||
2,000 | 0 | 100.0 | 13.0 | 3.6 | 5.2 | 100 | 80 | 17 | 26 | 99.7 | 7.3 | 4.9 | 5.7 | 100 | 68 | 49 | 47 | |
20 | 74.0 | 6.9 | - | - | 99 | 65 | - | - | 46.4 | 4.2 | - | - | 96 | 53 | - | - | ||
80 | 17.4 | 3.6 | - | - | 84 | 51 | - | - | 7.0 | 2.6 | - | - | 73 | 46 | - | - | ||
160 | 3.9 | 4.6 | - | - | 52 | 43 | - | - | 2.7 | 4.8 | - | - | 43 | 44 | - | - | ||
4,000 | 0 | 100.0 | 10.6 | 3.3 | 5.5 | 100 | 96 | 74 | 80 | 100.0 | 6.9 | 4.5 | 5.1 | 100 | 93 | 89 | 87 | |
20 | 70.8 | 2.2 | - | - | 100 | 83 | - | - | 42.2 | 1.8 | - | - | 100 | 79 | - | - | ||
80 | 15.5 | 2.8 | - | - | 99 | 82 | - | - | 7.9 | 2.9 | - | - | 97 | 83 | - | - | ||
160 | 7.4 | 3.8 | - | - | 93 | 82 | - | - | 5.6 | 4.1 | - | - | 91 | 83 | - | - |
Note: Number of replications = 1,000, number of bootstrap replication = 1,000.
DGP = data generating process; HAR = heterogeneous autoregressive.
Rejection rates (%) of the level 5% variance break tests
DGP | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Size | Power | Size | Power | |||||||||||||||
1,000 | 0 | 54.0 | 3.9 | 4.1 | 3.4 | 65 | 9 | 32 | 19 | 66.0 | 3.7 | 4.0 | 3.8 | 73 | 7 | 32 | 19 | |
20 | 5.0 | 4.3 | - | - | 13 | 8 | - | - | 5.1 | 4.2 | - | - | 10 | 7 | - | - | ||
80 | 2.4 | 4.7 | - | - | 5 | 8 | - | - | 2.4 | 4.8 | - | - | 4 | 7 | - | - | ||
160 | 0.4 | 5.7 | - | - | 1 | 8 | - | - | 0.2 | 5.8 | - | - | 1 | 8 | - | - | ||
cline2-22 | 2,000 | 0 | 55.6 | 3.1 | 4.9 | 3.1 | 79 | 16 | 61 | 45 | 66.9 | 2.9 | 4.9 | 3.1 | 83 | 13 | 61 | 45 |
HAR(7) | 20 | 4.0 | 2.3 | - | - | 21 | 11 | - | - | 4.1 | 2.6 | - | - | 18 | 9 | - | - | |
80 | 3.1 | 3.5 | - | - | 13 | 14 | - | - | 2.9 | 3.3 | - | - | 11 | 11 | - | - | ||
160 | 1.7 | 4.4 | - | - | 9 | 15 | - | - | 1.7 | 4.0 | - | - | 7 | 13 | - | - | ||
cline2-22 | 4,000 | 0 | 60.1 | 3.3 | 3.9 | 3.5 | 88 | 31 | 89 | 80 | 72.3 | 3.2 | 3.9 | 3.2 | 89 | 26 | 89 | 80 |
20 | 4.8 | 2.9 | - | - | 39 | 27 | - | - | 5.2 | 2.9 | - | - | 33 | 22 | - | - | ||
80 | 4.0 | 4.0 | - | - | 34 | 30 | - | - | 4.0 | 4.0 | - | - | 28 | 26 | - | - | ||
160 | 2.9 | 4.3 | - | - | 28 | 29 | - | - | 2.6 | 4.1 | - | - | 25 | 26 | - | - | ||
DGP | ||||||||||||||||||
Size | Power | Size | Power | |||||||||||||||
Historic HAR(3) | 1,000 | 0 | 98.5 | 7.4 | 4.1 | 2.7 | 98 | 7 | 30 | 16 | 47.1 | 4.0 | 4.1 | 2.8 | 64 | 8 | 31 | 16 |
20 | 38.7 | 6.5 | - | - | 38 | 8 | - | - | 14.2 | 4.5 | - | - | 25 | 10 | - | - | ||
80 | 3.4 | 4.9 | - | - | 4 | 5 | - | - | 2.2 | 4.6 | - | - | 6 | 7 | - | - | ||
160 | 0.1 | 4.0 | - | - | 0 | 4 | - | - | 0.7 | 3.7 | - | - | 1 | 7 | - | - | ||
2,000 | 0 | 99.6 | 6.0 | 3.9 | 2.3 | 100 | 7 | 60 | 41 | 60.1 | 3.9 | 4.2 | 2.8 | 77 | 15 | 60 | 42 | |
20 | 35.8 | 2.3 | - | - | 46 | 5 | - | - | 11.8 | 1.2 | - | - | 37 | 12 | - | - | ||
80 | 6.4 | 3.9 | - | - | 7 | 4 | - | - | 3.5 | 3.6 | - | - | 16 | 12 | - | - | ||
160 | 1.2 | 2.7 | - | - | 2 | 5 | - | - | 1.8 | 4.0 | - | - | 8 | 13 | - | - | ||
4,000 | 0 | 99.9 | 5.5 | 3.6 | 2.3 | 100 | 9 | 88 | 78 | 66.0 | 3.2 | 3.7 | 2.8 | 87 | 29 | 88 | 79 | |
20 | 40.9 | 2.1 | - | - | 46 | 3 | - | - | 14.9 | 2.2 | - | - | 55 | 20 | - | - | ||
80 | 7.3 | 2.4 | - | - | 11 | 5 | - | - | 5.3 | 3.7 | - | - | 33 | 26 | - | - | ||
160 | 2.0 | 3.2 | - | - | 5 | 5 | - | - | 2.1 | 3.1 | - | - | 25 | 25 | - | - |
Note: Number of replications = 1,000, number of bootstrap replication = 1,000.
DGP = data generating process; HAR = heterogeneous autoregressive.
Rejection rates (%) of the level 5% mean break tests
Size | Power | Size | Power | Size | Power | Size | Power | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
2 | 1,000 | 6.0 | 3.0 | 46 | 38 | 6.8 | 3.0 | 48 | 37 | 30.7 | 20.3 | 71 | 66 | 34.3 | 12.9 | 80 | 61 |
2,000 | 8.2 | 5.1 | 76 | 67 | 8.1 | 5.1 | 77 | 66 | 39.5 | 21.2 | 94 | 87 | 43.6 | 13.8 | 95 | 82 | |
4,000 | 7.4 | 4.1 | 96 | 95 | 7.6 | 3.9 | 96 | 94 | 39.5 | 18.4 | 100 | 99 | 43.7 | 11.7 | 100 | 97 | |
3 | 1,000 | 3.2 | 3.9 | 39 | 25 | 3.1 | 3.7 | 39 | 25 | 1.1 | 4.2 | 3 | 7 | 2.1 | 3.3 | 11 | 11 |
2,000 | 5.9 | 4.4 | 71 | 56 | 5.9 | 4.2 | 70 | 56 | 3.6 | 5.2 | 17 | 26 | 4.9 | 5.7 | 49 | 47 | |
4,000 | 5.1 | 4.2 | 95 | 89 | 5.1 | 4.2 | 95 | 89 | 3.3 | 5.5 | 74 | 80 | 4.5 | 5.1 | 89 | 87 | |
4 | 1,000 | 3.3 | 3.3 | 35 | 36 | 3.5 | 3.0 | 36 | 37 | 1.7 | 2.1 | 2 | 2 | 1.7 | 3.0 | 4 | 5 |
2,000 | 5.1 | 5.2 | 68 | 65 | 5.4 | 4.8 | 70 | 65 | 2.6 | 3.1 | 8 | 12 | 2.8 | 3.9 | 26 | 36 | |
4,000 | 4.4 | 3.7 | 95 | 94 | 4.5 | 3.9 | 95 | 94 | 2.7 | 4.0 | 57 | 72 | 3.4 | 4.3 | 83 | 88 |
Note: Number of replications = 1000, number of bootstrap replication = 1000.
HAR = heterogeneous autoregressive.
Rejection rates (%) of the level 5% variance break tests
Size | Power | Size | Power | Size | Power | Size | Power | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
2 | 1,000 | 5.3 | 3.6 | 31 | 25 | 5.2 | 3.7 | 31 | 26 | 5.3 | 3.1 | 31 | 23 | 5.5 | 3.3 | 31 | 25 |
2,000 | 4.5 | 4.0 | 59 | 53 | 4.7 | 3.7 | 59 | 53 | 4.8 | 3.5 | 58 | 49 | 5.2 | 3.9 | 59 | 50 | |
4,000 | 5.3 | 4.6 | 89 | 88 | 5.3 | 4.5 | 89 | 88 | 5.3 | 3.7 | 88 | 84 | 5.3 | 4.4 | 88 | 86 | |
3 | 1,000 | 4.1 | 3.4 | 32 | 19 | 4.0 | 3.8 | 32 | 19 | 4.1 | 2.7 | 30 | 16 | 4.1 | 2.8 | 31 | 16 |
2,000 | 4.9 | 3.1 | 61 | 45 | 4.9 | 3.1 | 61 | 45 | 3.9 | 2.3 | 60 | 41 | 4.2 | 2.8 | 60 | 42 | |
4,000 | 3.9 | 3.5 | 89 | 80 | 3.9 | 3.2 | 89 | 80 | 3.6 | 2.3 | 88 | 78 | 3.7 | 2.8 | 88 | 79 | |
4 | 1,000 | 4.9 | 3.7 | 31 | 26 | 5.2 | 3.7 | 31 | 26 | 5.5 | 3.1 | 28 | 21 | 5.3 | 3.4 | 29 | 22 |
2,000 | 4.6 | 3.9 | 59 | 53 | 4.7 | 4.0 | 59 | 53 | 4.6 | 2.5 | 56 | 47 | 4.6 | 3.6 | 57 | 49 | |
4,000 | 5.1 | 4.9 | 89 | 88 | 5.3 | 4.8 | 89 | 88 | 5.0 | 3.5 | 88 | 85 | 5.2 | 4.5 | 88 | 87 |
Note: Number of replications = 1000, number of bootstrap replication = 1000.
HAR = heterogeneous autoregressive.