We combine the integer-valued GARCH(1, 1) model with a generalized regime-switching model to propose a dynamic count time series model. Our model adopts Markov-chains with time-varying dependent transition probabilities to model dynamic count time series called the generalized regime-switching integer-valued GARCH(1, 1) (GRS-INGARCH(1, 1)) models. We derive a recursive formula of the conditional probability of the regime in the Markov-chain given the past information, in terms of transition probabilities of the Markovchain and the Poisson parameters of the INGARCH(1, 1) process. In addition, we also study the forecasting of the Poisson parameter as well as the cumulative impulse response function of the model, which is a measure for the persistence of volatility. A Monte-Carlo simulation is conducted to see the performances of volatility forecasting and behaviors of cumulative impulse response coefficients as well as conditional maximum likelihood estimation; consequently, a real data application is given.
Integer-valued time series models have attracted recent attention because big data sets of count time series, such as the number of transactions of some stock and incidences of a certain disease, are available in financial markets or epidemiology. These now need to be analyzed via a successful fitted model as sources for future predictions. Most time series data are affected by various market circumstances such as international policy and financial crisis. In particular, real world count time series data are greatly influenced by external factors, which can be represented as a multiple-regime model. This work is concerned with a successful model for count time series data with external multiple-regimes.
In this paper, we consider an integer-valued GARCH(1, 1) (INGARCH(1, 1)) model, which is a popular count time series model, and combine the model with Markov chain regime-switchings that can account for the external factors. The INGARCH model is proposed by Ferland
This work adopts the generalized regime-switching (GRS) of Gray (1996) to the INGARCH model of Ferland
In the present work, as a dynamic count time series model, we propose the generalized regime-switching integer-valued GARCH(1, 1) (GRS-INGARCH(1, 1)) process and discuss the forecasting, the cumulative impulse response function and the conditional maximum likelihood estimation (CMLE). As main theoretical results, we first derive a recursive formula of the conditional probability of Markov-chain state given the past information, in terms of transition probabilities and the Poisson parameters of the INGARCH(1, 1) process. Secondly under assumption of known parameters in INGARCH(1, 1), the forecasting of the Poisson parameter in the GRS-INGARCH(1, 1) process is investigated, and finally the cumulative impulse response function, which is a measure of the persistence of the volatility, is discussed. A Monte-Carlo study is conducted to see the time series plots of the GRS-INGARCH(1, 1) along with the volatility, and the performance of volatility forecasting and the behavior of the cumulative impulse response functions as well as the CMLE. A real data application is given to compare the INGARCH(1, 1) and the GRS-INGARCH(1, 1) process for the numbers of visitors in Hallasan Mountain National Park.
The remainder of the paper is organized as follows. In Section 2 we present the model and main results of forecasting and the cumulative impulse response function. In Section 3 a Monte-Carlo study is given and in Section 4 a real data analysis is presented. Proofs are drawn in Section 5.
An INGARCH (1, 1) process, introduced by Ferland
where
Now we apply a GRS-GARCH of Gray (1996) to the INGARCH model and consider the GRS-INGARCH( 1, 1) model in this work. To propose the GRS-INGARCH(1, 1) process, we adopt a two-state Markov chain {
In the GRS-INGARCH(1, 1) model, the parameter vector
The GRS-INGARCH(1, 1) model with (
where
Note that for the nonstationary GRS-INGARCH(1, 1) model with a regime having explosive values in volatility, the probabilities
The volatility, which is the conditional mean and the conditional variance,
where
where
It is easily shown that
In the next section we derive the recursive formula of the forecasting of
On behalf of the forecasting of the GRS-INGARCH(1, 1) process, we assume that parameters
In the following theorem we present the
where
Now we derive the representation for the cumulative impulse response function of the GRS-INGARCH( 1, 1) process. Impulse response function of a dynamic system is its output when presented with a brief input signal, called an impulse. An impulse response refers to the reaction of the dynamic system in response to some external change. Baillie
with
Letting
Note that {
and thus
we have
In the next section of a Monte-Carlo simulation, we observe the behaviors of
Recently, Cui and Wu (2016) proposed the CMLE for INGARCH models and established the strong consistency and asymptotic normality of the CMLE. We adopt their spirit of the CMLE for the case of unknown parameters. We discuss here the CMLE of the six parameters of the GRS-INGARCH(1, 1) models, but the explicit theoretical validity remains as a future study. Let
The CMLE
In this section a Monte-Carlo study is conducted to see time series plots of the GRS-INGARCH(1, 1) process and its volatility, and to obtain the performance of the forecasting and the cumulative impulse response. We first start with time series plot of the INGARCH(1, 1) model of Ferland
Figure 2 depicts the conditional probabilities
In Figures 3 and 4, we see the time series plots of the stationary GRS-INGARCH(1, 1) process and its volatility. Figure 3(a)–(d) use the constant transition probabilities
Now we consider the CMLE in Subsection 2.3. For the GRS-INGARCH(1, 1) models with two cases of
and
Now we discuss the forecasting, given in Theorem 2, under the assumption of known parameters of the INGARCH(1, 1). Tables 2 and 3 indicate the
Finally we observe the behaviors of the cumulative impulse response functions discussed in Lemma 1. Figure 7(a) depicts the plot of the cumulative impulse response functions for three cases: stationary (
As a real data application, we consider the number of visitors in Hallasan Mountain National Park in Jeju Island, during Jan. 2007–Dec. 2016, of which data are obtained from The Korea Tourism and Information System ( www.tour.go.kr). To deseasonalize the data we take the ratio of the monthly data to the minimum of each month during ten years. For instance, for ten data as of January (Jan), let Jan_{min} = min{
Now, to adopt a GRS-INGARCH(1, 1) model, we use two-state Markov chain with transition probability
We observe
Now we compute Pr(
Since
with
Hence we obtain
Given data
Two-step ahead forecast of
However, we estimate
For general
provided all
Authors are very grateful for valuable comments of two referee and an editor which improve the paper considerably. This work was supported by the National Research Foundation of Korea (NRF-2015-1006133).
Conditional maximum likelihood estimates for GRS-INGARCH(1, 1) with
0.4791 (0.0122) | 0.2217 (0.0438) | 0.2592 (0.0271) | 0.9587 (0.0104) | 0.4295 (0.0396) | 0.4661 (0.0331) | |
0.4853 (0.0205) | 0.2186 (0.0353) | 0.2687 (0.0214) | 0.9706 (0.0272) | 0.4167 (0.0309) | 0.4654 (0.0221) | |
1.9910 (0.0489) | 0.1271 (0.0363) | 0.4755 (0.0400) | 0.7931 (0.0489) | 0.2811 (0.0380) | 0.2795 (0.0401) | |
2.001 (0.0386) | 0.1405 (0.0105) | 0.4887 (0.0371) | 0.8026 (0.0399) | 0.2905 (0.0015) | 0.2933 (0.0383) |
Forecasting
1 | 0.434 | 11 | 0.448 | 21 | 0.439 | 31 | 0.432 | 41 | 0.438 |
2 | 0.438 | 12 | 0.449 | 22 | 0.438 | 32 | 0.431 | 42 | 0.437 |
3 | 0.441 | 13 | 0.449 | 23 | 0.437 | 33 | 0.432 | 43 | 0.438 |
4 | 0.444 | 14 | 0.450 | 24 | 0.435 | 34 | 0.434 | 44 | 0.437 |
5 | 0.446 | 15 | 0.450 | 25 | 0.434 | 35 | 0.438 | 45 | 0.440 |
6 | 0.447 | 16 | 0.448 | 26 | 0.433 | 36 | 0.442 | 46 | 0.443 |
7 | 0.448 | 17 | 0.445 | 27 | 0.432 | 37 | 0.443 | 47 | 0.444 |
8 | 0.449 | 18 | 0.443 | 28 | 0.431 | 38 | 0.441 | 48 | 0.446 |
9 | 0.447 | 19 | 0.441 | 29 | 0.431 | 39 | 0.441 | 49 | 0.447 |
10 | 0.446 | 20 | 0.440 | 30 | 0.432 | 40 | 0.439 | 50 | 0.446 |
Forecasting
1 | 0.255 | 11 | 0.278 | 21 | 0.256 | 31 | 0.248 | 41 | 0.253 |
2 | 0.268 | 12 | 0.277 | 22 | 0.264 | 32 | 0.248 | 42 | 0.255 |
3 | 0.276 | 13 | 0.283 | 23 | 0.261 | 33 | 0.258 | 43 | 0.249 |
4 | 0.281 | 14 | 0.279 | 24 | 0.257 | 34 | 0.270 | 44 | 0.259 |
5 | 0.283 | 15 | 0.272 | 25 | 0.255 | 35 | 0.276 | 45 | 0.267 |
6 | 0.284 | 16 | 0.263 | 26 | 0.253 | 36 | 0.277 | 46 | 0.272 |
7 | 0.281 | 17 | 0.261 | 27 | 0.252 | 37 | 0.268 | 47 | 0.277 |
8 | 0.273 | 18 | 0.264 | 28 | 0.248 | 38 | 0.264 | 48 | 0.276 |
9 | 0.268 | 19 | 0.262 | 29 | 0.250 | 39 | 0.261 | 49 | 0.271 |
10 | 0.275 | 20 | 0.256 | 30 | 0.251 | 40 | 0.254 | 50 | 0.266 |