Adding flexibility to Markov Switching models

Abstract

Abstract:

Very often time series are subject to abrupt changes in the level, which are generally represented by Markov Switching (MS) models, assuming that the level is constant within a certain state (regime). This is not a realistic framework because in the same regime the level could change with minor jumps with respect to a change of state; this is a typical situation in many economic time series such as the Gross Domestic Product (GDP) or the volatility of financial markets. We propose to make the state flexible, introducing a very general model which provides oscillations of the level of the time series within each state of the MS model; these movements are driven by a forcing variable. The new model allows for consideration of extreme jumps in a parsimonious way, without the adoption of a large number of regimes (in our examples the two-state MS models are used). Moreover, this model increases the interpretability and in particular the out-of-sample performance with respect to the most used alternative models. This approach can be applied in several fields, also using unobservable data. We show its advantages in three distinct applications, extending particular MS models, which involve macroeconomic variables, volatilities of financial markets and conditional correlations.

Keywords

abrupt changes Goodness of fit Hamilton filter smoothed changes time-varying parameters

1 Introduction

The nonlinear behaviour of many economic time series, characterized by abrupt level changes, has been the object of several studies in the last decades, favouring the development of switching regime models; see Hamilton (2016) for a review of this kind of models in a macroeconomic framework, Franses and van Dijk (2000) for financial time series. In particular, the level of the time series changes in unknown time points; this characteristic has favoured the great success of models with two or more regimes that are able to detect the unknown change points. Among them, we recall the Markov Switching Autoregressive (MS–AR) model of Hamilton (1989); Hamilton (1990), $^{1}$ ¹

1
The AR–MS model was originally proposed by Lindgren (1978), but its diffusion and success in the economic framework is due to the works of Hamilton.

the multiple change-point model of Chib (1998), the smooth transition model of Teräsvirta (1994) and their extensions.

The simple form of the MS model, its clear interpretation and the possibility to infer on the regime using the Hamilton (1990) filter, justify its enormous success in statistical and econometrical literature and its application in several fields of economics. In general, the MS model provides a good fit of data capturing the presence of abrupt changes, but the consideration of fixed coefficients within each state is a rigid constraint. This problem could be solved with a larger number of states, which involves computational problems (high computational time, convergence, identification, Markov chains with zero elements). Also, the identification of the number of states is an open problem; very often the practitioners adopt a two-state MS model to avoid loss of efficiency in estimation due to the small possible number of observations falling in a certain state. Similarly, a higher number of states might imply the absence of transitions from one state to another (there are no cases in which the state changes from regime $i$ to regime $j$ ), with the corresponding coefficients of the transition probability matrix identically equal to zero (see, e.g., Hamilton and Susmel (1994)).

A graphical example is helpful to understand the main motivations of this work. Let us observe the first differences of the quarterly US GDP series (in logs) in Figure 1. We can notice that the series is characterized by frequent oscillations, in particular in its first half, until 1984, with a few cases in which the series level is more than 4. These points characterize the boom in 1950 after the Second World War and the following quarters: Q1 71, Q2 78, Q4 80 and Q1 81. An MS model with two states can identify these dates, assigning them to the regime with the highest mean, which will be constant within the same regime. But it is evident that the level of these points changes.

Figure 1

Source: Federal Reserve Economic Data of St. Louis Bank.

It is likely that a better fit of the data could be obtained, providing the possibility to change the parameters within the regimes. In practice, in Figure 1, a certain flexibility could be obtained if we allow a change in the level within the regimes, with the chance to isolate in a single state the highest peaks of the series. For this purpose we propose a new model, called Flexible State Markov Switching (FSMS) model, considering time-varying coefficients within each state. The within-state dynamics of the coefficients is driven by forcing variables, which can also be non-observable. We have developed a very general framework, extending different MS models and providing examples of their applications.

This article is structured in the following way: in the next section we will describe the new model proposed, pointing out how the estimation procedure can follow the steps proposed in Hamilton (1990). Section 3 illustrates three examples of application of the FSMS model, dealing with the US GDP, the volatility of the NASDAQ 100 index and the conditional correlations of the Dow Jones Industrial Average index (DJ henceforth) components; in this section we will develop specific FSMS models and will compare them with MS models and other alternatives by indices of in-sample and out-of-sample performance. Some final remarks will conclude the article.

2 The flexible state Markov Switching model

Let $y_{t}$ be the variable object of study. It can be a single variable or a vector of variables; moreover, it is not necessary that $y_{t}$ is observed. In an MS framework, we consider different models for the different states; such models have the same structural form and differ only for the value of some coefficients (called the switching coefficients). Let $m_{i, t}$ be the model representing $y_{t}$ in state $i$ . Formally, in an MS model with two states we have

\begin{matrix} y_{t} = (1 - s_{t}) m_{0, t} + s_{t} m_{1, t}; \\ P = [\begin{matrix} p_{00} & p_{01} \\ p_{10} & p_{11} \end{matrix}], \end{matrix}

(2.1)

in which

s_{t} = 0, 1

represents the state at time

t

and

P

is the transition probability matrix relative to an ergodic Markov chain with elements

p_{ji} = \Pr (s_{t} = i | s_{t - 1} = j)

and when

i \neq j

p_{ji} = 1 - p_{ii}

We assume that, within each regime, the model can vary in a range of models which differ for the values of the switching parameters (representing the level of the series $y_{t}$ ) in the following way:

y_{t} = (1 - s_{t}) [m_{0, t}^{*} f_{0, t} + m_{1, t}^{*} (1 - f_{0, t})] + s_{t} [m_{1, t}^{*} f_{1, t} + m_{2, t}^{*} (1 - f_{1, t})] .

(2.2)

In practice, in state

0

, the model is a weighted mean of

m_{0, t}^{*}

and

m_{1, t}^{*}

, with weights derived from the function

f_{0, t}

ranging in

[0, 1]

; in state

1

the model is a weighted mean of

m_{1, t}^{*}

and

m_{2, t}^{*}

, with weights derived from the function

f_{1, t}

(again ranging in

[0, 1]

). The level increases when the model changes from

m_{0, t}^{*}

m_{1, t}^{*}

and from

m_{1, t}^{*}

m_{2, t}^{*}

, so we need feasible re-parameterizations to obtain this constraint. Each

m_{i, t}^{*}

(

i = 0, 1, 2

) depends on a set of common coefficients

ϕ

(including also

p_{00}

and

p_{11}

) and a set of specific coefficients

ϑ_{i}

(the switching coefficients). The functions

f_{0, t}

and

f_{1, t}

depend on a variable

x_{t}

(the forcing variable), not necessarily observed, and a set of coefficients

ζ_{h}

(

h = 0, 1

). They can assume different specifications; in our experience, good representations can be given by smooth transition functions and logistic functions. We call them within-state dynamic (wsd) functions.

The estimation of Equation 2.2 can be obtained extending the procedure proposed by Hamilton (1990), including $ζ_{0}$ and $ζ_{1}$ among the set of unknown coefficients. Denoting with $I_{t}$ the amount of information available at time $t$ and $π = (ϕ^{'}, ϑ_{0}^{'}, ϑ_{1}^{'}, ϑ_{2}^{'}, ζ_{0}^{'}, ζ_{1}^{'})^{'}$ the full set of parameters, the procedure consists of the following iterative steps:

Fix $\Pr (s_{0} = j)$ ; it can be given by the ergodic probability $p_{e}$ such that $P^{'} p_{e} = p_{e}$ ;

$\Pr (s_{t} = i, s_{t - 1} = j | I_{t - 1}) = p_{ji} \Pr (s_{t - 1} = j | I_{t - 1})$ ;

$f (y_{t} | I_{t - 1}, π) = \sum_{i = 1}^{2} \sum_{j = 1}^{2} \Pr (s_{t} = i, s_{t - 1} = j | I_{t - 1}) f (y_{t} | s_{t} = i, s_{t - 1} = j, I_{t - 1}, π)$ ;

$\Pr (s_{t} = i, s_{t - 1} = j | I_{t}) = \frac{\Pr (s_{t} = i, s_{t - 1} = j | I_{t - 1}) f (y_{t} | s_{t} = i, s_{t - 1} = j, I_{t - 1}, π)}{f (y_{t} | I_{t - 1}, π)}$ ;

$\Pr (s_{t} = i | I_{t}) = \sum_{j = 1}^{2} \Pr (s_{t} = i, s_{t - 1} = j | I_{t})$ ;

Iterate steps 2–5 from $t = 1$ until $t = T$ .

In practice the density of $y_{t}$ is a mixture of four densities $f (y_{t} | s_{t}, s_{t - 1}, I_{t - 1}, π)$ obtained from the four possible combinations of $s_{t}$ and $s_{t - 1}$ . The weights of each component of the mixture are given by the probabilities $\Pr (s_{t}, s_{t - 1} | I_{t - 1})$ , derived in step 2. The previous scheme is the basic Hamilton (1990) filter to obtain the filtered probabilities at each time; it is possible to obtain an inference on the regimes using the full available information $I_{T}$ from the Kim (1994) algorithm, starting from the results of the Hamilton filter. It consists in iterating the following steps, starting from $t = T$ :

The first smoothed probability $\Pr (s_{T} = i | I_{T})$ is obtained from the final iteration of the Hamilton filter;

From the same Hamilton filter derive the probabilities $\Pr (s_{t} = j | I_{t})$ and $\Pr (s_{t + 1} = i | I_{t}) = \sum_{j = 1}^{2} \Pr (s_{t + 1} = i, s_{t} = j | I_{t})$ ;

$\Pr (s_{t} = j, s_{t + 1} = i | I_{T}) = \frac{\Pr (s_{t + 1} = i | I_{T}) \Pr (s_{t} = j | I_{t}) p_{ji}}{\Pr (s_{t + 1} = i | I_{t})}$

$\Pr (s_{t} = j | I_{T}) = \sum_{i = 1}^{2} \Pr (s_{t} = j, s_{t + 1} = i | I_{T})$ ;

Iterate steps 2–4 from $t = T - 1$ to $t = 1$ .

The inference on the regime is obtained from the smoothed probabilities, assigning the observations with $\Pr (s_{t} = 0 | I_{T}) > 0.5$ to regime 0 (otherwise regime 1).

In the following section we will propose three different specifications of Equation (2.2), which extend three well-known MS models to include the flexible structure within the regimes.

3 Examples of applications

As mentioned before, Equation (2.2) is a very general form of the FSMS model and can include the extension of all MS models. In this section we propose the extension of three MS models developed in different applicative fields. In the first case, we extend the analysis of the US GDP by an MS–AR model, similar to the one adopted by Hamilton (1990); in the second case, we consider the analysis of the volatility of the NASDAQ 100 US index extending the Markov Switching Generalized AutoRegressive Conditional Heteroskedasticity (MS-GARCH) model (see, for example, Dueker, 1997); finally, we extend the analysis of the correlations of the assets compounding the DJ index extending the Regime Switching Dynamic Correlation (RSDC) model proposed by Pelletier (2006). It should be noticed that in the first case the variable analyzed (GDP) is observed, whereas in the last two applications, it is estimated by the model (i.e., the conditional variance in the second case and the conditional correlations in the third one). We will adopt as forcing variables two different quantities: the estimated lagged level of GDP in the first application and the filtered probabilities of state $0$ derived from the Hamilton filter in the second and third applications. It should be remarked that the forcing variables are not observed but are obtained as a sub-product of the maximum likelihood estimation procedure of the model. $^{2}$ ²

2
A similar solution was adopted by Otranto (2008) in developing MS models with transition probabilities driven by latent variables.

Finally, we will adopt both the smooth transition function and the logistic function as wsd functions; the differences are not relevant, so we will describe only one case for each example.

In all the examples, the FSMS model is compared with four competitors among the most used models in each framework; $^{3}$ ³

In the first two examples, devoted to the AR and GARCH models and their extensions, we have tried to estimate also smooth transition models (Teräsvirta (1994); Teräsvirta (2009)) and threshold models Tong, (1990), but their estimation provides results similar to the linear case, as they are not able to detect different regimes. For this reason we have not included them in the comparison..

the comparison is conducted both in in-sample and out-of-sample sets by specific loss functions measuring the forecasting performance rather than parsimony criteria (such as Akaike Information criterion (AIC) or Bayesian Information Criterion (BIC)), the properties of which are not formally established for nonlinear models.

^{4}

⁴

We thank the Associate Editor who suggested this approach.

3.1 FSMS–AR model

Let us suppose that the variable of interest $y_{t}$ is the first difference of the log of US GDP. The data used are shown in Figure 1, considering the span 1948–2012 for the in-sample evaluation and the span 2013–2014 for the out-of-sample one.

Let us consider the MS model Equation (2.1), adopting an AR(2) specification for the components $m_{0, t}$ and $m_{1, t}$ :

\begin{matrix} m_{s_{t}, t} = μ_{s_{t}} + ϕ_{1} (y_{t - 1} - μ_{s_{t - 1}}) + ϕ_{2} (y_{t - 2} - μ_{s_{t - 2}}) + ε_{t} s_{t} = 0, 1 \\ ε_{t} \sim IIN (0, σ^{2}) \end{matrix}

(3.1)

Adopting a similar specification for the FSMS model, the components

m_{0, t}^{*}

m_{1, t}^{*}

and

m_{2, t}^{*}

in Equation (2.2) will be:

m_{s_{t}, t}^{*} = μ_{s_{t}} + ϕ_{1} (y_{t - 1} - μ_{s_{t - 1}, t - 1}^{*}) + ϕ_{2} (y_{t - 2} - μ_{s_{t - 2}, t - 2}^{*}) + ε_{t} s_{t} = 0, 1, 2

(3.2)

where,

μ_{s_{t}, t}^{*} = \{\begin{matrix} μ_{0} f_{0, t} + μ_{1} (1 - f_{0, t}) & if s_{t} = 0 \\ μ_{1} f_{1, t} + μ_{2} (1 - f_{1, t}) & if s_{t} = 1 . \end{matrix}

The only switching coefficient is the mean of the process $μ_{s_{t}}$ ; to ensure the increase in the level when changing the state, we impose that $μ_{0} \leq μ_{1} \leq μ_{2}$ . Moreover, we can adopt as wsd functions in Equation (2.2) the smooth transition functions:

f_{i, t} = [1 + exp (- γ_{i} (μ_{t - 1}^{*} - c_{i}))]^{- 1}, with γ_{i} > 0 i = 0, 1,

(3.3)

where

μ_{t}^{*} = \Pr (s_{t} = 0 | I_{t}) μ_{0, t}^{*} + \Pr (s_{t} = 1 | I_{t}) μ_{1, t}^{*},

is the estimated mean at time

t

(

\Pr (s_{t} = i | I_{t})

i = 0, 1

is obtained at step 5 of the Hamilton filter-see section 2); we are simply hypothesizing that the most recent (estimated) value of the unconditional level of GDP can drive the movements in level within the states.

In terms of the notation used in section 2, we put $ϕ = (ϕ_{1}, ϕ_{2}, p_{00}, p_{11})^{'}$ , $ϑ_{0} = μ_{0}$ , $ϑ_{1} = μ_{1}$ , $ϑ_{2} = μ_{2}$ , $ζ_{0} = (γ_{0}, c_{0})^{'}$ and $ζ_{1} = (γ_{1}, c_{1})^{'}$ .

We also consider four competitive models:

The AR-KC (Known Change-point) model: It is a simple switching AR(2) model with known change-points; in practice, we use the official dating of the business cycle proposed by NBER $^{5}$ ⁵

Available at the website http://www.nber.org/cycles/cyclesmain.html.

to identify the observations belonging to regime 0 (recession) and regime 1 (growth). Formally, this model has the same form of Equation (3.1) but

s_{t}

is observed at each time

t

. Of course, using this kind of information, this model is favoured in terms of in-sample performance with respect to the models with unobservable regimes; moreover, it cannot be used in out-of-sample terms (we can merely assume that the successive regimes are the same as the last observation). Anyway, it is a useful benchmark to evaluate the in-sample performance of the other models;

The linear AR model: In this case all the parameters are constant;

The time-varying parameter (TVP) model: We used a state–space representation (see, e.g., Harvey, (1990)) with the coefficient $μ$ changing at each time; formally:

\begin{matrix} y_{t} = [\begin{matrix} ϕ_{1} & ϕ_{2} \end{matrix}] [\begin{matrix} y_{t - 1} \\ y_{t - 2} \end{matrix}] + [\begin{matrix} 1 & - ϕ_{1} & - ϕ_{2} \end{matrix}] ξ_{t} + e_{t}, \\ ξ_{t} = k (1 - ψ) + [\begin{matrix} ψ & 0 & 0 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{matrix}] ξ_{t - 1} + [\begin{matrix} v_{t} \\ 0 \\ 0 \end{matrix}] \end{matrix},

(3.4)

where

e_{t} \sim IIN (0, σ^{2})

v_{t} \sim IIN (0, σ_{ξ}^{2})

and

E (e_{t}, v_{s}) = 0

for each

t

and

s

. In this case, the mean of the series at time

t

is given by the first element of the vector

ξ_{t}

;

the MS–AR model Equation (3.1).

Table 1

Parameter estimates (standard errors in parentheses) and evaluation criteria of alternative models for the US GDP variations (in-sample set: Q1 1948–Q4 2012; out-of-sample set: Q1 2013–Q4 2014)

	$μ_{0}$	$μ_{1}$	$μ_{2}$	$ϕ_{1}$	$ϕ_{2}$	$σ$	$p_{00}$	$p_{11}$
AR-KC	0.373	1.816		0.412	$-$ 0.042	0.882
	(0.183)	(0.092)		(0.057)	(0.012)	(0.039)
AR	1.400			0.053	$-$ 0.001	0.895
	(1.078)			(0.051)	(0.001)	(0.038)
MS–AR	1.468	4.354		0.452	0.145	0.816	0.980	0.546
	(0.128)	(0.441)		(0.065)	(0.085)	(0.040)	(0.011)	(0.165)
FSMS–AR	$-$ 0.032	2.412	4.785	0.459	0.054	0.804	0.984	0.614
	(0.768)	(0.442)	(0.371)	(0.056)	(0.053)	(0.036)	(0.008)	(0.121)
	${pv}_{1}$	${pv}_{5}$	${pv}_{10}$	$Log - Lik$	$MSEi . s .$	$MSEo . s .$
AR–KC	0.319	0.000	0.000	$-$ 331.86	0.783 $^{*}$	0.299
AR	0.000	0.000	0.000	$-$ 435.24	1.210	0.220
TVP	0.086	0.003	0.003	$-$ 332.58	0.916	0.199
MS–AR	0.695	0.052	0.055	$-$ 339.33	0.913	0.211
FSMS–AR	0.651	0.243	0.096	$-$ 334.75	0.883 $^{*}$	0.185 $^{*}$

Source: Processing of Federal Reserve Economic Data of St. Louis Bank.

Notes: The symbol ${pv}_{i}$ indicates the p-value of the Ljung-Box statistic to verify the presence of residual autocorrelation at lag $i$ . $MSEi . s .$ and $MSEo . s .$ indicate the MSE calculated in the in-sample and in the out-of-sample set, respectively. The symbol $^{*}$ is put next to the best model in MSE terms and next to the models with an MSE not significantly different from the best one, based on a DM test at a $5 %$ nominal size.

Table 1 shows the estimation of the parameters of the alternative models, $^{6}$ ⁶

Most of the parameters of the TVP model (3.4) are different from the parameters of the four AR(2) models shown in Table 1, so we do not show them into the same table. They are (standard errors in parentheses): $ϕ_{1} = 0.534 (0.071)$ , $ϕ_{2} = - 0.071 (0.019)$ , $σ = 0.001 (0.031)$ , $k = 1.590 (0.110)$ , $ψ = 0.526 (0.046)$ , $σ_{ξ} = 0.838 (0.023)$ .

the p-values of the Ljung–Box statistic evaluated at lags 1, 5 and 10, the value of the Log-likelihood and the mean squared error (MSE) loss function, both for the in-sample and out-of-sample cases. Most of the parameters of the TVP model (Equation 3.4) are different from the parameters of the four AR(2) models shown in Table 1, so we do not show them into the same table. They are (standard errors in parentheses):

ϕ_{1}

= 0.534 (0.071),

ϕ_{2}

= 0.071 (0.019),

σ

= 0.001 (0.031), k = 1.590 (0.110),

ψ

= 0.526 (0.046),

σ_{ε}

= 0.838 (0.023). For the out-of-sample analysis, we perform the one-step-ahead forecast, adding one observation at each step and re-estimating the model. In practice, we suppose that the model is estimated when a new quarterly observation is available.

The estimation of the unconditional mean is equal to 1.40 in the linear AR model; it assumes value 1.47 in regime 0 and 4.35 in regime 1, if we adopt model MS, whereas using model FSMS, it varies in the ranges $[- 0.03, 2.41]$ and $[2.41, 4.78]$ in regime 0 and regime 1, respectively. It should be noticed that these means are different from the ones of the AR-KC model (0.37 in regime 0 and 1.82 in regime 1). This happens because the MS and FSMS models capture in regime 1 only the highest peaks of the series, whereas the AR-KC considers all the observations within growth periods.

Figure 2

Source: Processing of Federal Reserve Economic Data of St. Louis Bank.

In Figure 2 we have shown the two estimated wsd functions; it should be noticed that both functions are approximately constant around 0.43 and 0 respectively with abrupt jumps, reaching the maximum value 1 in correspondence to the highest peaks of GDP. This result implies that when the state is 0, the estimated mean is around 1.36 with small changes, whereas near to 4.78 (the estimated value of $μ_{2}$ ) when the state is 1. This will be clearer observing Figure 3, where the GDP is represented with the estimated means obtained by MS model (upper panel) and FSMS model (bottom panel). Differences between the unconditional means are evident in 1950, where the mean derived from the FSMS model follows the dynamics of the GDP series, whereas the mean estimated by the MS model is constant.

Figure 3

Source: Processing of Federal Reserve Economic Data of St. Louis Bank.

The second part of Table 1 compares the models by residual autocorrelation and forecasting performance. It is interesting to notice that only MS and FSMS models are not affected by residual autocorrelation, if we base the analysis on a Ljung–Box test at a $5 %$ nominal size; considering a $1 %$ size this result applies only to the FSMS model. We can infer that the Markovian dynamics is a more correct way to capture the nonlinearities present in the series with respect to other kinds of nonlinearities, represented by the TVP or AR–KC models.

The AR–KC model possesses the highest log-likelihood, but, as mentioned before, this information has to be considered with due care. More interesting are the results deriving from the MSE evaluation; in in-sample terms the best performance is shown from the AR-KC model and FSMS is the second best. Anyway, comparing the squared errors by the well-known Diebold–Mariano (DM) test (Diebold and Mariano, 1995) with the correction proposed by Harvey et al. (1997), we obtain that the predictions of the two models are not significantly different at a $5 %$ nominal size, so the FSMS model has the same in-sample performance (equal MSE) of the model with known change-points. In out-of-sample terms the FSMS model is clearly the best one and the DM statistic rejects the null hypothesis of equality of the prediction squared errors.

In practice, the comparison of the five models for the US GDP dataset seems to favour the FSMS model not only in terms of interpretability of the regimes but also by goodness of fit, diagnostic checking and forecasting performance.

3.2 FSMS–GARCH model

A recent application field of MS models concerns the analysis of the volatility of financial time series, in particular, inserting a Markovian dynamics in the coefficients of the GARCH model (Bollerslev, 1986).

Figure 4

Source: Oxford-Man Institute realized library.

Let us consider the series of the returns $^{7}$ ⁷

In financial analysis the return at time $t$ is the logarithm of the ratio between the price of the asset at time $t$ and the price of the asset at time $t - 1$ , multiplied by 100.

of the NASDAQ index from 2 January 2004 to 5 May 2015 (2840 daily observations); the gray line of Figure 4 illustrates the dynamics of this series, where the high volatility (i.e., large conditional variance) period between October 2007 and July 2009 is evident, corresponding to the world financial crisis.

^{8}

⁸

The dates on the $x$ axis of Figure 4 are put at the beginning of each year.

The model we have extended is a MS–GARCH(0,1) with asymmetric effects;

^{9}

⁹

We have estimated, both in MS and FSMS cases, the usual GARCH(1,1) model, but the coefficient of the lagged return was estimated equal to zero. The asymmetric effects are referred to the increase in the volatility level when the most recent return is negative; see Glosten et al. (1993).

calling

r_{t}

the series of the returns, we assume that the conditional distribution of

r_{t}

is given by:

r_{t} | I_{t - 1} \sim N (μ, y_{t})

and the conditional variance

y_{t}

follows the structure Equation (2.1) in the case of MS model and Equation (2.2) in the case of FSMS model. The components of the FSMS models are given by:

\begin{matrix} m_{s_{t}, t}^{*} = ω_{s_{t}} + β y_{s_{t - 1}, t - 1} + γ I (r_{t - 1} < 0) ε_{t - 1}^{2} & s_{t} = 0, 1, 2 and \\ f_{i, t} = \frac{\exp (a_{i} + b_{i} \Pr (s_{t - 1} = 0 | I_{t - 1})}{1 + \exp (a_{i} + b_{i} \Pr (s_{t - 1} = 0) | I_{t - 1})} & i = 0, 1, \end{matrix}

(3.5)

where

ε_{t} = r_{t} - μ

β

and

γ

are non-negative and

(β + γ / 2) < 1

;

I (\cdot)

is the indicator function which provides the asymmetric effects due to the sign of the return at time

t - 1

. In Equation (3.5), the wsd function is a logistic function driven by the filtered probabilities of state 0 obtained, step by step, from the Hamilton filter during the estimation procedure (step 5 of the filtering algorithm is illustrated in section 2).

Resuming, the parameters of the FSMS–GARCH model are: $ϕ = (β, γ, p_{00}, p_{11})^{'}$ , $ϑ_{0} = ω_{0}$ , $ϑ_{1} = ω_{1}$ , $ϑ_{2} = ω_{2}$ , $ζ_{0} = (a_{0}, b_{0})^{'}$ and $ζ_{1} = (a_{1}, b_{1})^{'}$ .

In MS–GARCH models (and, as a consequence, in FSMS models) a path dependence problem arises due to the non-observability of the volatility and the dependence of the state $s_{t}$ on all the past values: At the end of the $t$ -th step of the Hamilton filter, it would be necessary to keep track of all possible paths taken by the regime until time $t$ , making the model intractable. Several solutions were proposed in literature (see, e.g., Gray, 1996; Klaassen (2002); Haas et al., 2004). The proposal we adopt is similar to the one of Dueker, (1997), who extends the method of Kim (1994), developed in a state–space framework with a simple approximation. The Kim approximation consists in collapsing at each step of the Hamilton filter, the four possible values of the estimated volatility $y_{s_{t}, s_{t - 1}, t}$ into 2 values, by:

y_{s_{t}, t} = \frac{\sum_{j = 1}^{2} \Pr (s_{t} = i, s_{t - 1} = j | I_{t}) y_{i, j, t}}{\Pr (s_{t} = i | I_{t})} .

(3.6)

Kim (1994), using real data, and Gallo and Otranto (2015), via simulations in an asymmetric multiplicative error model framework, show that the Kim approximation provides good results with decreasing bias when the sample size increases. This approximation can also be used for the FSMS model.

In addition to the FSMS–GARCH model, we have estimated the following four models on the NASDAQ series: $^{10}$ ¹⁰

We also tried to use a Time Varying Normalized Variance Stabilisation (TV–NoVaS) method (see Politis, (2015)), a model free forecasting approach which is robust against structural breaks. In particular we used a rolling windows of 250 observations, the exponential scheme to detect the weights, the component of the conditional variance to forecast the squared returns (for details see Politis and Thomakos, (2012)). Unfortunately the results show a relatively poor performance when compared with the other models, so we will not describe them.

Exponential smoother variance (ESV) model: It is known as the Riskmetrics variance, established in 1989 by the J.P. Morgan holding company, largely diffused among practitioners. The dynamics of the $y_{t}$ depends on the parameter $0 \leq λ \leq 1$ :

y_{t} = λ y_{t - 1} + (1 - λ) ε_{t}^{2} .

J.P. Morgan suggests to fix the coefficient

λ

equal to 0.94. We will adopt the maximum likelihood estimate, which is equal to 0.980 (with standard error equal to 0.0005);

Asymmetric GARCH model: We adopt the linear relationship:

y_{t} = ω + β y_{t - 1} + γ I (r_{t - 1} < 0) ε_{t - 1}^{2};

Exponential GARCH (EGARCH) model: This model, proposed by Nelson (1991), is an alternative way to consider asymmetries in the volatility. Its logarithmical expression is given by:

\ln (y_{t}) = ω + α \frac{|ε_{t - 1}|}{\sqrt{y_{t - 1}}} + β \ln (y_{t - 1}) + γ \frac{ε_{t - 1}}{\sqrt{y_{t - 1}}} .

For this model we need the constraint

0 < β < 1

for stationarity, whereas the expected sign of

γ

is negative because, in general, the volatility increases in front of negative shocks (negative returns);

the MS–GARCH model: We adopt Equation (2.1), where $m_{s_{t}, t}$ is the same of $m_{s_{t}, t}^{*}$ in Equation (3.5), but $s_{t} = 0, 1$ .

We consider the full available span for the in-sample evaluation. For the out-of-sample evaluation we do not consider the last part of the series because it is a quiet period for the markets with small variations in prices (see the behaviour of the returns in Figure 4); as a consequence, the alternative models will estimate low levels of volatility and the results will be very similar. More interesting is the out-of-sample evaluation in turbulent periods such as the recent world crisis. For this reason we have considered 2008 as the out-of-sample span, estimating first the model using the data until 2007, forecasting the volatility for the first quarter of 2008; then by updating the data with the observations relative to the first quarter of 2008, we estimate the model and forecast volatility for the second quarter, and so on. In practice, we estimate each model four times and consider 246 out-of-sample forecasts.

In Table 2 the estimates of the common coefficients of the models belonging to the GARCH family are shown. The MS and FSMS models have similar coefficients, excluding the $ω$ coefficients. In state $0$ , the constant of the MS–GARCH model, is equal to 0.037, whereas in the FSMS–GARCH model it varies in the interval $[0.027, 0.124]$ ; the difference is more evident in state $1$ , where the constant coefficient is equal to 0.138 for the MS model, whereas it ranges in $[0.124, 0.909]$ in the FSMS model, with the possibility to capture the highest peaks of the turbulent periods.

Table 2

Parameter estimates (standard errors in parentheses) and evaluation criteria of alternative models for the volatility of the NASDAQ 100 index (in-sample set 2 January 2004–5 May 2015; out-of-sample set 2 January 2008–31 December 2008)

	$μ$	$ω_{0}$	$ω_{1}$	$ω_{2}$	$α$	$β$	$γ$	$p_{00}$	$p_{11}$
GARCH	0.024	0.035				0.893	0.107
	(0.018)	(0.002)				(0.005)	(0.005)
EGARCH	0.000	$-$ 0.115			0.141	0.976	$-$ 0.091
	(0.009)	(0.006)			(0.007)	(0.002)	(0.006)
MS–GARCH	0.014	0.037	0.138			0.873	0.127	0.999	0.994
	(0.020)	(0.003)	(0.009)			(0.007)	(0.007)	(0.000)	(0.000)
FSMS–GARCH	0.013	0.027	0.124	0.909		0.865	0.135	0.999	0.998
	(0.009)	(0.023)	(0.009)	(0.471)		(0.007)	(0.007)	(0.000)	(0.002)
	${pv}_{1}$	${pv}_{5}$	${pv}_{10}$	$Log - Lik$	$HMSEi . s .$	$HMSEo . s .$
ESV	0.227	0.390	0.801	$-$ 3933.09	1.649 $^{*}$	2.204
GARCH	0.522	0.530	0.932	$-$ 3766.68	1.659 $^{*}$	2.174
EGARCH	0.475	0.473	0.904	$-$ 3754.01	1.573 $^{*}$	2.299
MS–GARCH	0.541	0.505	0.890	$-$ 3743.43	1.571 $^{*}$	1.910
FSMS–GARCH	0.604	0.569	0.918	$-$ 3739.35	1.517 $^{*}$	1.727 $^{*}$

Source: Processing of Oxford-Man Institute data.

Notes: The symbol ${pv}_{i}$ indicates the p-value of the Ljung-Box statistic to verify the presence of residual autocorrelation at lag $i$ . $HMSEi . s .$ and $HMSEo . s .$ indicate the HMSE calculated in the in-sample and in the out-of-sample set respectively. The symbol $^{*}$ marks the best model in HMSE terms and the models with HMSE not significantly different from the best one, based on a DM test at a $5 %$ nominal size.

In Figure 5 we show the behaviour of both wsd functions with the inference on the regime obtained from the FSMS model. The wsd function $f_{0, t}$ is almost constant around 0.85 for a large part of the span considered, excluding the period of high turmoil (but this interval belongs to state 1) and two brief successive periods, the end of July 2010 and the end of 2011; the wsd function $f_{1, t}$ shows several variations during the turbulent period. The behaviour of these functions is reflected on the dynamics of the estimated coefficient $ω$ , illustrated in Figure 6, where it is compared with the same coefficient of the MS model. The inference on the regime for each time is obtained by the mode of the state derived from the smoothed probabilities. It is notable that the two models show very different coefficients during the turmoil periods (regime 1), with the FSMS model which follows the dynamics of the volatility with high values when it reaches the maximum variability (end of 2008 and first part of 2009; see also Figure 4) and a smoothed return to the values of state 0. Conversely, the MS model provides a constant level which underestimates the volatility level. Model MS includes two brief periods, the end of July 2010 and the end of 2011, in state 1, with the same level of the global crisis period, whereas FSMS assigns them to state 0, with a small increase, a smoothed change in volatility and a smoothed return to the previous level.

In the bottom part of Table 2 it is possible to observe that all the five estimated models do not reject the hypothesis of non-autocorrelation of residuals. Their comparison by loss functions cannot be conducted as in the previous example because the volatility $y_{t}$ is not observed. Starting from the hypothesis relative to the conditional distribution of $ε_{t}$ , having zero mean and conditional variance $y_{t}$ , we can consider the standardized conditional returns $\frac{ε_{t}}{\sqrt{y_{t}}}$ and compare them with the value 1. This is the approach followed by Bollerslev and Ghysels (1996), who proposed the Heteroskedasticity-adjusted MSE (HMSE) loss function:

HMSE = \sqrt{\frac{1}{H} \sum_{t} {(\frac{ε_{t}^{2}}{y_{t}} - 1)}^{2}},

where

H

indicates the size of the in-sample or out-of-sample set.

In Table 2 we can notice that the FSMS model shows the best in-sample and out-of-sample performance, but, using a DM test at a nominal size of $5 %$ , the first one is not significantly different from the other models, whereas the out-of-sample analysis shows that the mean of the heteroskedasticity-adjusted squared errors are significantly less than the other models.

Figure 5

Source: Processing of Oxford-Man Institute data.

Figure 6

Source: Processing of Oxford-Man Institute data.

3.3 FSMS–RSDC model

A recent strand of the literature is devoted to the analysis of the time-varying conditional correlation of a set of financial time series; this is due to the empirical evidence that financial time series are subject to co-movements and this relationship is more evident in the periods of high volatility (see, e.g., Longin and Solnik, 1995). Several models have been developed to represent the dynamic conditional correlation, adopting some re-parameterization to avoid the so-called curse of dimensionality: generally a large set of assets (or financial indices) are considered and the number of parameters is explosive. Feasible models were proposed by Engle (2002), Tse and Tsui (TT) (2002) and Silvennoinen and Teräsvirta (2015), to cite just a few. The increase in correlation for high volatility regimes can be developed inserting an MS dynamics in the conditional correlation models; this idea was developed, for example, by Billio and Caporin (2005), Pelletier (2006) and Otranto (2010). In particular, Pelletier (2006) proposed the simple RSDC model, where the full correlation matrix can switch from one regime to another, without path dependence problems; this model was extended by Bauwens and Otranto (2016), who provided the dependence of the state on some measure of volatility of a financial market. Referring to the notation in Equation (2.1), $y_{t}$ represents the conditional correlation matrix $R_{t}$ and $m_{i, t}$ is the conditional correlation matrix $R_{i}$ in state $i$ . Calling $n$ the number of time series considered, each conditional correlation matrix will contain $n (n - 1) / 2$ coefficients, requiring a re-parameterization to reduce the number of unknown parameters. We follow the proposal of Bauwens and Otranto (2016), adopting:

\begin{matrix} R_{i} = \overset{̅}{R} λ_{i} + I_{n} (1 - λ_{i}) i = 0, 1 \\ λ_{0} \in [0, 1], λ_{1} \in [1, 1 / {\overset{̅}{r}}_{\max}], \end{matrix}

(3.7)

where

I_{n}

is the

n \times n

identity matrix,

{\overset{̅}{r}}_{\max} (> 0)

is the maximum correlation coefficient in the sample correlation matrix

\overset{̅}{R}

. We extend the RSDC model to the FSMS case; we adopt a re-parameterization as Equation (3.7) with

i = 0, 1, 2

and

0 \leq λ_{0} \leq λ_{1} \leq 1

and

λ_{2} \in [1; 1 / {\overset{̅}{r}}_{\max}]

. Moreover, we put:

λ_{1}^{*} = 1 - (1 - λ_{1}) (1 - s_{t}) .

(3.8)

In terms of Equation (2.2), we have:

\begin{matrix} m_{0, t}^{*} = \overset{̅}{R} λ_{0} + I_{n} (1 - λ_{0}); \\ m_{1, t}^{*} = \overset{̅}{R} λ_{1}^{*} + I_{n} (1 - λ_{1}^{*}); and \\ m_{2, t}^{*} = \overset{̅}{R} λ_{2} + I_{n} (1 - λ_{2}) . \end{matrix}

(3.9)

The re-parameterization in Equation (3.8) provides the possibility to express the FSMS model in the general form of Equation (2.2), ensuring that in

m_{1, t}^{*}

the scale factor

λ_{1}^{*} = λ_{1}

, when

s_{t} = 0

and

λ_{1}^{*} = 1

, when

s_{t} = 1

The wsd functions are represented by smooth transition functions, using the filtered probability of the state $0$ as forcing variable:

f_{i, t} = {\{1 + exp [- γ_{i} (\Pr (s_{t - 1} = 0 | I_{t - 1}) - c_{i})]\}}^{- 1}, with γ_{i} > 0 i = 0, 1 .

(3.10)

The parameters included in the vector $π = (ϕ^{'}, ϑ_{0}^{'}, ϑ_{1}^{'}, ϑ_{2}^{'}, ζ_{0}^{'}, ζ_{1}^{'})^{'}$ are: $ϕ = (p_{00}, p_{11})^{'}$ , $ϑ_{0} = λ_{0}$ , $ϑ_{1} = λ_{1}$ , $ϑ_{2} = λ_{2}$ , $ζ_{0} = (γ_{0}, c_{0})^{'}$ and $ζ_{1} = (γ_{1}, c_{1})^{'}$ . We adopt the two-step estimation procedure by Engle (2002): In the first step we estimate $n$ univariate asymmetric GARCH(1,1) models to represent the conditional variances of each asset, and in the second step we estimate $π$ on the standardized conditional (degarched) residuals derived from the first step.

The dataset we have considered is formed by 28 of the 30 assets composing the DJ index from 2 January 2002 to 31 December 2015 (3525 daily returns for each series; source: Yahoo Finance). $^{11}$ ¹¹

We have excluded the series relative to the assets Nike, which starts in December 2002, and Visa, which starts in March 2008.

The observations up to 2014 are used for the in-sample analysis (3,272 daily returns), the remaining 252 observations for the out-of-sample analysis.

Bauwens and Otranto (2016) have proposed a wide class of conditional correlation models, depending on the volatility of the financial markets (they call this class volatility dependent conditional correlation (VDCC) models) and showed their good properties with respect to the other conditional correlation models in both in-sample and out-of-sample terms. They classify their models into three categories—dynamic VDCC, smooth transition VDCC, Markov switching VDCC. For the purpose of comparison, we select, within each category and among the models without volatility dependence, the model which obtained the best performance in the application of Bauwens and Otranto (2016) concerning the DJ components. $^{12}$ ¹²

The application proposed by Bauwens and Otranto (2016) concerns a different time span and different assets because the DJ basket has been recently updated. In the new basket the assets Apple, Goldman Sachs, Nike and Visa replace the assets Alcoa, AT&T, Bank of America and Hewlett-Packard, present in the dataset of Bauwens and Otranto (2016).

They are:

The dynamic conditional correlation model by TT (2002): The conditional correlation is expressed by:

R_{t} = (1 - a - b) \overset{̅}{R} + a Γ_{t - 1} + b R_{t - 1},

where

Γ_{t - 1}

is the sample correlation matrix calculated on the standardized residuals from time

t - n

to time

t - 1

;

The TT Time-Varying Volatility dependent (TT-TVV) model: It is a TT model with a time–varying coefficient, given by the following set of equations:

\begin{matrix} R_{t} = (1 - a_{t} - b) \overset{̅}{R} + a_{t} Γ_{t - 1} + b R_{t - 1}; \\ a_{t} = a_{0} + a_{1} f_{a, t}; \\ f_{a, t} = \frac{\exp (θ_{a, 0} + θ_{a, 1} v_{t})}{1 + \exp (θ_{a, 0} + θ_{a, 1} v_{t})}, \end{matrix}

where

v_{t}

represents a measure of the volatility of DJ at time

t

;

The STCC–V model: It is a smooth transition conditional correlation model, proposed by Silvennoinen and Teräsvirta (2015):

\begin{matrix} R_{t} = R_{0} f_{R, t} + R_{1} (1 - f_{R, t}), \\ f_{R, t} = [1 + exp (- γ v_{t} - c))]^{- 1}, with γ > 0, \end{matrix}

where

R_{0}

and

R_{1}

are given by Equation (3.7);

The TV–RSDC model: It is a RSDC model with time-varying probabilities, depending on the volatility measure:

p_{ii, t} = \frac{\exp (θ_{0, i} + θ_{1, i} v_{t})}{1 + \exp (θ_{0, i} + θ_{1, i} v_{t})}, i = 0, 1 .

Similarly to Bauwens and Otranto (2016), in the VDCC models the variable

v_{t}

is given by the one-step-ahead forecasts of the VXD index,

^{13}

¹³

VXD is a proxy of the volatility of the DJ index, constructed in the same way as the well-known VIX, which is referred to the Standard & Poor 500 index.

obtained by an AR(4) model. VXD is a proxy of the volatility of the DJ index, constructed in the same way as the well-known VIX, which is referred to the Standard & Poor 500 index.

Table 3

Parameter estimates (standard errors in parentheses) and evaluation criteria of dynamic conditional correlation models for the DJ dataset (in-sample set 2 January 2002–31 December 2014; out-of-sample set 2 January 2015–31 December 2015)

	$λ_{0}$	$λ_{1}$	$λ_{2}$	$p_{00}$	$p_{11}$
STCC	0.763	1.141
	(0.412)	(0.033)
TV–RSDC	0.306	1.108		0.438	0.879
	(0.027)	(0.005)		(0.070)	(0.009)
FSMS–RSDC	0.010	0.286	1.229	0.325	0.889
	(0.189)	(0.178)	(0.002)	(0.271)	(0.371)
	$Log - Lik$	$VPi . s .$	$VPo . s .$
TT	$-$ 22852.57	102.23	102.69
TT–TVV	$-$ 22721.01	106.76	103.01
STCC	$-$ 23743.52	101.18	101.33
TV–RSDC	$-$ 22005.12	100.00 $^{*}$	100.32
FSMS–RSDC	$-$ 21955.96	100.11 $^{*}$	100.00 $^{*}$

Source: Processing of Yahoo Finance data.

Notes: The values of $p_{00}$ and $p_{11}$ for the TV–RSDC model represent the averages of the corresponding time-varying probabilities in the sample considered; so the values in parentheses are their standard deviations and not the standard errors. $VPi . s .$ and $VPo . s .$ indicate the portfolio variance calculated in the in-sample and out-of-sample sets, respectively. The lowest variance is set to 100, so a number like (100 $+$ x) means that the corresponding model provides an x% higher portfolio variance than the model having the lowest variance. The symbol $^{*}$ marks the best model in VP terms and models with VP, not significantly different from the best one, based on a DM test at a $5 %$ nominal size.

In Table 3 we show the estimation results relative to the $λ$ coefficients and the transition probabilities. The $λ$ coefficients are higher in the STCC case and they can assume a range of possible values in the FSMS model: between 0.01 (near to the uncorrelation case) and 0.29 in state $0$ and between 1 and 1.23 in state $1$ . The average of the $p_{11, t}$ transition probability in the TV–RSDC model is very similar to $p_{11}$ in the FSMS–RSDC model and its slight variability (standard deviation equal to 0.009) shows that it is quite stable; conversely, the average of $p_{00, t}$ of TV–RSDC differs from $p_{00}$ in FSMS–RSDC and its standard deviation (0.07) shows a certain variability in its behaviour. The log-likelihood is lower for the FSMS model.

Of course, in this application it does not make sense to evaluate the performance of the models by MSE because the true correlation is unknown and there are no logical proxies. When dealing with financial markets, it is frequent to compare the performance of alternative models by minimum variance portfolio rather than statistical criteria. Following Engle and Colacito (2006), we consider a set of $n = 28$ expected returns, each one equal to $1 / \sqrt{(} n)$ and we construct two alternative portfolios with the $n$ assets with weights minimizing the portfolio variance (see Markowitz, 1959); using the same expected returns and the same conditional variance for both portfolios, the differences will be due only to the conditional correlation matrices. The best model will be the one with the minimum sample portfolio variance. The VP index in Table 3 represents the portfolio variance obtained from the five models, both in the in-sample and out-of-sample set, setting to 100 the minimum variance. TV–RSDC is the one with the minimum variance portfolio in the in-sample experiment, but the variance of the portfolio derived from the FSMS–RSDC model is only $0.11 %$ higher; moreover, verifying the null hypothesis of equal variance by the DM test, $^{14}$ ¹⁴

See Engle and Colacito (2006) for the methodological and practical tools to apply this test in this framework.

the variances of these two portfolios are not significantly different. Considering the out-of-sample performance, the FSMS model has the minimum portfolio variance, significantly different from all the other alternatives.

4 Final remarks

We have proposed an extension of the MS model to allow the switching coefficients to vary within the states, increasing the flexibility of the model; $^{15}$ ¹⁵

The GAUSS codes and the data for performing estimation in all three examples are available on request or can be downloaded at the website: https://sites.google.com/site/edoardootranto/home/publications.

it provides gains in interpretability of the dynamics of the time series and in the goodness of fit. We have illustrated our theory using a very general model, which includes a wide range of specifications, whereas the examples of section 3 want to show how this general framework can be adapted to specific MS models and applications.

In our applications we have considered as switching the coefficients that affect the levels of the time series (in particular, the mean in the GDP example, the constant parameter of the volatility in the NASDAQ example and the scale coefficients of the conditional correlation matrix in the DJ dataset). This is done because the analysis of the levels can be made in graphical terms and provides a better appreciation of the interpretability of the new model; anyway, it is possible to extend the analysis also considering the other coefficients of the models as switching and flexible, or splitting the set of switching coefficients in two subsets: a set of flexible switching parameters and a set of fixed switching parameters. In fact, these cases are consistent with Equation (2.2), representing the general FSMS model.

We have considered only 2-state FSMS models; the formal extension to a larger number of states is not difficult, but, as in the MS case, it involves some computational problems, in particular, when a few observations belong to a single state. Anyway, the flexibility added to the MS model, with the possibility to change the value of the switching coefficients within the regimes, is sufficient to capture extreme jumps that, in a classical MS model, would belong to a further regime.

The comparison with other models was made performing in-sample and out-of- sample experiments for each application. In general, using the DM test, we have noticed that the FSMS model belongs to a restricted class of models showing the best in-sample performance, whereas in out-of-sample terms, in all the applications it shows the lowest loss functions, significantly better than those of the alternative models.

In FSMS models the changes within the states are driven by selected forcing variables; a pleasing characteristic is that these variables are not necessarily observed, but they could be derived from the estimation process. For example, in the GARCH and RSDC cases we have used the filtered probabilities, derived from the Hamilton filter, to drive the wsd of the switching coefficients. This idea is particularly appealing because the filtered probabilities are strictly correlated to the movements within the states; moreover, it is the only practical solution when observed forcing variables are not available.

Footnotes

Acknowledgments

I would like to thank an associate editor and an anonymous referee for their detailed reading of the article and many suggestions which led to the clarification of the focus of the article. All remaining errors are mine.

This research was supported by University of Messina (Research & Mobility 2015 Project; project code RES_AND_MOB_2015_ DISTASO).

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Adding flexibility to Markov Switching models

Abstract

Abstract:

Keywords

1 Introduction

1 The AR–MS model was originally proposed by Lindgren (1978), but its diffusion and success in the economic framework is due to the works of Hamilton.

2 A similar solution was adopted by Otranto (2008) in developing MS models with transition probabilities driven by latent variables.

Parameter estimates (standard errors in parentheses) and evaluation criteria of alternative models for the US GDP variations (in-sample set: Q1 1948–Q4 2012; out-of-sample set: Q1 2013–Q4 2014)

Parameter estimates (standard errors in parentheses) and evaluation criteria of alternative models for the volatility of the NASDAQ 100 index (in-sample set 2 January 2004–5 May 2015; out-of-sample set 2 January 2008–31 December 2008)

Parameter estimates (standard errors in parentheses) and evaluation criteria of dynamic conditional correlation models for the DJ dataset (in-sample set 2 January 2002–31 December 2014; out-of-sample set 2 January 2015–31 December 2015)

Footnotes

Acknowledgments

References

1
The AR–MS model was originally proposed by Lindgren (1978), but its diffusion and success in the economic framework is due to the works of Hamilton.

2
A similar solution was adopted by Otranto (2008) in developing MS models with transition probabilities driven by latent variables.