Markov Switching International Capital Asset Pricing Model,an Emerging Market Case: Mexico

Abstract

The article shows how the international capital asset pricing model (ICAPM) with Markov regime switching can model the asset returns in the emerging market of Mexico. For most assets, although significant, the international risk premium factor is not subject to regime switching, but the domestic factor is. The probabilities of regimes are correlated with the volatility of assets. A GARCH(1,1) Markov regime switching model offers better adjustment than a non-GARCH.

JEL Classification: C58, F36, F65, G12, G15

Keywords

International asset pricing model Markov regime switching GARCH

Introduction

Mexico has advanced a long way in its process of economic opening started in the 1980s and the financial liberalisation, the last vigorously promoted since the beginning of the 1990s. In general, a great deal has been done in the financial markets liberalisation with a strong emphasis on stock market regulations to promote the income of foreign capital flows pari passu the globalisation process around the world. López-Herrera and Ortiz (2010) have shown evidence that as a consequence of the Mexican financial markets’ liberalisation, the local stock market returns have a significant exposure to the returns of the world capital market portfolio. López-Herrera, Ortiz and Cabello (2009) and López-Herrera, Santillán and Ortiz (2014) have shown that the Mexican stock returns are highly sensitive to the movements of the US and Canadian stock markets, both being mature and summing up high to the market capitalisation of the world capital markets. López-Herrera and Venegas-Martínez (2012) extend the evidence about NAFTA capital markets integration to the options and futures markets. So, it is to be expected that the Mexican assets exhibit a no zero exposure to international risk factors further to the local risk factors derived from local risk sources.

The emerging markets and, in particular, Mexico suffered the consequences of national and international crises on numerous occasions in the period of study, which threatened the national economies and caused stock market downturns: the Tequila crises in 1994, the Asia crises in 1997, the Russia crises in 1998, the Argentina crises in 1999, the dot-com in 2000, the American sub-prime crises in 2007 and the global credit crises from 2010 to 2012. The consequences in emerging markets were diverse. In Mexico, more often than not, they resulted in downturns in the expectations of economic growth of Mexican companies and the economy as a whole, which translated in negative returns from asset prices.

The sharp decline and sudden devaluation of the Mexican peso against the US dollar in December 1994 resulted in what became known as the Tequila crisis. A fixed exchange rate system, low quality of the low interest rate credit granted by Mexican banks during the previous period and post-election economic instability were the precursors of the crises. At the end of 1994, the reserves were insufficient to maintain the fixed exchange rate, so, in the last days of that month, the Mexican peso suffered a devaluation in spite of the official statement of the government who had previously assured the contrary. Investors were scared away, which further rose the level of risk. The investors were unwilling to buy the debt, issued by the Mexican government in order to roll over the debt as maturity was coming; so a default episode became a possibility. The banking system suffered from the crisis of confidence, further affecting the confidence of investors. The consequences of the crisis lasted till 1996 (Wilson, Saunders, & Caprio, 2000).

Drastic currency depreciation worsened the corporate capital structure and brought widespread financial turmoil in East Asia in 1997 (Ho, 2004). The Asia crisis of 1997 became contagious. Russia suffered an exchange rate speculative attack in 1998 which forced the devaluation of the rouble and the default in its public debt (CEMLA, 2003).

The privatisation of state-owned enterprises, success in the fight against inflation, the strength of its banking system, the opening of its economy and a stable currency caused investors and the IMF to see Argentina as a success story. Unfortunately, the successful initial results were not sustained in the long term. Argentina’s currency had a fixed one to one parity against the dollar, so that if the dollar gained value against other currencies, the Argentine peso too. This regime of exchange rate influenced in a negative way Argentina’s exports, but its imports continued at the same rate. In 2000, investors lost confidence in the economy as the country increased its account deficits. Substantial capital outflows were observed and a deep currency devaluation occurred as a direct result (Wucker, 2002).

As Mills (2001) points out, during the 1990s the capital markets channelled money without an adequate selection scheme and targeted the new enterprises related to the internet or dot-com(s) as they were known. The dot-com enterprises were based on new business models, so, as it could be expected, conventional business plans and financial measures did not fit them well. Nevertheless, the investors continued using the old-fashioned toolboxes pushing the created companies to a high speed in implementing strategies. Many companies were forced to grow even though they were not ready to augment their operations and in some cases, the funds mainly came for mass-media advertisement. As a result, overinvestment in the sector happened. When the Federal Reserve tightened the monetary policy at the beginning of 2000, the lack of funding showed up the vulnerabilities of the sector and the dot-com bubble burst.

The sub-prime crisis, which worsened between 2007 and the first half of 2009, was caused by too much floating rate credit granted during the period 2004–2006 in the USA, which became difficult to refinance when interest rates increased and property prices fell in mid-2006 (Zandi, 2009). During 2007, many mortgages fell into default and mortgage back securities, including junk bonds, lost value. Global investors reduced their holdings of mortgage-backed securities as a result of the decline in the ability and willingness of the private financial system to refinance debts (Simkovic, 2013). The requirements in lending became more restrictive, reducing the supply of credit and economic growth. This situation worsened during 2008 and 2009, when the American and European economies fell into a deep recession.

The lower international economic activity, mainly in the USA, resulted in economic activity in Mexico falling, although less than in other economies of advanced countries. As a result of the international crisis, the production in Mexico fell during the last quarter of 2008 and the first half of 2009, showing a contraction in domestic demand and slower growth in exports. External demand fell 8.8 per cent in the last quarter of 2008, adjusted for seasonality.

In the late 2009, investors feared a European sovereign debt crisis. The eurozone suffered imbalances in the real and financial sectors and an undermined monetary stability, which were attributed to divergence and lack of budgetary coordination between countries in that area (Stark, 2013). The lack of macroeconomic balances and sustainable fiscal policies were the roots of the problem. In some countries, the growth in labour compensation was far larger than the productivity gains, which resulted in large increases in unit labour costs and weakening of the competitiveness. Simultaneously, large account deficits and high levels of public and private indebtedness were observed as a consequence of large increases in the unregulated financial sector, domestic demand, credit and real estate prices. The crisis exploded. By May 2010, Greece had to be rescued by other member states. Leading European nations implemented a series of financial support measures as the European Financial Stability (EFSF) and the European Stability Mechanism (ESM). The European Central Bank (ECB) lowered its interest rates and offered cheap loans to maintain flows between European banks. By the end of 2012, the debt crisis had forced 5 out of 17 Eurozone countries to seek help from other nations. As a result of the crisis, direct investment from European countries into developed economies, including Mexico, had fallen significantly during 2012 (Valencia-Herrera & Rivera, 2015).

This article contributes to the asset pricing literature by studying the exposure of the returns of Mexican assets to the world capital markets risk and the local stock market, into a context of switching regimes to take in account the structural breaks induced by world and local market events like those depicted earlier. With this aim in mind, the article is structured in the following way. The next section is devoted to the discussion of the theoretical background that serves as base to our analysis. Next, we describe our data and expose the methodological issues that underpin the empirical estimations whose results are discussed in a following section. And, finally, we offer our concluding remarks.

Theoretical Background

The capital asset pricing model (CAPM) is useful to determine the market price for risk and the appropriate measure of risk for a single asset. The CAPM was developed by Sharpe (1963, 1964) and further developed by Mossin (1966), Lintner (1965, 1969) and Black (1972). The model proposes that the equilibrium rates of return on all risky assets are a function of the covariance with the market portfolio. The return on any asset must be equal to the risk-free rate of return plus a risk premium. The risk premium is the price of risk multiplied by the quantity of risk

E (R_{i}) = R_{f} + (E (R_{m}) - R_{f}) \frac{σ_{i m}}{σ_{m}^{2}},

(1)

where R_f is the risk free return, E(R_m) is the market equilibrium price for risk, σ_im is the covariance of the asset I with the market and σ²_m is the variance with the market.

Under the assumption that average past returns are equal to expected returns, the CAPM can be written as

R_{i t}^{e} = β_{i} γ_{1, t} + ε_{i t},

(2)

where R^e_it is the risk premium of the asset i over the risk free rate, γ₁ is the risk premium of the market rate of the return over the return of the free risk asset, R_mt – R_ft, and β_i is the sensibility of the risk premium of the asset i on the risk premium of the market.

The CAPM can be tested under the assumption that the market model is true, that is, the return on the ith asset is a linear function of a market portfolio proxy, like an equally weighted market portfolio or a market index and R^e_mt is the market risk premium,

R_{i t}^{e} = α_{i} + β_{i} R_{m t}^{e} + ε_{i t},

(3)

where the alpha α_i is the excess return on the asset i, over the CAPM return, that, in equilibrium, must be equal to zero for all assets.

The international CAPM (ICAPM) is an extension of CAPM when the capital market expands to take in account the world capital market. It is worth to note that such device has obtained some academic support (O’Brien & Dolde, 2000; Schramm & Wang, 1999; Stulz, 1996, 1999). The ICAPM adds the return on an international portfolio of stocks, for example, the MSCI world index.

An alternative formulation, the arbitrage pricing model (APT), developed by Ross (1976), requires that the returns of any stock be linearly related to a set of risk factors or indices as shown in the following equation

R_{i} = α_{i} + b_{i 1} I_{1} + b_{i 2} I_{2} + \dots + b_{i k} I_{k} + ε_{i},

(4)

where α_i is the expected level of return of stock i if all indices have a value of zero, I_j is the value of the jth index that impacts the stock i, b_ij is the sensitivity of stock i to the jth index, ε_i a random error term with mean equal to zero and variance equal to $σ_{e i}^{2}$ . The process of generating security returns must satisfy

E(ε_iε_j) = 0 for all i and j, where i ≠ j and

$E (ε_{i} (I_{j} - {\overset{─}{I}}_{j})) = 0$ for all stocks and indices.

The description corresponds to a multi-index model. Furthermore, if indices are uncorrelated to each other, the covariance between indices j and k equals $E ((I_{j} - {\overset{─}{I}}_{j}) (I_{k} - {\overset{─}{I}}_{k})) = 0$ for all indices, with j = 1,…, L and k = 1,…, L, j ! k, the description of the multi-index model simplifies. However, the condition is not necessary because a model with correlated indices can always be transformed to a model with uncorrelated ones.

The APT can be specified as an excess return formulation, including in the right hand side the asset risk premium, R^e_i,t, one of the factors is the market risk premium, $I_{1 t}^{e} = R_{m t}^{e}$ , and the other factors are self-financing portfolios, portfolios constructed by going long in some stocks and going short in other stocks with equal market value, so the total investment on those stocks is zero,

R_{i t}^{e} = α_{i} + b_{i m} I_{1 t}^{e} + b_{i 2} I_{2 t} + \dots + b_{i k} I_{k t} + ε_{i t}

(5)

In this formulation, in equilibrium, the excess return on each asset, α_i, must be equal to zero.

A formulation of the APT is the international APT model, in which the market index return, R^e_mt, is replaced with the return on an international portfolio of stocks, like the MSCI world index, and the other factors are self-financing portfolios.

R_{i t}^{e} = α_{i} + b_{i m} I_{1 t}^{e} + b_{i 2} I_{2 t} + \dots + b_{i k} I_{k t} + ε_{i t} .

(6)

A particular self-financing portfolio can be constructed with the residuals of the regression of a country return portfolio with the international return portfolio, R_ct. By definition, the expected value of the residuals E(R_ct) in a regression is equivalent to create a zero-investment portfolio, in which an equal amount is invested in positive return elementary assets and in negative elementary assets. In particular, if the errors are homoscedastic, the expected value becomes zero.¹

R_{i t}^{e} = α_{i} + b_{i m} R_{m t}^{e} + b_{i c} R_{c t} + ε_{i t} .

(7)

In the Schwert (1989) model, the returns may have a high or a low variance, resulting in two return distributions whose switching is conducted by a two-state Markov process. In Turner, Startz & Nelson (1989), the Markov switching is extended to the mean, so it and the variance may change of regime in an individual fashion or simultaneously. Hamilton and Susmel (1994) analyse a model in which volatility is governed by a process with sudden discrete changes; they also show that a Markov model is able to provide a better fit to the data in comparison with a GARCH model without regime switching.

The Hamilton (1989) approach suggests that Markov switching techniques can be used to model non-stationary time series. With this approach, a discrete-state Markov process defines the parameters of the model. We use an extension of Hamilton’s approach to study stock market returns. For the Markov switching model, which allows for switching means and variances, the following specification can be stated:

\begin{matrix} R_{t} = a_{0} (1 - S_{t}) + a_{| 1} S_{t} + b_{0} I_{t}^{I} (1 - S_{t}) + b_{| 1} I_{t}^{I} S_{t} + c_{0} I_{t}^{D} (1 - S_{t}) + \\ c_{| 1} I_{t}^{D} S_{t} + [d_{0} \log (σ_{0}) (1 - S_{t}) + d_{1} \log (σ_{1}) S_{t}] ε_{t}, \end{matrix}

(8)

where S_t is an unobserved state variable following a stochastic process leaded by a Markov chain that determines the j-state in which the process is found at time t, I^I_t is the international factor, I^D_t is the domestic factor and σ₀ and σ₁ are the standard deviations in regimes 1 and 2, respectively. It is worth to note that this specification allows that the effect of the international and local indices on returns to be asymmetric.

Particular extensions of these models are the Markov switching autoregressive heteroscedastic (SWARCH) models (Chen & Huang, 2007; Hamilton & Susmel, 1994; Leon Li & Hsiou-wei, 2004) which allows various parameters based on different states to control structural changes that allowed for kurtosis, tail-fatness and skewness problems.

The Markov switching model following a GARCH(1,1) process, that is, the time-varying variance changing according with the following equation:

σ_{t}^{2} = v (S_{t}) + α (S_{t}) r_{t - 1}^{2} + β (S_{t}) σ_{t - 1}^{2},

(9)

where S_t is once again the unobserved state j at the time t, and so the coefficients of GARCH equations are non constant but changing in a regime-dependent fashion. In order to have positive time-varying variances, υ(S_t) ≥ 0, α(S_t) ≥ 0, and β(S_t) ≥ 0 is required.

Data and Methodological Issues

Stock data is from Economatica. The 14 stocks that traded since 2 December 1991 till 12 November 2013 were selected: ALFAA, BIMBOA, CIDSAA, COMERCIUBC, FEMSAUBD, GCARSOA1, KIMBERA, KUOB, PENOLES, SORIANA, TLEVISACPO, TMMA, VITROA and WALMEXV. Non-trading days were omitted. The number of trading dates for each of the considering assets is between 3,861 to 5,511 days. Using the daily MXN-USD exchange rate, reported in Economatica, prices in MXN were converted to USD. Logarithmic daily returns were estimated. For each stock, the risk premium over the 12-month Libor rate reported by Bloomberg, used as a risk-free rate proxy was estimated.

As discussed previously, a Markov switching ICAPM with two factors (international and domestic factors) was estimated for each asset. The international factor is the MSCI world capital market portfolio risk premium with the 12-month Libor rate published by Bloomberg as proxy for the risk-free rate. The domestic factor is the residual return of the IPC (Mexico’s Bolsa index) obtained by regressing the IPC daily return measured in US dollars on the MSCI daily return.

Several studies have tested the ICAPM. For example, Adler and Solnik (1974), Agmon (1974) and Lessard (1974) estimate an ICAPM in which the betas are non-changing and are determined by global and domestic factors influencing the asset returns. In order to test the ICAPM, Harvey and Zhou (1993) and Ferson and Harvey (1994) included the Morgan Stanley world index as proxy for the world market portfolio. Bekaert and Harvey (1995) and Ramchand and Susmel (1998) analysed the beta coefficients of the ICAPM applying the Markov switching model, and they got a strong evidence in favour of state-varying beta coefficients. Aquino (2006) used a variance equality test of the ICAPM on Philippine stocks. More recently, Chen and Huang (2007) tested the ICAPM with regime-switching betas on stocks from four Pacific Rim countries.

In this article, the model was tested for regime switching in the constant, the international and domestic factors, and logarithm of the variances. The Durbin–Watson coefficient was used to test for autoregressive effects, in which case a correction for regime switching autocorrelation was introduced and tested for. The best fit model is reported for each asset.

Under the Markov switching modelling framework, there is an unobserved state S_t which is assumed as following a first-order Markov process. The implication of the Markov property is that the current regime, S_t = j, j = 1,…, m, depends only on the regime (or state) in which the process stayed one period ago, S_t-1 = i, that is, the transition probabilities can be defined as:

P (S_{t} = j | S_{t - 1} = i, S_{t - 2} = k, \dots) = P (S_{t} = j | S_{t - 1} i) = p_{j i},

\sum_{j = 1}^{m} p_{j i} = 1

(10)

which are m(m − 1) fixed parameters to be estimated.

f (y_t |s_t, Φ_t–1) is the probability density of observing y_t_, conditional on the state-variable S_t and the information set Ω_t–1, which, in turn, contains all the information available at time t – 1. We define the state regime probability as

P (S_{t} = j | Ω_{t}) = \frac{f (y_{t} | S_{t} = j, Ω_{t - 1}) P (S_{t} = j | Ω_{t - 1})}{\sum_{i}^{m} f (y_{t} | S_{t} = i, Ω_{t - 1}) P (S_{t} = i | Ω_{t - 1})}

P (S_{t} = j | Ω_{t - 1}) = \sum_{i = 1}^{m} p_{i j} P (S_{t - 1} = i | Ω_{t - 1}) .

(11)

So, we have the likelihood function to be maximised:

L_{T} = \sum_{t = 1}^{T} \log \sum_{j = 1}^{m} f (y_{t} | S_{t} = j, Ω_{t - 1}) P (S_{t} = j | Ω_{t - 1}) .

(12)

Following from the Bayes’ theorem, equation (13) shows the posterior probability of the regime variable S_t, conditional to all the information available at time t – 1:

P (S_{t} | Ω_{t - 1}) = P (S_{t} | y_{t}, Ω_{t - 1}) = \frac{f (y_{t} | S_{t}, Ω_{t - 1}) P (S_{t} | Ω_{t - 1})}{f (y_{t} | Ω_{t - 1})},

(13)

yielding the calculations that are necessary to provide the filtered regime probabilities, as a by-product of the maximum likelihood estimation procedure applied to a sample by means of a forward recursion initialised with an estimated initial value of the regime variable.

Based on the information available up to time t, the aforementioned recursion provides estimates for S_t. Therefore, filtered probabilities do not consider all the available information. That is, the technique used to get filtered probabilities offers only limited information because it does not make use of the complete information gathered at T. So, the use of all the information available at T allows to improve the inference about the regime at each point of the time and smoothed probabilities can be obtained. In summary, filtered probabilities P(S_t = j|Ω_t) are inferences about S_t considering only the relevant information up to time t and, at the other hand, smoothed probabilities P(S_t = j|Ω_T) are inferences about S_t employing all the information available in the whole sample, t = 1, 2,… T.

Analysis of Estimation Results

In Table 1, we can see the results from the analysis of the regime switching model. For most of the analysed stocks, the world market risk factor, PMSCI, is statistically a common factor, except for the case of KIMBERA, for which it is a regime switching factor. Similarly, for most of the stocks, the RIPC is statistically a regime switching factor, except for the cases of GCARSOA1, TMMA and WALMEXV. In the case of GCARSOA1, it is so, probably because of the strong composition of the Mexican IPC with GCARSOA1 shares. The logarithm of the volatility is statistically a regime switching variable for all stocks analysed in the sample.

For ALFAA (0.006 and 0.103) and VITROA (.058 and 0.001), the autoregressive coefficient was statistically different from cero and different in each of the two regimes. For CIDSAA (0.004), COMERCIUBC (0.0382), KIMBERA (0.051) and WALMEXV (0.034), the autoregressive coefficient was statistically different from cero, but it was statistically not different in each of the two regimes. The probabilities for each of the two states were statistically different from cero in all cases.

There is a relationship between probabilities and the volatility of the returns. Using as a volatility proxy the squared of the returns, we found that for all stocks, the probability of one of the regimes was positively correlated with the volatility proxy. Graphically, Figure 1 illustrates the probability positively correlated with the volatility proxy. For most of the stocks, a period in which the volatility proxy was higher, the probability of the regime positively associated with high volatility was higher. The relation was statistically significant.

As we can see in Table 2, the absolute value of the Pearson correlation coefficient was between 0.148 and 0.434 for all shares. The absolute value of the rho of Spearman was between 0.277 and 0.461 for all shares. The rho of Spearman has advantages for measuring correlation because it is a measure less sensitive than the Pearson correlation to outliers that can be in the tails of the samples.

Table 1.

Regime Switching Equation

Variable	ALFAA		BIMBOA		CIDSAA		COMERCIUBC		FEMSAUBD
Variable	Coef.	Std. Error	Coef.	Std. Error	Coef.	Std. Error	Coef.	Std. Error	Coef.	Std. Error
Regime 1
RIPC	0.779	(0.026)***	0.632	(0.040)***	0.517	(0.036)***	0.801	(0.024)***	1.372	(0.122)***
C	–0.0002	(0.00003)***							0.000	(0.000)***
AR(1)	0.006	(0.001)***
LOG(SIGMA)	–4.322	(0.036)***	–4.339	(0.091)***	–3.970	(0.044)***	–4.089	(0.040)***	–3.412	(0.112)***
Regime 2
RIPC	1.074	(0.067)***	0.904	(0.082)***	0.863	(0.107)***	1.529	(0.514)***	0.927	(0.028)***
C	–0.0000	(0.00000)***							–0.001	(0.000)***
AR(1)	0.103	(0.033)***
LOG(SIGMA)	–3.430	(0.057)***	–3.501	(0.133)***	–2.670	(0.072)***	–2.163	(0.435)***	–4.422	(0.038)***
Common
PMSCI	0.966	(0.013)***	0.803	(0.021)***	0.655	(0.043)***	0.9148	0.0292***	0.921	(0.017)***
C			–0.0026	(0.0004)***	–0.006	(0.001)***	–0.001	0.0006***
AR(1)					0.004	(0.001)***	0.0382	0.0178***
Transition Matrix Parameters
P11-C	3.656	(0.324)***	2.934	(0.597)***	2.804	(0.235)***	4.0955	0.3158***	1.308	(0.366)***
P21-C	–2.500	(0.334)***	–1.692	(0.400)***	–1.505	(0.274)***	–0.681	0.3748***	–3.095	(0.227)***
High Volatility Regime
	2		2		2		2		1

Variable	GCARSOA1		KIMBERA		KUOB		VITROA		WALMEXV
Variable	Coef.	Std. Error	Coef.	Std. Error	Coef.	Std. Error	Coef.	Std. Error	Coef.	Std. Error
Regime 1
PMSCI			0.745	(0.021)
RIPC			0.584	(0.029)	0.588	(0.033)	0.954	(0.069)
C	–0.000	(0.000)***							–0.002	(0.001)
AR(1)							0.058	(0.025)
LOG(SIGMA)	–2.120	(0.689)***	–4.675	(0.045)	–4.28	(0.056)	–3.14	(0.064)	–3.828	(0.055)
Regime 2
PMSCI			0.806	(0.030)
RIPC			0.972	(0.038)	0.962	(0.062)	0.791	(0.031)
C	–0.000	(0.000)***							–0.002	(0.000)
AR(1)							0.001	(0.000)
LOG(SIGMA)	–4.189	(0.048)***	–3.788	(0.040)	–3.24	(0.064)	–4.21	(0.042)	–4.568	(0.036)
Common
PMSCI	0.974	(0.013)***			0.766	(0.028)***	0.829	(0.027)***	0.860	(0.016)***
RIPC									1.035	(0.018)***
C	1.055	(0.022)***	–0.003	(0.000)***	–0.004	(0.000)***	–0.004	(0.000)**
AR(1)			0.051	(0.014)***					0.034	(0.008)***
Transition Matrix Parameters
P11-C	0.413	(0.219)***	2.533	(0.196)***	2.718	(0.306)***	1.236	(0.201)***	3.038	(0.425)***
P21-C	–4.689	(0.721) ***	–2.080	(0.217)***	–1.689	(0.275)***	–2.42	(0.172)**	–3.877	(0.400)***
High Volatility Regime
	1		2		2		1		1

Variable	PENOLES		SORIANAB		TLEVISACPO		TMMA
Variable	Coef.	Std. Error	Coef.	Std. Error	Coef.	Std. Error	Coef.	Std. Error
Regime 1
PMSCI							0.633	0.0644***
RIPC	0.482	(0.050)***	0.821	(0.040)***	0.866	(0.024)***
C	–0.002	(0.001)***	–0.002	(0.000)***	0.000	(0.000)***
LOG(SIGMA)	–4.164	(0.048)***	– 3.610	(0.036)***	–4.535	(0.024)***	–3.857	0.0837***
Regime 2
PMSCI							0.9368	0.2232***
RIPC	0.806	(0.077)***	0.782	(0.035)***	1.089	(0.037)***
C	—0.001	(0.001)	—0.001	(0.000)***	0.002	(0.000)***
LOG(SIGMA)	–3.210	(0.063)***	–4.381	(0.035)***	–3.646	(0.050)***	–2.407	0.1362***
Common
PMSCI	0.858	(0.029)***	0.868	(0.019)***	1.031	(0.012)***
RIPC							0.4724	0.0544***
C							–0.006	0.001***
AR(1)	0.078	(0.014)***
Transition Matrix Parameters
P11-C	2.624	(0.245)***	2.597	(0.257)***	3.327	(0.180)***	2.2393	0.2661***
P21-C	–1.580	(0.245)***	–3.138	(0.262)***	–2.373	(0.266)***	–0.749	0.2992**
High Volatility Regime
	2		1		2		2

Source: Authors’ own elaboration.

Notes: ***, ** and *, statistical significance at 99%, 95% and 90%. Standard errors are in parenthesis. Standard errors are robust, except for sigma, alpha, beta and probabilities, in which they are normal. Regime 1 has low volatility. Regime 2 has high volatility.

Figure 1.

Source: Authors’ own elaboration.

The stocks show statistically different alphas and betas at each regime, except for SORIANAB, which does not show statistical differences in those coefficients for the regimes. The low volatility regime is always stationary: The sum of alpha and beta is always lower than one in this regime. The high volatility regime is not always stationary. In 10 of the 13 cases in which alpha and beta are statistically different for each regime, the high volatility regime was not stationary; the sum of alpha and beta in the regime is higher than one, but only slightly for most cases, which can be compensated with the coefficients of the other stationary regime. Two possible exceptions are COMERCIUBC, which has alpha and beta coefficients that add up to 1.467, and VITROA, which has alpha and beta coefficients that add up to 1.15 in the high volatility regime.

Table 2.

Correlation Between Probability of Regime 2 and Volatility for Markov Switching Model

	Pearson	Spearman
ALFAA	0.349	0.317
BIMBOA	0.306	0.339
CIDSAA	0.384	0.461
COMERCIUBC	0.148	0.347
FEMSAUBD	–0.339	–0.345
GCARSOA1	–0.197	–0.277
KIMBERA	0.326	0.328
KUOB	0.434	0.379
PENOLES	0.464	0.388
SORIANAB	–0.322	–0.366
TLEVISACPO	0.340	0.333
TMMA	0.310	0.550
VITROA	–0.375	–0.355
WALMEXV	–0.279	–0.321

Source: Authors’ own elaboration.

Not only the volatility is higher in the high volatility regime, but the sensitivity to the local factor is also higher. In the high volatility regime, the beta coefficient for the local factor, RIPC, is larger than the one in the low volatility regime in all the stocks analysed. The sensibility to volatility is also higher in the high volatility regime than in the low one. The only observed exceptions were KIMBERA and ALFAA. KIMBERA has beta coefficients for volatility almost the same in both regimes. ALFAA stock has a different behaviour, the stock from a company which has been in financial difficulties during the period of analysis. ALFAA shows a positive relation of the low sensitivity regime with the high volatility and a relation of the high sensitivity regime with the low volatility. The coefficient for sigma in the low volatility regime is 0.001, slightly lower than in the high volatility regime, 0.002, but in the former regime all other coefficients are higher, RIPC, the constant, alpha and beta have coefficients 1.213, 0.002, 0.070 and 0.093 versus the coefficients in the later regime 0.722, –0.001, 0.029 and 0.928. Notice that in both regimes the GARCH regime is stationary, alpha and beta sum up to less than one.

Model estimates are improved with the GARCH(1,1) Markov switching specification whose estimation results are shown in Table 3. As Akaike and Bayesian information criteria show, all the estimated equations from the GARCH(1,1) Markov switching model show better (more negative) than estimates from the simple Markov switching model, see Table 4.

Table 3.

Regime Switching Equation (GARCH)

Variable	ALFAA		BIMBOA		CIDSAA		COMERCIUBC		FEMSAUBD
Variable	Coef.	Std. Error	Coef.	Std. Error	Coef.	Std. Error	Coef.	Std. Error	Coef.	Std. Error
Regime 1
RIPC	1.213	(0.104)***	0.660	(0.049)***	0.338	(0.046)***	0.772	(0.037)***	0.950	(0.025)***
C	0.002	(0.001)**					– 0.002	(< 0.001)***	–0.001	(< 0.001)***
SIGMA	0.001	(0.001)*	0.002	(0.000)***	0.003	(0.000)***	0.003	(< 0.001)***	0.002	(< 0.001)***
Alfa	0.070	(0.014)***	0.017	(0.007)**	0.011	(0.003)***	0.050	(0.009)***	0.027	(0.005)***
Beta	0.963	(0.007)***	0.963	(0.013)***	0.880	(0.028)***	0.866	(0.023)***	0.945	(0.010)***
Regime 2
RIPC	0.722	(0.037)***	1.074	(0.122)***	0.890	(0.059)***	0.937	(0.222)***	1.868	(0.222)***
C	–0.001	(0.000)**					0.005	(0.004)	0.000	(0.003)
SIGMA	0.002	(0.000)***	0.031	(0.002)***	0.004	(0.001)***	0.014	(0.002)***	0.014	(0.004)***
Alpha	0.029	(0.007)***	0.160	(0.073)**	0.147	(0.027)***	0.816	(0.175)***	0.224	(0.155)
Beta	0.928	(0.014)***	0.000	(0.000)	0.914	(0.015)***	0.657	(0.058)***	0.781	(0.112)***
Common
PMSCI	0.957	(0.019)***	0.806	(0.020)***	0.697	(0.032)***	0.904	(0.021)***	0.918	(0.016)***
C			–0.002	(0.000)***	–0.005	(0.001)***
AR(1)	0.040	(0.015)***
Transition Matrix Parameters
P11-C	0.617	(0.050)***	0.944	(0.014)***	0.640	(0.031)***	0.881	(0.023)***	0.941	(0.010)***
P22-C	0.856	(0.023)***	0.680	(0.096)***	0.663	(0.029)***	0.247	(0.051)***	0.261	(0.068)***

Variable	GCARSOA1		KIMBERA		KUOB		VITROA		WALMEXV
Variable	Coef.	Std. Error	Coef.	Std. Error	Coef.	Std. Error	Coef.	Std. Error	Coef.	Std. Error
Regime 1
PMSCI			0.761	(0.020)***
RIPC			0.527	(0.035)***	0.522	(0.038)***	0.751	(0.030)***
C									–0.002	(< 0.001)***
SIGMA	0.004	(< 0.001)***	0.002	(< 0.001)***	0.001	(< 0.001)***	0.003	(< 0.001)***	0.001	(< 0.001)***
Alfa	0.112	(0.013)***	0.043	(0.009)***	0.013	(0.004)***	0.030	(0.008)***	0.019	(0.003)***
Beta	0.835	(0.019)***	0.860	(0.023)***	0.957	(0.010)***	0.910	(0.023)***	0.967	(0.005)***
Regime 2
PMSCI			0.807	(0.036)***
RIPC			1.039	(0.044)***	1.049	(0.083)***	1.145	(0.083)***
C									–0.000	(0.001)
SIGMA	0.137	(0.782)	0.002	(0.001)***	0.002	(0.001)***	0.004	(0.001)***	0.003	(0.001)**
Alpha		(0.389)	0.075	(0.017)***	0.067	(0.014)***	0.095	(0.023)***	0.129	(0.062)**
Beta	0.795	(2.354)	0.945	(0.013)***	0.965	(0.008)***	0.957	(0.009)***	0.947	(0.027)***
Common
PMSCI	0.974	(0.020)***			0.755	(0.025)***	0.830	(0.026)***	0.878	(0.015)***
RIPC	1.042	(0.020)***							1.041	(0.018)***
C	–0.00	(< 0.001)	–0.003	(< 0.001)***	–0.004	(< 0.001)***	–0.004	(< 0.001)***
Transition Matrix Parameters
P11-C	0.997	(0.001)***	0.786	(0.029)***	0.815	(0.023)***	0.860	(0.016)***	0.917	(0.020)***
P22-C	0.036	(0.116)	0.672	(0.046)***	0.619	(0.045)***	0.537	(0.050)***	0.424	(0.082)***

Variable	PENOLES		SORIANAB		TLEVISACPO		TMMA
Variable	Coef.	Std. Error	Coef.	Std. Error	Coef.	Std. Error	Coef.	Std. Error
Regime 1
PMSCI							0.650	(0.060)***
RIPC	–0.003	(0.001)***	0.706	(0.062)***	0.887	(0.021)***
C	0.340	(0.052)***	–0.002	(< 0.001)***	0.000	(< 0.001)
SIGMA	0.003	(< .001)***	0.001	(< 0.001)***	0.001	(< 0.001)***	0.004	(< .001)***
Alfa	0.021	(0.007)***	0.021	(0.005)***	0.024	(0.004)***	0.045	(0.009)***
Beta	0.908	(0.026)***	0.964	(0.009)***	0.959	(0.006)***	0.876	(0.020)***
Regime 2
PMSCI							1.669	(0.379)***
RIPC		(0.001)	1.182	(0.164)***	1.782	(0.378)***
C	0.976	(0.052)***	–0.001	(0.001)	0.009	(0.002)***
SIGMA	0.001	(0.001)	0.004	(0.001)***	0.031	(0.002)***	0.011	(0.004)***
Alpha	0.047	(0.007)***			0.230	(0.160)	0.380	(0.116)***
Beta	0.973	(0.004)***			0.000	(< 0.001)	0.886	(0.034)***
Common
PMSCI	0.851	(0.027)***	0.857	(0.020)***	1.022	(0.017)***
RIPC							0.520	(0.048)***
C							–0.005	(0.001)***
AR(1)	0.085	(0.016)***
Transition Matrix Parameters
P11-C	0.800	(0.026)***	0.901	(0.026)***	0.952	(0.008)***	0.859	(0.020)***
P22-C	0.694	(0.050)***	0.677	(0.072)***	0.385	(0.071)***	0.258	(0.056)***

Source: Authors’ own elaboration.

As shown in Table 5, positive correlation is also observed between the probability of one of the regimes (two) and the volatility of the of the GARCH(1,1) Markov switching model. The Pearson correlation coefficient is between 0.112 and 0.466 and the Spearman correlation coefficient is between 0.220 and 0.578, which indicates that regime 2 corresponds to a high volatility state. Also notice that in all cases, the Spearman correlation coefficient is higher than the Pearson correlation coefficient because that coefficient is less sensitive to outliers, so frequently observed in stock markets from emerging economies.

The positive relation between probability of a regime and volatility, measured by the squared of the estimation error, is also observed in the graphs in the aforementioned Figures 1 and 2. Probability patterns has less fluctuations in the simple Markov switching model than in the GARCH(1,1) Markov switching model, which makes for a more simple interpretation of the relationship. However, when correlation is measured by the Pearson and Spearman correlation coefficients, the level of correlation is in most cases only slightly lower, and even in some cases higher in the GARCH(1,1) Markov switching model than in the simple one.

Table 4.

Comparison of Models Using Akaike and Bayesian Information Criteria

	Markov Switching		Garch(1,1) Markov Switching
	AIC	BIC	AIC	BIC
ALFAA	–5.234570	–5.221347	–5.29420189	–5.27737266
BIMBOA	–5.271188	–5.261193	–5.28958541	–5.27459198
CIDSAA	–4.220072	–4.205476	–4.32129237	–4.30183613
COMERCIUBC	–5.026505	–5.015673	–5.1290341	–5.11339006
FEMSAUBD	–5.492021	–5.481217	–5.54155649	–5.52595102
GCARSOA1	–5.365726	–5.356121	–5.44551381	–5.43230714
KIMBERA	–5.602991	–5.590976	–5.63852677	–5.62290926
KUOB	–4.916401	–4.906031	–4.98487295	–4.96931856
PENOLES	4.741022	–4.72852	–4.78365545	–4.76615182
SORIANAB	–5.207343	–5.196204	–5.23738666	–5.22377214
TLEVISACPO	–5.572798	–5.561961	–5.60996731	–5.5943136
TMMA	–3.849263	–3.832547	–3.93565639	–3.91058239
VITROA	–4.754940	–4.742523	–4.80538087	–4.7904835
WALMEXV	–5.749325	–5.738514	–5.78818396	–5.77377224

Source: Authors’ own elaboration.

Notes: AIC, Akaike information criterion; BIC, Bayesian information criterion.

Table 5.

Correlation Between Probability of Regime 2 and Volatility for GARCH (1,1) Markov Switching Model

	Pearson	Spearman
ALFAA	0.365	0.226
BIMBOA	0.368	0.352
CIDSAA	0.29	0.500
COMERCIUBC	0.112	0.280
FEMSAUBD	0.372	0.311
GCARSOA1	0.413	0.291
KIMBERA	0.351	0.262
KUOB	0.466	0.363
PENOLES	0.445	0.333
SORIANAB	0.401	0.356
TLEVISACPO	0.329	0.299
TMMA	0.383	0.578
VITROA	0.421	0.368
WALMEXV	0.267	0.220

Source: Authors’ own elaboration.

Figure 2.

Source: Authors’ own elaboration.

Conclusions and Recommendations

A contribution of the article is that it shows at the stock level how asset returns change in an emerging economy, such as Mexico, with a domestic and an international factor in a GARCH(1,1) model with Markov regime switching. The article shows how the sensibilities to a local factor and an international factor are different for a sample of stocks in the Mexican Stock Exchange. Most of the analysed stocks have an international non-switching factor and a domestic switching factor.

Emerging market economies, like the Mexican, have gone through repeated periods of instability. The behaviour of stocks changes during turbulent times. Stocks in unstable periods, when the volatility increases, have different behaviour from the one observed in stable periods, when the volatility is lower. For most stocks, the sensibilities are higher in the high volatility regime than in the low one. Most of the analysed stocks showed higher sensitivity to the local market factor and larger GARCH and arch coefficients in the high volatility regime than in the low one. The arch and GARCH parameters in the low volatility regime are always stationary, but not necessarily in the high volatility regime.

Footnotes

Note

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