Interactions between real economic and financial sides of the US economy in a regime-switching environment

Abstract

This objective of this study is to examine the linkages between real (economic) and financial variables in the United States in a regime-switching environment that accounts explicitly for high volatility in the stock market and high stress in financial markets. Since the linearity test shows that the linear model should be rejected, we employ the Markov-switching VECM to examine the same objective using the Bayesian Markov-chain Monte Carlo method. The regime-dependent impulse response function (RDIRF) highlights the increasing importance of the financial sector of the economy during stress periods. The responses and their fluctuations are significantly greater in the high-volatility regime than in the low-volatility regime.

Keywords:

• real and financial sectors
• Bayesian MCMC method
• regime-dependent impulse
• VIX
• FSI

JEL Classification:

• E10

Notes

¹ We have initially estimated models with policy uncertainty indexes. However, we were not able to find any significant impact from policy uncertainty index to other variables.

² This is evident by higher impulse responses in the high-volatility regime, as compared to the response in the low (noncrisis) volatility regime.

³ For the emerging markets, the literature also documents significant interactions between the oil price, economic activity and asset markets. Nikkinen et al. (2014) and Lin et al. (2014) examine the oil price shocks and asset markets, while Fernandez (2014) investigates the impact of macroeconomic factors on financial markets.

⁴ Due to data availability, monthly data are preferred in this study. Financial data may have different dynamics in higher frequencies and this are usually about volatility dynamics which is captured by many moments greater than one. This study focuses on the short-run and long-run interactions of the levels (conditional means or the first moment) of the variables and different (ultra) high moment dynamics should affect our results. Moreover, interactions in the first moments also imply interactions in higher moments, but is not necessarily vice versa.

⁵ Our sample period is restricted by the unavailability of St. Louis Fed Financial Stress Index (FSI) series before December. 1993.

⁶ The Litterman’s method allows nonstationary error processes and works for non-cointegrated series as well, while Chow and Lin (1971) allows only stationary error processes. We account for autocorrelation using an autoregressive process of order 1, AR(1).

⁷ For example, if VIX is 50, one can infer that the index options markets expect with a 68% probability the S&P 500 index to move up or down $\frac{50\mathrm{\%}}{\sqrt{12\mathrm{m}\mathrm{o}\mathrm{n}\mathrm{t}\mathrm{h}\mathrm{s}}}\approx 14\mathrm{\%}$ over the next 30-day period.

⁸ We thank an anonymous referee who suggested the robustness check for the unit root tests by allowing nonlinearity.

⁹ A deterministic time trend can be included as well.

¹⁰ Let T denote the LR statistic, m the number of coefficients in the conditional mean that vanish under the null hypothesis and q the number of transition probabilities that vanish under the null hypothesis, then the conventional LR test is: $P[{\chi }^2\left(m+q\right)>T]$. The approximate upper bound under the adjusted LR test is given by

$P\left[{\chi }^2\left(q\right)\text{>T}\right]+2{\text{T}}^{1/2}\text{exp}\left\{\left(\frac{q}{2}-0.5\right)\log \left(T\right)-\frac{\text{T}}{2}-\frac{q}{2}\log \left(2\right)-\log \left(\frac{q}{2}\right)\right\}.$

If the adjusted LR test statistic exceeds the approximate upper bound, then the null hypothesis of linear specification is rejected.

¹¹ We conduct the LR ratio test and use the Akaike and Schwartz information criteria on the number of regimes. The evidence supports that the number of regimes is two and not three. This result is available upon request.

¹² Our specification assumes constant and regime-independent cointegration vectors, while allows for the presence of the regime-dependent adjustment to the equilibrium. This specification is consistent with the nonlinear adjustment to the equilibrium examined in Savit (1988).

¹³ Although the equilibrium parameters (represented by matrix $\beta$) are state-independent, Camacho (2005) shows that the equilibrium errors follow an MS-VARM under the specification in Equation 4. Indeed, Equation 4 can be obtained from a model where the equilibrium errors follow an MS-VAR process. To our knowledge, there is no theoretical result that allows us to estimate the MS-VECM with a state-dependent $\beta$, i.e. $\beta \left(s_t\right).$

¹⁴ This study considers conditional mean interactions as its core objectives, but volatility transmission is not explicitly allowed as it is not part of this study’s goals. Allowing for volatility transmission would extensively complicate the model structure and it would also be difficult to estimate due to the presence of a large number of parameters. In relation to the importance of volatility transmission, our methodology follows the Granger causality approach, the most commonly used concept of volatility spillover in which the influence of a past shock of one series is emphasized on the current volatility of the other. Guidolin and Timmermann (2008) report that finding evidence of classical (ARCH effect is rather weak when the levels are modelled in a multivariate MS model. The MS models with a regime-switching variance have heteroscedastic conditional variance and may be rich enough to eliminate the ARCH effects in the residuals. Moreover, Cheung and Ng (1996) argue that the presence of volatility transmission has an ignorable impact on the analysis based on mean effects. Therefore, the impact of ignoring volatility transmission should have only an ignorable impact on our results.

¹⁵ We modify the MCMC method of Balcilar et al. (2015) and introduce a rejection sampling scheme where the draws that violate the restriction that second regime has greater variance than the first regime for all equations are rejected.

¹⁶ The details of the MCMC implementation is omitted to save space. We refer the interested reader to Balcilar et al. (2015) as our implementation is analogues to the MCMC procedure used there.

¹⁷ Further details can be found in Balcilar et al. (2015).

¹⁸ The BIC selects a restricted constant trend specification where a constant enters into the cointegration relationship. Both trace and maximal eigenvalue tests of Johansen (1988, 1991) indicate two cointegration vectors under the restricted constant specification.

¹⁹ We tested the number of regimes up to 3. The two-regime model is the preferred one by statistical tests. The results are available upon requests.