Stock market uncertainty and economic fundamentals: an entropy-based approach

Abstract

This study investigates the effects of stock market uncertainty on economic fundamentals, represented by economic activities and systemic risk, in China. To capture the uncertainty in the Chinese stock market precisely, we use the entropy measure through symbolic time-series analysis. The empirical findings reveal strong spillover effects from stock market uncertainty to economic fundamentals. Specifically, an uncertainty shock generates (i) a short-term decline in industrial production, (ii) a rapid drop and rebound in the composite leading indicator, and (iii) an increase in systemic risk. To understand these findings, we suggest and validate the transmission channel through changes in consumption and investment.

Keywords:

• Entropy
• Stock market uncertainty
• Economic activities
• Systemic risk

JEL Classification:

• C32
• E44

Disclosure statement

No potential conflict of interest was reported by the authors.

ORCID

D. Lee http://orcid.org/0000-0002-8247-9179

Notes

1 One may raise a possible endogeneity issue that a third factor (e.g., policy uncertainty) affects both stock market uncertainty and economic fundamentals. But this possibility does not harm or invalidate our analysis, because stock market uncertainty supposedly already reflects the shocks, not only from the stock market, but also from other areas of the economy. Indeed, stock market uncertainty has been widely used as a proxy for overall macroeconomic uncertainty (Alexopoulos and Cohen 2009, Bloom 2009).

2 The choice of a specific value $X^S$ can be arbitrary, because the entropy does not depend on the location but only on its density.

3 Changing the base of logarithm only causes a rescale of the corresponding coefficients, but does not affect the significance level of generalized spillover and SVAR analysis. Thus, using ‘ln’ or ‘${\log }_2$’ would not systematically affect our results. When we first introduce the concept of entropy, we borrow the general definition from Shannon (1948), which uses ‘ln’. In the analysis, e.g., generalized spillovers and SVARs, we take ‘${\log }_2$’ following Oh et al. (2015) that the normalized entropy is bounded in the range of $[0,1]$ and thus, useful for qualitative analysis.

4 In the case of STSA, the number of patterns is as large as $32(=2^5)$. For the raw return data, only 20+ return observations exist in a month. If we make an empirical distribution (histogram) from these 20+ observations, the number of bins cannot be large enough to meaningfully estimate the entropy.

5 Following Hansen (1994), we propose a GARCH model with the mean and variance equations specified as follows, $\begin{array}{rl}{r}_t& =\alpha +{ϵ}_t,\\ {\sigma }_t^2& ={\beta }_0+{\beta }_1({ϵ}_{t-1}^2-{\sigma }_{t-1}^2)+{\beta }_2{\sigma }_{t-1}^2,\end{array}$where $r_t$ is the daily stock return of the SSECI at time t and ${\sigma }_t^2$ is the conditional variance of the error term ${ϵ}_t$ at time t. Define the normalized error as $z_t\equiv {ϵ}_t/{\sigma }_t$. Instead of following a standard Gaussian distribution as specified in a traditional GARCH model, $z_t$ follows a skewed student's t distribution with the pdf: $g(z_t|\eta ,\kappa )=\left\{\begin{array}{ll}bc{\left(1+\frac{1}{\eta -2}{\left(\frac{b{z}_t+a}{1-\kappa }\right)}^2\right)}^{-(\eta +1)/2}& z_t<-a/b,\\ bc{\left(1+\frac{1}{\eta -2}{\left(\frac{b{z}_t+a}{1+\kappa }\right)}^2\right)}^{-(\eta +1)/2}& z_t\ge -a/b,\end{array}\right.$ where $2<\eta <\mathrm{\infty }$ and $-1<\kappa <1$. a, b, and c are given by $a=4\kappa c\left(\eta -2/\eta -1\right)$, $b^2=1+3{\kappa }^2-a^2$, and $c=\frac{\mathrm{\Gamma }\left(\frac{\eta +1}{2}\right)}{\sqrt{\pi (\eta -2)}\mathrm{\Gamma }\left(\frac{\eta }{2}\right)}$. We estimate parameters by the maximum likelihood method and obtain the daily volatilities ${\sigma }_t^2$. Monthly volatility is the average of daily volatilities in each month.

6 We calculate ΔIPI on a year-on-year basis, because it mitigates the periodic fluctuations and corrects the seasonality and holiday effect in the data (Sun 2012).

7 Owing to the Chinese New Year effect, the time series of ΔIPI in China is discontinuous, with January data missing since 2005. We apply the linear interpolation to complete the missing data (Osiewalski and Osiewalski 2012).

8 Bloom (2009) categorized all variables into three groups and ordered them as follows: the stock market factors (stock market level and uncertainty), the price-related variables (federal fund rate, consumer price index, average hourly earnings), and quantities (hours, employment, industrial production index). The rationale behind this ordering is that the uncertainty shock first influences the stock market, then the price-related variables, and finally the quantities.

9 The Hodrick-Prescott filter is a mathematical tool commonly used in macroeconomics to decompose a time series into short-term fluctuations and a long-term trend. The sensitivity of the trend to fluctuations can be adjusted by modifying a parameter λ. 6.25, 1600 and 129,600 are commonly suggested as a value for λ for annual, quarterly and monthly data, respectively (Ravn and Uhlig 2002).

10 For more details of the spillover effects, please see Appendix 4.

11 For more details of the IRFs, please see Appendix 4.

12 Including the stock market levels as the first variable in the state vector ensures that the effect of stock market levels is controlled (Bloom 2009).

13 We also try another restriction that an uncertainty shock has no long-run effect. See Appendix 4 for the results.

14 For heuristics, we present the scalar expression of Equation (11(11) $\begin{array}{rl}{y}_t& =A^{-1}{b}_0+A^{-1}{B}_1{y}_{t-1}+\cdots +A^{-1}{B}_K{y}_{t-K}+A^{-1}{u}_t\\ & \equiv c_0+C_1{y}_{t-1}+\cdots +C_K{y}_{t-K}+{ϵ}_t,\end{array}$(11) ) in case of $y_t=[P_t,Uncertaint{y}_t,R_t^F,\mathrm{\Delta }CP{I}_t,\mathrm{\Delta }IP{I}_t{]}^{\prime }$ for Table 5: $\begin{array}{rl}& \left[\begin{array}{c}{P}_t\\ Uncertaint{y}_t\\ R_t^F\\ \mathrm{\Delta }CP{I}_t\\ \mathrm{\Delta }IP{I}_t\end{array}\right]=\left[\begin{array}{c}{c}_{0,1}\\ c_{0,2}\\ c_{0,3}\\ c_{0,4}\\ c_{0,5}\end{array}\right]\\ & +\left[\begin{array}{ccccc}{C}_{1,11}& C_{1,12}& C_{1,13}& C_{1,14}& C_{1,15}\\ C_{1,21}& C_{1,22}& C_{1,23}& C_{1,24}& C_{1,25}\\ C_{1,31}& C_{1,32}& C_{1,33}& C_{1,34}& C_{1,35}\\ C_{1,41}& C_{1,42}& C_{1,43}& C_{1,44}& C_{1,45}\\ C_{1,51}& C_{1,52}& C_{1,53}& C_{1,54}& C_{1,55}\end{array}\right]\left[\begin{array}{c}{P}_{t-1}\\ Uncertaint{y}_{t-1}\\ R_{t-1}^F\\ \mathrm{\Delta }CP{I}_{t-1}\\ \mathrm{\Delta }IP{I}_{t-1}\end{array}\right]\\ & +\left[\begin{array}{ccccc}{C}_{2,11}& C_{2,12}& C_{2,13}& C_{2,14}& C_{2,15}\\ C_{2,21}& C_{2,22}& C_{2,23}& C_{2,24}& C_{2,25}\\ C_{2,31}& C_{2,32}& C_{2,33}& C_{2,34}& C_{2,35}\\ C_{2,41}& C_{2,42}& C_{2,43}& C_{2,44}& C_{2,45}\\ C_{2,51}& C_{2,52}& C_{2,53}& C_{2,54}& C_{2,55}\end{array}\right]\left[\begin{array}{c}{P}_{t-2}\\ Uncertaint{y}_{t-2}\\ R_{t-2}^F\\ \mathrm{\Delta }CP{I}_{t-2}\\ \mathrm{\Delta }IP{I}_{t-2}\end{array}\right]\\ & +\left[\begin{array}{c}{ϵ}_{t,1}\\ {ϵ}_{t,2}\\ {ϵ}_{t,3}\\ {ϵ}_{t,4}\\ {ϵ}_{t,5}\end{array}\right].\end{array}$ Since we want to examine the effects of stock market uncertainty on economic fundamentals, the equation of main interest is $\begin{array}{rl}\mathrm{\Delta }IP{I}_t& =c_{0,5}+C_{1,51}{P}_{t-1}+C_{2,51}{P}_{t-2}+C_{1,52}Uncertaint{y}_{t-1}\\ & +C_{2,52}Uncertaint{y}_{t-2}\\ & +C_{1,53}{R}_{t-1}^F+C_{2,53}{R}_{t-2}^F+C_{1,54}\mathrm{\Delta }CP{I}_{t-1}\\ & +C_{2,54}\mathrm{\Delta }CP{I}_{t-2}+C_{1,55}\mathrm{\Delta }IP{I}_{t-1}+C_{2,55}\mathrm{\Delta }IP{I}_{t-2}+{ϵ}_{t,5}.\end{array}$ Estimated coefficients in this equation are tabulated in column 1 of Table 5 for entropy type of uncertainty measure.

15 They were calculated on a year-on-year basis in order to mitigate the fluctuations and seasonality.