(iii) *Uncertainty innovation hypothesis*

The literature suggests that uncertainty causes higher stock market volatility. Liu and Zhang (2015) show that the inclusion of EPU helps to improve forecasting ability of existing volatility models; and Tsai (2017) reports that EPU has a predictive ability not only to explain local stock volatility but also to describe cross market volatility. Testing these phenomena involves examining Cov[*σ*2*t* ,Δ*η<sup>t</sup>*−<sup>1</sup>] = 0 and Cov[*σ*2*t* ,Δ*zt*−<sup>1</sup>] = 0. In terms of Equation (8), the null hypothesis tests joint significance of Δ*η<sup>t</sup>*−<sup>1</sup> = Δ*zt*−<sup>1</sup> = 0 in a variance equation, which can be examined by Lagrange Multiplier (LM) test using the chi-squared distribution.

#### **3. Description of Data and Variables**

The empirical analyses in this study cover the data of the world stock index and 15 individual country/market indices, which include G7: Canada (CA), France (FR), Germany (GM), Italy (IT), Japan (JP), the United Kingdom (UK), the United States (US); Asian-Pacific markets: Australia (AU), China (CN), Hong Kong (HK), India (IN), South Korea (KO) and Singapore (SG); South American markets: Brazil (BR) and Chile (CL). Since most global EPU data start from January 1997, the estimations mainly use a sample for the period from January 1997 to June 2016. However, the stated times of stock indices for China and Brazil are later than other markets. The stock indices (including the total return index (RI) as defined in Datastream includes dividends, interest, rights offerings and other distributions realized over a given month) are downloaded from the database of DataStream, and the EPU news indices are obtained from www.PolicyUncertainty.com provided by Baker et al. (2016) and Davis (2016). The U.S. EPU index is constructed from three underlying components: (i) newspaper coverage of policy-related economic uncertainty based on major local newspapers; (ii) the number of tax code provisions set to expire in future years; (iii) disagreement among economic forecasters, which is used as a proxy for uncertainty. In constructing the EPU, Baker et al. (2016) search the digital archives of each paper to obtain a monthly count of articles that contain the following terms: "uncertainty" or "uncertain"; "economic" or "economy"; and one or more of the terms "deficit", "the Fed," or "uncertainties" or its variants. They find this uncertainty index is reliable, unbiased, and consistent since the uncertainty index is highly correlated with a market's implied volatility, VIX (Whaley 2009), and closely related to other measures of policy uncertainty. Following Bekaert and Harvey (1995), the stock prices are measured using the U.S. dollar.<sup>4</sup>

Table 1 reports summary statistics of monthly stock returns for the G7 market (Panel A) and Asian-Pacific and Latin American (APLA) markets (Panel B). The statistics indicate that the U.S. market performs well as compared with the other advanced markets; Japan, on the other hand, displays a negative return and high volatility as indicated by the standard deviation. The statistics in Panel B

<sup>4</sup> Appendix A provides a description of a list of variables and data sources.

show that Chile, which has the highest return, performs very well, while China and South Korea, which have moderate returns, are relatively more volatile. In general, the returns in the group of APLA are much higher than those in G7 markets for the period of investigation.


**Table 1.** Summary statistics of monthly stock market returns: September 1990–June 2016.

To visualize the stock returns, the monthly stock returns are plotted in Figures 1 and 2. These time series evidently present some degree of comovements and capture the major turning points, especially for the G7 markets, suggesting that these series could be driven by some common factors.

**Figure 1.** Time series plots of the percentage of stock returns (vertical axis) vs. time for G7 and global markets.

**Figure 2.** Time series plots of the percentage of stock returns (vertical axis) vs. time for Asia-Pacific and Latin American markets.

Let us turn to the EPU series, which are plotted in Figure 3 for the G7 market and Figure 4 for the APLA markets. The time paths of G7 markets exhibit some degree of comovements over time, and their correlations with global EPU (GEPU) are in the range from 0.48 (for Italy) to 0.88 (for the U.S.). It is evident that the EPU index for the UK spiked during the time of Brexit. Similarly, correlation coefficients of EPUs shown in the APLA group in Figure 4 range from 0.67 (for India) to 0.98 (for Singapore); the high correlation may reflect some sort of global contagions as shocks occur in the global markets (Chiang et al. 2007; Forbes 2012). The time paths show that EPUs for China, Hong Kong and South Korea occasionally act more volatile.

**Figure 3.** Time series plots of EPU (vertical axis) vs. time for G7 and global markets.

**Figure 4.** Time series plots of EPU (vertical axis) vs. time for Asia-Pacific and Latin American markets.

#### **4. Test of Return Autocorrelations**

A simple approach to testing the EMH is to examine the autocorrelations at the lagged *s* periods by *t*-statistics or examine the joint significance for *ρs* = 0 for *s* = 1, 2, ... , S by *χ*2 distribution. Tables 2 and 3 report the autocorrelations of monthly stock market returns up to 12 orders for the G7 and APLA markets, respectively. Interestingly, only Canada and the U.S. lack autocorrelations, regardless of whether testing is on the individual lags or the lagged coefficients as a group. The results for Canada and the U.S. seem consistent with the EMH if we just look at the autocorrelation pattern. For the other markets, the evidence shows that at least some autocorrelations are significant, which leads to a rejection of the EMH. However, conclusions for the other G7 markets, with the exception of Italy, should be made with caution, since they do not display a consistent pattern, but rather exhibit significant autocorrelations in higher orders. This may result from a spurious correlation due to an omitted variable. If we look at the autocorrelations of the APLA markets in Table 3, five markets are statistically significant at the AR(1) term, which could result from price ceilings imposed by governmen<sup>t</sup> or some sort of market frictions.

In testing whether the volatility is independent, that is, Cov[*ε*2*t*,*ε*2*t*−*<sup>s</sup>*] = 0 for *s* = 0 as noted by Campbell et al. (1997), Table 4 reports the autocorrelations for absolute values of *Rmt*. Apparently, the null is uniformly rejected, as evidenced by the level of significance of the Ljung-Box Q-statistics up to 12 lags. Thus, the testing results show an absence of an independent assumption for volatility is invalid and sugges<sup>t</sup> some type of GARCH model may be considered to describe the return residuals.


**Table 2.** Autocorrelations of monthly stock market returns up to 12 orders: Group 7 markets.

Notes: This table examines stock market efficiency by testing the dependency of stock returns; *ρs* is the coefficient of autocorrelation with order *s* (*s* = 1, 2, ... , 12). *Q*12 is the Ljund-Box statistics for testing joint significance of 12 order lags. The numbers in the brackets are the *p*-values. The values in the first row are the estimated coefficients and in the second row are the *t*-statistics. \* indicates statistically significant at the 5% level.


**Table 3.** Autocorrelations of monthly stock market returns up to 12 orders: Asian-Pacific and Latin American markets.

Notes: This table tests the dependency of stock returns; *ρs* is the coefficient of autocorrelation up to the 12th order. *Q*12 is the Ljund-Box statistics for testing joint significance of 12 order lags. The numbers in the brackets are the *p*-values. The values in the first row are the estimated coefficients and in the second row are the *t*-statistics. \* indicates statistically significant at the 5% level or better.



Notes: The standard error for the estimated coefficient is 1/√*T* = 1/√310 = 0.05679, e.g., the *t*-statistic of *ρ*1 for CA is 0.231/0.05679 = 4.068. Testing for absence of autocorrelations of the absolute *Rmt* with 12 lags by *Q*(12), the null is uniformly rejected at the 1% level for all markets shown in *p*-values.

#### *Economies* **2019**, *7*, 7
