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Board of Governors of the Federal Reserve System
International Finance Discussion Papers
Number 956, November 2008 --- Screen Reader Version*

A Non-Random Walk Revisited: Short- and Long-Term Memory in Asset Prices

Paul S. Eitelman and Justin T. Vitanza^*

NOTE: International Finance Discussion Papers are preliminary materials circulated to stimulate discussion and critical comment. References in publications to International Finance Discussion Papers (other than an acknowledgment that the writer has had access to unpublished material) should be cleared with the author or authors. Recent IFDPs are available on the Web at http://www.federalreserve.gov/pubs/ifdp/. This paper can be downloaded without charge from the Social Science Research Network electronic library at http://www.ssrn.com/.

Abstract:

In this paper, we test for short and long memory in asset prices across 44 emerging and industrialized economies. Using methodology from Lo and MacKinlay (1988) and Lo (1991), we find that markets with a poor Sharpe ratio are more likely to reject the random walk than better performing markets. We also make a methodological contribution. Contrary to the Baillie (1996) criticism, our long memory analysis suggests that the choice of a truncation lag is not as important as one might initially believe. Tests that reject the null hypothesis tend to do so across any reasonable choice in lag.

Keywords: Random walk, long-range dependence, equities, commodities, exchange rates

JEL classification: E30, G14

1.  Introduction

This paper reexamines the possibility of both short- and long-term memory in asset returns. In a weak-form efficient market, all information contained in historical prices is instantaneously reflected in the current market price. This effectively precludes the opportunity to earn abnormal returns through a trend trading approach. However, some investment strategies are still dedicated to the research and exploitation of price trends. Given the continued use of technical analysis in financial economics, the random walk hypothesis deserves further empirical analysis. A random walk test indicates the existence of intertemporal dependence at some chosen lag. A rejection of the random walk hypothesis implies that a trend in asset prices, or memory, exists.

We make two main contributions to the literature. Using a uniquely large dataset¹ of commodities and stock market indices from 44 emerging and industrial economies, we are able to compare the behavior of international asset prices. We first perform a random walk test on each market over several lags. Next, we calculate the Sharpe ratio. A regression of a dummy variable indicating whether or not the random walk was rejected on the Sharpe ratio revealed that markets with poorer risk-adjusted returns are more likely to reject the random walk hypothesis. This finding suggests that contrarian investment strategies, which look for undervalued assets in depressed markets, may offer the prudent investor an opportunity to beat the returns of standard equity indices.

Second, we test for long memory in asset returns using the Modified Rescaled Range (R/S) statistic developed by Lo (1991). Some criticisms of this test cite that there is no optimal method for choosing a truncation lag. Although this criticism is theoretically valid, the results of our analysis suggest that the choice of a truncation lag often has little bearing on the significance of results. We find mixed evidence of long memory in international equity markets. Of the advaned markets, only Denmark, Japan, Norway, and Sweden exhibit long memory. In the U.S. dollar foreign exchange markets, regional factors appear to be at play. We find that four of the six Latin American currencies studied in this paper (Argentina, Brazil, Chile, and Colombia) show long memory, whereas we fail to reject the null hypothesis for every advanced market currency and for most countries across Emerging Asia, the Middle East, and Africa. Of the 22 commodities evaluated in this paper, only soybeans and cattle show signs of long memory.

1.1  A Brief History of Previous Literature

Financial time series analysis focuses on the intertemporal dependencies of asset prices. Early asset pricing models were based on the assumptions that returns are identically and independently distributed (iid) with a log-normal distribution. Mandelbrot (1963) was among the first to show that the normal distribution does not adequately capture the "fat tails", or leptokurtosis, of returns. Engle (1982) and Bollerslev (1986) then identified the presence of time-varying conditional volatility, or "volatility clustering", in daily equity returns with their AutoRegressive Conditional Heteroscedasticity (ARCH) class of models. Many subsequent studies have since confirmed these findings, and the assumption of iid-normal returns is now widely rejected in the literature.

The Random Walk Hypothesis posits that today's stock price is an unforecastable transformation of its previous value. Based on this argument, Malkiel (1973) and Sharpe (1991) conclude that passive, low cost index funds should outperform their actively managed counterparts by some function of the trading, administrative, and management expenses owed. More broadly, any effort to forecast asset prices that follow the random walk should be value-draining.

There is considerable debate among financial economists whether the random walk hypothesis holds. Early research into the random walk hypothesis assumed constant volatility in the residual term. Because of Engle's well-documented exposition on the conditional volatility of returns, a rejection of the random walk because of heteroscedasticity is of little interest. The more advanced Lo and MacKinlay (1988) heteroscedasticity-consistent methodology is invariant to ARCH processes.

Mandelbrot (1971) also discovered a new empirical property in the time series of asset prices: low frequency persistent temporal dependence (hereafter long memory). Long memory is commonly characterized by a slower-than-exponential rate of decay in the autocorrelation function. For example, Taylor (1986) and Ding, Granger and Engle (1993) found a hyperbolic rate of decay in the absolute value of asset returns. As pointed out in Lo (1991), long-term memory is troublesome because it is inconsistent with the continuous time stochastic processes employed in martingale methods for the pricing of options and futures contracts. Mandelbrot (1972) adapted a previously existing test to identify the presence of long memory. This test, rescaled range, was first introduced by the hydrologist Harold Hurst (1951) to identify long-term patterns in the discharge of the Nile River.

The rest of this paper is organized as follows: Section 2 describes the methodology and results for our short memory (random walk) analysis, as well as a discussion on the power of our chosen test statistic. Section 3 describes the methodology and results of our long memory analysis. Section 4 concludes our paper and discusses practical applications for our findings in financial economics. The appendices provide a more detailed view of our data and results.

2.  Short Memory

2.1  Methodology

First we denote $P_{t}$ as the index level at time $t$ and define the log-price process $X_{t}\equiv$ ln$P_{t}$. The conventional form of the random walk is then given as

$$X_{t}=\mu+X_{t-1}+\epsilon_{t}$$ (1)

where $\mu$ is an arbitrary drift parameter, and $\epsilon_{t}$ is the random error term. The standard null hypothesis, H$_{0}$, stipulates that the error terms $\epsilon_{t}$ are independently and identically distributed normal random variables with constant variance $\sigma_{o}^{2}$, or

$$_{0} \epsilon_{t} N(0,\sigma_{o}^{2})$$ (2)

However, given the well-documented fact that financial time series deviate from normality,² and the mounting evidence of volatility clustering shown first by Engle (1982) and Bollerslev (1986), a rejection of the random walk due to these factors would be of little interest. Therefore, we study the short memory processes of our variables using the Lo and MacKinlay (1988) variance ratio test. This test assumes that $X_{t}$ possesses uncorrelated increments, but it allows for more general forms of heteroscedasticity and relaxes the requirement for Gaussian increments.

The rest follows almost directly from the Lo and MacKinlay methodology.³ For $nq+1$ observations $\left[ X_{0},X_{1},...,X_{nq}\right]$, where $q$ is any integer greater than 1, we define the asymptotic estimators of the mean and variance of returns, respectively:

$$\hat{\mu}\equiv\frac{1}{nq}\overset{nq}{\underset{k=1}{\sum}}(X_{k} -X_{k-1})=\frac{1}{nq}(X_{nq}-X_{0})$$ (3)

$$\bar{\sigma}_{a}^{2}=\frac{1}{nq-1}\underset{k=1}{\overset{nq}{ {\displaystyle\sum} }}(X_{k}-X_{k-1}-\hat{\mu})^{2}$$ (4)

We then define $\bar{M}_{r}(q)$ as the variance ratio that obtains

$$\bar{M}_{r}(q)\overset{a}{=}\overset{q-1}{\underset{j=1}{\text{ }\sum}} \frac{2(q-j)}{q}\hat{\rho}(j)\text{,}$$ (5)

where $\hat{\rho}(j)$ is the autocorrelation coefficient estimator. Note that when $q=2$, $\bar{M}_{r}(2)$ simplifies to the first-order autocorrelation coefficient, $\hat{\rho}(1)$. Continuing from theorem 3 of Lo and MacKinlay (1988), we define $\hat{\delta}(j)$ and $\hat{\theta}(q)$ as the heteroscedasticity-consistent estimators of the asymptotic variances of $\hat{\rho}(j)$ and $\bar{M}_{r}(q)$, respectively

$$\hat{\delta}(j)=\frac{nq\underset{k=j+1}{\overset{nq}{\sum}}(X_{k} -X_{k-1}-\hat{\mu})^{2}(X_{k-j}-X_{k-j-1}-\hat{\mu})^{2}}{\left[ \overset {nq}{\underset{k=1}{\sum}}(X_{k}-X_{k-1}-\hat{\mu})^{2}\right] ^{2}}$$ (6)

$$\hat{\theta}(q)\equiv\underset{j=1}{\overset{q-1}{\sum}}\left[ \frac {2(q-j)}{q}\right] ^{2}\hat{\delta}(j)$$ (7)

The following test statistic $z^{\ast}(q)$ is asymptotically standard normal and is used to describe our empirical results in section 2.2. Note that q can be interpreted as the period for which the existence of memory is being tested.

$$z^{\ast}(q)\equiv\frac{\sqrt{nq}\bar{M}_{r}(q)}{\sqrt{\hat{\theta}(q)} }\overset{a}{\sim}N(0,1)$$ (8)

This test statistic is ideal for evaluating the random walk hypothesis because it accounts for the volatility clustering inherent in asset prices. However, since the effective sample size decreases as $q$ gets larger, small-sample biases will skew this statistic at sufficiently large lags (Lo and MacKinlay, 1989).

2.2  Results

The results of our random walk study are shown in Appendix 3. The dashed orange and red lines represent 95 and 99 percent rejection levels of the heteroscedasticity-consistent random walk hypothesis test, respectively. $\ z$*-values significantly greater than zero imply a positive persistence in the autocorrelation function for the time series at some lag, $q$; $z$*-values below the lower rejection band show a mean-reversion in the time series at that lag. This paper does not attempt to prove explanatory factors for any non-random processes identified herein; instead we reveal those areas where further fundamental analysis should prove fruitful for academics and practitioners. A few notable themes emerge.

In commodity markets, we find common trends within commodity groupings. Energy items, including crude oil, natural gas, and heating oil, exhibit mean reverting price behavior. This finding is consistent with recent work by Geman (2007). As in Bleaney and Greenaway (1993), we fail to reject the random walk hypothesis for each of the nine metals series in our study. The agricultural raw materials, namely cotton and rubber, show significant persistence in returns while the finished farm products are largely random. Soybean prices are the exception, however, as they are shown to deviate from the random walk.

We fail to reject the random walk hypothesis for U.S. foreign exchange markets. Our findings are consistent with Liu and He (1991), who used the same heteroscedasticity-consistent methodology. We find that the Yen-Dollar and Pound-Dollar rates each exhibit positive serial correlation, particularly for small $q$ values, whereas the random walk cannot be rejected for the Canadian-U.S. Dollar rate. Other notable currencies include the Australian dollar and the Euro, both of which seem to exhibit random price behavior.

We also fail to reject the random walk hypothesis for the S&P 500, a value-weighted index, at any conventional lag. Similarly, we cannot reject the random walk hypothesis for many other major international equity markets, including China, the United Kingdom, France, Germany, India, Brazil, Switzerland, South Korea, Spain, and Sweden. This is consistent with the findings of Campbell, Lo and MacKinlay (1997).⁴ They cannot reject the random walk hypothesis for a U.S. value-weighted index. However, the main finding of their paper is that they reject the random walk for the CRSP equal-weighted index. Accordingly, they suggest that firm size is a driving factor for the "randomness" of its security price. From our analysis, it appears that risk-adjusted returns are also an indicator of predictability. For example, by ordering markets according to their Sharpe ratios⁵, $S=\frac{\mu}{\sigma}$, we notice that the seven weakest equity markets in our sample: Czech Republic, Philippines, Japan, Thailand, Italy, and Argentina, all reject the random walk. This result is supported by economic intuition. Low risk-adjusted returns should decrease the demand for analyst coverage, which would result in weaker informational efficiency.

In Figure 2.2.1, we show a scatterplot of the Sharpe ratio ($S$) against the maximum value over all lags of the absolute value of $z^{\ast}(q).$ The $S$ was calculated using the return and standard deviation of returns over all available data. Sweden was immediately disregarded as an outlier.

Figure 2.2.1

Data for Figure 2.2.1: Non-Oil

	Zmax	Sharpe Ratio
Argentina	2.832	0.050
Brazil	0.697	0.126
Chile	6.145	0.139
Colombia	1.504	0.186
Mexico	1.881	0.139
Peru	4.131	0.158
China	1.944	0.070
Hong Kong	4.265	0.068
India	1.178	0.091
South Korea	2.060	0.055
Malaysia	2.530	0.055
Philippines	4.066	0.048
Russia	2.709	0.071
Singapore	3.769	0.064
Taiwan	3.963	0.049
Thailand	3.702	0.032
Israel	0.649	0.089
South Africa	1.153	0.106
Turkey	1.246	0.118
Australia	2.818	0.085
Belgium	0.943	0.076
Canada	2.811	0.090
Czech Republic	1.930	0.037
Denmark	2.234	0.107
France	2.063	0.064
Germany	1.802	0.074
Hungary	2.715	0.102
Italy	4.335	0.050
Japan	2.407	0.041
Netherlands	1.023	0.071
Norway	2.468	0.088
New Zealand	1.192	0.110
Poland	4.165	0.092
Spain	1.532	0.102
Switzerland	1.324	0.073
United Kingom	1.096	0.072
United States	0.896	0.089
Aluminum	1.669	0.168
Copper	0.950	0.113
Gold	0.940	0.158
Lead	1.742	0.089
Nickel	1.673	0.163
Platinum	2.468	0.112
Silver	0.815	0.167
Tin	1.066	0.181
Zinc	1.212	0.131
Cattle	1.978	0.051
Cocoa	5.905	0.159
Coffee	1.819	0.080
Corn	1.853	0.096
Cotton	2.802	0.093
Rubber	1.732	0.015
Soybeans	2.019	0.055
Sugar	1.178	0.106
Wheat	2.214	0.107

Data for Figure 2.2.1: Oil

	Zmax	Sharpe Ratio
Indonesia	1.331	0.067
Kuwait	3.449	0.268
Nigeria	2.144	0.166
Saudi Arabia	3.045	0.082
UAE	10.077	0.190
Venezuela	3.205	0.124
Crude Oil	2.266	0.024
Natural Gas	1.459	0.066
Heating Oil	2.288	0.019
Gasoline	2.217	0.121

We immediately noticed that energy commodities and OPEC-member equity markets tended to deviate from the majority of the data. Removing these points magnifies the relationship between risk-adjusted performance and forecastability, as shown in Figure 2.2.2.

Figure 2.2.2

Data for Figure 2.2.2: Non-Oil

	Zmax	Sharpe Ratio
Argentina	2.832	0.050
Brazil	0.697	0.126
Chile	6.145	0.139
Colombia	1.504	0.186
Mexico	1.881	0.139
Peru	4.131	0.158
China	1.944	0.070
Hong Kong	4.265	0.068
Indonesia	1.331	0.067
India	1.178	0.091
South Korea	2.060	0.055
Malaysia	2.530	0.055
Philippines	4.066	0.048
Russia	2.709	0.071
Singapore	3.769	0.064
Taiwan	3.963	0.049
Thailand	3.702	0.032
Israel	0.649	0.089
Nigeria	2.144	0.166
South Africa	1.153	0.106
Turkey	1.246	0.118
Australia	2.818	0.085
Belgium	0.943	0.076
Canada	2.811	0.090
Czech Republic	1.930	0.037
Denmark	2.234	0.107
France	2.063	0.064
Germany	1.802	0.074
Hungary	2.715	0.102
Italy	4.335	0.050
Japan	2.407	0.041
Netherlands	1.023	0.071
Norway	2.468	0.088
New Zealand	1.192	0.110
Poland	4.165	0.092
Spain	1.532	0.102
Switzerland	1.324	0.073
United Kingom	1.096	0.072
United States	0.896	0.089
Aluminum	1.669	0.168
Copper	0.950	0.113
Gold	0.940	0.158
Lead	1.742	0.089
Nickel	1.673	0.163
Platinum	2.468	0.112
Silver	0.815	0.167
Tin	1.066	0.181
Zinc	1.212	0.131
Cattle	1.978	0.051
Cocoa	5.905	0.159
Coffee	1.819	0.080
Corn	1.853	0.096
Cotton	2.802	0.093
Rubber	1.732	0.015
Soybeans	2.019	0.055
Sugar	1.178	0.106
Wheat	2.214	0.107

Table 2.2.1 shows the correlation between these series along with the coefficients and t-statistics of a regression of a dummy variable which indicates whether or not that market can reject the random walk with 95 confidence at some lag (i.e. $\max(\vert z^{\ast}(q)\vert)>1.96$) on the $S$. Though only the OLS estimates are reported, the results are robust to several modifications, including robust standard errors and iteratively reweighted least squares with Huber and biweight functions tuned to 95% efficiency.

Table 2.2.1

Assets	Correlation $(S,\max(\vert z^{\ast}(q)\vert))$	$\mathbf{\beta}$	t-stat
Non-Oil	$-0.13$	$-3.25$	$-2.05$

This preliminary evidence supports our hypothesis that there exists a positive relationship between risk-adjusted market performance and efficiency. Further research in this area would likely prove rewarding.

2.3  Power

In this section, we detail the relative power of the Lo and MacKinlay (1988) heteroscedasticity-consistent random walk test compared to the more restrictive random walk under homoscedastic increments. One should expect that, for random asset price innovations, both the homoscedastic and heteroscedastic methodologies will fail to reject the null hypothesis of a random walk and produce very similar results. Our simulation results confirm this. As an example, we include the first sample of the simulation in Figure 2.3.1.

Figure 2.3.1  IID-Normal Process

Data for Figure 2.3.1

The 95% critical values are plus-or-minus 1.96, and the 99% critical values are plus-or-minus 2.58. The critical values do not depend upon q.

q	Heteroscedastic Test - z	Homoscedastic Test - z
2	0.3525	0.3359
3	0.0054	0.0053
4	0.1036	0.1007
5	0.1968	0.1913
6	0.0716	0.0695
7	0.0979	0.0951
8	0.1497	0.1455
9	0.1839	0.1788
10	0.3196	0.3108
11	0.5123	0.4986
12	0.6368	0.6201
13	0.7741	0.7543
14	0.9282	0.9051
15	0.9984	0.9741
16	1.067	1.0415
17	1.153	1.1259
18	1.1982	1.1705
19	1.2584	1.2297
20	1.3096	1.2801
21	1.3671	1.3367
22	1.4003	1.3695
23	1.4225	1.3917
24	1.4097	1.3796
25	1.3973	1.3679
26	1.3619	1.3338
27	1.3196	1.2928
28	1.2664	1.2411
29	1.2283	1.204
30	1.1806	1.1576
31	1.1354	1.1136
32	1.0804	1.0599
33	1.0065	0.9876
34	0.9376	0.9202
35	0.8609	0.845
36	0.7751	0.761
37	0.6879	0.6754
38	0.6099	0.5989
39	0.5359	0.5262
40	0.4598	0.4516
41	0.3894	0.3824
42	0.3189	0.3133
43	0.2428	0.2385
44	0.1653	0.1623
45	0.0829	0.0814
46	0.0085	0.0084
47	-0.0566	-0.0556
48	-0.1214	-0.1193
49	-0.1746	-0.1715
50	-0.2181	-0.2143
51	-0.2655	-0.2609
52	-0.313	-0.3075
53	-0.3577	-0.3515
54	-0.3968	-0.3899
55	-0.436	-0.4284
56	-0.4679	-0.4597
57	-0.4996	-0.4909
58	-0.5262	-0.517
59	-0.5445	-0.535
60	-0.5626	-0.5528
61	-0.5827	-0.5726
62	-0.597	-0.5866
63	-0.6153	-0.6045
64	-0.6401	-0.6289
65	-0.6596	-0.648
66	-0.6736	-0.6617
67	-0.6871	-0.675
68	-0.696	-0.6837
69	-0.7108	-0.6982
70	-0.7274	-0.7145
71	-0.7369	-0.7238
72	-0.7452	-0.732
73	-0.7577	-0.7442
74	-0.7686	-0.7549
75	-0.7784	-0.7645
76	-0.7919	-0.7777
77	-0.8006	-0.7863
78	-0.8094	-0.7948
79	-0.8232	-0.8084
80	-0.8377	-0.8226
81	-0.8546	-0.8391
82	-0.8721	-0.8563
83	-0.8904	-0.8743
84	-0.9053	-0.8888
85	-0.9211	-0.9043
86	-0.9378	-0.9207
87	-0.9542	-0.9367
88	-0.9677	-0.9499
89	-0.9787	-0.9607
90	-0.9872	-0.969
91	-0.9989	-0.9804
92	-1.0098	-0.9911
93	-1.021	-1.0019
94	-1.0332	-1.0139
95	-1.0457	-1.0261
96	-1.0588	-1.0389
97	-1.0693	-1.0491
98	-1.077	-1.0566
99	-1.0837	-1.0631
100	-1.0901	-1.0693
101	-1.0978	-1.0768
102	-1.1037	-1.0825
103	-1.1094	-1.0881
104	-1.1142	-1.0928
105	-1.1135	-1.092
106	-1.1154	-1.0937
107	-1.1184	-1.0966
108	-1.1185	-1.0966
109	-1.1217	-1.0997
110	-1.1234	-1.1013
111	-1.1247	-1.1025
112	-1.1243	-1.102
113	-1.1255	-1.1031
114	-1.1259	-1.1034
115	-1.1282	-1.1055
116	-1.1327	-1.1099
117	-1.1343	-1.1114
118	-1.1337	-1.1107
119	-1.1311	-1.1081
120	-1.1276	-1.1046
121	-1.1246	-1.1016
122	-1.1237	-1.1006
123	-1.1207	-1.0976
124	-1.1189	-1.0957
125	-1.1156	-1.0925
126	-1.1101	-1.087
127	-1.1058	-1.0826
128	-1.101	-1.0779
129	-1.0941	-1.071
130	-1.0861	-1.0631
131	-1.0798	-1.0569
132	-1.0727	-1.0499
133	-1.0677	-1.0448
134	-1.064	-1.0411
135	-1.0608	-1.0379
136	-1.058	-1.0351
137	-1.0566	-1.0337
138	-1.0552	-1.0322
139	-1.0528	-1.0298
140	-1.0537	-1.0306
141	-1.0511	-1.028
142	-1.0464	-1.0233
143	-1.0429	-1.0198
144	-1.0419	-1.0187
145	-1.0396	-1.0164
146	-1.0381	-1.0149
147	-1.0379	-1.0146
148	-1.0385	-1.0151
149	-1.0403	-1.0167
150	-1.0432	-1.0195
151	-1.046	-1.0222
152	-1.0484	-1.0244
153	-1.0497	-1.0256
154	-1.0504	-1.0262
155	-1.0523	-1.028
156	-1.0544	-1.0299
157	-1.0567	-1.0321
158	-1.0563	-1.0316
159	-1.0561	-1.0313
160	-1.0563	-1.0314
161	-1.0572	-1.0323
162	-1.0591	-1.034
163	-1.06	-1.0348
164	-1.0593	-1.034
165	-1.0581	-1.0328
166	-1.0588	-1.0334
167	-1.0592	-1.0337
168	-1.0606	-1.0349
169	-1.06	-1.0343
170	-1.0595	-1.0337
171	-1.059	-1.0331
172	-1.0591	-1.0331
173	-1.0594	-1.0333
174	-1.0585	-1.0323
175	-1.0573	-1.0311
176	-1.0571	-1.0308
177	-1.0566	-1.0302
178	-1.0552	-1.0288
179	-1.0527	-1.0263
180	-1.0509	-1.0245
181	-1.0486	-1.0221
182	-1.0463	-1.0197
183	-1.0446	-1.018
184	-1.0416	-1.015
185	-1.0384	-1.0118
186	-1.0375	-1.0108
187	-1.038	-1.0113
188	-1.0386	-1.0118
189	-1.0397	-1.0127
190	-1.0411	-1.014
191	-1.0411	-1.0139
192	-1.0412	-1.0138
193	-1.0415	-1.0141
194	-1.042	-1.0145
195	-1.0424	-1.0148
196	-1.0439	-1.0162
197	-1.0446	-1.0167
198	-1.0465	-1.0185
199	-1.0471	-1.019
200	-1.0465	-1.0184
201	-1.0455	-1.0172
202	-1.0455	-1.0172
203	-1.0455	-1.0171
204	-1.0471	-1.0185
205	-1.0498	-1.0211
206	-1.0526	-1.0237
207	-1.0566	-1.0276
208	-1.0616	-1.0323
209	-1.0651	-1.0355
210	-1.0691	-1.0394
211	-1.0725	-1.0426
212	-1.0743	-1.0443
213	-1.0759	-1.0457
214	-1.0791	-1.0487
215	-1.0826	-1.0521
216	-1.0858	-1.055
217	-1.0878	-1.0569
218	-1.0889	-1.0579
219	-1.089	-1.0579
220	-1.0894	-1.0581
221	-1.0892	-1.0579
222	-1.0891	-1.0576
223	-1.0893	-1.0577
224	-1.0892	-1.0576
225	-1.0885	-1.0567
226	-1.0872	-1.0554
227	-1.086	-1.0542
228	-1.0843	-1.0524
229	-1.0812	-1.0493
230	-1.0773	-1.0454
231	-1.0731	-1.0412
232	-1.069	-1.0372
233	-1.0657	-1.0339
234	-1.0616	-1.0298
235	-1.0572	-1.0254
236	-1.0515	-1.0198
237	-1.045	-1.0134
238	-1.0384	-1.0068
239	-1.0313	-0.9999
240	-1.0239	-0.9926
241	-1.0164	-0.9853
242	-1.0098	-0.9788
243	-1.0032	-0.9722
244	-0.996	-0.9652
245	-0.9887	-0.958
246	-0.9812	-0.9507
247	-0.9729	-0.9425
248	-0.9646	-0.9344
249	-0.9561	-0.9261
250	-0.948	-0.9182
251	-0.9418	-0.912
252	-0.9361	-0.9065
253	-0.9308	-0.9013
254	-0.9259	-0.8964
255	-0.9212	-0.8917
256	-0.9159	-0.8865
257	-0.9114	-0.8821
258	-0.9067	-0.8775
259	-0.9031	-0.8739
260	-0.9004	-0.8712
261	-0.8976	-0.8683
262	-0.8953	-0.8661
263	-0.8949	-0.8655
264	-0.8958	-0.8663
265	-0.8973	-0.8677
266	-0.8987	-0.869
267	-0.9013	-0.8714
268	-0.9041	-0.874
269	-0.9074	-0.8771
270	-0.9108	-0.8803
271	-0.9142	-0.8835
272	-0.9189	-0.888
273	-0.9238	-0.8926
274	-0.9289	-0.8974
275	-0.9343	-0.9026
276	-0.9392	-0.9072
277	-0.9452	-0.9129
278	-0.9517	-0.919
279	-0.9585	-0.9255
280	-0.9654	-0.9321
281	-0.9724	-0.9387
282	-0.9783	-0.9444
283	-0.9832	-0.949
284	-0.9895	-0.9549
285	-0.9953	-0.9604
286	-1.0003	-0.9651
287	-1.0048	-0.9694
288	-1.0094	-0.9737
289	-1.0134	-0.9775
290	-1.0178	-0.9816
291	-1.0218	-0.9854
292	-1.0244	-0.9878
293	-1.027	-0.9902
294	-1.0288	-0.9918
295	-1.0304	-0.9932
296	-1.0314	-0.9941
297	-1.0322	-0.9947
298	-1.0324	-0.9948
299	-1.0333	-0.9956
300	-1.0342	-0.9963
301	-1.0335	-0.9955
302	-1.0323	-0.9943
303	-1.0313	-0.9932
304	-1.0302	-0.992
305	-1.0293	-0.991
306	-1.0278	-0.9895
307	-1.0256	-0.9873
308	-1.0236	-0.9852
309	-1.0224	-0.984
310	-1.0212	-0.9827
311	-1.0205	-0.9819
312	-1.0205	-0.9818
313	-1.0196	-0.9808
314	-1.0194	-0.9806
315	-1.0193	-0.9803
316	-1.019	-0.9799
317	-1.0182	-0.9791
318	-1.0181	-0.9789
319	-1.0181	-0.9787
320	-1.0179	-0.9785
321	-1.0187	-0.9791
322	-1.0191	-0.9793
323	-1.0196	-0.9798
324	-1.0207	-0.9807
325	-1.0216	-0.9814
326	-1.0213	-0.981
327	-1.0218	-0.9814
328	-1.0223	-0.9818
329	-1.0231	-0.9824
330	-1.0241	-0.9833
331	-1.0251	-0.9841
332	-1.0262	-0.9851
333	-1.0279	-0.9865
334	-1.0307	-0.9891
335	-1.0326	-0.9908
336	-1.034	-0.9921
337	-1.0345	-0.9924
338	-1.0346	-0.9924
339	-1.035	-0.9926
340	-1.0343	-0.9918
341	-1.0327	-0.9902
342	-1.0305	-0.988
343	-1.0291	-0.9865
344	-1.0279	-0.9852
345	-1.0267	-0.984
346	-1.0256	-0.9828
347	-1.0247	-0.9818
348	-1.0234	-0.9805
349	-1.022	-0.979
350	-1.0201	-0.977
351	-1.0183	-0.9753
352	-1.0161	-0.973
353	-1.0133	-0.9702
354	-1.0107	-0.9677
355	-1.0082	-0.9651
356	-1.0064	-0.9633
357	-1.0038	-0.9606
358	-1.0016	-0.9584
359	-0.9986	-0.9554
360	-0.9966	-0.9534
361	-0.9944	-0.9512
362	-0.9915	-0.9484
363	-0.9892	-0.9459
364	-0.9867	-0.9434
365	-0.984	-0.9408
366	-0.981	-0.9379
367	-0.9787	-0.9355
368	-0.9753	-0.9321
369	-0.9721	-0.929
370	-0.969	-0.9259
371	-0.9661	-0.923
372	-0.9626	-0.9195
373	-0.9598	-0.9168
374	-0.9571	-0.9141
375	-0.9542	-0.9112
376	-0.9514	-0.9084
377	-0.9489	-0.9059
378	-0.9465	-0.9035
379	-0.9437	-0.9007
380	-0.941	-0.898
381	-0.9381	-0.8951
382	-0.9351	-0.8922
383	-0.933	-0.89
384	-0.9305	-0.8876
385	-0.9283	-0.8854
386	-0.9264	-0.8834
387	-0.9247	-0.8817
388	-0.9233	-0.8802
389	-0.9214	-0.8783
390	-0.9188	-0.8758
391	-0.9169	-0.8739
392	-0.9149	-0.8718
393	-0.9132	-0.87
394	-0.9107	-0.8676
395	-0.9081	-0.865
396	-0.9054	-0.8623
397	-0.9031	-0.8601
398	-0.9016	-0.8585
399	-0.9006	-0.8574
400	-0.8996	-0.8563
401	-0.8996	-0.8562
402	-0.8992	-0.8558
403	-0.8989	-0.8554
404	-0.8987	-0.8551
405	-0.8984	-0.8547
406	-0.8989	-0.8551
407	-0.8993	-0.8554
408	-0.9004	-0.8563
409	-0.9016	-0.8573
410	-0.9022	-0.8578
411	-0.9029	-0.8583
412	-0.9038	-0.859
413	-0.9049	-0.86
414	-0.9056	-0.8605
415	-0.9063	-0.8611
416	-0.9071	-0.8618
417	-0.909	-0.8634
418	-0.9119	-0.8661
419	-0.9141	-0.8681
420	-0.917	-0.8707
421	-0.9198	-0.8732
422	-0.9239	-0.8771
423	-0.928	-0.8808
424	-0.9315	-0.884
425	-0.9346	-0.8869
426	-0.939	-0.8909
427	-0.9426	-0.8942
428	-0.9461	-0.8974
429	-0.9483	-0.8994
430	-0.9502	-0.9011
431	-0.9524	-0.903
432	-0.9535	-0.904
433	-0.9541	-0.9045
434	-0.9555	-0.9056
435	-0.9566	-0.9065
436	-0.9578	-0.9076
437	-0.9584	-0.908
438	-0.9597	-0.9091
439	-0.9606	-0.9098
440	-0.9625	-0.9116
441	-0.9647	-0.9135
442	-0.9667	-0.9153
443	-0.9686	-0.917
444	-0.9703	-0.9184
445	-0.9712	-0.9192
446	-0.9711	-0.919
447	-0.9706	-0.9184
448	-0.9702	-0.9179
449	-0.9694	-0.9171
450	-0.9695	-0.917
451	-0.9699	-0.9172
452	-0.9702	-0.9174
453	-0.97	-0.9171
454	-0.9703	-0.9173
455	-0.9701	-0.9169
456	-0.9695	-0.9163
457	-0.9686	-0.9153
458	-0.9675	-0.9142
459	-0.9662	-0.9128
460	-0.9647	-0.9112
461	-0.9631	-0.9096
462	-0.9619	-0.9083
463	-0.9604	-0.9069
464	-0.9585	-0.9049
465	-0.9558	-0.9023
466	-0.9529	-0.8994
467	-0.9503	-0.8968
468	-0.9474	-0.894
469	-0.9441	-0.8907
470	-0.9408	-0.8875
471	-0.9382	-0.8849
472	-0.9358	-0.8826
473	-0.9336	-0.8804
474	-0.9317	-0.8784
475	-0.9294	-0.8762
476	-0.9271	-0.8739
477	-0.9245	-0.8714
478	-0.9227	-0.8695
479	-0.9211	-0.8679
480	-0.9194	-0.8662
481	-0.9173	-0.864
482	-0.9155	-0.8622
483	-0.9142	-0.8609
484	-0.913	-0.8597
485	-0.9113	-0.858
486	-0.9091	-0.8557
487	-0.9071	-0.8537
488	-0.9053	-0.852
489	-0.9039	-0.8505
490	-0.9025	-0.8491
491	-0.9018	-0.8483
492	-0.9011	-0.8475
493	-0.901	-0.8473
494	-0.9007	-0.847
495	-0.9011	-0.8472
496	-0.9026	-0.8485
497	-0.9039	-0.8496
498	-0.9051	-0.8506
499	-0.9063	-0.8516
500	-0.9076	-0.8528

It is less obvious how the random walk tests will behave in the presence of time-varying volatility. Under GARCH(1,1) processes, the heteroscedasticity-consistent random walk test employed in this paper is shown to be robust to time-varying volatility and appropriately fails to reject the random walk, whereas the homoscedastic alternative is biased towards rejection. Figure 2.3.2 shows the first sample in our simulation and the rejection bias of the homoscedastic test.

Figure 2.3.2  GARCH(1,1) Process

Data for Figure 2.3.2

The 95% critical values are plus-or-minus 1.96, and the 99% critical values are plus-or-minus 2.58. The critical values do not depend upon q.

q	Heteroscedastic Test - z	Homoscedastic Test - z
2	1.3891	2.2013
3	1.0473	1.7368
4	1.0384	1.7609
5	0.8409	1.4337
6	0.581	0.9905
7	0.4987	0.849
8	0.4899	0.8321
9	0.5269	0.8926
10	0.5798	0.9796
11	0.6094	1.0269
12	0.5921	0.9951
13	0.5591	0.937
14	0.5276	0.8816
15	0.4889	0.8145
16	0.4127	0.6855
17	0.329	0.5451
18	0.267	0.4411
19	0.2221	0.3658
20	0.1918	0.3152
21	0.1822	0.2986
22	0.1677	0.2743
23	0.1259	0.2055
24	0.0816	0.1329
25	0.0265	0.0432
26	-0.039	-0.0633
27	-0.087	-0.1411
28	-0.1253	-0.2028
29	-0.1585	-0.2562
30	-0.1716	-0.277
31	-0.1792	-0.289
32	-0.1934	-0.3115
33	-0.2229	-0.3586
34	-0.2567	-0.4125
35	-0.2889	-0.4636
36	-0.3166	-0.5076
37	-0.3407	-0.5457
38	-0.3476	-0.556
39	-0.354	-0.5657
40	-0.3642	-0.5814
41	-0.3772	-0.6015
42	-0.4069	-0.6481
43	-0.4398	-0.6999
44	-0.4699	-0.7469
45	-0.5004	-0.7946
46	-0.5303	-0.8414
47	-0.548	-0.8686
48	-0.5693	-0.9015
49	-0.5915	-0.9358
50	-0.6093	-0.963
51	-0.6346	-1.0021
52	-0.6671	-1.0526
53	-0.6881	-1.0847
54	-0.7074	-1.1142
55	-0.7297	-1.1484
56	-0.7506	-1.1803
57	-0.7664	-1.2042
58	-0.7775	-1.2206
59	-0.7805	-1.2243
60	-0.7801	-1.2228
61	-0.7765	-1.216
62	-0.7738	-1.2108
63	-0.7782	-1.2168
64	-0.7809	-1.22
65	-0.7856	-1.2263
66	-0.7859	-1.2257
67	-0.7834	-1.2207
68	-0.7814	-1.2167
69	-0.7845	-1.2204
70	-0.788	-1.2248
71	-0.7931	-1.2316
72	-0.7984	-1.2387
73	-0.8015	-1.2425
74	-0.8039	-1.245
75	-0.8067	-1.2483
76	-0.8068	-1.2472
77	-0.8046	-1.2427
78	-0.8031	-1.2391
79	-0.802	-1.2363
80	-0.8022	-1.2355
81	-0.8052	-1.2389
82	-0.8076	-1.2413
83	-0.8107	-1.245
84	-0.8119	-1.2456
85	-0.8149	-1.2488
86	-0.8179	-1.2522
87	-0.8215	-1.2564
88	-0.8298	-1.2678
89	-0.8397	-1.2816
90	-0.8471	-1.2916
91	-0.8568	-1.305
92	-0.8675	-1.32
93	-0.8737	-1.3279
94	-0.8796	-1.3356
95	-0.8854	-1.3429
96	-0.8881	-1.3456
97	-0.8911	-1.3486
98	-0.8952	-1.3534
99	-0.8995	-1.3584
100	-0.9054	-1.3658
101	-0.9123	-1.3748
102	-0.9184	-1.3823
103	-0.9229	-1.3875
104	-0.928	-1.3937
105	-0.9344	-1.4018
106	-0.9404	-1.4092
107	-0.9447	-1.414
108	-0.9519	-1.4232
109	-0.9607	-1.4346
110	-0.9674	-1.443
111	-0.9732	-1.45
112	-0.9777	-1.455
113	-0.9828	-1.4609
114	-0.9892	-1.4687
115	-0.9988	-1.4813
116	-1.0093	-1.495
117	-1.0205	-1.51
118	-1.0295	-1.5215
119	-1.0347	-1.5273
120	-1.0357	-1.5271
121	-1.0392	-1.5303
122	-1.0418	-1.5323
123	-1.0427	-1.5318
124	-1.0432	-1.5307
125	-1.0418	-1.5269
126	-1.0384	-1.5201
127	-1.0353	-1.5137
128	-1.0323	-1.5076
129	-1.0293	-1.5014
130	-1.0258	-1.4945
131	-1.023	-1.4886
132	-1.0197	-1.4821
133	-1.0164	-1.4755
134	-1.0144	-1.4708
135	-1.013	-1.467
136	-1.0104	-1.4615
137	-1.0077	-1.4558
138	-1.006	-1.4517
139	-1.0036	-1.4464
140	-1.0019	-1.4423
141	-1.0001	-1.4378
142	-0.9968	-1.4314
143	-0.9928	-1.424
144	-0.9894	-1.4173
145	-0.9854	-1.4099
146	-0.9822	-1.4036
147	-0.9796	-1.3983
148	-0.9781	-1.3944
149	-0.9769	-1.391
150	-0.9761	-1.3882
151	-0.9745	-1.3842
152	-0.9717	-1.3786
153	-0.9687	-1.3727
154	-0.9656	-1.3666
155	-0.9625	-1.3606
156	-0.9603	-1.3559
157	-0.959	-1.3525
158	-0.9576	-1.3489
159	-0.9561	-1.345
160	-0.9549	-1.3418
161	-0.954	-1.3389
162	-0.9534	-1.3365
163	-0.9535	-1.335
164	-0.9535	-1.3335
165	-0.9527	-1.3306
166	-0.953	-1.3295
167	-0.9542	-1.3296
168	-0.955	-1.3291
169	-0.9554	-1.3281
170	-0.956	-1.3274
171	-0.9566	-1.3266
172	-0.9559	-1.324
173	-0.9556	-1.3222
174	-0.9555	-1.3204
175	-0.9561	-1.3197
176	-0.9573	-1.3198
177	-0.9578	-1.3189
178	-0.9572	-1.3165
179	-0.9565	-1.314
180	-0.956	-1.3119
181	-0.9549	-1.3088
182	-0.9535	-1.3054
183	-0.9521	-1.3019
184	-0.9506	-1.2984
185	-0.9482	-1.2936
186	-0.9463	-1.2895
187	-0.9447	-1.2858
188	-0.9429	-1.2819
189	-0.9401	-1.2767
190	-0.9377	-1.272
191	-0.9351	-1.2669
192	-0.9325	-1.262
193	-0.9306	-1.258
194	-0.9294	-1.2549
195	-0.9275	-1.2509
196	-0.9259	-1.2473
197	-0.9241	-1.2435
198	-0.9229	-1.2405
199	-0.9219	-1.2377
200	-0.9205	-1.2345
201	-0.919	-1.2311
202	-0.918	-1.2284
203	-0.9173	-1.226
204	-0.9167	-1.2239
205	-0.9162	-1.2218
206	-0.9152	-1.2192
207	-0.9142	-1.2165
208	-0.9135	-1.2143
209	-0.9126	-1.2117
210	-0.912	-1.2095
211	-0.9115	-1.2075
212	-0.9104	-1.2049
213	-0.909	-1.2017
214	-0.908	-1.199
215	-0.9078	-1.1975
216	-0.9078	-1.1961
217	-0.9073	-1.1942
218	-0.9069	-1.1923
219	-0.9065	-1.1905
220	-0.9062	-1.1889
221	-0.906	-1.1874
222	-0.9061	-1.1862
223	-0.9065	-1.1855
224	-0.9065	-1.1842
225	-0.9062	-1.1826
226	-0.9062	-1.1813
227	-0.9068	-1.1809
228	-0.9069	-1.1798
229	-0.9066	-1.1781
230	-0.9059	-1.176
231	-0.9047	-1.1733
232	-0.9039	-1.171
233	-0.9034	-1.1691
234	-0.9023	-1.1664
235	-0.9007	-1.1632
236	-0.899	-1.1598
237	-0.8962	-1.1551
238	-0.8933	-1.1501
239	-0.8901	-1.1448
240	-0.8873	-1.1401
241	-0.8838	-1.1343
242	-0.8802	-1.1286
243	-0.8764	-1.1226
244	-0.8726	-1.1166
245	-0.868	-1.1096
246	-0.8631	-1.1023
247	-0.8574	-1.0939
248	-0.8517	-1.0856
249	-0.846	-1.0771
250	-0.8404	-1.069
251	-0.8352	-1.0614
252	-0.8308	-1.0547
253	-0.8266	-1.0484
254	-0.8231	-1.0428
255	-0.8192	-1.0369
256	-0.8154	-1.0311
257	-0.8116	-1.0254
258	-0.8079	-1.0196
259	-0.8044	-1.0142
260	-0.8009	-1.0088
261	-0.7972	-1.0033
262	-0.7937	-0.9979
263	-0.791	-0.9935
264	-0.7889	-0.99
265	-0.7871	-0.9868
266	-0.7852	-0.9834
267	-0.7838	-0.9808
268	-0.7825	-0.9781
269	-0.781	-0.9754
270	-0.7799	-0.9731
271	-0.7787	-0.9707
272	-0.7784	-0.9694
273	-0.7782	-0.9682
274	-0.7783	-0.9675
275	-0.7786	-0.9669
276	-0.7784	-0.9658
277	-0.7787	-0.9653
278	-0.7789	-0.9647
279	-0.7792	-0.9641
280	-0.7803	-0.9646
281	-0.7817	-0.9655
282	-0.7834	-0.9667
283	-0.7844	-0.967
284	-0.786	-0.9681
285	-0.7881	-0.9699
286	-0.7899	-0.9712
287	-0.7915	-0.9722
288	-0.7932	-0.9735
289	-0.7945	-0.9743
290	-0.7963	-0.9756
291	-0.7979	-0.9767
292	-0.7996	-0.9779
293	-0.8014	-0.9792
294	-0.803	-0.9803
295	-0.8038	-0.9805
296	-0.8041	-0.98
297	-0.8044	-0.9795
298	-0.8052	-0.9796
299	-0.8065	-0.9803
300	-0.8076	-0.9808
301	-0.8084	-0.9809
302	-0.8083	-0.9799
303	-0.8078	-0.9786
304	-0.8071	-0.9768
305	-0.8053	-0.9739
306	-0.8031	-0.9703
307	-0.8008	-0.9667
308	-0.7986	-0.9633
309	-0.7971	-0.9606
310	-0.796	-0.9585
311	-0.7948	-0.9563
312	-0.7931	-0.9534
313	-0.7904	-0.9494
314	-0.7876	-0.9452
315	-0.7851	-0.9414
316	-0.7824	-0.9374
317	-0.7805	-0.9344
318	-0.7793	-0.9321
319	-0.7787	-0.9307
320	-0.7785	-0.9297
321	-0.7786	-0.9291
322	-0.7778	-0.9274
323	-0.777	-0.9256
324	-0.7756	-0.9233
325	-0.7744	-0.921
326	-0.7729	-0.9186
327	-0.772	-0.9168
328	-0.7711	-0.9149
329	-0.7705	-0.9135
330	-0.7701	-0.9122
331	-0.7691	-0.9104
332	-0.7679	-0.9082
333	-0.7666	-0.906
334	-0.7655	-0.904
335	-0.7637	-0.9011
336	-0.7618	-0.8982
337	-0.7602	-0.8956
338	-0.7588	-0.8933
339	-0.758	-0.8916
340	-0.7571	-0.8899
341	-0.756	-0.8879
342	-0.7546	-0.8856
343	-0.7536	-0.8838
344	-0.7531	-0.8825
345	-0.7532	-0.8819
346	-0.7534	-0.8814
347	-0.754	-0.8815
348	-0.7541	-0.8809
349	-0.7541	-0.8802
350	-0.7537	-0.8791
351	-0.7537	-0.8785
352	-0.753	-0.8771
353	-0.7526	-0.8759
354	-0.7521	-0.8746
355	-0.7523	-0.8742
356	-0.7529	-0.8743
357	-0.753	-0.8737
358	-0.7532	-0.8734
359	-0.7527	-0.8722
360	-0.7518	-0.8705
361	-0.7506	-0.8685
362	-0.748	-0.8648
363	-0.7458	-0.8616
364	-0.7431	-0.8579
365	-0.7401	-0.8538
366	-0.7369	-0.8495
367	-0.7342	-0.8457
368	-0.731	-0.8415
369	-0.7278	-0.8372
370	-0.7243	-0.8326
371	-0.721	-0.8282
372	-0.7176	-0.8236
373	-0.7149	-0.82
374	-0.7131	-0.8174
375	-0.7113	-0.8147
376	-0.7098	-0.8124
377	-0.7084	-0.8102
378	-0.7069	-0.8079
379	-0.7047	-0.8049
380	-0.7017	-0.8009
381	-0.6984	-0.7966
382	-0.6954	-0.7926
383	-0.6927	-0.7889
384	-0.69	-0.7853
385	-0.6874	-0.7818
386	-0.6849	-0.7785
387	-0.6827	-0.7754
388	-0.6799	-0.7717
389	-0.6768	-0.7677
390	-0.6739	-0.7639
391	-0.6714	-0.7606
392	-0.6691	-0.7574
393	-0.6668	-0.7543
394	-0.664	-0.7506
395	-0.6613	-0.747
396	-0.6589	-0.7438
397	-0.6565	-0.7407
398	-0.6541	-0.7375
399	-0.6516	-0.7342
400	-0.6491	-0.7309
401	-0.6474	-0.7284
402	-0.6464	-0.7269
403	-0.6459	-0.7258
404	-0.6459	-0.7253
405	-0.6454	-0.7243
406	-0.6454	-0.7238
407	-0.6451	-0.723
408	-0.6444	-0.7218
409	-0.6441	-0.7209
410	-0.6442	-0.7207
411	-0.6444	-0.7204
412	-0.6449	-0.7204
413	-0.6454	-0.7205
414	-0.646	-0.7207
415	-0.647	-0.7214
416	-0.6485	-0.7227
417	-0.6496	-0.7234
418	-0.6507	-0.7241
419	-0.652	-0.7252
420	-0.6533	-0.7262
421	-0.6544	-0.7269
422	-0.6557	-0.7279
423	-0.6566	-0.7285
424	-0.6569	-0.7284
425	-0.657	-0.7279
426	-0.6571	-0.7277
427	-0.6569	-0.727
428	-0.6559	-0.7254
429	-0.6545	-0.7234
430	-0.6531	-0.7214
431	-0.6515	-0.7192
432	-0.6502	-0.7173
433	-0.6489	-0.7155
434	-0.6476	-0.7136
435	-0.6457	-0.7111
436	-0.6441	-0.7089
437	-0.6424	-0.7065
438	-0.641	-0.7046
439	-0.64	-0.703
440	-0.6391	-0.7016
441	-0.6377	-0.6997
442	-0.6369	-0.6984
443	-0.6363	-0.6973
444	-0.6357	-0.6963
445	-0.6349	-0.695
446	-0.6341	-0.6937
447	-0.6332	-0.6923
448	-0.6326	-0.6913
449	-0.6318	-0.69
450	-0.6314	-0.6891
451	-0.6308	-0.6881
452	-0.6302	-0.687
453	-0.629	-0.6853
454	-0.6276	-0.6833
455	-0.6261	-0.6813
456	-0.6245	-0.6792
457	-0.6231	-0.6772
458	-0.6217	-0.6754
459	-0.6204	-0.6735
460	-0.6193	-0.672
461	-0.6182	-0.6704
462	-0.6173	-0.6691
463	-0.6164	-0.6677
464	-0.6148	-0.6656
465	-0.6129	-0.6632
466	-0.611	-0.6608
467	-0.6092	-0.6584
468	-0.6075	-0.6562
469	-0.6058	-0.654
470	-0.6035	-0.6512
471	-0.6019	-0.649
472	-0.6	-0.6467
473	-0.5977	-0.6438
474	-0.5956	-0.6412
475	-0.5929	-0.6379
476	-0.59	-0.6345
477	-0.5864	-0.6302
478	-0.5825	-0.6257
479	-0.579	-0.6215
480	-0.5753	-0.6173
481	-0.5715	-0.6129
482	-0.5677	-0.6085
483	-0.5639	-0.6041
484	-0.5602	-0.5997
485	-0.5567	-0.5956
486	-0.5524	-0.5907
487	-0.548	-0.5857
488	-0.5439	-0.581
489	-0.5402	-0.5767
490	-0.5367	-0.5727
491	-0.5339	-0.5694
492	-0.5313	-0.5663
493	-0.5285	-0.5631
494	-0.5256	-0.5597
495	-0.5229	-0.5565
496	-0.5209	-0.5541
497	-0.5196	-0.5525
498	-0.5188	-0.5513
499	-0.5182	-0.5503
500	-0.5175	-0.5493

More formally, we use Monte Carlo simulations to generate $10,000$ GARCH(1,1) processes⁶ for $X_{t}$, of the size $n=1000$, and then apply the homoscedastic and heteroscedastic random walk tests to the data. The results for several arbitrary lags, $q$, are presented in Table 2.3.1.

Table 2.3.1:  Percent of GARCH(1,1) Process Rejected

Confidence Level (φ)	q=2	q=4	q=8	q=16	q=32	q=64
φ=95%, Homoscedastic	12.41	12.35	11.57	11.08	9.85	8.12
φ=95%, Heteroscedastic	5.04	5.28	5.30	4.71	4.88	4.88
φ=98%, Homoscedastic	4.39	4.63	4.42	4.12	3.38	3.53
φ=99%, Heteroscedastic	0.83	1.07	1.35	1.41	1.58	2.03

The above table clearly shows that the homoscedastic test is biased towards rejection in the presence of time-varying volatility. This evidence supports our use of the heteroscedasticity-consistent test because it is the forecastability of price changes that is of primary interest in random walk analysis.

3.  Long Memory

Because a new type of test is needed for long memory analysis, we surveyed the literature for alternatives. Although powerful for theoretical modeling and volatility analysis, studies based upon the absolute value of returns (i.e. Ding, Granger, and Engle 1993; and Dacorogna et. al 1993) provide no intuition regarding the direction of a long memory cycle, only its existence. The rescaled range class of test statistics, however, are able to distinguish between positive strong dependence (persistence) and negative strong dependence (mean reversion).

3.1  Methodology

Again we denote $P_{t}$ as the index level at time $t$ and define the continuously compounded single period return from $t-1$ to $t$ as $X_{t} \equiv$ ln$P_{t}-$ln$P_{t-1}$. To avoid confusion with the notation from our random walk test in section 2, we denote the truncation lag, in days, as $\xi$. Our long memory methodology follows exactly from Lo (1991).

3.1.1  Classical Hurst-Mandelbrot Rescaled Range

The classical Hurst-Mandelbrot rescaled range statistic is

$$\tilde{Q}_{n}\equiv\frac{1}{s_{n}}\left[ \underset{1\leq k\leq n}{\text{Max} }\overset{k}{\underset{j=1}{\sum}}(X_{j}-\bar{X}_{n})-\underset{1\leq k\leq n}{\text{Min}}\underset{j=1}{\overset{k}{\sum}}(X_{j}-\bar{X}_{n})\right] \text{,}$$ (9)

where $s_{n}$ is the usual maximum likelihood standard deviation estimator

$$s_{n}\equiv\left[ \frac{1}{n}\underset{j=1}{\overset{n}{\sum}}(X_{j}-\bar {X}_{n})^{2}\right] ^{1/2}$$ (10)

When properly normalized, this statistic weakly converges to the range of the Brownian bridge.

$$\tilde{V}_{n}\Longleftarrow\frac{1}{\sqrt{n}}\tilde{Q}_{n}$$ (11)

The classical rescaled range statistic, however, does not distinguish between short- and long-term dependence. Lo (1991) shows that for short-term autocorrelation coefficients, $\hat{\rho}$, of 50 percent, the mean of $\tilde{V}_{n}$ may be biased upward by 73 percent, causing a rejection of the null hypothesis at any conventional significance level. Given this weakness and the short-term autoregressive properties of our data described in section 2.2, we selected the modified rescaled range statistic (Lo, 1991) for our long memory analysis because of its invariance to short-term serial correlation.

3.1.2  Lo's Modified Rescaled Range

The modified rescaled range statistic differs only in its denominator to achieve invariance to short memory processes:

$$Q_{n\text{ }}\equiv\frac{1}{\hat{\sigma}_{n}(\xi)}\left[ \underset{1\leq k\leq n}{\text{Max}}\overset{k}{\underset{j=1}{\sum}}(X_{j}-\bar{X} _{n})-\underset{1\leq k\leq n}{\text{Min}}\underset{j=1}{\overset{k}{\sum} }(X_{j}-\bar{X}_{n})\right]$$ (12)

where

$$\hat{\sigma}_{n}^{2}(\xi)\equiv\frac{1}{n}\overset{n}{\underset{j=1}{ {\textstyle\sum} }}(X_{j}-\bar{X}_{n})^{2}+\frac{2}{n}\overset{\xi}{\underset{j=1}{\sum}} \omega_{j}(\xi)\left[ \overset{n}{\underset{i=j+1}{\sum}}(X_{i}-\bar{X} _{n})(X_{i-j}-\bar{X}_{n})\right]$$ (13)

$\ \$

$$\ \ \ \ =\hat{\sigma}_{x}^{2}+2\underset{j=1}{\overset{\xi}{\sum}}\omega _{j}(\xi)\hat{\gamma}_{j},\qquad\omega_{j}(\xi)\equiv1-\frac{j}{\xi+1} ,\qquad\xi<n,$$ (14)

and $\hat{\sigma}_{x}^{2}$ and $\hat{\gamma}_{j}$ are the sample variance and autocovariance up to lag $\xi$ of $X$, respectively. Note that the modified rescaled range simplifies to the classical Hurst-Mandelbrot rescaled range statistic in the special case where $\xi=0$.

A main criticism of the modified R/S is that there is little guidance about how to select an optimal truncation lag, $\xi$. If $\xi$ is chosen too small, significant autocovariances may be ignored, thus biasing the $Q_{n\text{ }}$statistic. Also, if $\xi$ is too large relative to the sample size, Lo and MacKinlay (1989) show that the finite-sample distribution deviates significantly from its asymptotic limit.

Andrews (1991) provides an asymptotic data-dependent $\xi^{\ast}$

$$\xi^{\ast}=\left[ k_{n}\right] k_{n}\equiv\left( \frac {3n}{2}\right) ^{\frac{1}{3}}\cdot\left( \frac{2\hat{\rho}}{1-\hat{\rho} ^{2}}\right) ^{\frac{2}{3}}$$ (15)

where $\left[ k_{n}\right]$ is the greatest integer less than or equal to $k_{n}$, and $\hat{\rho}$ is the first-order autocorrelation coefficient. We show the Andrews (1991) data dependent $\xi^{\ast}$ in section 3.2 below; a broader range of $\xi$'s are graphed in Appendix 4. Although we report the Andrews $\xi^{\ast}$ in this paper, it has not been proven to be an optimal method for selecting $\xi$, and no known method currently exists.

In determining the significance of our results, it can be shown that the modified rescaled range statistic converges to the range of the Brownian bridge

$$\frac{1}{\sqrt{n}}Q_{n}\Longrightarrow V_{n}(\xi)$$ (16)

Following from Theorem 3.1 of Lo (1991), the cumulative distribution function (CDF) for the range of the Brownian bridge is given as

$$F_{V}(\upsilon)=1+2\overset{\infty}{\underset{k=1}{\sum}}(1-4k^{2}\upsilon ^{2})e^{-2(k\upsilon)^{2}}$$ (17)

Critical values of the two-tailed test can be obtained from the above equation; the most commonly used values are shown in Table 3.1.2.1.

Table 3.1.2.1  Critical Values of the Distribution F(v)

P(V<v)	0.005	0.025	0.050	0.500	0.950	0.975	0.995
v	0.721	0.809	0.861	1.223	1.747	1.862	2.098

Figure 3.1.2.1:  Cumulative Distribution Function of the Modified R/S

Data for Figure 3.1.2.1

v	F(v)
0.005	0.000
0.01	0.000
0.015	0.000
0.02	0.000
0.025	0.000
0.03	0.000
0.035	0.000
0.04	0.000
0.045	0.000
0.05	0.000
0.055	0.000
0.06	0.000
0.065	0.000
0.07	0.000
0.075	0.000
0.08	0.000
0.085	0.000
0.09	0.000
0.095	0.000
0.1	0.000
0.105	0.000
0.11	0.000
0.115	0.000
0.12	0.000
0.125	0.000
0.13	0.000
0.135	0.000
0.14	0.000
0.145	0.000
0.15	0.000
0.155	0.000
0.16	0.000
0.165	0.000
0.17	0.000
0.175	0.000
0.18	0.000
0.185	0.000
0.19	0.000
0.195	0.000
0.2	0.000
0.205	0.000
0.21	0.000
0.215	0.000
0.22	0.000
0.225	0.000
0.23	0.000
0.235	0.000
0.24	0.000
0.245	0.000
0.25	0.000
0.255	0.000
0.26	0.000
0.265	0.000
0.27	0.000
0.275	0.000
0.28	0.000
0.285	0.000
0.29	0.000
0.295	0.000
0.3	0.000
0.305	0.000
0.31	0.000
0.315	0.000
0.32	0.000
0.325	0.000
0.33	0.000
0.335	0.000
0.34	0.000
0.345	0.000
0.35	0.000
0.355	0.000
0.36	0.000
0.365	0.000
0.37	0.000
0.375	0.000
0.38	0.000
0.385	0.000
0.39	0.000
0.395	0.000
0.4	0.000
0.405	0.000
0.41	0.000
0.415	0.000
0.42	0.000
0.425	0.000
0.43	0.000
0.435	0.000
0.44	0.000
0.445	0.000
0.45	0.000
0.455	0.000
0.46	0.000
0.465	0.000
0.47	0.000
0.475	0.000
0.48	0.000
0.485	0.000
0.49	0.000
0.495	0.000
0.5	0.000
0.505	0.000
0.51	0.000
0.515	0.000
0.52	0.000
0.525	0.000
0.53	0.000
0.535	0.000
0.54	0.000
0.545	0.000
0.55	0.000
0.555	0.000
0.56	0.000
0.565	0.000
0.57	0.000
0.575	0.000
0.58	0.000
0.585	0.000
0.59	0.000
0.595	0.000
0.6	0.000
0.605	0.000
0.61	0.000
0.615	0.000
0.62	0.000
0.625	0.000
0.63	0.000
0.635	0.001
0.64	0.001
0.645	0.001
0.65	0.001
0.655	0.001
0.66	0.001
0.665	0.001
0.67	0.001
0.675	0.002
0.68	0.002
0.685	0.002
0.69	0.002
0.695	0.003
0.7	0.003
0.705	0.003
0.71	0.004
0.715	0.004
0.72	0.005
0.725	0.005
0.73	0.006
0.735	0.007
0.74	0.007
0.745	0.008
0.75	0.009
0.755	0.010
0.76	0.011
0.765	0.012
0.77	0.013
0.775	0.014
0.78	0.016
0.785	0.017
0.79	0.019
0.795	0.020
0.8	0.022
0.805	0.023
0.81	0.025
0.815	0.027
0.82	0.029
0.825	0.031
0.83	0.034
0.835	0.036
0.84	0.038
0.845	0.041
0.85	0.044
0.855	0.046
0.86	0.049
0.865	0.052
0.87	0.055
0.875	0.059
0.88	0.062
0.885	0.066
0.89	0.069
0.895	0.073
0.9	0.077
0.905	0.081
0.91	0.085
0.915	0.089
0.92	0.093
0.925	0.098
0.93	0.102
0.935	0.107
0.94	0.112
0.945	0.117
0.95	0.122
0.955	0.127
0.96	0.132
0.965	0.138
0.97	0.143
0.975	0.149
0.98	0.154
0.985	0.160
0.99	0.166
0.995	0.172
1	0.178
1.005	0.184
1.01	0.190
1.015	0.197
1.02	0.203
1.025	0.210
1.03	0.216
1.035	0.223
1.04	0.230
1.045	0.236
1.05	0.243
1.055	0.250
1.06	0.257
1.065	0.264
1.07	0.271
1.075	0.278
1.08	0.286
1.085	0.293
1.09	0.300
1.095	0.307
1.1	0.315
1.105	0.322
1.11	0.330
1.115	0.337
1.12	0.345
1.125	0.352
1.13	0.360
1.135	0.367
1.14	0.375
1.145	0.382
1.15	0.390
1.155	0.397
1.16	0.405
1.165	0.412
1.17	0.420
1.175	0.428
1.18	0.435
1.185	0.443
1.19	0.450
1.195	0.458
1.2	0.465
1.205	0.473
1.21	0.480
1.215	0.488
1.22	0.495
1.225	0.502
1.23	0.510
1.235	0.517
1.24	0.524
1.245	0.531
1.25	0.539
1.255	0.546
1.26	0.553
1.265	0.560
1.27	0.567
1.275	0.574
1.28	0.581
1.285	0.588
1.29	0.594
1.295	0.601
1.3	0.608
1.305	0.614
1.31	0.621
1.315	0.627
1.32	0.634
1.325	0.640
1.33	0.647
1.335	0.653
1.34	0.659
1.345	0.665
1.35	0.671
1.355	0.677
1.36	0.683
1.365	0.689
1.37	0.695
1.375	0.701
1.38	0.707
1.385	0.712
1.39	0.718
1.395	0.723
1.4	0.729
1.405	0.734
1.41	0.739
1.415	0.744
1.42	0.750
1.425	0.755
1.43	0.760
1.435	0.765
1.44	0.769
1.445	0.774
1.45	0.779
1.455	0.784
1.46	0.788
1.465	0.793
1.47	0.797
1.475	0.801
1.48	0.806
1.485	0.810
1.49	0.814
1.495	0.818
1.5	0.822
1.505	0.826
1.51	0.830
1.515	0.834
1.52	0.838
1.525	0.841
1.53	0.845
1.535	0.849
1.54	0.852
1.545	0.856
1.55	0.859
1.555	0.862
1.56	0.866
1.565	0.869
1.57	0.872
1.575	0.875
1.58	0.878
1.585	0.881
1.59	0.884
1.595	0.887
1.6	0.890
1.605	0.892
1.61	0.895
1.615	0.898
1.62	0.900
1.625	0.903
1.63	0.905
1.635	0.908
1.64	0.910
1.645	0.912
1.65	0.915
1.655	0.917
1.66	0.919
1.665	0.921
1.67	0.923
1.675	0.925
1.68	0.927
1.685	0.929
1.69	0.931
1.695	0.933
1.7	0.935
1.705	0.937
1.71	0.938
1.715	0.940
1.72	0.942
1.725	0.943
1.73	0.945
1.735	0.946
1.74	0.948
1.745	0.949
1.75	0.951
1.755	0.952
1.76	0.954
1.765	0.955
1.77	0.956
1.775	0.957
1.78	0.959
1.785	0.960
1.79	0.961
1.795	0.962
1.8	0.963
1.805	0.964
1.81	0.966
1.815	0.967
1.82	0.968
1.825	0.969
1.83	0.969
1.835	0.970
1.84	0.971
1.845	0.972
1.85	0.973
1.855	0.974
1.86	0.975
1.865	0.975
1.87	0.976
1.875	0.977
1.88	0.978
1.885	0.978
1.89	0.979
1.895	0.980
1.9	0.980
1.905	0.981
1.91	0.982
1.915	0.982
1.92	0.983
1.925	0.983
1.93	0.984
1.935	0.984
1.94	0.985
1.945	0.985
1.95	0.986
1.955	0.986
1.96	0.987
1.965	0.987
1.97	0.988
1.975	0.988
1.98	0.988
1.985	0.989
1.99	0.989
1.995	0.990
2	0.990
2.005	0.990
2.01	0.991
2.015	0.991
2.02	0.991
2.025	0.992
2.03	0.992
2.035	0.992
2.04	0.992
2.045	0.993
2.05	0.993
2.055	0.993
2.06	0.993
2.065	0.994
2.07	0.994
2.075	0.994
2.08	0.994
2.085	0.995
2.09	0.995
2.095	0.995
2.1	0.995
2.105	0.995
2.11	0.995
2.115	0.996
2.12	0.996
2.125	0.996
2.13	0.996
2.135	0.996
2.14	0.996
2.145	0.997
2.15	0.997
2.155	0.997
2.16	0.997
2.165	0.997
2.17	0.997
2.175	0.997
2.18	0.997
2.185	0.997
2.19	0.998
2.195	0.998
2.2	0.998
2.205	0.998
2.21	0.998
2.215	0.998
2.22	0.998
2.225	0.998
2.23	0.998
2.235	0.998
2.24	0.998
2.245	0.998
2.25	0.999
2.255	0.999
2.26	0.999
2.265	0.999
2.27	0.999
2.275	0.999
2.28	0.999
2.285	0.999
2.29	0.999
2.295	0.999
2.3	0.999
2.305	0.999
2.31	0.999
2.315	0.999
2.32	0.999
2.325	0.999
2.33	0.999
2.335	0.999
2.34	0.999
2.345	0.999
2.35	0.999
2.355	0.999
2.36	0.999
2.365	0.999
2.37	0.999
2.375	1.000
2.38	1.000
2.385	1.000
2.39	1.000
2.395	1.000
2.4	1.000
2.405	1.000
2.41	1.000
2.415	1.000
2.42	1.000
2.425	1.000
2.43	1.000
2.435	1.000
2.44	1.000
2.445	1.000
2.45	1.000
2.455	1.000
2.46	1.000
2.465	1.000
2.47	1.000
2.475	1.000
2.48	1.000
2.485	1.000
2.49	1.000
2.495	1.000
2.5	1.000
2.505	1.000
2.51	1.000
2.515	1.000
2.52	1.000
2.525	1.000
2.53	1.000
2.535	1.000
2.54	1.000
2.545	1.000
2.55	1.000
2.555	1.000
2.56	1.000
2.565	1.000
2.57	1.000
2.575	1.000
2.58	1.000
2.585	1.000
2.59	1.000
2.595	1.000
2.6	1.000
2.605	1.000
2.61	1.000
2.615	1.000
2.62	1.000
2.625	1.000
2.63	1.000
2.635	1.000
2.64	1.000
2.645	1.000
2.65	1.000
2.655	1.000
2.66	1.000
2.665	1.000
2.67	1.000
2.675	1.000
2.68	1.000
2.685	1.000
2.69	1.000
2.695	1.000
2.7	1.000
2.705	1.000
2.71	1.000
2.715	1.000
2.72	1.000
2.725	1.000
2.73	1.000
2.735	1.000
2.74	1.000
2.745	1.000
2.75	1.000
2.755	1.000
2.76	1.000
2.765	1.000
2.77	1.000
2.775	1.000
2.78	1.000
2.785	1.000
2.79	1.000
2.795	1.000
2.8	1.000
2.805	1.000
2.81	1.000
2.815	1.000
2.82	1.000
2.825	1.000
2.83	1.000
2.835	1.000
2.84	1.000
2.845	1.000
2.85	1.000
2.855	1.000
2.86	1.000
2.865	1.000
2.87	1.000
2.875	1.000
2.88	1.000
2.885	1.000
2.89	1.000
2.895	1.000
2.9	1.000
2.905	1.000
2.91	1.000
2.915	1.000
2.92	1.000
2.925	1.000
2.93	1.000
2.935	1.000
2.94	1.000
2.945	1.000
2.95	1.000
2.955	1.000
2.96	1.000
2.965	1.000
2.97	1.000
2.975	1.000
2.98	1.000
2.985	1.000
2.99	1.000
2.995	1.000
3	1.000

3.2  Results

The results of our long memory analysis are shown in detail in Appendix 4. Again, the dashed orange and red lines represent 95 and 99 percent rejection levels of the modified rescaled range hypothesis test, respectively. Significantly large $V$-values indicate positive strong dependence. $V$-values below the lower rejection band show negative strong dependence. Table 3.2.1 reports numerical results of our modified rescaled analysis using Andrews' (1991) data-dependent $\xi^{\ast}.$⁷

We find little evidence for long-term memory in global equity indices. With adjustments for short-term autocorrelations of up to 250 trading days, approximately one year, the null hypothesis of no long term dependence cannot be rejected⁸ for 36 of the 44 equity markets studied. Criticisms levied against the modified rescaled range by Baillie (1996), for example, cite that

...a major practical difficulty concerns the choice of $q[\xi]$ and how to distinguish between short range dependencies and long range dependencies.

Our graphical results in Appendix 4 do not support this claim. We fail to reject the null hypothesis at every integer lag for the United States, United Kingdom and many other emerging and industrialized equity markets. Furthermore, our findings are largely consistent with other analyses of the U.S. (Lo, 1991) and global (Chow et. al, 1996) equity markets.

In the foreign exchange markets, we find great disparity in the long memory evidence between emerging and industrialized economies. Contrary to Mulligan (2000),⁹ we find no evidence of long memory in the bilateral trade of U.S. dollars with other advanced market currencies. A likely reason for this disparity is Mulligan's use of monthly averages instead of point data, which may bias results in the presence of short-term dependece in the daily data. We reject the null hypothesis for many emerging economies, however, particularly in Latin America.

We find very little evidence of long memory in most commodity markets. However, Corrazza et. al. (1997) found fractal returns in the futures markets for soybeans, wheat, oats, and corn. We similarly found long memory in the spot returns of the soybeans market.

Table 3.2.1a. Results of the Modified R/S Test for Optimal Lag ξ - By Country

Country	Equity Indicies: ξ*	Equity Indicies: V(ξ*)	Foreign Exchange : ξ*	Foreign Exchange: V(ξ*)
Argentina	4	1.50	2	1.91*
Brazil	5	1.31	3	1.89*
Chile	14	1.75	3	1.88*
Colombia	7	1.62	0	1.91*
Mexico	7	1.38	0	1.00
Peru	11	2.31**	2	1.41
Venezuela	8	1.25
China	2	1.38
Hong Kong	4	1.37
Indonesia	12	1.66	6	1.46
India	6	1.45	9	2.12**
South Korea	6	1.88*	7	1.95*
Malaysia	8	1.50	1	1.35
Philippines	9	1.37	5	1.67
Russia	7	1.66	8	2.00*
Singapore	9	1.48	5	1.34
Taiwan	5	1.50
Thailand	7	1.79	4	1.57
Israel	1	0.98	2	2.05*
Kuwait	1	1.85
Nigeria	15	1.21	12	1.23
Saudi Arabia	2	2.20**
South Africa	4	1.43	1	2.01*
Turkey	4	1.47	5	1.36
UAE	2	2.16**
Australia	7	1.36	3	1.30
Belgium	8	1.85
Canada	9	0.99	2	1.64
Czech Republic	5	1.86	1	1.71
Denmark	3	2.02*
European Union			0	1.51
France	2	1.54
Germany	1	1.58
Hungary	5	1.39
Italy	11	1.61
Japan	7	1.95*	3	1.23
Netherlands	0	1.40
Norway	1	2.12**	0	1.43
New Zealand	5	1.00	2	1.45
Poland	6	1.32	3	1.79
Sweden	1	1.93*	0	1.48
Spain	2	1.63
Switzerland	2	1.48	2	1.38
UK	2	1.18	4	1.44
US	5	1.57

* and ** denote significane at the 95 and 99 percent confidence levels, respectively

Table 3.2.1b. Results of the Modified R/S Test for Optimal Lag ξ - By Commodity

Commodity	ξ*	V(ξ*)
Crude Oil	0	0.95
Gasoline	1	1.06
Natural Gas	0	0.92
Heating Oil	3	1.02
Copper	4	1.28
Aluminum	2	1.26
Lead	4	1.26
Nickel	1	1.30
Tin	4	1.54
Zinc	0	1.47
Gold	3	1.03
Platinum	2	1.39
Silver	2	1.00
Cotton	2	1.58
Rubber	13	1.21
Corn	0	1.65
Wheat	5	1.19
Soybeans	1	2.08*
Cattle	2	0.72**
Cocoa	2	0.90
Coffee	0	1.25
Sugar	4	1.57

* and ** denote significane at the 95 and 99 percent confidence levels, respectively

4.  Conclusion

This paper presents an extensive analysis of short- and long-term memory in the returns of international financial data. The random walk section attempts to answer the question, can asset prices be forecasted based on historical price information alone? Using a heteroscedasticity-consistent test, we conclude that financial data are broadly unpredictable with a few notable exceptions. First, energy prices exhibit mean-reverting price behavior. Second, markets with poorer risk-adjusted performance are more likely the reject the random walk. We expect that prolonged, weak performance in these markets has deterred analyst coverage, resulting in relatively low informational efficiency. This idea is consistent with contrarian investment strategies which seek to earn positive abnormal returns in depressed markets.

Much of the empirical literature on long memory concerns the rate of decay in the autocorrelation of volatility. Although important for applications in risk management, options and other derivative markets, long memory in volatility does not provide intuition regarding the direction of the cycle in returns. When Hurst (1951) set out to identify long memory patterns in the Nile River, for example, he was not concerned with identifying volatility in sediment discharge, but the prediction of flood (positive) and drought (negative) cycles to improve river management in the delta. We are concerned with similar questions in asset pricing. Can we identify long term cycles in asset prices to earn excess returns? We attempt to answer this question via a modernization of Hurst's test, the modified rescaled range statistic, and, in financial asset returns, we find very little evidence of long memory cycles. The most promising area for further long memory research appears to be the bilateral exchange of U.S. dollars with the currencies of Latin American economies.

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Appendix 1. Data

Appendix 1.1 Equity Indices

Daily equity index data for each country were downloaded from Bloomberg. We downloaded the fields "px_last" (last price) and "eqy_last_dps_gross" (dividends per share) to create a total return series according to the fundamental relationship:

$$P_{1}=\alpha_{1\text{,}}$$ (18)

$$P_{t}=P_{t-1}\left[ (\frac{\varphi_{t}}{\alpha_{t-1}})+(\frac{\alpha_{t} }{\alpha_{t-1}})\right]$$ (19)

We initialized the first observation of the total return index, $P_{1}$, to equal the basic index level, $\alpha_{1}$. Equation 19 was then used to construct the total return index series as the sum of the dividend and capital gains yields, respectively, where $\varphi_{t}$ are dividends per share. Because our Bloomberg download program only provided two decimal places, and indices are calculated with an arbitrary base (i.e. 1995=100), time series of countries experiencing extraordinary growth or crises were truncated to avoid rounding errors. For example, at the beginning of the dataset, some countries (like Peru), had index values at or near "0.00". Returns calculated on such low price levels were inherently undefined or extremely volatile, and thus not representative of the historical record. As a rule-of-thumb, we corrected for this by eliminating the first $t$ observations where $\alpha_{t}<100$, leaving a residual rounding error no greater than $\frac{\alpha_{t}}{20000}$.

Appendix Table 1.1

Country	Ticker	Description	Sample (mm/dd/yy)
Latin America: (AR) Argentina	MAR	Merval Index	01/03/00 - 03/07/08
Latin America: (BZ) Brazil	IBX	IBX Index	01/02/99 - 03/07/08
Latin America: (CL) Chile	IGPA	IPSA Stock Index	01/03/90 - 03/07/08
Latin America: (CO) Colombia	IGBC	IGBC Stock Index	07/04/01 - 03/07/08
Latin America: (MX) Mexico	MEXBOL	Bolsa Stock Index	01/03/89 - 03/07/08
Latin America: (PE) Peru	IGBVL	Peru Lima General Index	01/03/92 - 03/07/08
Latin America: (VE) Venezuela	IBVC	IBC Stock Index	01/03/94 - 03/07/08
Emerging Asia: (CH) China	SHCOMP	Shanghai Composite Index	01/04/95 - 03/07/08
Emerging Asia: (HK) Hong Kong	HSI	Hang Seng Stock Index	11/25/69 - 03/07/08
Emerging Asia: (ID) Indonesia	JCI	Jakarta Stock Index	04/05/83 - 03/07/08
Emerging Asia: (IN) India	SENSEX	Sensex Stock Index	04/04/79 - 03/07/08
Emerging Asia: (KO) South Korea	KOSPI	KOSPI Stock Index	01/05/80 - 03/07/08
Emerging Asia: (MA) Malaysia	KLCI	Kuala Lumpur Composite	01/04/77 - 03/07/08
Emerging Asia: (PH) Philippines	PCOMP	PSEI Stock Index	01/05/87 - 03/07/08
Emerging Asia: (RU) Russia	RTSI$	Russian Trading System	09/04/95 - 03/07/08
Emerging Asia: (SI) Singapore	SESALL	All-Equities Stock Index	01/03/75 - 03/07/08
Emerging Asia: (TA) Taiwan	TWSE	TWSE Stock Index	01/05/73 - 03/07/08
Emerging Asia: (TH) Thailand	SET	Bangkok SET Stock Index	07/03/87 - 03/07/08
Middle East and Africa: (IS) Israel	TA-25	Tel Aviv 25 Stock Index	01/05/92 - 03/07/08
Middle East and Africa: (KW) Kuwait	KWSEIDX	Kuwait SE Unweighed Index	06/18/01 - 03/07/08
Middle East and Africa: (NG) Nigeria	NGSEINDX	Nigeria Stock Exchange	01/05/98 - 03/07/08
Middle East and Africa: (SA) Saudi Arabia	SASEIDX	Tadawul All-Shares	01/29/94 - 03/07/08
Middle East and Africa: (SF) South Africa	JALSH	JSE All-Shares Stock Index	07/03/95 - 03/07/08
Middle East and Africa: (TK) Turkey	XU100	Istanbul National 100 Index	01/03/90 - 03/07/08
Middle East and Africa: (UA) UAE	DFMGI	Dubai General Stock Index	01/03/04 - 03/07/08
Industrialized Economies: (AL) Australia	AS30	All Ordinaries Stock Index	01/02/80 - 03/07/08
Industrialized Economies: (BE) Belgium	BELSTK	Belgian All-Shares	10/04/88 - 03/07/08
Industrialized Economies: (CA) Canada	SPTSX	S&P Toronto Stock Index	01/04/77 - 03/07/08
Industrialized Economies: (CZ) Czech Republic	PX	PX Stock Index	09/20/94 - 03/07/08
Industrialized Economies: (DN) Denmark	KAX	OMX Copenhagen	01/02/96 - 03/07/08
Industrialized Economies: (FR) France	SBF250	SBF 250 Stock Index	01/02/91 - 03/07/08
Industrialized Economies: (GE) Germany	HDAX	HDAX Stock Index	01/04/88 - 03/07/08
Industrialized Economies: (HU) Hungary	BUX	Budapest Stock Exchange	01/03/91 - 03/07/08
Industrialized Economies: (IT) Italy	ITSMBANC	Bacchi Stock Index	01/03/73 - 03/07/08
Industrialized Economies: (JA) Japan	TPX	TOPIX Stock Index	01/06/70 - 03/07/08
Industrialized Economies: (NE) Netherlands	AEX	Amsterdam Exchanges Index	01/04/83 - 03/07/08
Industrialized Economies: (NO) Norway	OSEAX	Oslo All-Shares Stock Index	01/02/96 - 03/07/08
Industrialized Economies: (NZ) New Zealand	NZSE	All Ordinaries Stock Index	03/31/92 - 03/07/08
Industrialized Economies: (PL) Poland	WIG	WSE WIG Index	04/23/91 - 03/07/08
Industrialized Economies: (SD) Sweden	SWSMAFFR	Affarsvarlden Stock Index	01/04/00 - 03/07/08
Industrialized Economies: (SP) Spain	MADX	Madrid Stock Index	02/19/93 - 03/07/08
Industrialized Economies: (SZ) Switzerland	SMI	Swiss Market Stock Index	07/04/88 - 03/07/08
Industrialized Economies: (UK) United Kingdom	ASX	FTSE All-Shares Stock Index	01/03/85 - 03/07/08
Industrialized Economies: (US) United States	SPX	S&P 500 Index	01/05/70 - 03/07/08

Appendix 1.2 Commodities

The commodity data are daily closing prices in the spot market. Rubber price data is from Bloomberg, and all other commodities are from the Commodity Research Bureau, Final Markets database.

Appendix Table 1.2

Commodity	Source10	Description	Sample (mm/dd/yy)
Energy: Crude Oil	NYMEX	West Texas Intermediate crude oil, Cushing OK	01/02/85 - 03/07/08
Energy: Natural Gas	NYMEX	Natural gas, Henry Hub	10/29/93 - 03/07/08
Energy: Heating Oil	NYMEX	Heating oil #2, fuel oil	09/02/03 - 03/07/08
Energy: Gasoline	NYMEX	Gasoline, unleaded, regular non-oxygenated	09/02/03 - 03/07/08
Metals: Aluminum	LME	Aluminum, high grade	09/01/03 - 03/07/08
Metals: Copper	NYMEX/COMEX	Copper high grade, scrap #2 wire	09/02/03 - 03/07/08
Metals: Gold	NYMEX/COMEX	Gold	09/01/03 - 03/07/08
Metals: Lead	USGS	Lead pig	09/02/03 - 03/07/08
Metals: Nickel	LME	Nickel	09/01/03 - 03/07/08
Metals: Platinum	NYMEX/COMEX	Platinum	09/02/03 - 03/07/08
Metals: Silver	NYMEX/COMEX	Silver	09/01/03 - 03/07/08
Metals: Tin	USGS	Tin straights, composite	09/02/03 - 03/07/08
Metals: Zinc	AMM	Zinc, prime western, domestic	09/02/03 - 03/07/08
Agriculture: Cattle	CME	Live cattle, choice average, Texas/Oklahoma	09/02/03 - 03/07/08
Agriculture: Cocoa	NYBOT	Cocoa, Ivory Coast	09/02/03 - 03/07/08
Agriculture: Coffee	USDA	Coffee, Brazilian	09/02/03 - 03/07/08
Agriculture: Corn	CBOT	Corn, #2 yellow	10/25/01 - 03/07/08
Agriculture: Cotton	USDA	Cotton, 1-1/16	09/02/03 - 03/07/08
Agriculture: Rubber	MARB	Standard Rubber #20	07/20/01 - 03/07/08
Agriculture: Soybean	CBOT	Soybeans, #1 yellow	09/02/03 - 03/07/08
Agriculture: Sugar	NYBOT	Sugar #11, world raw	09/02/03 - 03/07/08
Agriculture: Wheat	KCBOT	Wheat, #2 hard winter	09/02/03 - 03/07/08

Appendix 1.3 Foreign Exchange Rates

Nominal bilateral exchange rates are in units of foreign currency per U.S. dollar. The data for Australia, Brazil, Canada, Denmark, the European Monetary Union, India, Japan, Malaysia, Mexico, New Zealand, Norway, Singapore, South Africa, South Korea, Sweden, Switzerland, Taiwan, Thailand and the United Kingdom are the 12 PM rates from the Federal Reserve Bank of New York.¹¹ All other exchange rate data are closing prices from Bloomberg. Countries currently employing a fixed or pegged exchange rate regime are ignored in this study because calculations of temporal dependence are meaningless under fixed prices. Currency classifications are as defined in the 2006 IMF Annual Report on Exchange Arrangements and Exchange Restrictions.

Appendix Table 1.3

Country	Currency	Classification	Sample (mm/dd/yy)
Latin America: (AR) Argentina	Peso	Managed Float	02/11/02 - 03/07/08
Latin America: (BZ) Brazil	Real	Independent Float	01/15/99 - 03/07/08
Latin America: (CL) Chile	Peso	Independent Float	09/03/99 - 03/07/08
Latin America: (CO) Colombia	Peso	Managed Float	09/26/99 - 03/07/08
Latin America: (MX) Mexico	Peso	Independent Float	08/02/96 - 03/07/08
Latin America: (PE) Peru	Nuevo Sol	Managed Float	09/01/97 - 03/07/08
Emerging Asia: (ID) Indonesia	Rupiah	Managed Float	08/15/97 - 03/07/08
Emerging Asia: (IN) India	Rupee	Managed Float	01/02/73 - 03/07/08
Emerging Asia: (KO) South Korea	Won	Independent Float	04/13/81 - 03/07/08
Emerging Asia: (MA) Malaysia	Ringgit	Managed Float	01/22/05 - 03/07/08
Emerging Asia: (PH) Philippines	Peso	Independent Float	03/16/98 - 03/07/08
Emerging Asia: (RU) Russia	Ruble	Managed Float	01/06/99 - 03/07/08
Emerging Asia: (SI) Singapore	Dollar	Managed Float	01/02/81 - 03/07/08
Emerging Asia: (TH) Thailand	Baht	Managed Float	07/09/97 - 03/07/08
Middle East and Africa: (IS) Israel	Sheqel	Independent Float	12/18/91 - 03/07/08
Middle East and Africa: (NG) Nigeria	Naira	Managed Float	01/04/99 - 03/07/08
Middle East and Africa: (SF) South Africa	Rand	Independent Float	01/06/71 - 03/07/08
Middle East and Africa: (TK) Turkey	New Lira	Independent Float	02/28/01 - 03/07/08
Industrialized Economies: (AL) Australia	Dollar	Independent Float	01/02/75 - 03/07/08
Industrialized Economies: (CA) Canada	Dollar	Independent Float	01/04/71 - 03/07/08
Industrialized Economies: (CZ) Czech Republic	Koruna	Managed Float	01/16/96 - 03/07/08
Industrialized Economies: (EU) European Union12	Euro	Independent Float	01/04/99 - 03/07/08
Industrialized Economies: (JA) Japan	Yen	Independent Float	01/04/71 - 03/07/08
Industrialized Economies: (NO) Norway	Krone	Independent Float	01/04/71 - 03/07/08
Industrialized Economies: (NZ) New Zealand	Dollar	Independent Float	01/04/71 - 03/07/08
Industrialized Economies: (PL) Poland	Zloty	Independent Float	04/24/96 - 03/07/08
Industrialized Economies: (SD) Sweden	Krona	Independent Float	01/04/71 - 03/07/08
Industrialized Economies: (SZ) Switzerland	Franc	Independent Float	01/04/71 - 03/07/08
Industrialized Economies: (UK) United Kingdom	Pound	Independent Float	01/04/71 - 03/07/08

Appendix 2.  Summary Statistics

Appendix 2.1.  Equity Indices

Country	Sample Size	Mean	Standard Deviation	Skewness	Excess Kurtosis	Jarque-Bera Statistic
Argentina	1940	0.051	2.18	0.17	5.41	2377.82**
Brazil	2879	0.120	2.01	0.61	16.31	32072.80**
Chile	4358	0.069	0.78	0.16	3.82	2666.04**
Colombia	1528	0.167	1.51	-0.06	14.38	13165.80**
Mexico	4644	0.106	1.57	-0.09	5.68	6253.92**
Peru	3866	0.127	1.46	0.22	6.70	7259.93**
Venezuela	3224	0.133	1.82	-0.13	13.38	24045.15**
China	3121	0.060	1.84	0.55	21.33	59297.77**
Hong Kong	9042	0.057	1.85	-0.37	12.39	58083.83**
Indonesia	5795	0.059	1.57	4.00	97.61	2316205.03**
India	5639	0.051	1.59	-0.06	4.61	5005.44**
Korea	6620	0.043	1.62	-0.16	4.59	5847.42**
Malaysia	7316	0.036	1.40	-0.61	32.44	321301.57**
Philippines	5079	0.027	1.67	0.10	9.49	19048.88**
Russia	2995	0.078	2.70	-0.42	6.50	5357.90**
Singapore	7951	0.036	1.21	-1.32	34.04	386172.07**
Taiwan	8165	0.046	1.76	-0.23	3.77	4903.65**
Thailand	4803	0.029	1.70	0.02	6.23	7773.66**
Israel	2301	0.063	1.74	-0.17	3.70	1325.61**
Kuwait	736	0.228	1.27	-0.50	5.97	1122.54**
Nigeria	2186	0.081	0.86	-0.98	15.52	22303.69**
Saudi Arabia	2025	0.063	1.62	-0.94	15.32	20113.67**
South Africa	3054	0.050	1.20	-0.74	7.86	8144.32**
Turkey	4403	0.163	2.93	0.02	3.48	2220.95**
UAE	627	0.252	2.31	0.14	3.94	408.59**
Australia	6978	0.036	0.95	-4.53	127.01	4714368.66**
Belgium	4679	0.030	0.89	0.05	7.11	9847.77**
Canada	7602	0.035	0.85	-0.99	13.53	59242.83**
Czech Republic	3218	0.014	1.21	-0.22	3.09	1307.13**
Denmark	2975	0.043	0.94	-0.51	2.55	933.22**
France	4234	0.030	1.14	-0.21	3.45	2130.20**
Germany	4965	0.038	1.30	-0.56	6.98	10339.31**
Hungary	4163	0.074	1.60	-0.79	13.69	32927.86**
Italy	8545	0.027	1.26	-0.46	5.22	10001.44**
Japan	9027	0.017	1.04	-0.52	11.62	51222.52**
Netherlands	6275	0.035	1.28	-0.32	8.22	17771.00**
Norway	2981	0.037	1.19	-0.53	3.38	1560.75**
New Zealand	3902	0.041	0.80	0.00	8.70	12299.44**
Poland	3348	0.053	1.90	-0.17	7.36	7572.63**
Sweden	1995	-0.004	1.41	-0.11	2.42	490.24**
Spain	3641	0.045	1.17	-0.36	3.22	1649.38**
Switzerland	4824	0.029	1.13	-0.52	7.03	10142.03**
UK	5864	0.033	0.96	-0.81	11.71	34140.18**
US	9317	0.040	0.99	-1.43	34.83	474220.14**

* and ** represent significance at the 95 and 99 percent confidence levels, respectively

Appendix 2.2.  Commodities

Commodity	Sample Size	Mean	Standard Deviation	Skewness	Excess Kurtosis	Jarque-Bera Statistic
Crude Oil	5630	0.016	2.45	-1.22	18.93	85435.26**
Gasoline	1086	0.090	2.96	-0.15	7.08	2271.70**
Natural Gas	3405	0.080	4.89	0.70	24.74	87144.91**
Heating Oil	1086	0.114	2.33	0.26	1.06	63.64**
Copper	1086	0.109	1.99	-0.50	3.10	480.71**
Aluminum	1114	0.074	1.45	-0.25	2.11	219.03**
Lead	1087	0.109	2.03	-0.37	5.89	1597.30**
Nickel	1114	0.117	2.65	-0.35	3.32	535.69**
Tin	1097	0.111	1.69	-0.40	3.79	685.51**
Zinc	1097	0.079	2.00	-0.50	2.60	356.57**
Gold	1166	0.078	1.11	-0.57	2.58	385.74**
Platinum	1097	0.086	1.25	-0.02	5.48	1372.32**
Silver	1164	0.104	2.04	-1.41	9.79	5032.06**
Cotton	1087	0.017	1.73	0.29	11.91	6433.98**
Rubber	1548	0.088	0.87	-0.28	2.48	416.12**
Corn	1537	0.059	1.76	0.15	1.23	103.49**
Wheat	1092	0.088	2.28	-0.49	64.15	187269.43**
Soybeans	1090	0.057	1.75	-0.82	4.35	983.71**
Cattle	1108	0.022	1.43	0.24	3.64	622.91**
Cocoa	1086	0.044	1.73	-0.24	3.39	530.14**
Coffee	1085	0.084	1.98	0.30	4.31	857.82**
Sugar	1086	0.071	1.91	-0.13	2.27	236.46**

* and ** represent significance at the 95 and 99 percent confidence levels, respectively

Appendix 2.3.  Foreign Exchange Rates

Country	Sample Size	Mean	Standard Deviation	Skewness	Excess Kurtosis	Jarque-Bera Statistic
Argentina	1544	0.030	1.15	3.18	102.82	682741.37**
Brazil	2218	-0.004	1.07	0.54	13.98	18173.92**
Chile	2175	-0.007	0.54	0.07	2.14	414.97**
Colombia	2163	-0.003	0.53	0.06	6.49	3803.08**
Mexico	2808	0.010	0.51	0.87	8.19	8195.30**
Peru	543	0.049	0.30	-0.01	8.73	1726.16**
Indonesia	2695	0.036	1.85	1.71	36.17	148223.94**
India	8477	0.017	0.47	4.71	129.04	5912848.31**
Korea	6427	0.003	0.58	3.45	158.07	6704119.60**
Malaysia	638	-0.027	0.25	-0.05	3.24	279.5911**
Philippines	2561	0.001	0.56	-7.75	218.71	5130106.64**
Russia	2355	0.002	0.37	1.98	75.65	563129.97**
Singapore	6571	-0.008	0.33	-0.80	15.94	70261.65**
Thailand	2587	-0.002	0.73	-0.51	15.62	26420.83**
Israel	3819	0.010	0.43	1.47	25.24	102758.53**
Nigeria	2263	0.011	1.47	0.27	16.27	24986.60**
South Africa	8933	0.028	0.83	1.93	53.89	1086480.52**
Turkey	1832	0.018	1.24	1.43	23.55	42949.21**
Australia	8015	0.002	0.62	4.62	128.27	5523084.78**
Canada	8977	-0.001	0.32	-0.01	3.82	5445.18**
Czech Republic	3122	-0.016	0.69	0.53	8.09	8663.15**
European Union	2226	-0.013	0.58	-0.02	0.85	67.46**
Japan	8963	-0.013	0.62	-0.41	5.94	13425.10**
Norway	8967	-0.004	0.59	0.36	7.98	24018.13**
New Zealand	8949	0.002	0.67	2.72	66.20	1644907.54**
Poland	3049	-0.006	0.64	0.28	2.82	1049.42**
Sweden	8967	-0.002	0.59	1.02	19.68	146297.76**
Switzerland	8969	-0.016	0.71	0.01	3.77	5317.29**
UK	8969	0.001	0.57	0.14	4.20	6608.25**

* and ** represent significance at the 95 and 99 percent confidence levels, respectively

Appendix 3.  Random Walk Hypothesis Tests

For all figures in Appendix 3, the 95% critical values are plus-or-minus 1.96, and the 99% critical values are plus-or-minus 2.58. The critical values do not depend upon q.

Data for Appendix 3 - Random Walk Hypothesis Tests: Equity (Russia - South Korea)

q	Argentina	Brazil	Chile	Colombia	Mexico	Peru	Venezuela	China	Hong Kong	Indonesia	India	South Korea
2	2.807	-0.636	4.594	0.646	1.146	1.037	1.787	0.346	2.983	-0.260	-0.482	-0.218
3	2.761	0.361	5.598	0.938	1.127	2.411	1.843	0.440	3.942	-0.006	0.410	0.043
4	2.832	0.412	5.982	1.209	1.733	2.864	1.975	0.823	4.265	0.199	0.682	0.831
5	2.526	0.628	6.145	1.435	1.826	3.102	1.882	0.861	4.140	0.408	0.985	1.190
6	2.161	0.688	6.141	1.470	1.881	3.065	1.929	0.673	3.982	0.579	1.072	1.470
7	2.012	0.697	6.079	1.504	1.858	2.982	2.010	0.689	3.785	0.713	1.006	1.529
8	1.927	0.668	5.978	1.501	1.832	2.876	1.951	0.797	3.616	0.794	0.962	1.619
9	2.041	0.605	5.862	1.440	1.840	2.893	1.957	0.849	3.393	0.855	1.014	1.764
10	2.170	0.574	5.726	1.385	1.787	2.938	1.959	0.976	3.329	0.882	1.074	1.743
11	2.180	0.572	5.528	1.351	1.758	2.948	2.024	1.069	3.241	0.897	1.085	1.788
12	2.186	0.619	5.359	1.278	1.773	3.052	2.089	1.158	3.162	0.895	1.172	1.796
13	2.112	0.599	5.180	1.193	1.671	3.135	2.095	1.202	3.019	0.892	1.178	1.769
14	2.019	0.586	5.070	1.107	1.608	3.205	2.096	1.202	2.906	0.886	1.155	1.754
15	1.984	0.611	4.989	1.033	1.594	3.286	2.134	1.213	2.845	0.887	1.163	1.748
16	1.984	0.630	4.942	0.936	1.571	3.371	2.153	1.173	2.782	0.885	1.118	1.740
17	1.946	0.639	4.875	0.853	1.521	3.408	2.187	1.177	2.703	0.860	1.035	1.680
18	1.901	0.633	4.833	0.754	1.471	3.420	2.216	1.176	2.595	0.853	0.977	1.642
19	1.824	0.644	4.796	0.653	1.433	3.405	2.246	1.183	2.478	0.851	0.881	1.623
20	1.719	0.643	4.768	0.574	1.359	3.386	2.293	1.204	2.365	0.836	0.803	1.584
21	1.614	0.645	4.737	0.519	1.286	3.397	2.379	1.185	2.263	0.830	0.732	1.577
22	1.539	0.629	4.718	0.452	1.249	3.391	2.436	1.203	2.159	0.825	0.683	1.567
23	1.492	0.608	4.700	0.391	1.213	3.414	2.491	1.246	2.056	0.831	0.649	1.588
24	1.446	0.569	4.692	0.357	1.131	3.436	2.522	1.286	1.977	0.833	0.639	1.596
25	1.375	0.524	4.683	0.343	1.045	3.437	2.550	1.338	1.890	0.837	0.611	1.617
26	1.266	0.475	4.668	0.324	0.971	3.446	2.572	1.396	1.797	0.848	0.609	1.635
27	1.160	0.422	4.647	0.289	0.876	3.435	2.621	1.411	1.693	0.855	0.641	1.655
28	1.063	0.373	4.636	0.247	0.782	3.435	2.676	1.402	1.602	0.865	0.676	1.702
29	0.983	0.334	4.643	0.209	0.681	3.404	2.729	1.411	1.519	0.878	0.703	1.730
30	0.931	0.300	4.664	0.169	0.574	3.384	2.771	1.403	1.454	0.891	0.743	1.740
31	0.894	0.273	4.688	0.142	0.502	3.378	2.827	1.402	1.402	0.906	0.784	1.763
32	0.874	0.252	4.704	0.117	0.433	3.378	2.877	1.400	1.349	0.922	0.816	1.782
33	0.855	0.243	4.725	0.094	0.359	3.390	2.919	1.419	1.308	0.946	0.822	1.796
34	0.825	0.243	4.748	0.076	0.307	3.407	2.969	1.431	1.281	0.964	0.827	1.812
35	0.790	0.246	4.777	0.064	0.251	3.438	3.018	1.447	1.258	0.985	0.818	1.830
36	0.755	0.251	4.808	0.058	0.213	3.473	3.053	1.460	1.226	0.994	0.815	1.848
37	0.722	0.240	4.847	0.052	0.180	3.519	3.093	1.460	1.204	1.019	0.800	1.848
38	0.695	0.227	4.890	0.067	0.144	3.559	3.114	1.463	1.194	1.039	0.790	1.856
39	0.668	0.218	4.929	0.090	0.123	3.600	3.130	1.488	1.199	1.067	0.794	1.867
40	0.646	0.211	4.970	0.115	0.110	3.639	3.152	1.504	1.206	1.086	0.792	1.878
41	0.626	0.211	5.005	0.141	0.099	3.677	3.170	1.561	1.223	1.105	0.774	1.900
42	0.604	0.221	5.041	0.166	0.102	3.710	3.178	1.634	1.239	1.130	0.757	1.924
43	0.584	0.245	5.077	0.192	0.102	3.744	3.190	1.676	1.264	1.156	0.735	1.945
44	0.571	0.264	5.114	0.211	0.113	3.780	3.201	1.706	1.283	1.175	0.719	1.958
45	0.561	0.284	5.142	0.221	0.129	3.834	3.205	1.725	1.304	1.197	0.717	1.978
46	0.564	0.299	5.161	0.227	0.141	3.883	3.200	1.753	1.324	1.223	0.701	1.992
47	0.553	0.314	5.171	0.235	0.166	3.932	3.201	1.780	1.342	1.243	0.705	1.997
48	0.551	0.323	5.178	0.251	0.186	3.994	3.204	1.811	1.367	1.266	0.711	2.014
49	0.566	0.332	5.181	0.269	0.212	4.036	3.197	1.845	1.387	1.285	0.717	2.028
50	0.587	0.335	5.189	0.290	0.234	4.074	3.181	1.905	1.399	1.302	0.724	2.038
51	0.621	0.339	5.196	0.310	0.249	4.113	3.162	1.916	1.408	1.328	0.737	2.048
52	0.665	0.352	5.206	0.340	0.271	4.131	3.133	1.944	1.422	1.331	0.749	2.060

Data for Appendix 3 - Random Walk Hypothesis Tests: Equity (Malaysia - Turkey)

q	Malaysia	Philippines	Russia	Singapore	Taiwan	Thailand	Israel	Kuwait	Nigeria	Saudi Arabia	South Africa	Turkey
2	1.007	2.410	1.032	2.867	2.688	2.712	-0.649	2.306	-0.214	1.693	0.276	0.680
3	0.917	3.335	1.501	3.407	3.760	3.218	-0.102	2.523	0.523	1.388	0.649	1.114
4	1.275	3.930	1.865	3.591	3.955	3.698	0.028	2.950	0.759	1.541	0.930	1.246
5	1.411	4.062	1.789	3.556	3.963	3.702	0.054	3.083	0.817	1.598	0.995	1.198
6	1.660	4.066	1.914	3.636	3.936	3.625	0.152	3.107	0.979	1.608	1.120	1.149
7	1.838	3.973	1.975	3.660	3.724	3.619	0.355	2.943	0.947	1.570	1.153	0.997
8	2.026	3.911	2.045	3.766	3.698	3.602	0.336	3.049	0.955	1.454	1.019	0.880
9	2.219	3.841	2.014	3.769	3.604	3.477	0.251	3.169	1.003	1.377	0.984	0.783
10	2.380	3.715	2.026	3.654	3.547	3.308	0.164	3.273	1.100	1.355	0.982	0.732
11	2.474	3.554	2.029	3.575	3.444	3.200	0.105	3.402	1.151	1.393	0.940	0.642
12	2.514	3.447	2.039	3.541	3.351	3.081	0.088	3.449	1.187	1.476	0.966	0.593
13	2.530	3.324	2.079	3.493	3.294	2.958	0.070	3.374	1.133	1.529	0.918	0.555
14	2.514	3.232	2.129	3.443	3.228	2.873	0.069	3.298	1.158	1.595	0.928	0.536
15	2.478	3.123	2.161	3.341	3.158	2.794	0.106	3.233	1.176	1.656	0.964	0.530
16	2.436	3.018	2.153	3.212	3.085	2.702	0.119	3.187	1.177	1.673	0.963	0.528
17	2.415	2.934	2.177	3.113	3.046	2.632	0.102	3.141	1.191	1.705	0.942	0.552
18	2.376	2.818	2.226	3.019	2.982	2.549	0.099	3.118	1.216	1.728	0.946	0.563
19	2.314	2.696	2.272	2.934	2.909	2.433	0.048	3.150	1.218	1.749	0.906	0.555
20	2.271	2.564	2.345	2.857	2.899	2.310	0.020	3.194	1.225	1.790	0.823	0.526
21	2.228	2.414	2.393	2.778	2.877	2.181	-0.010	3.231	1.220	1.846	0.730	0.513
22	2.185	2.266	2.418	2.720	2.853	2.024	-0.027	3.221	1.237	1.908	0.626	0.476
23	2.142	2.134	2.449	2.673	2.821	1.887	-0.014	3.231	1.265	1.951	0.538	0.418
24	2.106	2.011	2.474	2.633	2.781	1.773	0.001	3.248	1.295	1.988	0.455	0.349
25	2.073	1.893	2.486	2.589	2.735	1.659	-0.029	3.227	1.335	1.991	0.392	0.297
26	2.035	1.769	2.493	2.546	2.691	1.551	-0.042	3.147	1.366	2.020	0.352	0.244
27	1.997	1.650	2.483	2.513	2.651	1.451	-0.047	3.137	1.421	2.068	0.315	0.197
28	1.942	1.533	2.466	2.467	2.596	1.360	-0.053	3.130	1.450	2.113	0.270	0.158
29	1.897	1.395	2.463	2.425	2.535	1.300	-0.059	3.132	1.472	2.161	0.219	0.124
30	1.852	1.292	2.458	2.379	2.483	1.246	-0.061	3.148	1.490	2.215	0.145	0.088
31	1.819	1.181	2.468	2.338	2.446	1.213	-0.083	3.134	1.518	2.257	0.097	0.059
32	1.803	1.080	2.485	2.311	2.407	1.187	-0.096	3.132	1.553	2.293	0.053	0.053
33	1.780	0.991	2.503	2.283	2.373	1.176	-0.118	3.138	1.587	2.332	0.019	0.041
34	1.768	0.914	2.532	2.264	2.333	1.184	-0.128	3.126	1.611	2.386	-0.009	0.041
35	1.760	0.848	2.561	2.248	2.279	1.189	-0.121	3.106	1.650	2.411	-0.032	0.038
36	1.745	0.798	2.600	2.221	2.233	1.193	-0.101	3.105	1.690	2.460	-0.056	0.032
37	1.739	0.773	2.629	2.190	2.180	1.196	-0.086	3.099	1.714	2.530	-0.087	0.024
38	1.741	0.752	2.647	2.162	2.134	1.199	-0.067	3.093	1.747	2.563	-0.108	0.020
39	1.738	0.733	2.661	2.139	2.088	1.202	-0.053	3.101	1.774	2.587	-0.114	0.023
40	1.742	0.710	2.668	2.120	2.044	1.212	-0.040	3.192	1.817	2.646	-0.109	0.021
41	1.749	0.686	2.673	2.096	2.010	1.217	-0.029	3.172	1.858	2.708	-0.099	0.022
42	1.752	0.672	2.680	2.069	1.979	1.233	-0.031	3.106	1.890	2.758	-0.085	0.031
43	1.760	0.662	2.686	2.046	1.957	1.254	-0.046	3.118	1.927	2.780	-0.081	0.047
44	1.764	0.651	2.681	2.018	1.932	1.271	-0.056	3.113	1.961	2.797	-0.070	0.061
45	1.774	0.638	2.685	2.004	1.916	1.293	-0.064	3.077	1.994	2.819	-0.066	0.086
46	1.792	0.623	2.695	1.993	1.904	1.313	-0.069	3.080	2.028	2.856	-0.060	0.110
47	1.799	0.610	2.700	1.974	1.888	1.329	-0.065	3.111	2.050	2.896	-0.055	0.136
48	1.804	0.600	2.709	1.960	1.873	1.346	-0.066	3.151	2.076	2.941	-0.051	0.161
49	1.809	0.586	2.705	1.954	1.855	1.361	-0.058	3.197	2.098	2.974	-0.039	0.180
50	1.818	0.577	2.693	1.957	1.841	1.370	-0.041	3.312	2.118	3.017	-0.027	0.197
51	1.828	0.569	2.682	1.964	1.835	1.372	-0.031	3.346	2.133	3.033	-0.016	0.212
52	1.835	0.564	2.677	1.972	1.836	1.376	-0.019	3.281	2.144	3.045	-0.012	0.221

Data for Appendix 3 - Random Walk Hypothesis Tests: Equity (United Arab Emirates - Netherlands)

q	UAE	Australia	Belgium	Canada	Czech Republic	Denmark	France	Germany	Hungary	Italy	Japan	Netherlands
2	2.654	1.758	-0.883	2.124	0.111	0.486	-2.063	-1.802	0.907	2.272	0.366	-1.023
3	3.184	2.479	-0.218	2.269	0.811	1.028	-1.359	-0.859	1.805	3.108	0.538	-0.083
4	3.252	2.818	0.171	2.636	1.553	1.175	-0.677	-0.268	2.366	4.058	1.067	0.224
5	3.129	2.757	0.229	2.788	1.563	1.194	-0.475	-0.119	2.553	4.335	1.304	0.303
6	3.113	2.648	0.235	2.811	1.720	1.196	-0.389	-0.050	2.715	4.249	1.511	0.299
7	3.055	2.399	0.308	2.794	1.930	1.122	-0.278	0.085	2.701	4.002	1.492	0.410
8	2.977	2.211	0.353	2.757	1.849	0.928	-0.234	0.180	2.596	3.607	1.594	0.431
9	2.979	2.002	0.390	2.642	1.674	0.917	-0.216	0.188	2.539	3.318	1.720	0.450
10	3.332	1.836	0.462	2.593	1.626	0.936	-0.192	0.214	2.359	3.180	1.851	0.485
11	3.556	1.681	0.479	2.530	1.504	0.945	-0.141	0.294	2.240	3.029	1.930	0.481
12	3.843	1.549	0.509	2.441	1.494	0.992	-0.018	0.381	2.176	2.901	1.980	0.533
13	4.029	1.447	0.511	2.372	1.379	0.984	0.012	0.413	2.073	2.795	1.999	0.526
14	4.211	1.370	0.508	2.351	1.313	1.021	0.078	0.464	2.034	2.840	2.002	0.516
15	4.416	1.312	0.496	2.308	1.308	1.073	0.136	0.501	1.979	2.925	2.034	0.519
16	4.561	1.272	0.481	2.274	1.281	1.123	0.164	0.533	1.949	2.980	2.088	0.513
17	4.681	1.238	0.483	2.225	1.221	1.140	0.182	0.549	1.925	2.990	2.133	0.492
18	4.857	1.205	0.492	2.211	1.158	1.155	0.180	0.575	1.895	3.049	2.188	0.474
19	5.020	1.169	0.507	2.187	1.129	1.174	0.180	0.584	1.860	3.095	2.230	0.438
20	5.187	1.133	0.511	2.145	1.080	1.194	0.161	0.586	1.822	3.081	2.240	0.393
21	5.372	1.098	0.511	2.123	1.057	1.212	0.150	0.588	1.782	3.046	2.254	0.353
22	5.546	1.069	0.511	2.106	1.051	1.235	0.134	0.595	1.756	3.008	2.271	0.313
23	5.766	1.036	0.537	2.110	1.106	1.299	0.143	0.621	1.731	2.985	2.304	0.296
24	5.972	1.007	0.559	2.118	1.108	1.380	0.162	0.642	1.719	2.981	2.353	0.280
25	6.198	0.987	0.572	2.137	1.107	1.462	0.192	0.672	1.729	2.987	2.357	0.270
26	6.427	0.955	0.597	2.154	1.113	1.551	0.226	0.707	1.745	2.979	2.371	0.268
27	6.607	0.931	0.604	2.188	1.141	1.620	0.260	0.739	1.768	2.993	2.380	0.259
28	6.769	0.909	0.623	2.211	1.168	1.681	0.297	0.776	1.797	3.020	2.373	0.260
29	6.957	0.889	0.621	2.219	1.184	1.738	0.315	0.796	1.837	3.029	2.363	0.249
30	7.150	0.873	0.616	2.235	1.199	1.768	0.322	0.792	1.867	3.034	2.340	0.222
31	7.342	0.847	0.616	2.260	1.216	1.787	0.337	0.799	1.898	3.049	2.328	0.198
32	7.532	0.827	0.621	2.257	1.232	1.815	0.350	0.805	1.942	3.072	2.318	0.182
33	7.704	0.804	0.614	2.258	1.244	1.837	0.358	0.814	1.960	3.091	2.313	0.165
34	7.877	0.787	0.626	2.263	1.255	1.868	0.379	0.831	1.993	3.103	2.307	0.163
35	8.025	0.772	0.638	2.260	1.257	1.886	0.395	0.841	2.013	3.104	2.280	0.156
36	8.172	0.751	0.641	2.255	1.253	1.894	0.407	0.847	2.035	3.105	2.268	0.143
37	8.315	0.738	0.650	2.256	1.261	1.908	0.419	0.844	2.047	3.105	2.255	0.133
38	8.447	0.726	0.656	2.257	1.256	1.919	0.426	0.838	2.062	3.096	2.245	0.125
39	8.575	0.717	0.673	2.260	1.247	1.937	0.440	0.842	2.067	3.094	2.244	0.123
40	8.687	0.706	0.690	2.271	1.251	1.956	0.453	0.838	2.077	3.109	2.243	0.124
41	8.819	0.705	0.709	2.276	1.250	1.973	0.471	0.839	2.080	3.125	2.245	0.129
42	8.954	0.702	0.728	2.279	1.249	1.987	0.486	0.843	2.092	3.147	2.254	0.140
43	9.079	0.693	0.743	2.283	1.257	2.009	0.499	0.837	2.093	3.162	2.266	0.154
44	9.212	0.683	0.765	2.286	1.267	2.036	0.518	0.838	2.094	3.182	2.278	0.176
45	9.334	0.662	0.785	2.288	1.285	2.065	0.543	0.844	2.100	3.212	2.296	0.201
46	9.437	0.646	0.806	2.291	1.304	2.100	0.569	0.847	2.110	3.230	2.318	0.229
47	9.570	0.626	0.825	2.296	1.323	2.142	0.593	0.849	2.114	3.238	2.334	0.253
48	9.685	0.611	0.848	2.294	1.355	2.180	0.617	0.858	2.126	3.251	2.350	0.275
49	9.781	0.605	0.874	2.294	1.373	2.208	0.645	0.864	2.128	3.277	2.368	0.297
50	9.870	0.604	0.898	2.299	1.387	2.224	0.670	0.871	2.127	3.308	2.382	0.314
51	9.972	0.606	0.918	2.297	1.425	2.230	0.692	0.875	2.126	3.330	2.394	0.329
52	10.077	0.606	0.943	2.282	1.458	2.234	0.719	0.880	2.120	3.360	2.407	0.345

Data for Appendix 3 - Random Walk Hypothesis Tests: Equity (Norway - United States)

q	Norway	New Zealand	Poland	Sweden	Spain	Switzerland	United Kingom	United States
2	0.939	0.554	2.280	-1.379	-1.532	-1.324	0.357	-0.896
3	0.922	0.730	2.363	-0.933	-0.390	-0.679	0.889	-0.718
4	1.115	1.027	2.578	-0.618	0.097	-0.205	1.096	-0.450
5	1.480	1.192	2.379	-0.593	0.382	0.050	1.059	-0.408
6	1.605	1.126	2.412	-0.541	0.377	0.066	0.996	-0.432
7	1.613	0.974	2.433	-0.341	0.412	0.189	0.995	-0.341
8	1.567	0.741	2.536	-0.335	0.380	0.223	0.842	-0.314
9	1.558	0.554	2.642	-0.303	0.302	0.150	0.668	-0.414
10	1.634	0.396	2.696	-0.321	0.298	0.166	0.527	-0.424
11	1.662	0.273	2.820	-0.312	0.262	0.176	0.417	-0.458
12	1.715	0.229	2.906	-0.259	0.312	0.183	0.375	-0.469
13	1.711	0.171	2.983	-0.182	0.291	0.163	0.316	-0.469
14	1.762	0.137	3.055	-0.044	0.296	0.156	0.268	-0.426
15	1.825	0.117	3.109	0.070	0.304	0.157	0.229	-0.400
16	1.883	0.164	3.126	0.137	0.296	0.157	0.188	-0.386
17	1.906	0.156	3.110	0.197	0.274	0.169	0.160	-0.378
18	1.936	0.120	3.086	0.243	0.259	0.169	0.134	-0.381
19	1.963	0.082	3.044	0.301	0.260	0.140	0.109	-0.405
20	1.966	0.025	3.036	0.345	0.246	0.114	0.074	-0.427
21	1.938	-0.065	3.051	0.375	0.238	0.093	0.012	-0.442
22	1.929	-0.170	3.082	0.409	0.230	0.093	-0.028	-0.446
23	1.926	-0.258	3.131	0.466	0.237	0.122	-0.048	-0.425
24	1.910	-0.318	3.172	0.511	0.228	0.159	-0.054	-0.384
25	1.904	-0.355	3.215	0.560	0.218	0.199	-0.059	-0.358
26	1.912	-0.380	3.252	0.614	0.200	0.237	-0.066	-0.342
27	1.930	-0.381	3.297	0.669	0.194	0.277	-0.084	-0.321
28	1.952	-0.393	3.348	0.738	0.199	0.309	-0.088	-0.302
29	1.964	-0.410	3.397	0.791	0.196	0.323	-0.096	-0.285
30	1.956	-0.431	3.449	0.826	0.187	0.321	-0.104	-0.271
31	1.957	-0.437	3.502	0.862	0.196	0.324	-0.110	-0.259
32	1.970	-0.441	3.560	0.893	0.202	0.326	-0.117	-0.267
33	1.974	-0.451	3.612	0.929	0.208	0.327	-0.131	-0.271
34	1.986	-0.450	3.651	0.968	0.228	0.336	-0.135	-0.264
35	1.999	-0.445	3.697	0.998	0.239	0.344	-0.145	-0.268
36	2.016	-0.454	3.732	1.019	0.241	0.343	-0.167	-0.269
37	2.034	-0.456	3.764	1.037	0.239	0.342	-0.187	-0.263
38	2.053	-0.440	3.796	1.057	0.239	0.343	-0.221	-0.247
39	2.076	-0.424	3.835	1.089	0.249	0.346	-0.247	-0.233
40	2.113	-0.416	3.882	1.120	0.272	0.342	-0.271	-0.217
41	2.152	-0.400	3.925	1.158	0.293	0.347	-0.292	-0.208
42	2.186	-0.377	3.975	1.186	0.320	0.350	-0.313	-0.206
43	2.224	-0.351	4.020	1.207	0.340	0.350	-0.337	-0.205
44	2.261	-0.334	4.054	1.232	0.365	0.358	-0.354	-0.193
45	2.298	-0.314	4.087	1.257	0.390	0.373	-0.361	-0.179
46	2.333	-0.305	4.110	1.286	0.414	0.387	-0.363	-0.164
47	2.363	-0.305	4.131	1.318	0.436	0.402	-0.365	-0.145
48	2.392	-0.306	4.149	1.338	0.461	0.425	-0.364	-0.135
49	2.418	-0.310	4.159	1.355	0.486	0.446	-0.360	-0.123
50	2.436	-0.316	4.165	1.379	0.505	0.469	-0.356	-0.101
51	2.453	-0.321	4.164	1.401	0.522	0.487	-0.358	-0.082
52	2.468	-0.326	4.163	1.425	0.545	0.505	-0.363	-0.070

Data for Appendix 3 - Random Walk Hypothesis Tests: Commodity (Crude Oil - Platinum)

q	Crude Oil	Natural Gas	Heating Oil	Gasoline	Aluminum	Copper	Gold	Lead	Nickel	Silver	Tin	Platinum
2	-2.035	-1.265	-1.996	-1.519	-0.044	-0.950	0.560	1.188	-0.230	0.104	0.630	1.040
3	-2.266	-1.302	-2.288	-2.217	0.966	-0.683	0.788	1.555	-0.657	-0.122	0.945	0.522
4	-1.969	-0.920	-2.035	-2.081	1.398	-0.411	0.831	1.422	-0.395	0.060	1.066	0.493
5	-1.312	-0.744	-2.015	-1.871	1.669	-0.262	0.940	1.287	-0.266	0.029	0.735	0.714
6	-1.066	-0.636	-2.042	-1.582	1.586	-0.264	0.872	1.181	-0.279	-0.126	0.397	1.019
7	-0.956	-0.663	-2.054	-1.595	1.554	-0.525	0.761	1.268	-0.248	-0.365	0.066	1.178
8	-0.956	-0.820	-2.055	-1.704	1.465	-0.665	0.674	1.411	-0.181	-0.530	-0.137	1.110
9	-0.835	-0.842	-1.943	-1.714	1.423	-0.685	0.600	1.690	-0.071	-0.610	-0.238	1.239
10	-0.792	-0.935	-1.992	-1.688	1.327	-0.725	0.589	1.742	0.086	-0.626	-0.294	1.346
11	-0.724	-0.987	-2.032	-1.654	1.249	-0.842	0.442	1.588	0.188	-0.745	-0.357	1.390
12	-0.773	-1.070	-2.029	-1.666	1.224	-0.917	0.361	1.484	0.244	-0.800	-0.393	1.516
13	-0.753	-1.144	-2.033	-1.659	1.130	-0.919	0.255	1.451	0.385	-0.813	-0.419	1.587
14	-0.757	-1.192	-2.025	-1.698	1.091	-0.910	0.193	1.425	0.468	-0.815	-0.438	1.651
15	-0.703	-1.223	-1.976	-1.660	1.047	-0.856	0.158	1.441	0.567	-0.726	-0.423	1.717
16	-0.683	-1.215	-1.967	-1.598	1.003	-0.802	0.114	1.455	0.643	-0.647	-0.398	1.773
17	-0.674	-1.216	-1.964	-1.560	0.975	-0.719	0.098	1.444	0.762	-0.517	-0.380	1.839
18	-0.660	-1.234	-1.959	-1.544	0.946	-0.605	0.127	1.391	0.857	-0.374	-0.319	1.875
19	-0.640	-1.186	-1.939	-1.466	0.900	-0.547	0.121	1.364	0.924	-0.281	-0.287	1.868
20	-0.664	-1.160	-1.912	-1.413	0.853	-0.527	0.096	1.371	0.971	-0.236	-0.232	1.890
21	-0.684	-1.157	-1.900	-1.412	0.811	-0.492	0.079	1.361	1.007	-0.207	-0.202	1.902
22	-0.698	-1.144	-1.886	-1.406	0.767	-0.501	0.023	1.334	1.047	-0.200	-0.226	1.898
23	-0.689	-1.140	-1.875	-1.393	0.742	-0.490	-0.033	1.310	1.110	-0.209	-0.250	1.913
24	-0.705	-1.147	-1.874	-1.371	0.722	-0.469	-0.088	1.287	1.171	-0.210	-0.290	1.941
25	-0.727	-1.178	-1.864	-1.339	0.676	-0.447	-0.146	1.274	1.237	-0.232	-0.324	1.965
26	-0.754	-1.212	-1.854	-1.324	0.645	-0.409	-0.176	1.271	1.295	-0.237	-0.327	2.012
27	-0.767	-1.231	-1.841	-1.315	0.625	-0.385	-0.218	1.297	1.337	-0.275	-0.350	2.050
28	-0.792	-1.261	-1.819	-1.309	0.591	-0.354	-0.287	1.330	1.372	-0.315	-0.372	2.087
29	-0.829	-1.282	-1.813	-1.307	0.560	-0.346	-0.351	1.355	1.389	-0.337	-0.408	2.115
30	-0.862	-1.296	-1.801	-1.289	0.525	-0.357	-0.404	1.366	1.373	-0.362	-0.453	2.137
31	-0.889	-1.315	-1.792	-1.264	0.488	-0.367	-0.443	1.370	1.370	-0.365	-0.483	2.168
32	-0.899	-1.325	-1.791	-1.243	0.458	-0.371	-0.460	1.376	1.352	-0.368	-0.495	2.194
33	-0.913	-1.333	-1.791	-1.217	0.431	-0.394	-0.463	1.384	1.359	-0.365	-0.496	2.204
34	-0.930	-1.334	-1.790	-1.184	0.406	-0.409	-0.461	1.408	1.383	-0.368	-0.489	2.240
35	-0.945	-1.336	-1.789	-1.161	0.373	-0.441	-0.449	1.432	1.395	-0.369	-0.480	2.280
36	-0.953	-1.344	-1.785	-1.149	0.339	-0.475	-0.445	1.458	1.409	-0.388	-0.479	2.325
37	-0.961	-1.342	-1.790	-1.130	0.301	-0.505	-0.441	1.472	1.415	-0.421	-0.478	2.364
38	-0.972	-1.344	-1.795	-1.109	0.252	-0.535	-0.431	1.471	1.419	-0.446	-0.478	2.372
39	-0.974	-1.361	-1.806	-1.100	0.197	-0.554	-0.420	1.456	1.402	-0.469	-0.478	2.381
40	-0.975	-1.377	-1.824	-1.083	0.151	-0.559	-0.409	1.436	1.406	-0.478	-0.477	2.383
41	-0.972	-1.390	-1.842	-1.074	0.104	-0.562	-0.389	1.413	1.414	-0.501	-0.486	2.394
42	-0.967	-1.406	-1.855	-1.056	0.060	-0.575	-0.362	1.393	1.429	-0.509	-0.479	2.406
43	-0.962	-1.432	-1.857	-1.048	0.015	-0.580	-0.345	1.352	1.446	-0.509	-0.453	2.394
44	-0.960	-1.451	-1.869	-1.043	-0.037	-0.592	-0.335	1.294	1.453	-0.512	-0.429	2.374
45	-0.951	-1.455	-1.871	-1.019	-0.075	-0.606	-0.325	1.238	1.460	-0.519	-0.411	2.363
46	-0.940	-1.459	-1.849	-0.998	-0.103	-0.624	-0.301	1.183	1.476	-0.521	-0.383	2.362
47	-0.931	-1.448	-1.846	-0.972	-0.128	-0.640	-0.276	1.132	1.500	-0.520	-0.358	2.364
48	-0.914	-1.431	-1.838	-0.939	-0.139	-0.647	-0.251	1.094	1.524	-0.525	-0.343	2.366
49	-0.896	-1.406	-1.820	-0.905	-0.124	-0.654	-0.213	1.056	1.566	-0.523	-0.327	2.380
50	-0.876	-1.371	-1.806	-0.856	-0.098	-0.664	-0.172	1.032	1.604	-0.516	-0.306	2.409
51	-0.862	-1.344	-1.794	-0.835	-0.066	-0.669	-0.139	0.997	1.635	-0.500	-0.280	2.440
52	-0.848	-1.315	-1.772	-0.810	-0.031	-0.683	-0.109	0.960	1.673	-0.501	-0.262	2.468

Data for Appendix 3 - Random Walk Hypothesis Tests: Commodity (Silver - Sugar)

q	Zinc	Cattle	Cocoa	Coffee	Corn	Cotton	Rubber	Soybeans	Sugar	Wheat
2	-0.638	1.630	5.417	0.083	-1.853	1.565	-0.589	-0.971	-1.178	1.785
3	-0.579	1.978	5.864	0.189	-1.724	1.818	-1.706	-1.518	-1.036	1.110
4	-0.314	1.871	5.905	0.688	-1.564	1.863	-1.732	-1.853	-1.081	0.864
5	-0.197	1.527	5.551	1.030	-1.683	1.674	-1.452	-1.901	-0.993	0.863
6	-0.323	1.115	5.035	1.350	-1.724	1.547	-1.388	-1.906	-0.932	0.960
7	-0.491	0.631	4.534	1.317	-1.691	1.465	-1.515	-1.825	-0.962	0.984
8	-0.562	0.298	4.175	1.425	-1.711	1.570	-1.284	-1.822	-0.954	1.038
9	-0.631	-0.084	3.845	1.527	-1.686	1.688	-1.117	-1.814	-1.012	1.032
10	-0.717	-0.225	3.575	1.538	-1.698	1.862	-1.113	-1.976	-0.988	1.020
11	-0.906	-0.334	3.399	1.582	-1.653	2.029	-1.079	-2.019	-0.943	1.112
12	-1.027	-0.422	3.289	1.600	-1.597	2.168	-1.088	-2.004	-0.827	1.162
13	-1.129	-0.456	3.237	1.634	-1.575	2.252	-1.119	-1.975	-0.749	1.171
14	-1.212	-0.497	3.196	1.652	-1.546	2.288	-1.197	-2.004	-0.587	1.185
15	-1.197	-0.496	3.124	1.714	-1.532	2.333	-1.212	-1.991	-0.447	1.290
16	-1.178	-0.471	3.025	1.746	-1.515	2.396	-1.190	-1.940	-0.337	1.359
17	-1.138	-0.428	2.916	1.776	-1.477	2.481	-1.203	-1.876	-0.282	1.404
18	-1.047	-0.340	2.797	1.819	-1.463	2.577	-1.233	-1.827	-0.228	1.465
19	-0.991	-0.257	2.676	1.803	-1.457	2.644	-1.283	-1.743	-0.176	1.543
20	-0.965	-0.177	2.549	1.749	-1.439	2.697	-1.320	-1.693	-0.171	1.616
21	-0.935	-0.122	2.432	1.713	-1.417	2.735	-1.336	-1.633	-0.172	1.688
22	-0.905	-0.098	2.333	1.671	-1.379	2.769	-1.368	-1.619	-0.212	1.749
23	-0.899	-0.084	2.261	1.604	-1.322	2.798	-1.383	-1.582	-0.231	1.778
24	-0.903	-0.047	2.189	1.525	-1.283	2.802	-1.406	-1.551	-0.255	1.824
25	-0.942	-0.032	2.099	1.430	-1.252	2.789	-1.421	-1.565	-0.264	1.886
26	-0.971	-0.018	2.012	1.352	-1.212	2.767	-1.427	-1.571	-0.275	1.934
27	-1.018	0.001	1.905	1.259	-1.174	2.728	-1.441	-1.557	-0.285	1.978
28	-1.053	0.033	1.778	1.173	-1.153	2.698	-1.451	-1.572	-0.313	1.987
29	-1.079	0.061	1.647	1.103	-1.135	2.650	-1.463	-1.579	-0.350	1.984
30	-1.102	0.105	1.541	1.042	-1.125	2.598	-1.458	-1.571	-0.385	1.979
31	-1.112	0.162	1.455	0.990	-1.124	2.545	-1.449	-1.562	-0.427	2.002
32	-1.120	0.217	1.389	0.923	-1.127	2.474	-1.437	-1.556	-0.470	2.057
33	-1.103	0.270	1.349	0.857	-1.127	2.401	-1.425	-1.516	-0.507	2.104
34	-1.088	0.332	1.316	0.787	-1.123	2.353	-1.428	-1.474	-0.556	2.137
35	-1.053	0.403	1.284	0.740	-1.104	2.319	-1.418	-1.432	-0.600	2.161
36	-1.035	0.467	1.257	0.674	-1.090	2.274	-1.409	-1.411	-0.634	2.198
37	-1.026	0.517	1.242	0.618	-1.073	2.228	-1.406	-1.397	-0.666	2.214
38	-1.021	0.554	1.234	0.563	-1.049	2.206	-1.393	-1.390	-0.685	2.210
39	-1.011	0.548	1.223	0.505	-1.042	2.191	-1.379	-1.376	-0.704	2.205
40	-0.998	0.527	1.205	0.453	-1.051	2.165	-1.362	-1.383	-0.728	2.198
41	-1.002	0.497	1.182	0.404	-1.056	2.127	-1.347	-1.383	-0.750	2.204
42	-0.997	0.449	1.143	0.355	-1.061	2.087	-1.337	-1.392	-0.765	2.201
43	-0.989	0.408	1.106	0.324	-1.056	2.040	-1.333	-1.403	-0.781	2.190
44	-0.992	0.366	1.069	0.285	-1.050	1.987	-1.332	-1.414	-0.798	2.183
45	-1.000	0.326	1.033	0.256	-1.038	1.934	-1.322	-1.413	-0.819	2.170
46	-1.001	0.305	0.992	0.231	-1.031	1.882	-1.310	-1.412	-0.826	2.153
47	-0.996	0.294	0.939	0.215	-1.034	1.842	-1.304	-1.399	-0.821	2.138
48	-0.995	0.273	0.878	0.200	-1.045	1.808	-1.295	-1.387	-0.810	2.121
49	-0.981	0.264	0.822	0.179	-1.050	1.776	-1.289	-1.359	-0.800	2.106
50	-0.963	0.258	0.780	0.172	-1.052	1.758	-1.284	-1.329	-0.786	2.101
51	-0.940	0.257	0.742	0.168	-1.046	1.745	-1.277	-1.305	-0.765	2.081
52	-0.938	0.262	0.710	0.170	-1.042	1.729	-1.274	-1.287	-0.742	2.064

Data for Appendix 3 - Random Walk Hypothesis Tests: USD Bilateral Exchange Rate (Argentina - Russia)

q	Argentina	Brazil	Chile	Colombia	Mexico	Peru	Indonesia	India	South Korea	Malaysia	Philippines	Russia
2	0.190	-1.755	1.564	0.634	-2.025	-0.109	0.430	-2.637	-0.318	-0.724	-1.266	-1.290
3	0.438	-1.221	1.615	1.579	-1.018	0.539	0.373	-1.955	0.767	-0.515	-0.640	-0.727
4	0.856	-0.794	1.989	2.104	-0.810	0.738	0.483	-1.620	0.974	-0.503	-0.504	-0.389
5	0.994	-0.636	2.137	2.268	-0.805	1.066	0.607	-1.218	1.214	-0.238	-0.252	-0.323
6	1.111	-0.459	2.463	2.235	-0.766	1.056	0.657	-0.946	1.312	0.065	-0.147	-0.160
7	1.113	-0.747	2.478	2.118	-0.758	1.048	0.803	-0.649	1.418	0.041	-0.002	-0.148
8	0.961	-0.650	2.510	2.043	-0.776	0.960	0.873	-0.485	1.464	-0.024	0.125	-0.058
9	1.006	-0.512	2.555	1.900	-0.877	0.886	0.901	-0.329	1.513	0.005	0.211	0.055
10	1.137	-0.477	2.403	1.931	-0.897	0.829	0.900	-0.259	1.514	0.030	0.329	0.169
11	1.105	-0.395	2.275	1.970	-0.952	0.839	0.915	-0.103	1.520	-0.060	0.399	0.210
12	1.171	-0.334	2.234	2.009	-0.944	0.799	0.875	-0.025	1.529	-0.116	0.525	0.150
13	1.146	-0.266	2.195	2.023	-1.014	0.797	0.845	0.060	1.512	-0.123	0.601	0.189
14	1.144	-0.225	2.234	2.030	-1.022	0.820	0.809	0.139	1.493	-0.095	0.674	0.177
15	1.110	-0.185	2.276	2.016	-1.059	0.838	0.760	0.216	1.463	-0.125	0.716	0.175
16	1.028	-0.134	2.301	1.955	-1.065	0.816	0.726	0.327	1.420	-0.200	0.743	0.199
17	0.920	-0.103	2.327	1.913	-1.083	0.784	0.719	0.409	1.379	-0.183	0.773	0.248
18	0.905	-0.053	2.352	1.892	-1.092	0.761	0.679	0.490	1.343	-0.155	0.782	0.295
19	0.869	0.014	2.402	1.884	-1.054	0.738	0.628	0.543	1.301	-0.145	0.777	0.315
20	0.770	0.081	2.442	1.905	-1.036	0.724	0.576	0.599	1.266	-0.154	0.753	0.340
21	0.739	0.135	2.479	1.948	-0.974	0.689	0.509	0.655	1.232	-0.166	0.711	0.390
22	0.717	0.180	2.512	1.994	-0.936	0.679	0.482	0.699	1.198	-0.201	0.666	0.437
23	0.710	0.230	2.517	2.032	-0.899	0.672	0.388	0.741	1.178	-0.237	0.641	0.488
24	0.680	0.272	2.496	2.080	-0.868	0.672	0.292	0.790	1.158	-0.283	0.610	0.546
25	0.643	0.309	2.455	2.113	-0.849	0.646	0.249	0.831	1.141	-0.309	0.585	0.601
26	0.617	0.347	2.434	2.101	-0.859	0.629	0.246	0.857	1.133	-0.362	0.554	0.648
27	0.601	0.387	2.388	2.090	-0.852	0.582	0.224	0.888	1.121	-0.455	0.523	0.680
28	0.600	0.413	2.338	2.086	-0.854	0.524	0.201	0.920	1.111	-0.549	0.498	0.697
29	0.595	0.421	2.279	2.083	-0.847	0.463	0.181	0.944	1.101	-0.622	0.491	0.723
30	0.586	0.420	2.207	2.101	-0.851	0.380	0.158	0.966	1.086	-0.696	0.500	0.743
31	0.576	0.415	2.149	2.131	-0.844	0.307	0.139	0.987	1.072	-0.743	0.506	0.763
32	0.561	0.412	2.103	2.153	-0.822	0.238	0.143	1.007	1.052	-0.807	0.510	0.779
33	0.542	0.407	2.061	2.193	-0.814	0.189	0.139	1.031	1.032	-0.851	0.528	0.803
34	0.502	0.403	2.035	2.235	-0.804	0.115	0.137	1.066	1.014	-0.884	0.551	0.823
35	0.469	0.399	2.016	2.269	-0.785	0.075	0.146	1.079	0.994	-0.935	0.571	0.828
36	0.449	0.396	1.988	2.306	-0.760	0.043	0.156	1.125	0.980	-0.981	0.598	0.839
37	0.427	0.392	1.955	2.329	-0.739	-0.016	0.167	1.167	0.970	-1.041	0.611	0.850
38	0.410	0.382	1.938	2.359	-0.706	-0.050	0.177	1.220	0.961	-1.061	0.635	0.865
39	0.401	0.377	1.923	2.384	-0.671	-0.060	0.157	1.275	0.960	-1.088	0.654	0.875
40	0.380	0.370	1.907	2.415	-0.624	-0.064	0.129	1.318	0.962	-1.110	0.684	0.876
41	0.367	0.366	1.882	2.441	-0.584	-0.059	0.113	1.370	0.967	-1.116	0.711	0.895
42	0.354	0.362	1.863	2.471	-0.539	-0.054	0.089	1.420	0.976	-1.092	0.746	0.904
43	0.328	0.358	1.851	2.509	-0.506	-0.042	0.046	1.461	0.986	-1.064	0.763	0.922
44	0.307	0.353	1.830	2.542	-0.477	-0.025	-0.008	1.504	0.994	-1.044	0.781	0.939
45	0.282	0.350	1.813	2.575	-0.452	-0.005	-0.058	1.539	1.001	-1.029	0.792	0.957
46	0.260	0.344	1.799	2.604	-0.433	-0.003	-0.107	1.575	1.008	-1.011	0.806	0.979
47	0.243	0.339	1.785	2.628	-0.418	-0.007	-0.161	1.603	1.014	-0.996	0.821	0.997
48	0.227	0.331	1.783	2.651	-0.403	-0.026	-0.207	1.631	1.021	-0.967	0.838	1.014
49	0.216	0.324	1.783	2.662	-0.384	-0.036	-0.253	1.664	1.027	-0.944	0.857	1.027
50	0.216	0.315	1.786	2.672	-0.373	-0.042	-0.290	1.698	1.031	-0.907	0.876	1.042
51	0.198	0.305	1.808	2.681	-0.360	-0.054	-0.322	1.731	1.032	-0.884	0.894	1.056
52	0.189	0.295	1.828	2.688	-0.350	-0.064	-0.356	1.757	1.032	-0.889	0.914	1.061

Data for Appendix 3 - Random Walk Hypothesis Tests: USD Bilateral Exchange Rate (Singapore - Norway)

q	Singapore	Thailand	Israel	Nigeria	South Africa	Turkey	Australia	Canada	Czech Republic	Euro Area	Japan	Norway
2	0.570	0.004	1.172	-2.602	-0.183	-0.252	-0.757	0.549	1.011	0.214	2.472	0.944
3	1.329	0.326	2.052	-2.702	0.182	0.330	-0.189	0.842	1.276	0.016	3.550	1.332
4	1.484	0.684	2.387	-2.630	0.462	0.983	-0.426	0.571	0.909	0.097	3.954	1.304
5	1.691	0.872	2.572	-2.628	0.708	0.929	-0.181	0.550	0.807	0.288	4.218	1.305
6	1.808	0.956	2.652	-2.552	0.806	0.739	-0.140	0.443	0.962	0.588	4.216	1.491
7	1.750	1.153	2.619	-2.450	0.797	0.572	-0.023	0.418	1.009	0.681	4.111	1.413
8	1.739	1.220	2.580	-2.361	0.862	0.384	0.027	0.471	1.084	0.721	3.998	1.359
9	1.716	1.147	2.635	-2.239	0.962	0.262	0.035	0.455	1.043	0.700	3.913	1.410
10	1.634	1.179	2.585	-2.246	0.957	0.285	0.027	0.470	1.036	0.699	3.843	1.437
11	1.575	1.080	2.498	-2.246	0.958	0.331	-0.013	0.422	1.002	0.678	3.681	1.494
12	1.545	1.034	2.356	-2.195	1.003	0.401	-0.062	0.353	0.992	0.668	3.604	1.499
13	1.491	0.968	2.178	-2.195	1.055	0.435	-0.099	0.300	0.989	0.631	3.584	1.486
14	1.439	0.970	2.046	-2.125	1.100	0.459	-0.095	0.302	1.028	0.591	3.617	1.481
15	1.363	0.935	1.926	-2.114	1.164	0.517	-0.063	0.313	1.055	0.535	3.627	1.472
16	1.313	0.874	1.832	-2.113	1.177	0.532	-0.067	0.278	1.058	0.446	3.623	1.454
17	1.260	0.806	1.751	-2.098	1.178	0.516	-0.043	0.318	1.055	0.396	3.632	1.441
18	1.184	0.716	1.705	-2.111	1.194	0.512	-0.035	0.345	1.033	0.362	3.639	1.428
19	1.115	0.668	1.637	-2.115	1.214	0.499	-0.034	0.391	1.014	0.344	3.637	1.395
20	1.024	0.633	1.594	-2.120	1.212	0.484	-0.075	0.416	0.974	0.303	3.610	1.344
21	0.950	0.597	1.568	-2.111	1.192	0.471	-0.095	0.452	0.958	0.284	3.609	1.325
22	0.869	0.530	1.552	-2.088	1.165	0.431	-0.122	0.496	0.963	0.258	3.596	1.323
23	0.807	0.491	1.550	-2.070	1.118	0.392	-0.129	0.542	0.979	0.261	3.557	1.329
24	0.734	0.401	1.516	-2.053	1.071	0.342	-0.143	0.594	0.976	0.237	3.504	1.307
25	0.656	0.342	1.494	-2.051	1.050	0.251	-0.153	0.629	0.968	0.206	3.459	1.278
26	0.617	0.275	1.475	-2.040	1.018	0.190	-0.162	0.644	0.978	0.203	3.420	1.253
27	0.560	0.143	1.454	-2.029	0.970	0.148	-0.166	0.646	0.972	0.170	3.363	1.211
28	0.513	0.046	1.426	-2.015	0.929	0.120	-0.154	0.656	0.979	0.153	3.282	1.176
29	0.471	-0.063	1.396	-2.001	0.873	0.088	-0.153	0.659	0.981	0.144	3.191	1.145
30	0.437	-0.185	1.375	-1.987	0.817	0.045	-0.145	0.658	0.975	0.151	3.098	1.120
31	0.417	-0.258	1.327	-1.975	0.768	-0.007	-0.121	0.665	0.967	0.175	3.020	1.102
32	0.397	-0.279	1.267	-1.959	0.745	-0.064	-0.100	0.650	0.963	0.188	2.944	1.075
33	0.378	-0.304	1.226	-1.951	0.732	-0.128	-0.081	0.642	0.964	0.198	2.881	1.061
34	0.369	-0.302	1.179	-1.932	0.719	-0.200	-0.058	0.642	0.983	0.231	2.821	1.051
35	0.361	-0.311	1.146	-1.921	0.718	-0.257	-0.031	0.653	0.992	0.271	2.758	1.050
36	0.365	-0.316	1.143	-1.894	0.729	-0.315	0.002	0.661	0.998	0.300	2.712	1.054
37	0.380	-0.300	1.141	-1.869	0.736	-0.361	0.030	0.676	1.000	0.330	2.678	1.056
38	0.385	-0.272	1.157	-1.853	0.743	-0.389	0.071	0.685	1.003	0.366	2.656	1.050
39	0.396	-0.248	1.173	-1.830	0.751	-0.410	0.116	0.693	1.010	0.407	2.651	1.040
40	0.405	-0.233	1.201	-1.805	0.766	-0.421	0.155	0.702	1.013	0.452	2.641	1.032
41	0.419	-0.219	1.238	-1.767	0.782	-0.420	0.190	0.720	1.020	0.500	2.635	1.027
42	0.436	-0.201	1.293	-1.740	0.804	-0.423	0.220	0.728	1.020	0.546	2.638	1.028
43	0.452	-0.188	1.352	-1.718	0.822	-0.436	0.250	0.729	1.027	0.587	2.636	1.032
44	0.473	-0.177	1.407	-1.698	0.845	-0.450	0.278	0.739	1.033	0.626	2.630	1.036
45	0.487	-0.168	1.443	-1.680	0.865	-0.462	0.306	0.749	1.034	0.663	2.619	1.036
46	0.505	-0.162	1.477	-1.662	0.885	-0.457	0.333	0.762	1.042	0.707	2.612	1.044
47	0.524	-0.157	1.502	-1.637	0.898	-0.449	0.367	0.784	1.051	0.746	2.611	1.051
48	0.544	-0.163	1.509	-1.625	0.912	-0.436	0.397	0.804	1.062	0.781	2.616	1.056
49	0.559	-0.179	1.519	-1.609	0.938	-0.425	0.427	0.824	1.075	0.824	2.623	1.071
50	0.571	-0.184	1.524	-1.594	0.968	-0.414	0.457	0.845	1.091	0.861	2.639	1.082
51	0.584	-0.192	1.531	-1.584	0.997	-0.409	0.485	0.871	1.111	0.904	2.662	1.096
52	0.594	-0.199	1.543	-1.569	1.025	-0.414	0.512	0.890	1.123	0.937	2.685	1.104

Data for Appendix 3 - Random Walk Hypothesis Tests: USD Bilateral Exchange Rate (New Zealand - United Kingdom)

q	New Zealand	Poland	Sweden	Switzerland	United Kingdom
2	0.039	-0.166	0.252	1.933	1.696
3	0.806	-0.034	1.072	2.194	1.620
4	0.898	0.352	1.369	2.187	1.452
5	1.077	0.454	1.855	2.372	1.564
6	1.200	0.654	2.304	2.513	1.783
7	1.293	0.777	2.576	2.492	1.902
8	1.373	0.777	2.692	2.348	2.008
9	1.354	0.727	2.680	2.340	2.115
10	1.315	0.646	2.691	2.369	2.119
11	1.222	0.583	2.714	2.346	2.047
12	1.211	0.518	2.719	2.310	1.997
13	1.264	0.461	2.702	2.313	1.942
14	1.339	0.428	2.714	2.326	1.903
15	1.404	0.386	2.716	2.325	1.874
16	1.458	0.366	2.689	2.329	1.900
17	1.534	0.348	2.708	2.323	1.916
18	1.597	0.368	2.741	2.293	1.899
19	1.611	0.382	2.742	2.265	1.880
20	1.597	0.347	2.730	2.219	1.879
21	1.581	0.320	2.749	2.199	1.905
22	1.570	0.310	2.774	2.176	1.925
23	1.561	0.308	2.793	2.138	1.936
24	1.564	0.300	2.806	2.106	1.930
25	1.568	0.308	2.810	2.070	1.911
26	1.582	0.323	2.799	2.039	1.883
27	1.585	0.338	2.764	1.994	1.865
28	1.602	0.361	2.732	1.937	1.851
29	1.610	0.386	2.703	1.879	1.839
30	1.631	0.409	2.667	1.836	1.821
31	1.675	0.449	2.644	1.812	1.794
32	1.721	0.481	2.610	1.779	1.762
33	1.763	0.515	2.589	1.747	1.722
34	1.790	0.564	2.584	1.727	1.691
35	1.833	0.622	2.586	1.712	1.667
36	1.884	0.664	2.596	1.701	1.647
37	1.931	0.691	2.617	1.687	1.636
38	1.994	0.705	2.635	1.680	1.629
39	2.062	0.714	2.644	1.671	1.617
40	2.123	0.726	2.662	1.668	1.606
41	2.191	0.736	2.680	1.668	1.605
42	2.245	0.741	2.710	1.674	1.603
43	2.299	0.741	2.739	1.682	1.601
44	2.343	0.739	2.771	1.697	1.599
45	2.392	0.739	2.797	1.712	1.591
46	2.436	0.737	2.835	1.731	1.582
47	2.496	0.736	2.868	1.740	1.571
48	2.546	0.728	2.896	1.748	1.560
49	2.581	0.720	2.933	1.764	1.564
50	2.619	0.724	2.962	1.775	1.572
51	2.653	0.731	2.991	1.787	1.582
52	2.684	0.741	3.014	1.797	1.592

Appendix 4.  Long Memory Hypothesis Tests

For all figures in Appendix 4, the 95% critical values are 0.809 (lower) and 1.862 (upper), and the 99% critical values are 0.721 (lower) and 2.098 (upper). The critical values do not depend upon xi.

Data for Appendix 4 - Long Memory Hypothesis Tests: Equity (Argentina - South Korea)

xi	Argentina	Brazil	Chile	Colombia	Mexico	Peru	Venezuela	China	Hong Kong	Indonesia	India	South Korea
0	1.511	1.330	2.294	1.744	1.411	2.614	1.394	1.376	1.392	1.878	1.494	1.865
1	1.506	1.307	2.148	1.675	1.398	2.511	1.348	1.378	1.397	1.764	1.479	1.860
2	1.504	1.303	2.067	1.641	1.399	2.468	1.330	1.378	1.386	1.708	1.471	1.861
3	1.501	1.306	2.008	1.621	1.394	2.441	1.308	1.373	1.369	1.679	1.460	1.864
4	1.503	1.311	1.960	1.612	1.389	2.420	1.281	1.372	1.363	1.667	1.454	1.878
5	1.498	1.319	1.920	1.618	1.384	2.405	1.263	1.374	1.355	1.665	1.453	1.892
6	1.481	1.328	1.888	1.626	1.381	2.393	1.252	1.380	1.347	1.667	1.455	1.901
7	1.467	1.331	1.860	1.632	1.379	2.379	1.242	1.386	1.340	1.669	1.459	1.905
8	1.457	1.329	1.836	1.638	1.380	2.359	1.230	1.394	1.335	1.670	1.459	1.911
9	1.445	1.319	1.813	1.641	1.377	2.334	1.215	1.400	1.327	1.668	1.457	1.911
10	1.437	1.308	1.793	1.642	1.375	2.308	1.206	1.407	1.319	1.664	1.456	1.906
11	1.432	1.300	1.775	1.646	1.371	2.287	1.199	1.411	1.312	1.657	1.457	1.903
12	1.427	1.295	1.758	1.645	1.366	2.266	1.192	1.415	1.303	1.648	1.457	1.902
13	1.422	1.291	1.741	1.639	1.361	2.243	1.186	1.418	1.293	1.638	1.455	1.898
14	1.417	1.286	1.723	1.631	1.354	2.222	1.181	1.416	1.286	1.631	1.454	1.892
15	1.414	1.283	1.707	1.622	1.347	2.204	1.178	1.415	1.281	1.624	1.452	1.886
16	1.412	1.282	1.693	1.613	1.342	2.188	1.176	1.412	1.276	1.616	1.449	1.880
17	1.411	1.281	1.681	1.603	1.338	2.172	1.173	1.409	1.272	1.608	1.446	1.874
18	1.411	1.280	1.670	1.591	1.334	2.154	1.171	1.409	1.269	1.600	1.443	1.868
19	1.412	1.277	1.659	1.582	1.329	2.138	1.170	1.407	1.266	1.592	1.440	1.864
20	1.411	1.273	1.649	1.575	1.324	2.124	1.170	1.409	1.265	1.584	1.437	1.861
21	1.409	1.270	1.640	1.570	1.320	2.111	1.169	1.409	1.264	1.577	1.435	1.855
22	1.408	1.265	1.631	1.565	1.315	2.101	1.168	1.410	1.263	1.570	1.433	1.850
23	1.409	1.262	1.623	1.558	1.312	2.091	1.167	1.412	1.261	1.563	1.430	1.843
24	1.411	1.259	1.616	1.552	1.309	2.082	1.167	1.416	1.259	1.555	1.427	1.836
25	1.416	1.256	1.609	1.547	1.306	2.074	1.166	1.418	1.258	1.548	1.424	1.831
26	1.419	1.254	1.602	1.543	1.303	2.066	1.166	1.422	1.257	1.541	1.422	1.826
27	1.420	1.252	1.596	1.539	1.301	2.057	1.165	1.425	1.256	1.535	1.420	1.822
28	1.423	1.251	1.590	1.536	1.299	2.046	1.163	1.427	1.255	1.529	1.418	1.817
29	1.427	1.249	1.584	1.532	1.298	2.036	1.161	1.427	1.255	1.524	1.416	1.811
30	1.431	1.248	1.578	1.527	1.298	2.028	1.160	1.428	1.254	1.519	1.415	1.805
31	1.433	1.247	1.573	1.522	1.298	2.021	1.159	1.428	1.254	1.515	1.414	1.801
32	1.434	1.245	1.568	1.520	1.297	2.016	1.157	1.428	1.253	1.511	1.413	1.797
33	1.436	1.245	1.564	1.518	1.296	2.012	1.157	1.426	1.253	1.507	1.413	1.794
34	1.438	1.244	1.559	1.516	1.295	2.008	1.156	1.424	1.252	1.504	1.413	1.791
35	1.438	1.245	1.554	1.515	1.295	2.004	1.157	1.421	1.251	1.501	1.413	1.787
36	1.439	1.246	1.550	1.515	1.294	2.001	1.157	1.418	1.251	1.499	1.412	1.783
37	1.441	1.247	1.547	1.514	1.294	1.998	1.157	1.417	1.250	1.496	1.412	1.778
38	1.443	1.248	1.545	1.513	1.293	1.995	1.158	1.416	1.250	1.494	1.411	1.774
39	1.444	1.250	1.542	1.513	1.293	1.992	1.158	1.414	1.250	1.493	1.410	1.770
40	1.445	1.252	1.539	1.514	1.293	1.990	1.158	1.412	1.250	1.491	1.409	1.766
41	1.444	1.253	1.535	1.514	1.292	1.987	1.158	1.409	1.250	1.489	1.407	1.762
42	1.444	1.253	1.532	1.514	1.291	1.984	1.157	1.406	1.250	1.486	1.404	1.758
43	1.445	1.253	1.528	1.513	1.290	1.981	1.157	1.404	1.249	1.484	1.402	1.755
44	1.445	1.252	1.525	1.512	1.290	1.978	1.156	1.402	1.249	1.482	1.399	1.752
45	1.445	1.252	1.522	1.512	1.289	1.975	1.156	1.399	1.248	1.480	1.396	1.749
46	1.445	1.252	1.519	1.510	1.289	1.972	1.155	1.397	1.247	1.479	1.393	1.746
47	1.446	1.253	1.516	1.510	1.289	1.969	1.154	1.395	1.246	1.478	1.390	1.743
48	1.447	1.253	1.514	1.509	1.290	1.965	1.154	1.394	1.245	1.477	1.387	1.739
49	1.447	1.253	1.512	1.509	1.290	1.962	1.153	1.392	1.244	1.476	1.384	1.736
50	1.447	1.253	1.511	1.509	1.290	1.959	1.152	1.391	1.243	1.475	1.381	1.733
51	1.447	1.253	1.509	1.508	1.290	1.956	1.150	1.390	1.242	1.474	1.379	1.729
52	1.447	1.253	1.509	1.507	1.290	1.953	1.149	1.388	1.241	1.473	1.376	1.726
53	1.447	1.251	1.507	1.507	1.290	1.950	1.147	1.386	1.240	1.472	1.374	1.723
54	1.449	1.250	1.506	1.506	1.289	1.946	1.145	1.384	1.240	1.471	1.372	1.719
55	1.452	1.248	1.505	1.507	1.289	1.942	1.144	1.382	1.239	1.471	1.370	1.716
56	1.455	1.247	1.503	1.508	1.288	1.938	1.143	1.380	1.239	1.471	1.368	1.714
57	1.457	1.247	1.502	1.509	1.288	1.933	1.142	1.379	1.238	1.471	1.366	1.711
58	1.460	1.247	1.501	1.512	1.288	1.929	1.142	1.378	1.237	1.470	1.364	1.709
59	1.462	1.247	1.500	1.514	1.289	1.924	1.141	1.378	1.237	1.470	1.363	1.707
60	1.465	1.247	1.499	1.517	1.290	1.920	1.140	1.378	1.237	1.470	1.362	1.705
61	1.468	1.246	1.498	1.519	1.291	1.915	1.139	1.377	1.236	1.469	1.361	1.704
62	1.471	1.245	1.498	1.521	1.292	1.911	1.139	1.376	1.236	1.469	1.361	1.703
63	1.472	1.245	1.497	1.523	1.294	1.908	1.138	1.375	1.237	1.468	1.361	1.701
64	1.474	1.244	1.496	1.525	1.295	1.904	1.137	1.375	1.237	1.467	1.360	1.699
65	1.476	1.243	1.495	1.526	1.295	1.901	1.136	1.375	1.236	1.467	1.360	1.697
66	1.478	1.243	1.494	1.528	1.296	1.897	1.135	1.375	1.236	1.466	1.359	1.695
67	1.480	1.243	1.492	1.529	1.296	1.894	1.134	1.374	1.235	1.465	1.359	1.694
68	1.481	1.244	1.491	1.530	1.296	1.891	1.133	1.373	1.234	1.464	1.358	1.692
69	1.482	1.243	1.489	1.532	1.296	1.887	1.132	1.372	1.233	1.463	1.357	1.690
70	1.483	1.242	1.488	1.533	1.296	1.883	1.130	1.371	1.233	1.462	1.356	1.688
71	1.484	1.241	1.486	1.534	1.296	1.879	1.129	1.370	1.232	1.461	1.355	1.686
72	1.485	1.240	1.484	1.536	1.296	1.875	1.127	1.370	1.231	1.460	1.355	1.683
73	1.485	1.239	1.482	1.537	1.295	1.872	1.126	1.370	1.230	1.460	1.355	1.682
74	1.486	1.238	1.481	1.539	1.295	1.868	1.124	1.370	1.229	1.459	1.354	1.681
75	1.486	1.237	1.479	1.540	1.295	1.865	1.122	1.370	1.228	1.458	1.354	1.680
76	1.486	1.236	1.477	1.542	1.296	1.862	1.120	1.370	1.227	1.457	1.354	1.678
77	1.487	1.235	1.475	1.543	1.296	1.859	1.118	1.370	1.226	1.456	1.354	1.677
78	1.487	1.235	1.474	1.546	1.296	1.856	1.116	1.369	1.225	1.455	1.354	1.676
79	1.488	1.234	1.472	1.548	1.296	1.853	1.114	1.369	1.225	1.454	1.355	1.675
80	1.488	1.234	1.471	1.550	1.296	1.850	1.112	1.368	1.224	1.454	1.355	1.674
81	1.489	1.233	1.470	1.553	1.296	1.848	1.110	1.367	1.223	1.453	1.355	1.673
82	1.489	1.233	1.468	1.556	1.297	1.845	1.109	1.366	1.223	1.453	1.355	1.672
83	1.490	1.232	1.467	1.559	1.297	1.842	1.107	1.365	1.222	1.453	1.355	1.671
84	1.490	1.232	1.465	1.563	1.297	1.840	1.105	1.363	1.221	1.453	1.356	1.670
85	1.490	1.231	1.464	1.566	1.297	1.838	1.103	1.362	1.221	1.453	1.356	1.669
86	1.490	1.230	1.462	1.570	1.297	1.836	1.102	1.360	1.220	1.453	1.356	1.668
87	1.491	1.229	1.461	1.574	1.297	1.833	1.100	1.359	1.220	1.453	1.356	1.667
88	1.492	1.228	1.459	1.577	1.297	1.831	1.098	1.358	1.220	1.453	1.356	1.666
89	1.494	1.228	1.457	1.581	1.297	1.829	1.096	1.357	1.220	1.453	1.356	1.665
90	1.495	1.227	1.456	1.584	1.297	1.827	1.094	1.355	1.220	1.453	1.356	1.664
91	1.496	1.227	1.454	1.587	1.298	1.825	1.092	1.354	1.220	1.453	1.356	1.663
92	1.498	1.226	1.452	1.591	1.298	1.824	1.090	1.354	1.220	1.453	1.356	1.662
93	1.500	1.226	1.451	1.595	1.298	1.822	1.088	1.354	1.220	1.453	1.356	1.660
94	1.502	1.225	1.449	1.597	1.299	1.821	1.086	1.353	1.220	1.453	1.356	1.659
95	1.503	1.224	1.448	1.600	1.299	1.819	1.084	1.352	1.220	1.453	1.356	1.658
96	1.505	1.224	1.446	1.604	1.300	1.818	1.082	1.352	1.220	1.453	1.356	1.657
97	1.506	1.223	1.444	1.608	1.301	1.816	1.079	1.351	1.220	1.453	1.356	1.657
98	1.508	1.223	1.443	1.611	1.302	1.814	1.077	1.350	1.220	1.453	1.356	1.656
99	1.509	1.222	1.441	1.614	1.302	1.812	1.075	1.349	1.220	1.453	1.355	1.656
100	1.510	1.222	1.440	1.618	1.303	1.810	1.072	1.348	1.220	1.454	1.355	1.655
101	1.512	1.222	1.439	1.622	1.304	1.808	1.070	1.347	1.221	1.454	1.355	1.654
102	1.513	1.221	1.437	1.626	1.305	1.806	1.068	1.346	1.221	1.454	1.355	1.653
103	1.514	1.221	1.436	1.630	1.305	1.804	1.066	1.345	1.221	1.454	1.355	1.652
104	1.517	1.221	1.434	1.635	1.306	1.802	1.064	1.344	1.221	1.454	1.355	1.651
105	1.519	1.222	1.433	1.639	1.307	1.801	1.062	1.343	1.222	1.454	1.355	1.650
106	1.521	1.222	1.431	1.644	1.308	1.799	1.060	1.343	1.222	1.454	1.356	1.649
107	1.522	1.222	1.430	1.648	1.308	1.797	1.058	1.342	1.223	1.454	1.356	1.648
108	1.523	1.223	1.429	1.652	1.309	1.795	1.055	1.341	1.224	1.454	1.356	1.647
109	1.524	1.223	1.427	1.656	1.309	1.794	1.054	1.340	1.224	1.453	1.356	1.645
110	1.525	1.224	1.426	1.660	1.309	1.792	1.052	1.339	1.225	1.453	1.356	1.644
111	1.527	1.225	1.425	1.663	1.309	1.789	1.050	1.338	1.225	1.453	1.355	1.643
112	1.528	1.225	1.424	1.666	1.310	1.787	1.048	1.337	1.226	1.453	1.355	1.642
113	1.529	1.226	1.422	1.668	1.311	1.786	1.046	1.335	1.227	1.453	1.355	1.640
114	1.531	1.227	1.421	1.671	1.312	1.784	1.045	1.334	1.227	1.453	1.355	1.639
115	1.533	1.228	1.420	1.673	1.313	1.782	1.043	1.333	1.228	1.453	1.354	1.638
116	1.535	1.229	1.419	1.675	1.314	1.780	1.042	1.332	1.228	1.453	1.354	1.637
117	1.537	1.230	1.418	1.678	1.316	1.779	1.040	1.331	1.229	1.453	1.354	1.635
118	1.539	1.231	1.416	1.679	1.317	1.777	1.039	1.330	1.229	1.452	1.354	1.634
119	1.542	1.232	1.415	1.681	1.319	1.776	1.037	1.329	1.230	1.452	1.354	1.633
120	1.544	1.233	1.414	1.682	1.320	1.774	1.036	1.327	1.230	1.452	1.354	1.632
121	1.548	1.234	1.413	1.683	1.321	1.773	1.034	1.326	1.231	1.451	1.354	1.630
122	1.551	1.235	1.412	1.684	1.323	1.772	1.033	1.325	1.232	1.451	1.354	1.629
123	1.555	1.237	1.411	1.685	1.324	1.770	1.032	1.324	1.232	1.451	1.354	1.627
124	1.559	1.238	1.410	1.686	1.325	1.769	1.030	1.322	1.233	1.450	1.354	1.626
125	1.564	1.239	1.409	1.686	1.326	1.767	1.029	1.321	1.234	1.450	1.353	1.625
126	1.569	1.241	1.408	1.687	1.328	1.765	1.028	1.320	1.235	1.450	1.353	1.624
127	1.574	1.242	1.407	1.689	1.329	1.764	1.026	1.318	1.235	1.450	1.353	1.623
128	1.580	1.244	1.406	1.690	1.331	1.762	1.025	1.317	1.236	1.450	1.353	1.622
129	1.585	1.245	1.405	1.692	1.332	1.761	1.024	1.316	1.237	1.449	1.353	1.621
130	1.590	1.247	1.404	1.694	1.334	1.760	1.022	1.315	1.238	1.449	1.353	1.620
131	1.595	1.248	1.403	1.696	1.335	1.759	1.021	1.314	1.239	1.448	1.353	1.618
132	1.600	1.249	1.402	1.699	1.337	1.757	1.019	1.313	1.240	1.448	1.353	1.617
133	1.605	1.251	1.401	1.701	1.339	1.756	1.018	1.313	1.241	1.448	1.353	1.615
134	1.610	1.252	1.400	1.704	1.341	1.755	1.017	1.312	1.242	1.447	1.353	1.614
135	1.614	1.254	1.399	1.707	1.342	1.754	1.015	1.312	1.243	1.447	1.352	1.612
136	1.619	1.255	1.398	1.709	1.344	1.753	1.014	1.312	1.244	1.446	1.352	1.611
137	1.624	1.256	1.397	1.712	1.346	1.751	1.012	1.312	1.245	1.446	1.352	1.609
138	1.628	1.258	1.396	1.714	1.348	1.750	1.011	1.312	1.246	1.445	1.352	1.608
139	1.632	1.259	1.395	1.716	1.351	1.749	1.009	1.312	1.247	1.445	1.351	1.607
140	1.635	1.261	1.394	1.719	1.353	1.749	1.008	1.312	1.248	1.444	1.351	1.605
141	1.639	1.262	1.393	1.721	1.355	1.748	1.006	1.312	1.248	1.444	1.351	1.603
142	1.642	1.263	1.391	1.724	1.358	1.747	1.005	1.312	1.249	1.443	1.350	1.602
143	1.645	1.264	1.390	1.726	1.360	1.747	1.003	1.312	1.250	1.443	1.350	1.601
144	1.648	1.266	1.389	1.729	1.363	1.746	1.002	1.312	1.251	1.442	1.350	1.600
145	1.650	1.267	1.387	1.732	1.366	1.745	1.001	1.312	1.251	1.442	1.350	1.599
146	1.653	1.268	1.386	1.734	1.368	1.744	1.000	1.312	1.252	1.441	1.350	1.598
147	1.655	1.269	1.385	1.737	1.371	1.743	0.999	1.311	1.252	1.440	1.350	1.597
148	1.657	1.270	1.383	1.738	1.373	1.742	0.997	1.312	1.253	1.440	1.349	1.595
149	1.659	1.271	1.382	1.739	1.376	1.741	0.996	1.312	1.253	1.439	1.349	1.594
150	1.660	1.272	1.380	1.741	1.378	1.740	0.995	1.312	1.254	1.438	1.349	1.592
151	1.662	1.273	1.379	1.742	1.380	1.739	0.993	1.312	1.254	1.437	1.348	1.591
152	1.664	1.274	1.378	1.744	1.382	1.738	0.992	1.313	1.254	1.437	1.348	1.589
153	1.665	1.275	1.376	1.745	1.384	1.736	0.991	1.313	1.255	1.436	1.348	1.588
154	1.666	1.276	1.375	1.747	1.386	1.735	0.989	1.313	1.255	1.435	1.347	1.586
155	1.668	1.277	1.373	1.749	1.388	1.734	0.988	1.314	1.256	1.434	1.347	1.585
156	1.669	1.277	1.372	1.751	1.390	1.732	0.987	1.314	1.256	1.434	1.346	1.584
157	1.670	1.278	1.371	1.753	1.392	1.731	0.986	1.314	1.257	1.433	1.346	1.582
158	1.671	1.279	1.369	1.755	1.394	1.730	0.985	1.314	1.257	1.432	1.346	1.581
159	1.672	1.280	1.368	1.756	1.397	1.728	0.984	1.314	1.258	1.432	1.346	1.579
160	1.673	1.281	1.367	1.758	1.399	1.726	0.982	1.314	1.258	1.431	1.345	1.578
161	1.674	1.282	1.365	1.759	1.401	1.725	0.981	1.314	1.258	1.431	1.345	1.577
162	1.675	1.282	1.364	1.760	1.403	1.723	0.980	1.314	1.258	1.430	1.345	1.575
163	1.676	1.282	1.362	1.760	1.405	1.722	0.979	1.314	1.259	1.430	1.345	1.574
164	1.677	1.283	1.361	1.761	1.407	1.720	0.977	1.314	1.259	1.429	1.345	1.573
165	1.678	1.283	1.360	1.762	1.408	1.718	0.976	1.314	1.259	1.428	1.345	1.572
166	1.679	1.283	1.358	1.762	1.410	1.717	0.974	1.314	1.259	1.427	1.345	1.570
167	1.681	1.284	1.357	1.763	1.411	1.715	0.973	1.314	1.259	1.426	1.345	1.569
168	1.683	1.284	1.355	1.764	1.413	1.713	0.971	1.314	1.259	1.425	1.345	1.567
169	1.685	1.284	1.354	1.765	1.415	1.712	0.970	1.314	1.259	1.423	1.345	1.566
170	1.688	1.284	1.353	1.767	1.416	1.710	0.968	1.314	1.259	1.422	1.345	1.564
171	1.690	1.284	1.351	1.768	1.418	1.708	0.967	1.313	1.259	1.420	1.345	1.563
172	1.692	1.283	1.350	1.770	1.420	1.707	0.965	1.313	1.259	1.419	1.345	1.561
173	1.694	1.283	1.349	1.771	1.421	1.705	0.964	1.313	1.259	1.417	1.345	1.560
174	1.697	1.283	1.347	1.772	1.422	1.703	0.963	1.313	1.259	1.416	1.345	1.558
175	1.698	1.283	1.346	1.774	1.423	1.701	0.961	1.313	1.259	1.415	1.346	1.557
176	1.700	1.283	1.344	1.775	1.425	1.699	0.960	1.313	1.259	1.414	1.346	1.555
177	1.702	1.284	1.343	1.776	1.426	1.697	0.959	1.313	1.259	1.413	1.346	1.554
178	1.704	1.284	1.341	1.777	1.427	1.695	0.958	1.314	1.260	1.412	1.346	1.553
179	1.706	1.284	1.340	1.777	1.429	1.693	0.957	1.314	1.260	1.411	1.346	1.552
180	1.708	1.285	1.338	1.778	1.430	1.691	0.956	1.314	1.260	1.409	1.346	1.551
181	1.711	1.285	1.337	1.779	1.431	1.688	0.955	1.314	1.260	1.408	1.346	1.550
182	1.712	1.286	1.335	1.780	1.433	1.686	0.954	1.314	1.260	1.407	1.346	1.549
183	1.714	1.287	1.334	1.781	1.434	1.684	0.952	1.314	1.259	1.405	1.347	1.548
184	1.716	1.287	1.333	1.782	1.435	1.682	0.951	1.314	1.259	1.404	1.347	1.547
185	1.718	1.288	1.331	1.782	1.437	1.680	0.950	1.315	1.259	1.402	1.347	1.546
186	1.720	1.288	1.330	1.782	1.438	1.678	0.949	1.315	1.259	1.401	1.347	1.545
187	1.721	1.289	1.329	1.782	1.439	1.676	0.948	1.315	1.259	1.399	1.348	1.544
188	1.724	1.290	1.327	1.782	1.440	1.674	0.948	1.315	1.258	1.398	1.348	1.543
189	1.725	1.290	1.326	1.781	1.440	1.672	0.947	1.315	1.258	1.396	1.348	1.542
190	1.727	1.291	1.325	1.780	1.441	1.670	0.946	1.314	1.257	1.395	1.348	1.541
191	1.729	1.291	1.323	1.779	1.441	1.668	0.945	1.314	1.257	1.393	1.348	1.540
192	1.731	1.292	1.322	1.778	1.442	1.666	0.944	1.314	1.256	1.391	1.348	1.539
193	1.732	1.292	1.321	1.777	1.443	1.664	0.943	1.314	1.256	1.390	1.348	1.538
194	1.734	1.292	1.319	1.776	1.444	1.663	0.942	1.313	1.255	1.389	1.348	1.537
195	1.737	1.293	1.318	1.775	1.444	1.661	0.941	1.313	1.255	1.387	1.348	1.536
196	1.739	1.293	1.317	1.774	1.445	1.659	0.941	1.312	1.254	1.386	1.348	1.535
197	1.741	1.293	1.315	1.773	1.445	1.657	0.940	1.311	1.254	1.385	1.348	1.534
198	1.742	1.293	1.314	1.772	1.446	1.655	0.939	1.311	1.253	1.383	1.348	1.533
199	1.743	1.294	1.313	1.771	1.446	1.653	0.938	1.310	1.252	1.382	1.348	1.531
200	1.745	1.294	1.312	1.770	1.447	1.651	0.937	1.309	1.252	1.381	1.348	1.530
201	1.747	1.294	1.311	1.769	1.447	1.650	0.936	1.308	1.251	1.379	1.348	1.529
202	1.749	1.294	1.309	1.768	1.448	1.648	0.936	1.307	1.250	1.378	1.348	1.528
203	1.751	1.294	1.308	1.767	1.448	1.646	0.935	1.306	1.250	1.377	1.349	1.526
204	1.752	1.294	1.307	1.765	1.448	1.644	0.934	1.305	1.249	1.375	1.349	1.525
205	1.754	1.294	1.306	1.764	1.449	1.642	0.934	1.305	1.248	1.374	1.349	1.524
206	1.756	1.294	1.305	1.762	1.449	1.640	0.933	1.304	1.247	1.373	1.349	1.523
207	1.758	1.294	1.304	1.761	1.449	1.639	0.932	1.303	1.247	1.372	1.350	1.521
208	1.760	1.293	1.303	1.759	1.449	1.637	0.932	1.302	1.246	1.370	1.350	1.520
209	1.762	1.293	1.301	1.758	1.450	1.635	0.931	1.302	1.245	1.369	1.350	1.519
210	1.764	1.292	1.300	1.756	1.450	1.633	0.930	1.301	1.244	1.367	1.351	1.518
211	1.766	1.292	1.299	1.754	1.450	1.631	0.930	1.300	1.244	1.366	1.351	1.517
212	1.768	1.291	1.298	1.752	1.451	1.629	0.929	1.300	1.243	1.364	1.351	1.516
213	1.769	1.290	1.297	1.751	1.451	1.627	0.928	1.299	1.242	1.363	1.351	1.514
214	1.771	1.289	1.295	1.750	1.451	1.626	0.928	1.299	1.241	1.361	1.352	1.514
215	1.772	1.289	1.294	1.748	1.451	1.624	0.927	1.298	1.241	1.360	1.352	1.513
216	1.774	1.288	1.293	1.747	1.451	1.622	0.927	1.298	1.240	1.359	1.352	1.512
217	1.776	1.287	1.292	1.746	1.451	1.620	0.926	1.298	1.239	1.357	1.352	1.511
218	1.777	1.287	1.291	1.744	1.451	1.618	0.925	1.298	1.239	1.356	1.353	1.510
219	1.777	1.286	1.290	1.743	1.451	1.616	0.925	1.297	1.238	1.355	1.353	1.509
220	1.778	1.285	1.289	1.742	1.451	1.614	0.924	1.297	1.237	1.353	1.353	1.508
221	1.779	1.285	1.288	1.741	1.451	1.612	0.923	1.297	1.237	1.352	1.353	1.507
222	1.781	1.284	1.287	1.740	1.451	1.610	0.923	1.296	1.236	1.351	1.353	1.505
223	1.782	1.284	1.286	1.739	1.451	1.608	0.922	1.296	1.236	1.349	1.354	1.504
224	1.784	1.283	1.285	1.738	1.451	1.606	0.922	1.296	1.235	1.348	1.354	1.504
225	1.786	1.282	1.284	1.738	1.451	1.604	0.921	1.295	1.235	1.347	1.354	1.503
226	1.787	1.282	1.283	1.737	1.451	1.602	0.920	1.295	1.234	1.346	1.355	1.502
227	1.788	1.281	1.282	1.737	1.451	1.600	0.920	1.295	1.234	1.344	1.355	1.501
228	1.790	1.281	1.282	1.736	1.451	1.598	0.919	1.294	1.233	1.343	1.355	1.500
229	1.791	1.280	1.281	1.735	1.450	1.596	0.919	1.294	1.233	1.342	1.356	1.500
230	1.793	1.280	1.280	1.734	1.450	1.594	0.918	1.294	1.232	1.341	1.356	1.499
231	1.795	1.279	1.279	1.733	1.449	1.592	0.918	1.294	1.232	1.339	1.356	1.498
232	1.797	1.279	1.279	1.732	1.449	1.590	0.917	1.293	1.231	1.338	1.356	1.497
233	1.798	1.278	1.278	1.731	1.448	1.588	0.917	1.293	1.230	1.337	1.356	1.497
234	1.800	1.278	1.277	1.730	1.448	1.585	0.916	1.292	1.230	1.336	1.356	1.496
235	1.801	1.278	1.277	1.728	1.447	1.583	0.916	1.292	1.229	1.335	1.356	1.495
236	1.802	1.277	1.276	1.727	1.447	1.581	0.915	1.291	1.228	1.334	1.356	1.494
237	1.803	1.277	1.275	1.726	1.446	1.579	0.915	1.291	1.228	1.333	1.356	1.494
238	1.803	1.277	1.275	1.725	1.446	1.577	0.914	1.290	1.227	1.332	1.356	1.493
239	1.804	1.277	1.274	1.723	1.445	1.575	0.914	1.290	1.226	1.331	1.356	1.492
240	1.803	1.276	1.274	1.722	1.444	1.573	0.914	1.289	1.226	1.329	1.356	1.491
241	1.802	1.276	1.273	1.721	1.443	1.571	0.913	1.289	1.225	1.328	1.355	1.490
242	1.801	1.276	1.272	1.720	1.443	1.569	0.913	1.288	1.225	1.327	1.355	1.490
243	1.800	1.276	1.272	1.718	1.442	1.567	0.913	1.287	1.224	1.326	1.355	1.489
244	1.799	1.276	1.271	1.717	1.441	1.565	0.912	1.286	1.224	1.325	1.355	1.488
245	1.798	1.276	1.271	1.716	1.441	1.564	0.912	1.286	1.223	1.324	1.355	1.487
246	1.797	1.276	1.270	1.715	1.440	1.562	0.912	1.285	1.223	1.324	1.355	1.487
247	1.796	1.276	1.270	1.714	1.439	1.560	0.911	1.284	1.223	1.323	1.355	1.486
248	1.794	1.276	1.269	1.713	1.439	1.558	0.911	1.283	1.222	1.322	1.355	1.485
249	1.791	1.276	1.268	1.712	1.438	1.556	0.911	1.282	1.222	1.321	1.355	1.485
250	1.791	1.276	1.268	1.712	1.438	1.556	0.911	1.282	1.222	1.321	1.355	1.485

Data for Appendix 4 - Long Memory Hypothesis Tests: Equity (Malaysia - Turkey)

xi	Malaysia	Philippines	Russia	Singapore	Taiwan	Thailand	Israel	Kuwait	Nigeria	Saudi Arabia	South Africa	Turkey
0	1.624	1.439	1.770	1.622	1.605	1.896	0.982	1.841	1.446	2.205	1.550	1.557
1	1.575	1.409	1.727	1.595	1.577	1.852	0.981	1.840	1.326	2.203	1.492	1.507
2	1.538	1.396	1.704	1.569	1.541	1.825	0.987	1.835	1.275	2.203	1.454	1.485
3	1.522	1.383	1.688	1.543	1.518	1.805	0.990	1.834	1.253	2.185	1.438	1.478
4	1.508	1.381	1.676	1.525	1.501	1.794	1.003	1.814	1.250	2.178	1.432	1.468
5	1.503	1.383	1.661	1.511	1.491	1.793	1.017	1.811	1.251	2.172	1.432	1.466
6	1.499	1.382	1.648	1.498	1.483	1.796	1.028	1.806	1.250	2.168	1.431	1.469
7	1.498	1.374	1.639	1.486	1.472	1.795	1.034	1.800	1.247	2.151	1.433	1.471
8	1.495	1.366	1.632	1.475	1.462	1.792	1.036	1.792	1.241	2.129	1.430	1.471
9	1.493	1.359	1.622	1.466	1.450	1.786	1.033	1.783	1.234	2.123	1.426	1.469
10	1.492	1.353	1.612	1.456	1.437	1.780	1.029	1.774	1.227	2.109	1.423	1.463
11	1.490	1.344	1.602	1.446	1.422	1.775	1.027	1.766	1.220	2.087	1.420	1.458
12	1.488	1.332	1.591	1.436	1.408	1.764	1.027	1.760	1.215	2.069	1.420	1.452
13	1.487	1.319	1.581	1.426	1.395	1.754	1.026	1.753	1.210	2.049	1.417	1.446
14	1.483	1.306	1.571	1.417	1.382	1.743	1.025	1.735	1.206	2.029	1.413	1.440
15	1.480	1.295	1.563	1.408	1.371	1.733	1.024	1.717	1.202	2.014	1.407	1.435
16	1.477	1.284	1.557	1.401	1.364	1.724	1.023	1.700	1.198	2.002	1.402	1.430
17	1.474	1.273	1.554	1.394	1.357	1.716	1.024	1.686	1.195	1.994	1.399	1.428
18	1.471	1.265	1.551	1.387	1.351	1.709	1.024	1.672	1.193	1.987	1.396	1.426
19	1.468	1.258	1.549	1.381	1.346	1.703	1.024	1.660	1.190	1.977	1.394	1.425
20	1.465	1.251	1.547	1.376	1.340	1.698	1.024	1.649	1.188	1.968	1.392	1.424
21	1.462	1.246	1.544	1.371	1.335	1.693	1.024	1.639	1.186	1.962	1.392	1.422
22	1.458	1.241	1.539	1.366	1.331	1.689	1.025	1.631	1.184	1.960	1.393	1.419
23	1.454	1.236	1.535	1.361	1.327	1.684	1.026	1.625	1.182	1.959	1.394	1.419
24	1.450	1.231	1.529	1.356	1.323	1.680	1.026	1.620	1.180	1.958	1.393	1.417
25	1.445	1.226	1.524	1.352	1.320	1.678	1.027	1.616	1.178	1.956	1.390	1.414
26	1.442	1.222	1.519	1.347	1.316	1.677	1.027	1.613	1.177	1.954	1.387	1.413
27	1.438	1.219	1.515	1.342	1.313	1.678	1.027	1.610	1.175	1.951	1.383	1.412
28	1.434	1.216	1.512	1.337	1.310	1.677	1.028	1.608	1.174	1.947	1.380	1.412
29	1.430	1.212	1.508	1.332	1.308	1.675	1.028	1.604	1.173	1.944	1.377	1.411
30	1.427	1.208	1.504	1.327	1.306	1.672	1.027	1.600	1.172	1.942	1.374	1.412
31	1.425	1.205	1.501	1.323	1.304	1.670	1.026	1.597	1.171	1.940	1.372	1.412
32	1.422	1.202	1.498	1.319	1.303	1.668	1.026	1.594	1.170	1.938	1.370	1.414
33	1.420	1.200	1.495	1.315	1.301	1.667	1.026	1.590	1.169	1.935	1.368	1.414
34	1.416	1.198	1.493	1.311	1.299	1.665	1.026	1.586	1.168	1.934	1.367	1.414
35	1.413	1.197	1.491	1.307	1.297	1.665	1.028	1.582	1.166	1.935	1.367	1.415
36	1.410	1.196	1.489	1.303	1.295	1.664	1.029	1.579	1.164	1.936	1.369	1.417
37	1.407	1.194	1.486	1.299	1.294	1.664	1.030	1.575	1.163	1.937	1.371	1.418
38	1.404	1.192	1.484	1.295	1.292	1.664	1.032	1.572	1.162	1.938	1.373	1.420
39	1.401	1.190	1.483	1.291	1.291	1.664	1.034	1.569	1.160	1.936	1.374	1.423
40	1.399	1.188	1.482	1.288	1.289	1.665	1.035	1.566	1.158	1.933	1.375	1.424
41	1.396	1.186	1.482	1.284	1.287	1.666	1.036	1.564	1.156	1.929	1.376	1.426
42	1.394	1.184	1.481	1.281	1.286	1.666	1.037	1.561	1.153	1.926	1.376	1.427
43	1.391	1.182	1.479	1.278	1.285	1.666	1.038	1.559	1.151	1.924	1.377	1.429
44	1.388	1.181	1.477	1.275	1.283	1.667	1.040	1.556	1.149	1.921	1.376	1.429
45	1.385	1.179	1.476	1.272	1.281	1.667	1.041	1.552	1.147	1.918	1.376	1.429
46	1.382	1.178	1.475	1.270	1.279	1.668	1.043	1.549	1.145	1.916	1.376	1.429
47	1.379	1.177	1.475	1.267	1.278	1.670	1.044	1.546	1.142	1.913	1.376	1.429
48	1.376	1.176	1.474	1.265	1.276	1.671	1.046	1.543	1.140	1.909	1.376	1.430
49	1.373	1.176	1.473	1.262	1.275	1.672	1.048	1.539	1.138	1.904	1.377	1.431
50	1.370	1.175	1.471	1.260	1.274	1.672	1.051	1.535	1.136	1.899	1.377	1.432
51	1.367	1.174	1.469	1.258	1.274	1.672	1.053	1.531	1.135	1.894	1.377	1.434
52	1.365	1.173	1.468	1.255	1.273	1.672	1.055	1.528	1.133	1.889	1.377	1.435
53	1.362	1.172	1.467	1.253	1.272	1.672	1.057	1.524	1.132	1.883	1.377	1.436
54	1.360	1.171	1.466	1.251	1.271	1.671	1.058	1.521	1.131	1.877	1.377	1.438
55	1.358	1.170	1.465	1.249	1.270	1.671	1.060	1.518	1.129	1.871	1.376	1.438
56	1.356	1.169	1.463	1.247	1.269	1.672	1.061	1.515	1.128	1.866	1.375	1.439
57	1.354	1.169	1.462	1.246	1.268	1.672	1.063	1.513	1.127	1.861	1.374	1.440
58	1.352	1.169	1.461	1.244	1.266	1.673	1.064	1.511	1.125	1.856	1.373	1.441
59	1.350	1.168	1.460	1.243	1.265	1.674	1.066	1.509	1.125	1.851	1.372	1.441
60	1.348	1.168	1.458	1.242	1.264	1.674	1.067	1.506	1.124	1.847	1.372	1.442
61	1.346	1.167	1.457	1.240	1.263	1.675	1.067	1.504	1.124	1.842	1.372	1.442
62	1.345	1.167	1.456	1.239	1.261	1.675	1.068	1.502	1.123	1.837	1.373	1.442
63	1.343	1.167	1.454	1.238	1.260	1.675	1.068	1.500	1.123	1.833	1.374	1.443
64	1.342	1.166	1.452	1.237	1.259	1.675	1.069	1.498	1.123	1.828	1.374	1.444
65	1.341	1.166	1.451	1.236	1.258	1.674	1.070	1.497	1.123	1.824	1.374	1.444
66	1.340	1.165	1.449	1.235	1.257	1.673	1.071	1.495	1.123	1.819	1.374	1.444
67	1.339	1.165	1.448	1.234	1.256	1.672	1.072	1.494	1.122	1.815	1.373	1.443
68	1.338	1.164	1.446	1.233	1.255	1.672	1.072	1.492	1.122	1.810	1.372	1.443
69	1.338	1.164	1.445	1.232	1.254	1.671	1.072	1.489	1.121	1.806	1.372	1.442
70	1.338	1.164	1.444	1.231	1.253	1.670	1.072	1.487	1.121	1.802	1.371	1.441
71	1.337	1.163	1.444	1.231	1.252	1.669	1.072	1.485	1.121	1.798	1.369	1.441
72	1.337	1.163	1.443	1.230	1.251	1.668	1.071	1.484	1.120	1.795	1.368	1.441
73	1.336	1.163	1.442	1.230	1.250	1.667	1.071	1.482	1.120	1.791	1.367	1.441
74	1.336	1.163	1.441	1.230	1.250	1.667	1.071	1.480	1.120	1.789	1.367	1.441
75	1.335	1.163	1.441	1.229	1.249	1.667	1.071	1.478	1.119	1.787	1.366	1.441
76	1.335	1.163	1.441	1.229	1.248	1.668	1.072	1.476	1.119	1.785	1.366	1.441
77	1.334	1.163	1.441	1.229	1.247	1.668	1.072	1.475	1.119	1.783	1.366	1.441
78	1.333	1.162	1.441	1.229	1.246	1.668	1.073	1.473	1.119	1.781	1.366	1.441
79	1.332	1.162	1.440	1.229	1.245	1.667	1.074	1.472	1.119	1.778	1.366	1.441
80	1.332	1.161	1.439	1.228	1.244	1.667	1.075	1.471	1.118	1.776	1.366	1.440
81	1.331	1.161	1.438	1.228	1.243	1.667	1.076	1.470	1.118	1.773	1.367	1.439
82	1.330	1.161	1.437	1.228	1.242	1.667	1.077	1.469	1.117	1.770	1.367	1.439
83	1.329	1.160	1.436	1.228	1.240	1.668	1.079	1.468	1.117	1.767	1.367	1.439
84	1.329	1.160	1.435	1.227	1.239	1.668	1.081	1.467	1.116	1.765	1.366	1.439
85	1.329	1.160	1.434	1.227	1.239	1.669	1.082	1.466	1.115	1.763	1.367	1.439
86	1.328	1.160	1.432	1.227	1.238	1.670	1.084	1.465	1.114	1.760	1.367	1.439
87	1.328	1.160	1.430	1.227	1.237	1.671	1.085	1.464	1.113	1.758	1.368	1.439
88	1.328	1.161	1.429	1.227	1.237	1.673	1.087	1.463	1.112	1.756	1.368	1.439
89	1.328	1.161	1.427	1.227	1.236	1.674	1.088	1.462	1.111	1.754	1.369	1.438
90	1.328	1.161	1.426	1.227	1.236	1.675	1.091	1.461	1.110	1.751	1.369	1.438
91	1.328	1.161	1.424	1.227	1.235	1.676	1.093	1.459	1.109	1.748	1.370	1.438
92	1.328	1.162	1.422	1.227	1.234	1.678	1.096	1.458	1.109	1.745	1.371	1.439
93	1.328	1.162	1.420	1.227	1.234	1.679	1.099	1.458	1.108	1.743	1.373	1.439
94	1.327	1.163	1.418	1.227	1.233	1.681	1.102	1.457	1.107	1.739	1.374	1.440
95	1.327	1.163	1.416	1.227	1.232	1.683	1.104	1.456	1.107	1.736	1.376	1.441
96	1.326	1.164	1.414	1.227	1.232	1.686	1.107	1.455	1.107	1.733	1.379	1.442
97	1.326	1.164	1.411	1.227	1.231	1.688	1.109	1.454	1.106	1.730	1.381	1.442
98	1.325	1.165	1.409	1.227	1.231	1.690	1.112	1.453	1.106	1.727	1.383	1.443
99	1.325	1.165	1.407	1.227	1.230	1.692	1.114	1.451	1.106	1.723	1.386	1.444
100	1.325	1.166	1.405	1.227	1.230	1.694	1.116	1.450	1.105	1.719	1.389	1.444
101	1.324	1.167	1.403	1.227	1.229	1.697	1.118	1.449	1.105	1.716	1.392	1.445
102	1.324	1.168	1.401	1.227	1.229	1.699	1.121	1.448	1.105	1.713	1.394	1.445
103	1.324	1.169	1.400	1.227	1.228	1.702	1.123	1.446	1.104	1.710	1.397	1.445
104	1.324	1.169	1.398	1.227	1.228	1.705	1.125	1.445	1.104	1.706	1.401	1.445
105	1.324	1.170	1.397	1.227	1.227	1.708	1.126	1.443	1.104	1.703	1.404	1.446
106	1.324	1.171	1.396	1.227	1.227	1.711	1.129	1.442	1.104	1.699	1.407	1.446
107	1.324	1.172	1.395	1.227	1.227	1.715	1.130	1.441	1.104	1.696	1.410	1.446
108	1.324	1.173	1.394	1.227	1.227	1.718	1.132	1.440	1.104	1.693	1.413	1.447
109	1.324	1.174	1.392	1.228	1.226	1.722	1.134	1.438	1.104	1.690	1.416	1.448
110	1.324	1.174	1.391	1.227	1.226	1.725	1.135	1.437	1.104	1.687	1.419	1.449
111	1.324	1.175	1.390	1.227	1.226	1.728	1.137	1.436	1.104	1.684	1.422	1.450
112	1.324	1.176	1.389	1.227	1.225	1.731	1.138	1.435	1.104	1.681	1.425	1.451
113	1.324	1.176	1.387	1.227	1.225	1.734	1.139	1.433	1.103	1.678	1.428	1.452
114	1.324	1.177	1.386	1.227	1.225	1.736	1.140	1.432	1.103	1.675	1.430	1.453
115	1.324	1.177	1.385	1.227	1.224	1.739	1.141	1.431	1.102	1.672	1.433	1.455
116	1.324	1.178	1.383	1.227	1.224	1.741	1.142	1.430	1.101	1.670	1.436	1.457
117	1.324	1.178	1.382	1.226	1.224	1.744	1.144	1.429	1.101	1.667	1.439	1.458
118	1.324	1.179	1.381	1.226	1.224	1.747	1.145	1.427	1.100	1.665	1.442	1.460
119	1.323	1.180	1.379	1.226	1.224	1.749	1.147	1.426	1.100	1.662	1.445	1.462
120	1.323	1.180	1.378	1.226	1.223	1.752	1.149	1.426	1.099	1.660	1.447	1.463
121	1.323	1.181	1.378	1.226	1.223	1.754	1.150	1.425	1.098	1.657	1.450	1.464
122	1.323	1.182	1.377	1.226	1.223	1.756	1.152	1.424	1.097	1.655	1.453	1.466
123	1.322	1.182	1.376	1.226	1.223	1.759	1.153	1.423	1.096	1.652	1.456	1.467
124	1.322	1.183	1.375	1.226	1.223	1.761	1.155	1.423	1.095	1.650	1.458	1.469
125	1.322	1.183	1.374	1.226	1.222	1.763	1.157	1.423	1.094	1.648	1.460	1.470
126	1.322	1.184	1.374	1.226	1.222	1.766	1.159	1.423	1.093	1.645	1.462	1.472
127	1.322	1.185	1.373	1.226	1.222	1.768	1.160	1.423	1.093	1.643	1.464	1.474
128	1.322	1.185	1.373	1.226	1.222	1.771	1.162	1.422	1.092	1.641	1.465	1.475
129	1.322	1.186	1.372	1.226	1.222	1.773	1.163	1.422	1.090	1.638	1.467	1.477
130	1.321	1.186	1.371	1.226	1.222	1.776	1.164	1.421	1.089	1.636	1.469	1.478
131	1.321	1.187	1.371	1.226	1.222	1.778	1.164	1.421	1.088	1.634	1.470	1.480
132	1.321	1.187	1.370	1.226	1.222	1.781	1.165	1.420	1.086	1.631	1.472	1.481
133	1.321	1.188	1.370	1.226	1.221	1.784	1.165	1.420	1.085	1.629	1.473	1.483
134	1.321	1.188	1.370	1.226	1.221	1.787	1.166	1.420	1.084	1.626	1.475	1.485
135	1.321	1.189	1.370	1.226	1.221	1.790	1.168	1.419	1.083	1.624	1.477	1.486
136	1.321	1.189	1.370	1.226	1.221	1.793	1.169	1.419	1.082	1.622	1.478	1.488
137	1.322	1.189	1.370	1.226	1.222	1.796	1.170	1.419	1.081	1.619	1.480	1.489
138	1.322	1.190	1.369	1.226	1.222	1.798	1.172	1.419	1.080	1.617	1.482	1.491
139	1.322	1.190	1.369	1.227	1.222	1.801	1.173	1.418	1.079	1.615	1.484	1.492
140	1.322	1.191	1.369	1.227	1.222	1.803	1.174	1.418	1.078	1.612	1.486	1.494
141	1.322	1.191	1.368	1.227	1.222	1.806	1.175	1.418	1.077	1.610	1.487	1.496
142	1.322	1.192	1.368	1.227	1.222	1.808	1.176	1.418	1.076	1.608	1.489	1.498
143	1.322	1.193	1.367	1.228	1.222	1.809	1.177	1.418	1.075	1.605	1.490	1.499
144	1.322	1.193	1.367	1.228	1.222	1.811	1.178	1.418	1.074	1.603	1.492	1.501
145	1.322	1.194	1.366	1.228	1.222	1.813	1.180	1.418	1.073	1.601	1.495	1.502
146	1.322	1.194	1.366	1.228	1.222	1.815	1.181	1.418	1.073	1.599	1.497	1.503
147	1.322	1.195	1.366	1.229	1.222	1.817	1.183	1.418	1.072	1.597	1.499	1.505
148	1.322	1.195	1.366	1.229	1.222	1.819	1.184	1.418	1.071	1.594	1.502	1.506
149	1.322	1.196	1.366	1.229	1.222	1.821	1.186	1.418	1.071	1.592	1.504	1.508
150	1.322	1.196	1.365	1.229	1.222	1.822	1.187	1.419	1.070	1.590	1.507	1.509
151	1.321	1.197	1.365	1.230	1.222	1.824	1.189	1.419	1.070	1.588	1.509	1.510
152	1.321	1.197	1.365	1.230	1.222	1.825	1.191	1.420	1.069	1.586	1.511	1.511
153	1.320	1.198	1.364	1.230	1.222	1.826	1.194	1.420	1.069	1.584	1.513	1.512
154	1.320	1.199	1.364	1.231	1.222	1.827	1.196	1.421	1.068	1.582	1.515	1.513
155	1.320	1.199	1.364	1.231	1.222	1.829	1.198	1.421	1.067	1.580	1.516	1.513
156	1.319	1.200	1.363	1.231	1.222	1.830	1.201	1.422	1.066	1.578	1.518	1.514
157	1.319	1.201	1.363	1.231	1.222	1.832	1.203	1.422	1.065	1.576	1.520	1.515
158	1.318	1.202	1.362	1.231	1.222	1.833	1.205	1.423	1.065	1.575	1.522	1.515
159	1.318	1.203	1.361	1.232	1.222	1.834	1.207	1.423	1.064	1.573	1.524	1.516
160	1.318	1.204	1.361	1.232	1.222	1.835	1.210	1.423	1.063	1.571	1.526	1.517
161	1.317	1.205	1.360	1.232	1.222	1.835	1.211	1.423	1.062	1.569	1.528	1.517
162	1.317	1.205	1.359	1.232	1.223	1.836	1.213	1.423	1.061	1.567	1.530	1.518
163	1.316	1.206	1.359	1.232	1.223	1.837	1.215	1.423	1.060	1.566	1.532	1.518
164	1.316	1.207	1.358	1.233	1.223	1.837	1.216	1.423	1.059	1.564	1.533	1.519
165	1.315	1.208	1.357	1.233	1.223	1.837	1.218	1.423	1.058	1.562	1.535	1.519
166	1.314	1.209	1.357	1.233	1.223	1.837	1.220	1.423	1.058	1.560	1.536	1.519
167	1.313	1.210	1.356	1.233	1.224	1.838	1.221	1.423	1.057	1.558	1.537	1.519
168	1.312	1.210	1.355	1.233	1.224	1.838	1.223	1.423	1.056	1.556	1.539	1.519
169	1.311	1.211	1.355	1.233	1.224	1.838	1.224	1.423	1.055	1.554	1.539	1.520
170	1.310	1.212	1.354	1.233	1.225	1.839	1.224	1.424	1.054	1.552	1.540	1.520
171	1.310	1.213	1.353	1.233	1.225	1.839	1.225	1.424	1.053	1.550	1.542	1.520
172	1.309	1.214	1.352	1.233	1.226	1.839	1.225	1.425	1.053	1.548	1.542	1.520
173	1.308	1.215	1.351	1.233	1.226	1.839	1.226	1.425	1.052	1.546	1.543	1.520
174	1.308	1.215	1.350	1.233	1.227	1.839	1.226	1.425	1.051	1.544	1.544	1.521
175	1.307	1.216	1.349	1.233	1.227	1.839	1.227	1.425	1.051	1.542	1.546	1.521
176	1.306	1.217	1.348	1.233	1.228	1.839	1.227	1.425	1.050	1.540	1.547	1.521
177	1.306	1.217	1.347	1.234	1.228	1.839	1.227	1.425	1.049	1.538	1.548	1.522
178	1.305	1.218	1.346	1.234	1.229	1.839	1.228	1.426	1.049	1.536	1.549	1.522
179	1.305	1.219	1.345	1.234	1.229	1.839	1.228	1.426	1.048	1.534	1.550	1.523
180	1.304	1.219	1.344	1.234	1.229	1.839	1.228	1.426	1.047	1.532	1.552	1.523
181	1.304	1.220	1.343	1.235	1.230	1.838	1.229	1.426	1.047	1.530	1.553	1.524
182	1.303	1.220	1.343	1.235	1.230	1.838	1.229	1.426	1.046	1.528	1.555	1.525
183	1.302	1.220	1.342	1.235	1.231	1.838	1.229	1.426	1.046	1.525	1.557	1.525
184	1.302	1.221	1.341	1.235	1.231	1.837	1.229	1.427	1.045	1.523	1.558	1.526
185	1.301	1.221	1.340	1.236	1.231	1.837	1.229	1.427	1.045	1.521	1.559	1.526
186	1.300	1.222	1.339	1.236	1.232	1.837	1.228	1.427	1.045	1.519	1.560	1.527
187	1.299	1.222	1.339	1.236	1.232	1.836	1.228	1.428	1.044	1.517	1.561	1.527
188	1.299	1.222	1.338	1.237	1.233	1.836	1.229	1.428	1.044	1.515	1.562	1.528
189	1.298	1.223	1.338	1.237	1.233	1.836	1.229	1.429	1.043	1.513	1.563	1.528
190	1.298	1.223	1.337	1.237	1.233	1.836	1.229	1.429	1.043	1.511	1.564	1.528
191	1.297	1.224	1.337	1.237	1.234	1.835	1.229	1.429	1.043	1.509	1.565	1.528
192	1.296	1.224	1.336	1.238	1.234	1.835	1.229	1.429	1.042	1.507	1.565	1.529
193	1.296	1.224	1.336	1.238	1.234	1.834	1.229	1.430	1.042	1.505	1.566	1.529
194	1.295	1.225	1.336	1.238	1.235	1.833	1.229	1.430	1.041	1.503	1.565	1.529
195	1.294	1.225	1.335	1.238	1.235	1.832	1.230	1.430	1.041	1.501	1.565	1.530
196	1.294	1.226	1.335	1.239	1.235	1.832	1.230	1.430	1.040	1.499	1.565	1.530
197	1.293	1.226	1.335	1.239	1.236	1.831	1.230	1.430	1.040	1.497	1.565	1.531
198	1.293	1.226	1.335	1.239	1.236	1.831	1.230	1.430	1.039	1.495	1.564	1.531
199	1.292	1.227	1.335	1.239	1.236	1.830	1.230	1.431	1.039	1.493	1.564	1.532
200	1.291	1.227	1.334	1.240	1.236	1.830	1.230	1.431	1.038	1.491	1.564	1.532
201	1.291	1.228	1.334	1.240	1.236	1.829	1.230	1.432	1.038	1.490	1.564	1.533
202	1.290	1.228	1.334	1.240	1.236	1.829	1.230	1.433	1.037	1.488	1.564	1.533
203	1.289	1.229	1.334	1.241	1.237	1.828	1.230	1.434	1.037	1.486	1.564	1.534
204	1.289	1.229	1.334	1.241	1.237	1.828	1.231	1.434	1.037	1.484	1.563	1.534
205	1.288	1.230	1.333	1.241	1.237	1.827	1.232	1.435	1.036	1.482	1.563	1.535
206	1.288	1.230	1.333	1.242	1.237	1.826	1.232	1.435	1.036	1.481	1.562	1.535
207	1.288	1.230	1.332	1.242	1.237	1.825	1.233	1.435	1.035	1.479	1.562	1.535
208	1.287	1.230	1.332	1.242	1.237	1.825	1.234	1.436	1.035	1.477	1.561	1.535
209	1.287	1.231	1.332	1.243	1.237	1.823	1.234	1.436	1.034	1.476	1.561	1.535
210	1.286	1.231	1.331	1.243	1.237	1.822	1.235	1.436	1.034	1.474	1.560	1.535
211	1.285	1.231	1.331	1.243	1.237	1.821	1.235	1.436	1.034	1.473	1.560	1.534
212	1.285