Twenty years of market capitalisation: the statistical properties of a flagship WFE indicator

Published by: The WFE Research Team Feb 2020

WFE statistics are an important asset for the WFE. They are widely used among researchers, practitioners, journalists and other stakeholders; they are frequently cited in news, reports and academic papers. Given the crucial role the production and dissemination of statistics plays in the WFE value proposition, we believe the WFE team should also provide insights into the statistical properties of our indicators. This piece focuses on a flagship one: equity market capitalisation. This article is intended to review how this indicator can be analysed and/or implemented in wider analyses based on its statistical properties.

To study the statistical properties of equity market capitalisation we obtained monthly time series for the period January 2000 – December 2019 (twenty years of data, for a maximum of 240 observations) for all exchanges reported by the WFE. We collected a total of 114 time series. Sample sizes varied between exchanges as several markets started reporting at a later date. To ensure accuracy of our statistical analyses, we retained time series for exchanges that have at least five years of data (60 observations). This left us with 90 individual time series. We further excluded exchanges the statistics of which were subsequently consolidated into broader exchange group statistics (such as those for individual exchanges comprising Nasdaq Nordic or Euronext), which left us with 75 individual time series. We analysed time series in local currency (as opposed to USD) to avoid considering exchange rate dynamics into the analyses.

The first step in our analysis was to perform a visual inspection of time series plots to infer their statistical properties. A visual inspection allowed us to conclude that most individual time series were characterised by a clear linear trend, either positive or negative (Group 1). A smaller group of exchanges (generally located in Europe) did not show any clear linear trend in their time series (Group 2). Occasionally, time series were characterised by structural breaks  [1]  likely due to changes in market or macroeconomic conditions, though they could typically be bucketed in either Group 1 or 2. A time series pattern from each group is reported below:

Figure 1: Time series pattern of an exchange from Group 1 (with a linear trend)

 

Figure 2: Time series pattern of an exchange from Group 2 (without a linear trend)

After having visually inspected the data, we decided to determine whether the individual time series of market capitalisation are stationary (i.e. oscillating around a finite mean) or characterised by a unit root (i.e., at a basic level, their mean changes over time). This is important as the statistical properties of a non-stationary time series are different from usual: for example, first and second moments (the mean and the variance) of non-stationary time series cannot be computed in a meaningful way, while correlation and regression analyses between non-stationary time series have a misleading interpretation. To meaningfully study the relation between time series characterised by non-stationarity, one has to perform different kinds of analyses (i.e. cointegration).

To conclude whether our time series were stationary, we compared and contrasted three different statistical tests: the augmented Dickey-Fuller test, the Phillips-Perron test, and the GLS Dickey Fuller test. Given that time series of market capitalisation were likely to be autocorrelated (i.e. this month’s value might be correlated with the previous months’ values), and that this correlation could influence the results of the tests, we introduced past values of the series in the tests to take this correlation into account. To choose the number of past values (lags) we ran another statistical test.   [2] In many cases, the test suggested that the number of past values to be introduced in the unit root tests was equal to one.

The test results are displayed in Table 1 below:

Exchange Observations Trend Chosen lag (AIC) Result
ASX Australian Securities Exchange 238 Yes 1 Unit root with a trend
Abu Dhabi Securities Exchange 92 No 1 Unit root no trend
Amman Stock Exchange 136 Yes 7 Unit root with a trend
Athens Stock Exchange 226 No 1 Unit root no trend
B3 234 Yes 5 Unit root with a trend
BME Spanish Exchanges 238 No 1 Unit root no trend
BRVM 70 Yes 1 Unit root with a trend
BSE India Limited 201 Yes 2 Unit root with a trend
Bahrain Bourse 54 Yes 5 Unit root with a trend
Barbados Stock Exchange 63 Yes 4 Unit root with a trend
Beirut Stock Exchange 65 Yes 1 Unit root with a trend
Bermuda Stock Exchange 188 No 1 Unit root no trend
Bolsa Mexicana de Valores 238 Yes 1 Unit root with a trend
Bolsa Nacional de Valores 70 No 1 Unit root no trend
Bolsa de Comercio de Santiago 238 No 1 Unit root no trend
Bolsa de Valores de Colombia 178 No 1 Unit root no trend
Bolsa de Valores de Lima 237 Yes 2 Unit root with a trend
Bolsa de Valores de Panama 70 Yes 1 Unit root with a trend
Bolsa y Mercados Argentinos 225 Yes 14 Unit root with a trend
Borsa Istanbul 195 Yes 8 Unit root with a trend
Bourse de Casablanca 106 Yes 2 Unit root with a trend
Bucharest Stock Exchange 94 Yes 1 Unit root with a trend
Budapest Stock Exchange 212 No 3 Unit root no trend
Bursa Malaysia 238 Yes 1 Unit root with a trend
CEESG - Vienna 235 No 4 Unit root no trend
Chittagong Stock Exchange 70 No 1 Unit root no trend
Colombo Stock Exchange 238 Yes 1 Unit root with a trend
Cyprus Stock Exchange 154 No 13 Unit root with a trend
Deutsche Boerse AG 237 Yes 1 Unit root with a trend
Dhaka Stock Exchange 70 No 1 Unit root no trend
Dubai Financial Market 62 No 9 Unit root no trend
Euronext 226 Yes 1 Unit root with a trend
Euronext Dublin 224 No 4 Unit root no trend
Hanoi Stock Exchange 70 No 1 Unit root no trend
Hochiminh Stock Exchange 77 Yes 6 Unit root with a trend
Hong Kong Exchanges and Clearing 238 Yes 1 Stationary around a trend
Indonesia Stock Exchange 236 Yes 1 Unit root with a trend
Jamaica Stock Exchange 67 Yes 4 Unit root with a trend
Japan Exchange Group 70 No 1 Unit root no trend
Johannesburg Stock Exchange 228 Yes 10 Unit root with a trend
Kazakhstan Stock Exchange 106 Yes 1 Unit root with a trend
Korea Exchange 238 Yes 1 Unit root with a trend
LSE Group 130 Yes 1 Unit root with a trend
Luxembourg Stock Exchange 233 No 6 Unit root no trend
Malta Stock Exchange 118 Yes 1 Unit root with a trend
Moscow Exchange 93 Yes 2 Unit root with a trend
Muscat Securities Market 118 No 1 Unit root no trend
NYSE 227 No 11 Unit root with a trend
NZX Limited 94 Yes 1 Unit root with a trend
Nasdaq - US 229 Yes 10 Unit root with a trend
Nasdaq Nordic and Baltics 178 Yes 1 Unit root with a trend
National Stock Exchange of India 201 Yes 2 Unit root with a trend
Nigerian Stock Exchange 70 No 1 Unit root no trend
Oslo Bors 236 Yes 3 Unit root with a trend
Palestine Exchange 82 Yes 1 Unit root with a trend
Philippine Stock Exchange 235 Yes 4 Unit root with a trend
Qatar Stock Exchange 82 No 1 Unit root no trend
SIX Swiss Exchange 238 Yes 1 Unit root with a trend
Saudi Stock Exchange (Tadawul) 124 Yes 7 Unit root with a trend
Shanghai Stock Exchange 202 Yes 1 Unit root with a trend
Shenzhen Stock Exchange 200 Yes 3 Unit root with a trend
Singapore Exchange 238 No 1 Unit root no trend
Stock Exchange of Mauritius 164 Yes 3 Unit root with a trend
TMX Group 238 Yes 1 Unit root with a trend
Taipei Exchange 94 Yes 1 Unit root with a trend
Taiwan Stock Exchange 238 Yes 1 Unit root with a trend
Tehran Stock Exchange 227 No 12 Unit root no trend
Tel-Aviv Stock Exchange 237 No 2 Unit root no trend
The Egyptian Exchange 166 No 1 Unit root no trend
The Stock Exchange of Thailand 238 Yes 1 Unit root with a trend
Tunis Stock Exchange 71 Yes 1 Unit root with a trend
Ukrainian Exchange 65 Yes 1 Unit root with a trend
Warsaw Stock Exchange 235 No 4 Unit root no trend
Zagreb Stock Exchange 69 Yes 2 Unit root with a trend

 

As evident from Table 1, market capitalisation was overwhelmingly characterised by a unit root. The only time series for which the unit root hypothesis was rejected is Hong Kong Exchanges and Clearing, for which the null of a unit root was rejected at the 10% level.

The academic literature has pointed out that in presence of structural breaks, traditional unit root tests might perform poorly. Given that our time series contained the global financial crisis, we were mindful that structural breaks might affect our results. We therefore cross-checked our results against the Clemente, Montanes and Reyes unit root test, which accounts for the presence of up to two structural breaks, and the Zivot and Andrews test, which accounts for the presence of one structural break. While the tests allowed us to detect the presence of structural breaks (see figure 3 below), we found that our times series are overwhelmingly characterised by unit roots.

Figure 3: Structural break in the Athens Stock Exchange

This result should be kept in mind when computing first and second moments (the mean and the variance) of the series, or when performing correlation analyses between those series, given that the computation of all these statistics would be meaningless in the presence of non-stationarity. To correctly utilise market capitalisation statistics, users should take the first difference or the log difference of this indicator, which would likely be stationary and hence characterised by the usual statistic properties [3] or perform analyses that are appropriate for non-stationary time series (cointegration).  

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[1] A structural is an unexpected change in the intercept term or the slope coefficient of a regression model.

[2] We chose the autoregressive model that minimises the Akaike Information Criterion.

[3] This is true if the series is integrated of order one. While this typically turns out to be true for most financial stochastic processes, in principle one should test stationarity on the differentiated series as well.