Visibility graph analysis of economy policy uncertainty indices
aa r X i v : . [ q -f i n . S T ] J u l Visibility graph analysis of economy policy uncertainty indices
Peng-Fei Dai a , Xiong Xiong a,b , Wei-Xing Zhou c,d,e, ∗ a College of Management and Economics, Tianjin University, Tianjin 300072, China b China Center for Social Computing and Analytics, Tianjin University, Tianjin 300072, China c Department of Finance, East China University of Science and Technology, Shanghai 200237, China d Research Center for Econophysics, East China University of Science and Technology, Shanghai 200237, China e Department of Mathematics, East China University of Science and Technology, Shanghai 200237, China
Abstract
Uncertainty plays an important role in the global economy. In this paper, the economic policy uncertainty (EPU)indices of the United States and China are selected as the proxy variable corresponding to the uncertainty of nationaleconomic policy. By adopting the visibility graph algorithm, the four economic policy uncertainty indices of theUnited States and China are mapped into complex networks, and the topological properties of the correspondingnetworks are studied. The Hurst exponents of all the four indices are within [0 . , L ( N ) = .
626 ln N + . Keywords:
Econophysics; Economic policy uncertainty; Complex network; Visibility graph
PACS:
1. Introduction
Nowadays, the development of world economy has been highly globalized, and the ties between different economiesare getting stronger and stronger. The internal factors and external environment that affect economic development arechanging over time. Hence, uncertainty has become a new normal that the global economy needs to face. Particularly,economic policy uncertainty (EPU) is related to the development of financial markets and even the harmony and sta-bility of the society. The research on economic policy uncertainty becomes a hot topic in the fields of macro-economyand macro-finance.The study on economic policy uncertainty mainly focuses on theoretical research and econometric analysis. P´astorand Veronesi developed a general equilibrium model to analyze how changes in government policy choice affect stockprices [1]. In their model, two types of uncertainty are considered, the political uncertainty and the impact uncertainty.P´astor and Veronesi further set up a general equilibrium model of government policy choice to study the relationshipbetween political uncertainty and stock risk premium [2]. In their study, some simple exploratory empirical analyseswere conducted, where the economic policy uncertainty index constructed by Baker et al. was adopted as the proxyvariable of policy uncertainty [3]. Segal et al. decomposed the aggregate uncertainty into ‘good’ and ‘bad’ volatilitycomponents, associated with positive and negative innovations to macroeconomic growth [4]. According to theirtheoretical analysis, the two kinds of uncertainty have opposite effects on economic growth and asset prices. Bakeret al. developed a new index of economic policy uncertainty, and made empirical analyses of the impact of economic ∗ Corresponding author.
Email address: [email protected] (Wei-Xing Zhou )
Preprint submitted to Elsevier July 28, 2020 olicy uncertainty on micro-finance and macro-economy from the perspectives of firm levels and macro levels [3].Jes´us et al. studied empirically the effects of changes in uncertainty about future fiscal policy on the aggregateeconomic activity [5]. Brogaard and Detzel used a search-based measure to capture economic policy uncertainty for21 economies, and found economic policy uncertainty has a significant effect on the contemporaneous market returnsand volatility [6]. Dakhlaoui and Aloui found strong evidence of a time-varying correlation between US economicuncertainty and stock market volatility [7]. What is worth mentioning is that the studies of the EPU index attract theinterest of a large number of scholars since Baker et al. introduced the concept [3].Alternatively, it is possible to study the EPU time series from the view angle of complex networks. Indeed,quite a few methods have been developed to convert time series into complex networks [8–11]. Zhang and Smallconnected time series with complex networks, by constructing complex networks from pseudoperiodic time series,and investigate the relationship between the topology of the constructed network and dynamics of the raw time series[12, 13]. Yang et al. and Gao et al proceeded to construct segment correlation networks [14, 15]. Xu et al. designed themethod to map time series into nearest neighbor networks [16]. Marvan et al. introduced the concept of recurrencenetworks [17, 18]. Lucasa et al. introduced the visibility graph (VG) algorithm for time series [19], and variousexpansions of the visibility graph algorithm have been proposed [20–34]. At the same time, in-depth analyses of theseVG-based methods and their applications appeared in different fields [35–52].In this paper, we convert the time series of EPU indices into complex networks by adopting the visibility graphalgorithm and study the topological properties of the constructed complex networks. The remainder of the paper isorganized as follows. Section 2 describes the data we analyze. Section 3 presents briefly the visibility graph algorithm.In Section 4, the topological properties of the uncertainty networks are studied in detail. Section 5 concludes.
2. Data description
The EPU index is the proxy variable of economic policy uncertainty. Baker et al. constructed the index and updatethe data regularly on a website ( ) [3]. To measure policy-relatedeconomic uncertainty of the United States, they constructed an index from three types of underlying components. Thefirst component quantifies the newspaper coverage of policy-related economic uncertainty. The second componentreflects the number of federal tax code provisions set to expire in future years. The third component uses disagreementamong economic forecasters as a proxy for uncertainty. To measure the EPU for China, they constructed a scaledfrequency count of articles about policy-related economic uncertainty in the South China Morning Post (SCMP), oneof Hong Kong’s leading English-language newspapers. The method follows the news-based EPU index for the UnitedStates.We retrieved the EPU indices of the United States and China, the world’s two largest economies. The EPU indicesof the USA include monthly data, from January 1985 to November 2018, and daily data, from 1 January 1985 to 11November 2018. The monthly data comprise the EPU index and the news-based EPU index. In contrast, the dailydata are only news-based EPU index. The EPU index of China contains only the monthly data, and its range is fromJanuary 1985 to November 2018. Figure 1 illustrates the temporal evolution of the four EPU indices.Table 1 shows the summary statistics of the EPU indices, including their mean, median, minimum, maximum,standard deviation, skewness and kurtosis. According to Table 1, all the three mean values of the US EPU indicesare close to the benchmark value of 100, while that of China is 43.75% above the benchmark value. By comparingthe standard deviations of the monthly EPU indices in the two countries, we can find that the volatility of China’sEPU inde is much higher than that of the United States. The monthly EPU index of the United States (EPU-US-M)ranges from 57.21 to 245.13, while the news-based monthly US EPU index (EPU-US-News-M) varies from 44.78 to283.67. China’s monthly EPU index (EPU-CN-M) varies widely, from 9.067 to 694.849. Furthermore, the data in theUnited States cover a larger time range than that in China. This shows that from January 1995 to November 2018, thevariation of EPU index in the United States is more stable than that in China and the economic policy uncertainty ofChina is higher and more volatile than that of the United States. Analysing only the statistical results of the UnitedStates, we find that the news-based EPU-US-News-M is almost the same as the EPU index in terms of mean, standarddeviation, skewness and kurtosis. 2
985 1990 1995 2000 2005 2010 2015 20200100200300400500600700800 E P U i nd e x (a) E P U i nd e x (b) E P U i nd e x (c) E P U i nd e x (d) Figure 1: Temporal evolution of the four EPU indices: (a) daily EPU index of the USA (EPU-US-D), (b) monthly EPU index of the USA (EPU-US-M), (c) news-based monthly EPU index of the USA (EPU-US-News-M), and (d) monthly EPU index of China (EPU-CN-M).Table 1: Summary statistics of the EPU indices.
Variable Observation Mean Median Minimum Maximum Std. dev. Skewness KurtosisEPU-US-D 12368 100.93 83 .
63 3 .
32 719.07 68 .
43 1.86 5.95EPU-US-M 407 108.11 102 .
20 57 .
20 245.13 31 .
30 0.96 0.86EPU-US-News-M 407 111.26 102 .
02 44 .
78 283.67 40 .
10 1.27 1.95EPU-CN-M 286 143.75 104 .
43 9 .
07 694.85 117 .
16 2.22 5.323 . The visibility algorithm
The visibility graph algorithm maps a time series { y i = y ( t i ) } i = ,..., N into a visibility graph G [19], where each datavalue ( y i ) in the time series corresponds to a data point ( t i , y i ), which is viewed as a node in the visibility graph. Anytwo nodes in the visibility graph are connected if they are visible to each other. Mathematically, two points ( t i , y i ) and( t j , y j ) are visible to each other if and only if: y j − y n t j − t n > y j − y i t j − t i , (1)for any t n ∈ ( t i , t j ).According to the construction manner, each point is at least connected to its two neighbor points. Thereforeevery visibility graph is connected. In this basic setting, the edges in the network are not weighted, and there are nodirections for the edges. That is, the constructed networks are unweighted and undirected.
4. Empirical analysis
The Hurst exponent can be used to measure whether a time series is long-range correlated or not and is also animportant indicator to describe the fractal characteristics of the time series. Time series can be classified into threecategories due to their auto-correlation features quantified by their Hurst exponents: A time series is anti-persistent if0 < H < .
5, uncorrelated if H = .
5, or persistent if 0 . < H <
1. We adopt the derended fluctuation analysis (DFA)to estimate the Hurst exponent of each EPU index, where quadratic polynomial is employed [53–55]. For the sake ofillustration, the scaling behavior curves of the four EPU indices are presented in Fig. 2. (a) (b) (c) (d) Figure 2: The scaling behavior curves of the four EPU indices: (a) EPU-US-D, (b) EPU-US-M, (c) EPU-US-News-M, and (d) EPU-CN-M, where n is the timescale and F ( n ) is the fluctuation function. The Hurst exponent is 0.835 for the EPU-US-D index, 0.915 for the EPU-US-M index, 0.801 for the EPU-US-News-M index, and 0.924 for the EPU-CN-M index. It shows that all the EPU indices exhibit very strong persistence,which can be seen in Fig. 1 through eyeballing. It means that the economic policy uncertainty is persistent in eachcountry. 4 .2. Degree distributions
The degree k of a node in a network is the number of nodes connected to it. As shown in Table 2, the minimumdegree k min of each EPU network is 2, indicating that the endpoints of the time series are visible to at least oneadditional point other than their neighbor nodes. For the networks converted from the monthly EPU series, themaximum degrees k max is close to 56 and the average degree h k i is close to 8. In contrast, for the daily data, themaximum degree is much larger, but the average degree is smaller. Table 2: Topological statistics of the four EPU networks. k is the degree, γ is tail exponent of the degree distribution, C is the clustering coefficient,and r is degree-degree correlation coefficient. Network h k i k i , max k i , min γ C C i , max C i , min r EPU-US-M 7.86 56 2 1.99 0.76 1 0.11 0.08EPU-US-News-M 7.97 56 2 2.13 0.77 1 0.11 0.04EPU-CN-M 8.06 59 2 1.83 0.76 1 0.13 0.07EPU-US-D 6.99 158 2 2.78 0.77 1 0.05 0.12We estimate the empirical degree distributions of the four EPU networks, as illustrated in Fig. 3. It shows that allthe distributions have power-law tails: p ( k ) ∼ k − γ , (2)where γ is the power-law tail exponent. The tail exponents are presented in Table 2. The goodness-of-fit values of thefour linear regressions are 0.8703 in Fig. 3(a), 0.9184 in Fig. 3(b), 0.9057 in Fig. 3(c), 0.9539 in Fig. 3(d) respectively.Therefore, the EPU networks are scale-free. -4 -3 -2 -1 (a)(a) -4 -3 -2 -1 (b)(b) -4 -3 -2 -1 (c)(c)(c) -5 -4 -3 -2 -1 (d)(d) Figure 3: The degree distributions of the networks converted from the four EPU time series: (a) EPU-CN-M, (b) EPU-US-M, (c) EPU-US-News-M,and (d) EPU-US-D. .3. Clustering coefficient The clustering coefficient measures the cliquishness of a node in a network [56]. For any node V in a network, itsclustering coefficient is defined as follows: C i = E i k i ( k i − , (3)where k i is the degree of node i and E i is the number of edges between i ’s neighbor points. We obtain all the C i values.According to Table 2, the maximum clustering coefficient is 1, which is somehow trivial, showing that there are localcliques. The minimum clustering coefficients are close and small for the three networks mapped from the monthlyindices, and that from the daily indices is smaller.The clustering coefficient of the network is defined as the average clustering coefficient of all nodes: C = h C i i i = N N X i = C i , (4)which measures the probability of triadic closure in the network. If C =
1, the network is a complete graph, where anytwo nodes of the network are connected to each other. According to Table 2, the average cluster clustering coefficientis large (about 0.76) for all the EPU networks. This is consistent with the strongly persistent feature of the indices. Italso shows that the edges of the EPU networks are dense.
Mixing pattern is an important topological statistic of complex networks, where the assortativity refers to thefact that nodes in a network are more likely to be connected with nodes of comparable degrees [57]. In assortativenetworks, nodes with high degrees are more likely to connect to other nodes also with high degrees, while nodes withlow degrees are more likely connected to low-degree nodes. Similarly, nodes connected to nodes with lower degreemay also have lower degree. The mixing pattern is generally quantified by the Pearson correlation coefficient betweennode degrees: r = M − P i j i k i − h M − P i ( j i + k i ) i M − P i (cid:16) j i + k i (cid:17) − h M − P i ( j i + k i ) i (5)where M is the number of all edges in the network, j i and k i are the degrees of the endpoint at the i th edge. If r iszero, the network has no assortative mixing. If r is positive, the network exhibits assortative mixing. If r is negative,the network shows disassortative mixing.Table 2 lists the assortativity coefficients of the four economic policy uncertainty networks. All the assortativecoefficients are positive but small, which implies that the four networks exhibit weak assortative mixing patterns. Theassortativity coefficient of EPU-US-D is larger than the other three ones. In the network corresponding to higher-frequency economic policy uncertainty index, there exists relatively stronger assortativity. In terms of economicpolicy uncertainty, the networks from monthly data have only weak assortativity. In networks from the visibilityalgorithm, the larger the value of EPU index is, the higher degree of the corresponding node will be to a large extend.A considerable degree of economic policy uncertainty in the economic environment will not disappear quickly in theshort term. Hence, there will be a period of high economic policy uncertainty which explains why the day-basedassortativity coefficient is higher than the month-based assortativity coefficient. A network with the small-world property usually has two main characteristics [56]. Firstly, the average shortestpath length L ( N ) of a network fulfills: L ( N ) ∼ ln N (6)where N is the number of nodes in the network, and L ( N ) is calculated as follows: L ( N ) = N ( N − X d ( i , j ) (7)6here d ( i , j ) is the shortest path length (or distance) between any two nodes i and j . Secondly, the clustering coeffi-cient of the network should be large. Figure 4: Dependence of the average shortest path length on the total number of nodes for the EPU network converted from the daily US data.
Fig. 4 shows the dependence of the average shortest path length of the network on the total number of nodes forthe EPU network converted from the daily US data. We do not analyze the networks from the monthly data becausetheir sizes are not large enough. A linear regression gives that L ( N ) = .
626 ln N + .
5. Conclusions
This paper utilizes the EPU indices as a proxy variable of economic policy uncertainty. The data shows that Chinahas much higher and more volatile economic policy uncertainty than the United States. The Hurst exponents of theEPU index time series in the United States and China have been calculated, which are found to be strongly long-rangecorrelated. It suggests that the economic policy uncertainty is persistent in different economies.The EPU indices are mapped into complex networks by the visibility graph algorithm. We studied the topologicalproperties of these networks. By investigating the degree distributions, the clustering coefficients, the mixing pat-terns, and the shortest path lengths, we unveiled that the EPU networks are scale-free, densely connected and weaklyassortative. The EPU network constructed from the daily US index also exhibit evident small-world features.Our research highlights the possibility to study the EPU from the view angle of complex networks. For single timeseries, we can convert them into networks using different methods. The visibility graph approach used in this workis only one of them. Different mapping methods result in different networks. It is thus possible to unveil differentstructural properties that may correspond to different temporal features of the time series. We can also consider cross-sectional methods such as the random matrix theory, which aims at understanding the global and regional behavior ofeconomic policy uncertainty. These issues can be investigated in the future research.
Acknowledgements
This work was supported by the National Natural Science Foundation of China (Grants No. 71532009, U1811462and No. 71790594), the Fundamental Research Funds for the Central Universities, and Tianjin Development Programfor Innovation and Entrepreneurship.
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