The Network Factor of Equity Pricing:
A Signed Graph Laplacian Approach
Ajim Uddin
∗Martin Tuchman School of Management New Jersey Institute of Technology
Abstract
Firms in the equity market are interconnected to each other. This interconnection network is dynamic, complex, and its influence on equity prices is difficult to capture.
This paper designs a signed graph Laplacian approach to construct a network factor, Z-score, and tests the implication of network on equity prices. Z-score captures the network evolution of the U.S. equity market over time. Larger Z-scores indicate greater changes in the market network, and Z-scores spikes during major events such as the COVID-19. Incorporating the network factor into conventional asset pricing models improves return predictability. Cross-sectional analysis shows that firms with positive sensitivity to the network change have lower expected returns. The results are robust in time-series estimates, two-pass cross-sectional regression, and three-pass estimator.
These findings suggest that the network factor measured by the Z-score is not a weak factor and produces negative and significant risk premia.
Keywords— Dynamic Equity Network; Signed Graph; Laplacian Spectrum; Economic and Financial Crisis; Asset Pricing.
1 Introduction
Firms are connected through multiple types of networks. These network connections play an important role in how information and shocks transmit from one firm to another or to the whole system (Cohen & Frazzini, 2008; Grullon et al., 2014; Herskovic, 2018). A single firm may generate a large impact on the market, such as the collapse of Lehman Brothers during the 2008 financial crisis. Similarly, market changes can affect individual firms as well as the interconnection between firms. For example, the COVID-19 has accelerated the adoption of digital technologies and changed supply-chain interactions. The underlying network that dictates this interaction is not static, instead dynamically changing over time. Furthermore, the direction and magnitude of an individual firm’s exposure to this dynamic network changes vary across firms. Empirical evidence suggests that firms’ network exposures are associated with systematic risk, reduce diversification power (Billio et al., 2016), and warrant a centrality risk premium (Buraschi & Porchia, 2012). Meanwhile, most traditional asset pricing models mainly focus on firm-specific and market/macro factors, overlooking the indispensable interconnection among firms. In this work, I propose a novel approach for incorporating network information into asset pricing models.
The main challenge to learn the implication of network changes in asset prices is modeling and quantifying the network changes. The network representation inspired by graph theory is popular with wide applicability in other domains such as computer science and information system (Huang et al., 2020; Kipf & Welling, 2017; Li et al., 2017). Unfortunately, those techniques are mainly for static and unsigned graph. Directly applying those techniques to model equity market network may produce suboptimal results. Firms enter into or exit from the market and change their business models, capital structures, and supply chains. As a result, relationships among firms continually evolve. In addition, market shocks can affect the structure of the equity market network. A static graph built by aggregating all available information cannot capture such time-varying information.
Previous studies use the correlation of firms’ historical returns on a rolling basis to capture network dynamism (Chi et al., 2010; Mantegna, 1999; Namaki et al., 2011; Tumminello et al., 2010).
However, these works apply unsigned network frameworks, which ignore the positive/negative signs by either using a distance function or using the absolute values of the correlation. Such application
(a) December 2019 (b) April 2020 (c) December 2020
Figure 1: The network structure of S&P-500 stocks surrounding the COVID-19. Green (red) links represent the positive (negative) edge between two firms.
overlooks the core idea that a positive correlation indicates similarity and co-movement, while a negative correlation indicates the opposite.
Figure 1 and 2 demonstrate the importance of modeling dynamic network changes and incor- porating positive and negative connections in representing the equity market network. During the normal economic condition, the market consists of a mixture of positive and negative edges (1a and 1c). However, on the dawn of the Covid-19 pandemic (1b), most negative edges disappear, positive edges multiply, and the market forms a ball-shaped structure. The presence of negative edges conveys information that an unsigned graph is not able to reveal. For example, as shown in Figure 2:(a)-(c), the smallest Eigenvalue (the rightmost λ in the parentheses) of all unsigned graph Laplacians is zero and unable to offer any information to identify different structures of networks.
Differently, for the signed graph Laplacian, only the balanced networks in Figure 2e (network with two sub-clusters) contain the zero Eigenvalue. Therefore, incorporating negative linkage helps to partition a network into multiple sub-networks (clusters), with positive edges signifying intra-cluster cohesion and negative edges serving as inter-cluster bridges. As shown in Figure 1c, the negative links are able to bisect S&P-500 stock network into two subgroups and positive links bond the firms within the groups.
In this paper, I apply a generalized Laplacian matrix of Gallier (2016) with the modified Lapla- cian for a signed graph to handle both positive and negative network connections. Particularly, the diagonal degree matrix in the modified Laplacian sums the absolute values of the pair-wise correlations for each firm, and the affinity (weight) matrix contributes to retaining the signs of
(a) Connected network λ = (5, 5, 5, 5, 0)
(b) Star network λ = (5, 1, 1, 1, 0)
(c) Cyclic network λ = (3.6, 3.6, 1.4, 1.4, 0)
(d) Signed connected network λ = (6.37, 5, 5, 3, 0.63)
(e) Signed balanced network λ = (5, 5, 3, 3, 0)
(f) Signed cyclic network λ = (4, 2.62, 2.62, 0.38, 0.38)
Figure 2: Graph representations based on the Laplacian spectrum. Green (red) links repre- sent the positive (negative) edge between two nodes. λ denotes the Laplacian spectra (the Eigenvalues of the Laplacian matrix) of the graph. Intuitively, different network topologies have different connectivities and are accompanied by a distinct set of Eigenvalues.
correlations (details in Section 2.1). This modification ensures the positive-semidefinite property of the Laplacian matrix and the benefits therein.1 I then use the signed Laplacian spectra (the Eigenvalues of the Laplacian matrix) to encode the network structure of the equity market at each time point. The Laplacian spectra of a graph essentially represent the frequency domain of dis- crete networks and directly link to the global structures, properties, and motif of a network. As a result, the encoded representation of the network structure also incorporates the changing market condition, economic environment, and uncertainties in a macro context. Moreover, to capture the dynamic evolution, I construct the network factor “Z-score” by measuring the difference between the current network state and network states of previous months.
1A detailed discussion of using Eigenvalues and spectrum to represent heterogeneous networks and the advantages can be found in Jin and Zafarani (2020).
I evaluate the importance and implication of the network factor in equity pricing by examining the factor in relation to the market, macro, and other asset pricing factors. The empirical analysis provides several interesting findings. First, Z-score aligns well with significant market events, such as the 1987 Black Monday, 2008 financial crisis, and COVID-19. Those major events generate con- siderable and asymmetric impacts on different firms, significantly changing the network structure.
Moreover, the correlations of Z-score with VIX and EPU are only 0.15 and 0.19, respectively. This finding suggests that Z-score, to a large extent, reveals information different from volatility and uncertainty.
Second, I incorporate the proposed network factor into conventional asset pricing models and show that network is an important equity pricing factor. I follow the general practice in the literature and perform factor model comparison tests to assess whether adding network factor improves pricing models (Ahmed et al., 2018; Giglio & Xiu, 2021; Hou et al., 2020). Both two- pass cross-sectional regression (Fama & MacBeth, 1973) and three-pass estimator (Giglio & Xiu, 2021) produce significant and negative risk premia for the network factor. The time-series R2 for the network factor is much higher than other nontradable factors like macro-finance factors of Ludvigson and Ng (2009) and consumption growth factor of Malloy et al. (2009). The Wald-test for the null hypothesis that network is a weak factor is also rejected. Signifying that network is an important pricing factor for equity risk premia. In addition, time-series tests indicate that network factor indeed enhances return predictability and reduces mispricing. Results are robust for different asset portfolios with multiple performance metrics.
Third, cross-sectional analysis shows that firms with the positive (negative) sensitivity to net- work changes have lower (higher) future returns. If a given firm reacts to the change of network in the same direction, it suggests that this firm has the capability to adjust quickly and tends to be less vulnerable. Investors, therefore, treasure such adaptability. As a result, the demand for those stocks increases, prices increase, and the expected returns decrease. In addition, I document that this negative relation is more significant when the market network experiences substantial changes.
The proposed methodology for the network factor and its associated empirical findings con- tribute to the existing literature in a number of ways. First, this paper proposes a novel way of evaluating the effects of network dynamics on financial markets. Current works predominantly use
historical return correlation (Di Cerbo & Taylor, 2020; Namaki et al., 2011; Tumminello et al., 2010), industry sector similarity (Herskovic, 2018), and customer-supplier network (Herskovic et al., 2020) to represent financial networks and mainly analyze how information flows among firms and institutions in times of crisis and pandemic situations (Elliott et al., 2014; Le et al., 2020; Nobi et al., 2014; Rogers & Veraart, 2013; R¨onnqvist & Sarlin, 2015; So et al., 2020). In contrast, to construct the network factor, I apply the state-of-the-art Laplacian spectrum analysis technique in the equity market to represent the network. In addition to discussing the importance of negative signs in equity networks, I demonstrate how to effectively incorporate this information in equity network modeling. The general framework for the signed Laplacian spectrum detailed in this pa- per can also extend to analyze the network structure of other financial markets where negative connections exist.
Second, I explicitly construct a network index Z-score to track the aggregated changes at the market level rather than the pairwise correlation at the firm level. Z-score provides direct informa- tion to market participants about the network state of the equity market. As a result, it can be considered complementary to the existing macro indices and asset pricing factors.
Third, with the empirical analysis, I show that network is an important pricing factor and uncover how firms respond to the change of network. The result also suggests that the exposure to network changes is positive for big firms, whereas network changes negatively influence small firms.
These findings can provide useful insight about portfolio diversification to investors.
The remainder of the paper proceeds as follows. Section 2 first presents the equity market network representation method for the signed graph and then provides details for constructing the network factor. Section 3 discusses the data and the results from empirical analysis. This section provides a thorough examination of the network factor’s application and validity and evaluates its significance in equity pricing. Finally, Section 4 summarizes the findings and provides conclusions.
2 Methodology
2.1 Equity Market Network Representation with Dynamic Signed Graph
The network structure of the U.S. equity market at time t can represent by a weighted graph Gt= (Vt, Et, Wt), where each node v ∈ Vt represents a firm, each edge eij ∈ Et shows a connection between firms i and j at time t, and Wt∈ RN ×N is a weighted adjacency matrix representing the quantitative proximity among firms at time t with weights wij for all eij ∈ Et, otherwise wij = 0.
The network structure in each time point (t) represents the market state at that time.2 A proper representation of this network and the efficient extraction of network information are necessary to understand the equity market as a whole (Di Cerbo & Taylor, 2020; Namaki et al., 2011).
An efficient representation of the equity market network over time will reveal how information dissipates in the network and how firms react to financial crises, and major economic and political events.
Graph Laplacian and the network spectrum based analysis are heavily applied in computer science and social science to represent and extract information from networks. It has its origin from the graph spectrum theory (Chung & Graham, 1997) and can reveal the connectivity, local and global structures, and motifs of networks (Jin & Zafarani, 2020). Encouraged by their success, in this work, I apply the Laplacian spectrum to extract useful properties of the equity network. For a simple (unweighted, undirected with no multiple edges incident to the same two vertices) graph G = (V, E) with V is the set of nodes |V | = N , edges E ⊆ V × V , and adjacency matrix A ∈ RN ×N given by aij = 1 if i and j are adjacent and aij = 0 otherwise, the Laplacian matrix L can define as follows:
Lij =
deg(i) i = j
−1 if i and j are adjacent (i ∼ j)
0 otherwise
2The time subscript is dropped if it is clear that it does not influence the basic understanding of the described procedure.
where deg(i) is the degree of node i, i.e., the number of edges incident to node i. Let, D ∈ RN ×N is a diagonal matrix with the diagonal elements as (deg(1), ..., deg(n)), L is rewritten as
L = D − A (1)
The eigenvalues 0 = λ0 ≤ λ1≤ · · · ≤ λN −1of the Laplacian matrix L constitute graph’s spectrum.
L is symmetric and positive-semidefinite; that is, λi ≥ 0 for all i (Chung & Graham, 1997). The eigenvalues of the Laplacian matrix reveal the critical properties of the graph. The smallest non- zero eigenvalue of L represents the first spectral gap. The second smallest eigenvalue of L represents the algebraic connectivity of the graph (Fiedler, 1973). When the graph is connected, the algebraic connectivity (Fiedler value) is the same as the first spectral gap. The eigenvectors associated with the K smallest eigenvalues of the graph Laplacian capture critical information content and provide a low dimensional graph embedding (Belkin & Niyogi, 2003).3
A weighted undirected graph G defines weight matrix W ∈ RN ×N, where wij = wji if i and j are adjacent, and wij = 0 if i is not adjacent to j in G. Dii = P
j∼iwij is the degree of i. The unnormalized Laplacian matrix of a weighted graph G is similar to equation 1, as L = D − W . The positive-semidefinite property of the graph Laplacian spectrum also holds when the weighted graph is non-negative (Chung & Graham, 1997).
Following the literature, I use the correlation of historical returns as a measurement of firms’
proximity and capture the U.S. equity market network (Puliga et al., 2014; Sandhu et al., 2016).
Unlike previous studies, instead of removing negative edges or using absolute values, I use both positive and negative correlations to construct a signed network (see details about network con- struction in the Appendix A.1). I calculate the modified signed Laplacian matrix following Kunegis et al. (2010) and Gallier (2016) as ¯L = ¯D − W , where ¯D is:
D¯ii=
N
X
j=1
|Wij| (2)
Figure 3 shows a simple example of how the adjacency matrix, degree matrix, and Laplacian matrix are constructed from a signed graph based on the method.
3Graph embedding is the process of representing an entire graph in a vector space.
Figure 3: An undirected weighted signed graph, its adjacency matrix, degree matrix, and Laplacian matrix calculated based on the proposed model.
The modified diagonal degree matrix ¯D ensures that in a signed, weighted, undirected graph G, the modified signed Laplacian matrix ¯L is positive-semidefinite. That is, the eignevalues λ0· · · λN −1 of ¯L is non-negative. This can be proved with the incidence matrix of the signed G. For a signed graph with weights, following Biyikoglu et al. (2007) the |E|×|V | oriented weighted graph incidence matrix can define as follows:
Si∼j,v =
+p|Wi,j| if v = i
−sgn(Wij)p|Wij| if v = j
0 otherwise
(3)
The diagonal and off-diagonal entries of the product S>S ∈ RV ×V are:
(S>S)ii=X
j∼i
|Wij|
(S>S)ij = −Wij
(4)
where S> is the matrix transpose of S. This shows S>S = ¯L. The eigendecomposition of ¯L generates N eigenvectors xi ∈ RV and their corresponding eigenvalues λi can be shown as:
λi = x>i Lx¯ i
= x>i S>Sxi
= (Sxi)>(Sxi)
(5)
as the inner product of Sx with itself is ≥ 0. ∀1 ≤ i ≤ N, λi ≥ 0. Therefore, equation 5 ensures that the Laplacian of weighted graph ¯L is also positive-semidefinite with non-negative eigenvalues.
The network representation for equity market at each time point t is constructed by calculating the Laplacian spectrum λt ∈ RK of Gt by taking the K lowest eigenvalues of ¯L after solving the generalized eigenvalue problem ¯Lx = λx. An efficient representation of the network structure does not require all N eigenvalues. A large enough K lowest eigenvalues of the Laplacian matrix are sufficient to capture almost all necessary graph properties for building an efficient embedding (Belkin & Niyogi, 2003; Fiedler, 1973). In addition, the Laplacian eigenmaps theory suggests that the eigenvectors associated with the largest eigenvalues encode high-frequency changes among nodes and noise. Therefore, ignoring the (N − K) largest eigenvalues disentangles the noise from the useful network information (Belkin & Niyogi, 2003). The details of selecting hyper-parameter K and the sensitivity of the network factor with respect to K are provided in the Appendix A.2.
2.2 Constructing the Network Factor Z-score
The equity market network embedding at each time point provides a static representation of the market structure of that time. To track the dynamic evolution of the network over time, I compare the current network embedding with the benchmark, i.e., normal graph behavior during a context window (M ), and the difference is captured as Z-score. In computer science literature, Z-score is a widely used measure for graph change point detection (Akoglu & Faloutsos, 2010; Huang et al., 2020; Id´e & Kashima, 2004). To calculate Z-score, I perform the following three-step procedure on the Laplacian spectra in the context window M :
• For each time period t, construct a context matrix Ct ∈ RK×M by concatenating previous M spectrum vectors of time t
Ct= [−→ λt−M
−
→λt−M +1 · · · −→
λt−1], (6)
where M is the window size and −→
λt represents the normalized Laplacian spectrum at time t with the K lowest eigenvalues of L.
• Apply Singular value decomposition (SVD) on Ctas Ct= UtΣtVt> to obtain the left singular matrix Ut, singular value matrix Σt, and the right singular matrix Vt at time t. Use first left
singular vector U:1,t as the current context vector −→
λt = U:1,t. The current context vector represents the network’s normal behavior in the time window ending at t. This process is also equivalent to obtain the weighted average of M vectors in the rolling window (Akoglu
& Faloutsos, 2010).
• Calculate the difference between current network structure−→
λtand normal network structure
−
→λt as:
Zt= 1 −
−
→λ>t −→ λt
||−→ λt||2||−→
λt||2
= 1 −−→ λ>t−→
λt= 1 − cosθ (7)
Equation 7 shows that Z-score essentially calculates the cosine distance between−→
λ and−→ λ with a bound [0 ≤ Z ≤ 1]. A high Z-score reflects a significant evolution of the current spectrum from the normal spectrum. The choice of M depends on the application objective and the property of a network. A large M will capture the impact of market evolution and the business cycle, whereas a small M will help identifying the impact of shocks in the market. A sensitivity analysis of different M is presented in the Appendix A.2.
3 Empirical Analysis
In this section, I apply the proposed singed Laplacian based network factor to the U.S. equity market data. First, I provide the details of the data and the time-series of the network factor for the period 1960 to 2020 calculated using the methodology described in Section 2. Second, I discuss the relevance of the network factor in the broader macroeconomic environment and, with a battery of tests, show its significance in equity pricing. Finally, I evaluate firms’ exposure to network risk and discuss the implications.
3.1 Data
To capture network structure and compute monthly Z-score, I use the daily returns from the Center for Research in Security Prices (CRSP) of all stocks listed in the NYSE, AMEX, and NASDAQ.
The time period considered is from January 1960 to December 2020, totaling 720 months. Z-score
is computed at the end of each month based on the daily returns of all available stocks during the month. For all calculation window, I remove the assets with missing values to ensure that all return data in the given window have complete values. After this filtering, there are about 30,000 firms in total and on average, about 3,800 firms in each month. In the study period, the minimum number of firms for a given month is for March 1960 with 716 firms, and the maximum for December 1997 with 5400 firms. In all analyses, equity returns are after risk free rates, which is the one-month Treasury bill rates.
For constructing the initial signed network, I use the correlation among firms’ historical market returns. For each month, I calculate the end-of-month correlation ρij using daily returns in that month. The top panel in figure 4 shows the average correlation coefficient among studied stock returns in a given month. On average, the mean correlation among stocks is 0.10. However, in the later years of the studied period, the mean correlation coefficient increased to almost 0.16.
The middle and bottom panels in figure 4 show the time-trend for the fractions of positive edges and negative edges, respectively. The spikes in positive edges are associated with the spikes in mean correlation. Figure 5 shows the distribution of positive and negative edges during the studied period. In the equity network, among all possible connections, on average, the percentage of positive edges is 12% and negative edges is 2.5%. Compared to negative edges, the distribution of positive edges has a long tail. In some months, the percentage of positive edges reaches 70%.
The significance of the proposed network factor is evaluated in comparison to other factor models in asset pricing literature. Following literature, I include multiple important and well-acknowledged factors in my analysis. These are: Capital Asset Pricing Model (CAPM) of Lintner (1969) and Sharpe (1964) that use the value-weighted market return; Fama and French (1993) three-factor (FF3) model that extends CAPM with the size (SMB) and value (HML) factors; Carhart (1997) four-factor (FFC) model that incorporates momentum factor to the FF3; P´astor and Stambaugh (2003) liquidity-factor (FFPS) model that adds liquidity to the FF3; Fama and French (2015) five- factor (FF5) model that adds operating profitability (RMW) and investment (CMA) to the FF3 (Fama & French, 2015); industrial production growth (IP); Ludvigson and Ng (2009) macro-finance factor (LN) that uses principal components of 279 macro-finance variables. Following Giglio and Xiu (2021), I only use the first three principal components. He et al. (2017) intermediary capital
1960 1970 1980 1990 2000 2010 2020
Time
0.0 0.2 0.4
Mean Correlation
1960 1970 1980 1990 2000 2010 2020
Time
0.25 0.50
Positve Edges
1960 1970 1980 1990 2000 2010 2020
Time
0.02
0.04 Negative Edges
Figure 4: The timeline of mean correlation between equity returns, the fraction of positive edges in a given month, and the fraction of negative edges in a given month, respectively.
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
Positive Edges
0 10 20 30 40 50 60
Frequency
0.00 0.05 0.10 0.15 0.20 0.25
Negative Edges
0 10 20 30 40 50 60 70
Frequency
Figure 5: Histogram of the fraction of positive and negative edges between January 1960 and December 2020
factor; and Malloy et al. (2009) consumption based factor.4
For test assets, I consider 173 anomaly portfolios. These anomaly portfolios are the value- weighted monthly excess returns on: the 30 IND (industry) portfolios, 25 size-AC (accruals) port- folios, 25 size-β (market beta) portfolios, 25 size-RVar (residual variance) portfolios, 35 size-CI (ab- normal capital investment) portfolios, 25 size-NI (abnormal profitability) portfolios, and 8 D10-1
4The data for CAPM, FF3, FF4, and FF5 are obtained from Kenneth French’s website; for liquidity, from Lubos Pastor’s website; for IP, from the Federal Reserve Bank of St. Louis; for LN, Sydney Ludvigson’s website; for intermediary capital, from Asaf Manela’s website; for consumption based factor, from Toby Moskowitz’s website.
(high minus low decile) portfolios. These portfolios capture a vast cross-section of return anoma- lies, pose a greater challenge to existing asset pricing models, and are often used as the benchmark portfolios for evaluating and comparing asset pricing models (Ahmed et al., 2018; Giglio & Xiu, 2021; Hou et al., 2020). In addition, the data for these portfolios is easily accessible from Kenneth French’s website.5
I consider multiple market and macroeconomic indicators to evaluate the significance of network factors in relation to the macroeconomic environment. These include the monthly S&P-500 index returns (S&P), monthly Russell-3000 index returns (RUT), Chicago Board Options Exchange’s volatility index (VIX), monthly industrial production total index (INDPRO), monthly consumer price index (CPI), monthly unemployment rate (UNRATE), and monthly U.S. economic policy uncertainty index (EPU).6
3.2 Network Factor and Macroeconomic Environment
The graph Laplacian provides essential information about the network structure at a specific time point. The Z-score from the graph Laplacian spectrum aligns with major events in the past fifty years. Figure 6 exhibits the Z-score over time in solid lines, major events labeled by the red star, and financial crisis periods are highlighted in gray blocks. The Z-score spikes during 1971 President Nixon’s announcement to break up Bretton Woods, 1987 Black Monday, 2001 Dot-com bubble, 2008 financial crisis, 2020 COVID-19, and other major events, which have generated apparent impacts on the market network.
Important to note, in the calculated Z-score, the effects of sudden shocks, such as Black Monday, 09/11, and the 2016 U.S. election, are more prominent than the prolonged recession and economic stagnation. When the market experiences recession for an extended period, the change in the Laplacian spectrum from one month to the next month is low. In such volatile periods, the volatility becomes the “normal” market behavior and this is why the magnitude in network change shrinks.7
5For a detailed description of all portfolio construction methodology, please see Kenneth French’s website:
https://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data library.html.
6The data for S&P, RUT, and VIX are obtained from Yahoo Finance; for INDPRO, CPI, and UNRATE, from the Federal Reserve Bank of St. Louis; and for EPU, from the economic policy uncertainty website:
https://www.policyuncertainty.com/.
7This behavior can also attribute to the use of daily return data for calculating monthly Z-score. During
Figure 6: Z-score along with major events from Jan-1969 to Dec-2020. Financial crisis periods are highlighted in gray blocks and major events are marked by red star.
Table 1: Network Factor and Macroeconomic Indicators Correlation Matrix
S&P VIX RUT EPU INDPRO CPI UNRATE
VIX −0.65
RUT 0.81 −0.59
EPU −0.28 0.24 −0.26
INDPRO −0.07 0.06 −0.04 −0.06
CPI −0.03 0.05 0.01 0.05 0.16
UNRATE 0.09 −0.08 0.08 0.01 −0.70 −0.17
Network −0.23 0.15 −0.23 0.19 0.27 0.04 −0.56
Note: This table reports the correlation matrix of the proposed network factor and other macroeconomic indicators.
Next, I examine whether the network indicator (Z-score) is highly correlated with other macro indicators or reveals additional information complementary to existing macro indicators. Table 1 reports the correlation matrix of Z-score and other macro indicators. There is a negative relation with S&P, RUT, and UNRATE, and a positive relation with VIX, INDPRO, and EPU. This is the experiment, I find that the Z-score changes driven by sudden shocks are less prominent once I switch to use monthly returns to construct monthly network indicator. In this case, changes due to the long- term recession become more prominent and severe. For brevity, I did not report the results of correlation calculation based on monthly return data. Results are available on upon request.
Table 2: Asset Pricing Factors Correlation Matrix
MKT SMB HML RMW CMA Mom. Liq. LN1 LN2 LN3 Int. cap. IP gr. Cons. gr.
SMB 0.29
HML −0.22 −0.03 RMW −0.21 −0.34 0.08 CMA −0.37 −0.10 0.68 −0.02 Mom. −0.16 −0.06 −0.20 0.11 −0.03 Liq. −0.02 −0.01 0.05 0.03 0.03 −0.03 LN1 −0.09 −0.05 −0.06 0.04 0.03 −0.08 −0.03 LN2 0.28 0.13 −0.08 0.00 −0.08 −0.14 0.00 0.02 LN3 0.01 −0.01 0.01 0.03 −0.02 0.00 −0.08 0.01 −0.01 Int. cap. 0.74 0.11 0.01 −0.16 −0.19 −0.26 −0.01 −0.04 0.18 0.08
IP gr. −0.04 −0.01 0.05 −0.03 0.00 0.05 0.00 −0.36 −0.12 0.03 −0.03
Cons. gr. 0.03 0.14 0.04 −0.07 −0.02 0.00 0.08 −0.19 −0.09 −0.06 0.00 0.57
Network −0.22 −0.14 −0.08 −0.03 0.06 0.06 0.02 0.12 −0.13 0.01 −0.12 0.26 0.14
Note: This table reports the bivariate correlation between asset pricing factors including the proposed network factor.
intuitive as Z-scores indicate the magnitude of network changes and can be identified as a measure of network uncertainty and risk. When market volatility (VIX) and policy uncertainty (EPU) are high, firms tend to adjust more, and consequently, the network structure changes more. However, the magnitude of positive correlations is only 0.15 and 0.19 for VIX and EPU, respectively. This finding suggests that Z-score reflect information other than volatility and uncertainty, complement- ing current market indicators.
3.3 Network Factor for Equity Pricing
Here I evaluate the significance of the proposed network factor in relation to existing asset pricing models. To begin with, I first analyze correlation statistics. Table 2 reports the bivariate time-series correlation between all studied asset pricing factors, including the network factor. The network factor has a high and negative correlation with the market (-0.22) and a high and positive correlation with industrial production growth (0.26). Except for the CMA, the network factor has a negative correlation coefficient with all other FF5 factors. However, the magnitude is only significant for SMB with -0.14. Among the other non-traded factors, the network factor is significant for LN
first and second principal components, industrial production growth and aggregate consummation growth. The magnitude of all these correlation coefficients is almost similar. More importantly, the network factor is not highly correlated with any other existing assets pricing factor. This indicates that network factor captures additional information from the market that is not captured by existing factors.
Although correlation statistics reported in Tables 1 and 2 indicate additional and complemen- tary information of network factor, it still needs formal statistical tests to validate the significance of network factor in asset pricing. Therefore, I perform a battery of tests to compare and contrast the network factor’s significance with other existing factors. These include Fama-MacBeth cross- sectional regression (Fama & MacBeth, 1973), Giglio and Xiu three-pass estimator (Giglio & Xiu, 2021), and time-series return predictability (Ahmed et al., 2018).
3.3.1 Two-Pass Cross-Sectional Estimator
The Fama MacBeth two-pass cross-sectional regression estimates a factor risk premium in two-step (Fama & MacBeth, 1973). First, it estimates the assets risk exposure β by performing a time series regression of each asset’s excess return onto the factor as:
Rt= α + βft+ t, t = 1, ..., T.
where ft is the factor at time t and Rt is a vector of returns on N test assets at time t. Second, it estimates the risk premium by performing a cross-sectional regression of average returns on the estimated β.
bγ = ( bβ>β)b −1βb>R,
A rolling time-series regression can estimate changing β throughout the sample period. For example, Fama and MacBeth (1973) use prior 5-year rolling-regressions to estimate beta for month t. One can also use two-pass cross-regression regression to estimate full-sample β. In this subsection, I use the latter approach for simplicity. A rolling-regression approach with changing β is used for analyzing firms’ exposure to network factor in subsection 3.4.
Table 3 reports the results from two-pass cross-sectional regression for the proposed network
Table 3: Two-Pass Regression: Empirical Results
No control w/Rm w/FF3
Factors Avg. Ret. γ stderr γ stderr γ stderr
Market 0.57 0.64∗∗∗ (0.18) 0.64∗∗∗ (0.18) 0.58∗∗∗ (0.17)
SMB 0.23 0.66∗∗∗ (0.19) 0.15 (0.12) 0.07 (0.08)
HML 0.25 −1.33∗∗∗ (0.44) 0.48∗∗∗ (0.13) 0.48∗∗∗ (0.13)
Momentum 0.69 −2.66∗∗∗ (0.75) 0.27 (0.29) 1.34∗∗∗ (0.26)
RMW 0.25 −0.12 (0.14) 0.00 (0.14) 0.27∗∗ (0.11)
CMA 0.26 −0.63∗∗∗ (0.18) 0.29∗∗∗ (0.10) 0.29∗∗∗ (0.09)
Liquidity −0.02∗∗ (0.01) 0.00 (0.00) −0.01 (0.00)
Interm. cap. 1.23∗∗∗ (0.39) 1.95∗∗∗ (0.54) −0.07 (0.49)
IP growth −2.67∗∗∗ (0.74) −0.03 (0.11) −0.24∗∗ (0.11)
LN PC1 0.58∗∗∗ (0.17) 0.47∗∗ (0.19) 0.19 (0.17)
LN PC2 0.15 (0.12) 0.11∗∗∗ (0.04) 0.20∗ (0.12)
LN PC3 0.48∗∗∗ (0.13) −0.08∗∗ (0.03) 0.05 (0.11)
Cons. growth 1.35∗∗∗ (0.38) 0.13 (0.17) −0.03 (0.11)
Network −0.22∗∗∗ (0.06) −0.14∗∗∗ (0.04) −0.08∗∗∗ (0.03)
Network (M=12) −0.24∗∗∗ (0.07) −0.15∗∗∗ (0.04) −0.08∗∗∗ (0.03) Note: This table reports the risk premia estimates for each factor using two-pass cross- sectional regression with no control factor in the model, with the market as control, and with the Fama-French three factors as control, respectively; “Avg. Ret.” is the time-series average return of the tradable factors; ***, **, * are significant at the 1%, 5%, and 10%
significance level, respectively.
factor and other traditional factors. The test includes all 173 portfolio (n = 173) over 680 months (T = 680). The first column reports the time-series average return for the tradable factors. This is the model-free estimator for a factor risk premia and is only available for tradable factors. For each factor, I estimate the risk premium without any additional control factors (No control), controlling for the market return (w/R − m), and controlling for the Fama-French three factors (w/FF3). In addition, I analyze two versions of the network factor. ‘Network’ is the proposed network factor with M = 36 in the context matrix Ct of equation 6. ‘Network (M=12)’ is a alternative network factor with M = 12.
For most factors, the estimated risk premia are closely comparable to the values reported in Giglio and Xiu (2021). For example, in my analysis, the time-series average return for Market factor is 57bp, the risk premia with no-control is 64bp, adding the market induces no changes, and adding SMB and HML gives 58bp. In Giglio and Xiu (2021) the reported values are 50bp, 59bp, 59bp, and 49bp, respectively. For SMB, my result for time-series average is 23bp, risk premium with no-control is 66bp, adding the market gives 15bp, and adding FF3 gives 7bp. In comparison to Giglio and Xiu (2021) reported values of 23bp, 63bp, 16bp and 13bp, respectively. However, for a few factors, the difference is visible. For example, according to my analysis, the risk premia estimator for intermediary capital factor is 123bp without controls, 195bp after controlling for the market, and -7bp after controlling for FF3. In comparison, Giglio and Xiu (2021) report 73bp, -18bp, and 10bp, respectively.8
The factor of interest network factor is significant in all three two-pass model specifications at 1% level of significance. The network factor risk premium without controls is -24bp. After controlling for the market return, the risk premium is -14bp, and the risk premium with controlling for the market, SMB, and HML return is -8bp. The result is robust with a shorter context window (M=12). In this case, the risk premia are 24bp, 15bp, and 8bp, respectively. In comparison to other non-tradable factors, the proposed network factor is much more significant. Controlling for the market, SMB, and HML, I find none of the liquidity, intermediary capital, LN PC1, LN PC3, and consumption growth factors are significant. The industrial production growth factor is significant at 5%, and the LN PC2 factor is significant at 10% significance level. The insignificance of risk premia of these factors according to two-pass cross-sectional regression is also reported in Giglio and Xiu (2021).
3.3.2 Three-Pass Estimator
Although the evidence from two-pass cross-sectional regression signifies the importance of network factor in equity pricing, the Fama and MacBeth (1973) technique often criticized for its associated bias. The two-pass cross-sectional regression is affected by omitted variable bias in both time-
8This difference is expected as the data used in these two works is slightly different. In my analysis, I use 173 portfolios whereas Giglio and Xiu (2021) use 202 portfolios. The time period for my analysis is from January 1960 to December 2020, whereas in Giglio and Xiu (2021), it is from July 1963 to December 2015.
series and cross-sectional steps (Giglio & Xiu, 2021). In order to avoid the omitted variable bias, Giglio and Xiu (2021) propose a three-pass method that produces valid estimates even when not all factors in the model are specified or observed. In this Subsection, I apply this three-pass estimator to corroborate two-pass regression results for validating the proposed network factor.
The recently developed three-pass estimator of Giglio and Xiu (2021) can overcome omitted variable bias of two-pass regressions and mimicking-portfolio estimator by identifying rotation in- variant risk premium of an observed factor. In addition, it is also a powerful tool for identifying measurement error in an observed factor and detecting spurious or useless factor (Giglio & Xiu, 2021). Therefore, the three-pass estimator is a better choice to evaluate the usefulness of my proposed network factor in the asset pricing factor universe.
As the name suggests, the risk premium for a factor in the three-pass estimator is estimated using three steps. First, identify latent factors by performing the principal component analysis (PCA) on the matrix n−1T−1R¯>R. Second, obtain the risk premia of the estimated latent factors¯ by performing a cross-sectional ordinary least square (OLS) regression on average returns. Third, identify the relation between observed factor gt and the estimated latent factors by performing a time-series regression of gt onto the PC’s. This step also removes the measurement error from gt. In the three-pass model, the risk premium of the observed factor is estimated as:
ˆ
γg = ¯G bV>( ˆV bV>)−1( bβ>β)b−1βb>r,¯ (8)
where G is the factor of interest, bV = T1/2(ξ1 : ξ2 : ... : ξpˆ)> are the latent factors calculated from the eigenvactors ξ1, ξ2 : ...ξpˆ corresponding to the largest ˆp eigenvalues, and bβ = T−1R b¯V> are the factor loadings.
In addition to estimating the risk premium of the network factor in light of three-pass regression of Giglio and Xiu (2021), I also analyze two other significance tests proposed in Giglio and Xiu (2021), time-series R2 for observable factor R2g and Wald test for a weak g. The R2g measures the signal-to-noise ratio of the observed factor g and is calculated as:
R2g = η bbV bV>bη>
G ¯¯G> ,
Table 4: Three-Pass Regression: Empirical Results
Factors γ stderr R2g p-value
Market 0.58∗∗∗ (0.17) 99.25 0.00
SMB 0.22∗ (0.12) 97.92 0.00
HML 0.20∗∗ (0.10) 63.91 0.00
Momentum 0.60∗∗∗ (0.22) 71.09 0.00
RMW 0.08 (0.06) 49.01 0.00
CMA 0.08 (0.07) 53.36 0.00
Liquidity −0.23∗ (0.13) 3.90 0.08
Interm. Cap 0.65∗∗∗ (0.23) 60.52 0.00
IP growth −0.01 (0.01) 0.97 0.25
LN PC1 0.28∗ (0.17) 2.07 0.01
LN PC2 0.12 (0.15) 5.48 0.00
LN PC3 0.06 (0.10) 2.22 0.13
Cons. growth 0.01 (0.01) 2.90 0.00
Network −0.02∗∗∗ (0.00) 7.70 0.03
Network (M=12) −0.02∗∗∗ (0.00) 7.03 0.04
Note: This table reports the risk premia estimates for each factor using three pass estima- tor; the R2 of the projection of factors onto the latent factors; and the p-value of the test that factor is weak; ***, **, * are significant at the 1%, 5%, and 10% significance level, respectively.
where η is the estimator ¯b G bV>( bV bV>)−1 of the time-series regression between observed factor and the latent factors. The Wald test evaluates the null hypothesis that an observed factor g is weak.
Therefore, the Wald test allows me to examine whether the proposed network factor is weak or strong and validate the necessity of incorporating it in the asset pricing model. Giglio and Xiu (2021) showed that the parameters and test statistics of three-pass estimator posses asymptotic property and as n, T −→ ∞ the values converges. For detailed theorems and mathematical proof, I refer interested readers to Giglio and Xiu (2021).
Table 4 reports the results from three-pass regression. Following Giglio and Xiu (2021), I use seven principle components as latent factors. Similar to two-pass cross-sectional results, the risk premia estimators from most of the factors are closely comparable to the result reported in Giglio
and Xiu (2021). The risk premium of the proposed network factor is -2bp and significant at 1%
level of significance. Among all the competing tradable and nontradable factors, the risk premia of only market, momentum, and intermediary capital are significant at 1% level of significance. The risk premium of HML is significant at 5% significant level, and risk premia of SMB, liquidity, and LN PC1 are significant at 10% significance level.
The R2g of the network factor is 7.70%. This is higher than other nontradable factors like liquidity, IP growth, all three macro factors, and consumption growth factor. In comparison to tradable factors, the R2g of network factor is very low. Giglio and Xiu (2021) also report that the Rg2 for nontradable factor is much lower and, for some cases, below 1%. They attribute this finding with less measurement error for tradable factors and associated noise of nontradable factors. The Wald test for the null hypothesis that the network factor is weak is rejected at 5% significance level.
The result for the network factor is also robust for a shorter context window. The risk premium of the network (M=12) is also -2bp and significant at 1% level of significance, R2g is 7.03%, and the null hypothesis for the weak factor is rejected at 5% significance level.
3.3.3 Anomaly Reduction in Time-Series Return Prediction
In previous two subsections (3.3.1 and 3.3.2), I show the validity and significance of the network factor using two-pass cross-sectional regression and three-pass estimation. In this subsection, I ex- amine the ability of the network factor in reducing return prediction error to reaffirm the usefulness of the network factor in the asset pricing factor universe. Literature has shown that risk factors can be associated with market returns, size, book-to-market ratio, profitability, investment, liquidity, and so forth (Fama & French, 1993, 2015; P´astor & Stambaugh, 2003; Sharpe, 1964). Whether the network can be an additional risk factor on top of those conventional pricing factors is unexplored.
If my proposed network factor Z-score, is a valid complimentary pricing factor, return prediction anomalies should reduce after adding Z-score to the benchmark pricing models.
Following Ahmed et al. (2018), I perform time-series tests to evaluate the network factor. The test assets include all 173 portfolios. Five asset pricing models are used for comparing return predictability with versus without the network factor: the CAPM (Lintner, 1969; Sharpe, 1964);
Fama-French three-factor (FF3) model that extends CAPM with the size and value factors (Fama
& French, 1993); Carhart four-factor (FFC) model that incorporates momentum factor into the FF3 (Carhart, 1997); P´astor and Stambaugh liquidity-factor (FFPS) model that adds liquidity into the FF3 (P´astor & Stambaugh, 2003); and Fama-French five-factor (FF5) model that adds the profitability and investment along with the FF3 (Fama & French, 2015).
For each portfolio, I run the time-series regression of each pricing model with and without the network factor Z-score. The regressions specification of FF3, FFC, FFPS, and FF5 with Z-score are as follows:
ri,t+1e = αi,F F 3+Z + βi,M KTM KTt+ βi,SM BSM Bt+ βi,HM LHM Lt+ βi,ZZt+ i,t
ri,t+1e = αi,F F C+Z+ βi,M KTM KTt+ βi,SM BSM Bt+ βi,HM LHM Lt+ βi,U M DU M Dt+ βi,ZZt+ i,t ri,t+1e = αi,F F P S+Z+ βi,M KTM KTt+ βi,SM BSM Bt+ βi,HM LHM Lt+ βi,LIQLIQt+ βi,ZZt+ i,t ri,t+1e = αi,F F 5+Z + βi,M KTM KTt+ βi,SM BSM Bt+ βi,HM LHM Lt+ βi,RM WRM Wt
+ βi,CM ACM At+ βi,ZZt+ i,t
where rei = ri− rf is the excess return over risk free rate of asset i. M KT , SM B, HM L, RM W , CM A are the market, size, value, profitability, and investment factors of Fama and French (1993, 2015), respectively. U M D is the momentum factor of Carhart (1997), LIQ is the liquidity factor of P´astor and Stambaugh (2003), and Z is the network factor Z-score. Subscripts i indicates distinct assets and t indicates time. To avoid the contemporaneous effect of time-series regression, I use next month’s excess return as the dependent variable of the time-series analysis.
I apply a set of evaluation metrics following asset pricing literature (Ahmed et al., 2018; Fama
& French, 1993, 2016, 2018; Hou et al., 2015, 2020). First, I measure the average absolute value of alphas denoted as A|ai|, and the average absolute value of alphas normalized by average return A|α¯ri
i|. These two values capture the deviation of actual returns from the model prediction. A smaller value of A|αi| and A|α¯ri
i| indicates better predictability. Next, I analyze the average R2 value of the time series regression adjusted for the degree of freedom, denoted as A(R2). A higher value of A(R2) suggests better model prediction. Following Fama and French (2018) and Ahmed et al. (2018), I use the maximum squared Sharpe ratio of alphas for the test assets Sh2(α) as the fifth performance measure. Sh2(α) indicates the squared Sharpe ratio improvement from exploiting
mispricing by a given factor model. Therefore, a lower value of Sh2(α) indicates better model performance (Ahmed et al., 2018).
Table 5 summarizes the comparison results of model performance with versus without Z-score on anomaly portfolios. The reported values are the cross-sectional average of time-series regression for each group of portfolios. For example, Panel A reports the average result for time-series regression on the next month’s return of 30 industry portfolios. The results suggest that by adding network factor (model+Z), the pricing performance of each model is significantly improved, with smaller A|αi|, A|αr¯i
i|, and Sh2(α) and higher A(R2). Especially, FF5+Z and FFPS+Z consistently are ranked as the top two superior models. The average absolute value of alphas is the lowest in the FFPS+Z model for 25 Size-AC portfolios, 25 Size-β portfolios, and 35 Size-NI portfolios, and in the FF5+Z model for 30 IND portfolios, 25 Size-CI portfolios, 25 Size-RVar portfolios, and 8 D10-1 portfolios. The average absolute value of the alpha standardized by excess return A|α¯ri
i| is the lowest in the FF5+Z for all portfolio groups except 25 Size-RVar portfolios.
The improvement of test matrices by the inclusion of network factors is also significant. In Panel A for 30 IND portfolios, compared with FFPS, FFPS+Z reduces A|αi| and A|α¯ri
i| by 1%, and compared to FF5, FF5+Z reduces A|αi| and A|αr¯i
i| by 6%. The adjusted R2 of the next month’s excess return prediction is improved by 19% (FFFS+Z) and 21% (FF5+Z). The model return prediction improvement by including the network factor is also visible in the maximum squared Sharpe ratio of alphas for the test assets Sh2(α). The Sh2(α) improves by 1% in FFC+Z compared to FFC and 0.8% in FF5+Z compared to FF5.
The time-series results exhibit similar patterns in the other six portfolios groups. In 25 Size-AC portfolios, the A|αi| reduction from FF5 to FF5+Z is 5% and the R2 improvement is 29%. In 25 Size-β portfolios, the improvement of R2 is 30% in FF3+Z over FF3, 29% in FFC+Z over FFC, 23% in FFPS+Z over FFPS, and 25% in FF5+Z over FF5. For the 25 Size-CI portfolios, compared to FFPS, FFPS+Z reduces the average absolute alpha by 1.4% and improves the adjusted R2 by 22%. FF5+Z reduces the average absolute alpha by 6% and improves adjusted R2 by 34% in comparison with FF5. In 35 Size-NI portfolios, the scaled absolute alpha A|αr¯i
i| reduce by 27% in FF3+Z from FF3, 26% in FFC+Z from FFC, 20 in FFPS+Z from FFPS, and 29% from FF5+Z from FF5. For the 25 Size-RVar portfolios, compared to FFPS, FFPS+Z reduces the A|αi| by 1%
Table 5: Performance of Factor Models in Time-Series (Absolute) Tests Model A|αi| A|α¯ri
i| A(R2) Sh2(α)
Panel A. 30 IND Portfolios
CAPM 0.6020 0.9473 0.0014 0.0692 FF3 0.5397 0.8400 0.0317 0.0713 FF3+Z 0.5156 0.7996 0.0379 0.0707 FFC 0.5885 0.9257 0.0331 0.0696 FFC+Z 0.5632 0.8834 0.0392 0.0689 FFPS 0.4725 0.7849 0.0333 0.0757 FFPS+Z 0.4648 0.7710 0.0396 0.0755 FF5 0.4839 0.7621 0.0345 0.0620 FF5+Z 0.4558 0.7153 0.0417 0.0615 Panel B. 25 Size-AC Portfolios
CAPM 0.6973 0.9389 0.0024 0.1342 FF3 0.6440 0.8663 0.0324 0.1273 FF3+Z 0.6169 0.8300 0.0409 0.1260 FFC 0.6740 0.9127 0.0329 0.1324 FFC+Z 0.6456 0.8745 0.0413 0.1309 FFPS 0.5323 0.7992 0.0344 0.1539 FFPS+Z 0.5238 0.7859 0.0421 0.1538 FF5 0.5839 0.7841 0.0338 0.1334 FF5+Z 0.5523 0.7413 0.0437 0.1321 Panel C. 25 Size-β Portfolios
CAPM 0.7054 0.9399 0.0025 0.1078 FF3 0.6445 0.8577 0.0338 0.1043 FF3+Z 0.6167 0.8208 0.0439 0.1030 FFC 0.6766 0.9063 0.0346 0.1140 FFC+Z 0.6474 0.8675 0.0446 0.1125 FFPS 0.5543 0.7957 0.0355 0.1229 FFPS+Z 0.5463 0.7838 0.0437 0.1232 FF5 0.5795 0.7737 0.0359 0.0921 FF5+Z 0.5469 0.7305 0.0476 0.0907 Panel D. 25 Size-CI Portfolios
CAPM 0.7079 0.9402 0.0024 0.1864
Model A|αi| A|α¯ri
i| A(R2) Sh2(α)
FF3 0.6478 0.8589 0.0341 0.1786 FF3+Z 0.6198 0.8203 0.0444 0.1764 FFC 0.6787 0.9033 0.0347 0.1795 FFC+Z 0.6493 0.8628 0.0450 0.1771 FFPS 0.5615 0.7915 0.0365 0.2122 FFPS+Z 0.5533 0.7781 0.0445 0.2125 FF5 0.5858 0.7755 0.0358 0.1687 FF5+Z 0.5531 0.7306 0.0478 0.1660 Panel E. 35 Size-NI Portfolios
CAPM 0.6767 0.9253 0.0027 0.2785 FF3 0.6196 0.8395 0.0329 0.2705 FF3+Z 0.5927 0.7982 0.0417 0.2687 FFC 0.6519 0.8898 0.0338 0.2846 FFC+Z 0.6237 0.8464 0.0426 0.2825 FFPS 0.5259 0.7554 0.0356 0.2879 FFPS+Z 0.5186 0.7452 0.0427 0.2880 FF5 0.5602 0.7607 0.0347 0.2508 FF5+Z 0.5287 0.7125 0.0449 0.2488 Panel F. 25 Size-RVar Portfolios
CAPM 0.6994 0.9667 0.0027 0.2387 FF3 0.6481 0.9355 0.0356 0.2420 FF3+Z 0.6226 0.9201 0.0461 0.2403 FFC 0.6880 1.0413 0.0362 0.2356 FFC+Z 0.6612 1.0254 0.0466 0.2337 FFPS 0.5889 0.8565 0.0373 0.2649 FFPS+Z 0.5822 0.8507 0.0454 0.2653 FF5 0.5912 0.9184 0.0377 0.2435 FF5+Z 0.5614 0.8996 0.0497 0.2416 Panel G. 8 D10-1 Portfolios
CAPM 0.6142 0.9505 0.0015 0.0323 FF3 0.5498 0.8505 0.0425 0.0302 FF3+Z 0.5274 0.8158 0.0518 0.0290 FFC 0.6108 0.9473 0.0451 0.0324
Model A|αi| A|α¯ri
i| A(R2) Sh2(α)
FFC+Z 0.5874 0.9110 0.0543 0.0310 FFPS 0.5275 0.8242 0.0462 0.0279 FFPS+Z 0.5201 0.8125 0.0561 0.0277 FF5 0.5126 0.7908 0.0428 0.0290 FF5+Z 0.4864 0.7503 0.0535 0.0275 Note: This table reports the summary statis- tics for time-series tests of Z-score contribution as a factor with other established factor mod- els. The reported values are the cross-sectional average of each group of test assets portfolios.
Smaller A|αi|, A|αr¯i
i|, and Sh2(α), and higher A(R2) indicate better model performance.
and improves the adjusted A(R2) by 21%; compared to FF5, FF5+Z reduces the A|αi| by 5% and improves adjusted A(R2) by 31%. Finally, for eight D10-1 portfolios, the best adjusted R2 value of 5.6% per month is observed with the FFPS+Z model, a 21% improvement over the original FFPS model. I also document that FF5+Z achieves the lowest maximum Sharpe ratio of Alphas of 0.0275 per month with a 5% improvement over the FF5 model.
The time-series tests provide useful insight regarding the factor model in assets return predic- tion. Although the explanation power of cross-sectional return variability of the factor models is significant (as shown in Section 3.3.1 and 3.3.2), when it comes to predicting future returns, the error rate is too high. There is two probable explanation. The first one is the misspecification of the factor models. It is normally assumed that asset pricing models are generally misspecified as they are merely an approximation of true data generating process (Ahmed et al., 2018; Kan et al., 2013). The misspecification problem is intensified when these models are used for predicting future returns. The second one is the use of linear models to capture nonlinear and time-varying patterns in asset returns. The distribution of asset returns exhibits both nonlinear and time-varying patterns (Bansal & Yaron, 2004; He & Krishnamurthy, 2013). In contrast, the factors in traditional factor
with linear techniques like time-series regression, they tend to produce sub-optimal results. Recent studies suggest that efficiently capturing nonlinearity using machine learning techniques like au- toencoder or deep neural network can improve the predictive capacity of factor models (Gu et al., 2020; Jiang et al., 2020; Uddin & Yu, 2020). The construction framework of the proposed network factor allows me to capture the dynamism and time-varying pattern in the network structure of the equity market. Therefore, the improvement offered by the network factor in time-series tests can partially be attributed to this characteristic.
3.4 Return Predictability with Cross Sectional Regression
Testing pricing models verifies whether the network factor can be a useful pricing factor but cannot shows how the network factor affects equity prices, i.e., positively or negatively. To better under- stand the implication of Z-score on equity prices, I further perform Fama-MacBeth cross-sectional regressions with market, size, value, momentum, as well as network factor. The analysis in this Section differs from the analysis in Section 3.3.1 in two aspects. First, instead of using portfolio returns as test assets, I use all individual stock returns here. This includes all the listed stocks in the NYSE, AMEX, and NASDAQ. Using individual stocks as test assets allow us to understand how an individual company reacts to change in the equity market network structure. Second, instead of a single β, I estimate changing β ’s. I first compute the sensitivity β for each factor using a 60-month rolling window with a minimum requirement of 15 months and then regress next-month future stock excess returns on those β ’s at each month. The independent variables are standardized by their monthly standard deviations to make results more reliable.
The coefficient of βZ (β of Z-score) can be either positive or negative. On the one hand, if a given firm responds to the change of network in the opposite direction, it can offer an option as investment hedging against market conditions. Investors appreciate such hedging option and the demand of those stocks increases, which results in price increase and return decrease. If so, firms with low (negative) βZ (β of Z-score) are expected to have lower future returns, i.e., the coefficient of βZ should be positive. On the other hand, if a given firm reacts to the change of network in the same direction, it suggests that this firm is able to adjust to market changes quickly, less vulnerable with lower risks especially when macro environment is volatile. Investors therefore treasure such
Table 6: Fama-Machbath Regression on Next Month Excess Return Variables 1. Full-Sample 2. Low-Z 3. High-Z
βZ −0.047∗ 0.034 −0.130∗∗∗
(−1.665) (1.073) (−3.088)
βM KT 0.000 −0.049 0.049
(−0.005) (−0.450) (0.394)
βSM B 0.115 0.142 0.087
(1.290) (1.171) (0.738)
βHM L 0.143∗ 0.227∗ 0.057
(1.887) (1.909) (0.505)
βU M D −0.153∗∗ −0.132 −0.176∗∗
(−2.478) (−1.490) (−2.210) Constant 0.677∗∗∗ 0.556∗∗∗ 0.801∗∗∗
(4.281) (2.743) (4.116) Observations 3,355,977 1,722,132 1,633,845
R-squared 0.036 0.036 0.036
Number of groups 651 329 322
Note: This table reports the average value of Fama-Macbeth regressions. The dependent variable is the next-month excess returns. Independent variables are the sensitivity (β) of Z- score (βZ), Fama-French three factors (βM KT, βSM B, βHM L) and momentum factor (βU M D).
Model (1) is for full sample period. Model (2) is when Z-score is low (below the median), and model (3) represents values when Z-score is high (above the median). Newey-West adjusted t-statistics are in parentheses. ***, **, * are significant at the 1%, 5%, and 10% significance level, respectively.
adaptability and require lower compensation. Given this situation, firms with high (positive) βZ tend to have lower future returns, i.e., the coefficient of βZ should be negative.
In Table 6 model (1), the coefficient of βZ(β of Z-score) is significantly negative at 10%. In other words, stocks with great adjustment are valued more than hedging option, and their future returns are lower. I then divide the time periods into two groups: below the median Z-score (periods with small or no network changes) and above the median Z-score (periods with large network changes).
In model (2), when network changes are minor, the significance of βZ disappears. Conversely, in model (3) when there are big changes in network structure, the impact of the Z-score on firms’ next
