Dynamic network and own effects on abnormal returns: evidence from China’s stock market

This paper addresses the question of how to model the process of abnormal returns on individual stocks. It postulates a framework, where abnormal returns are generated by a process which features two autoregressive components, one stock-specific and one related to network effects. This process deviates from customary ones in that the parameters are specific to each stock/firm, that the autoregressive process is explicitly modelled instead of using cumulative abnormal returns over a pre-specified window, and that network effects are present. Abandoning either one of those deviations is rejected by data on Chinese stocks in 2018 and 2019, an episode which is significant for an abnormal stock-market returns analysis, as it was characterized by numerous tariff-setting events related to the “trade war” between the USA and China.

announcements of trade or investment liberalizations or de-liberalizations. Examples processes which can be used to model abnormal returns. Section 5 summarizes the data sources as well as the variables used in the analysis. Section 6 provides a condensed discussion of the main findings, and the last section concludes with a brief summary.

Estimating abnormal returns
Let us use i = 1, . . . , F to denote firms, e = 1, . . . , E to denote event dates, and t = 1, . . . , T to denote time which are typically days. Then, changes in firms' stockmarket returns are typically modelled as a function of time-invariant, firm-specific characteristics and time-specific common characteristics which also matter in a firm specific way. Using r i,e+t to denote the stock-market return of i at day e + t, m e+t for the (common) market return, and ex e+t for the import-share-weighted nominal exchange rate of China, a customary model of stock-market returns is where α i is a stock-specific constant, ζ 1i and ζ 2i are stock-specific slope parameters, and [w 1 , w 2 ] is the estimation window for the event study. Examples of estimates of this and similar equations are found in O' Hara and Shaw (1990), Breinlich (2014), Moser and Rose (2014), Moenninghoff et al. (2015), Alfaro et al. (2017), Dewenter and Riddick (2018), or Egger and Zhu (2019). Note that everything in Eq. (1) varies by stock/firm so that the results are quite hard to summarize and report. However, a large branch of the literature is less interested in the parameters (α i , ζ 1i , ζ 2i ) but in the residual, u i,e+t , which is reflective of the abnormal returns of a stock or company. In what follows, we will focus on estimates of u i,e+t using an estimation window of [e − 610, e − 361] (i.e., t ∈ [−610, −361]), 1 and consider customary processes determining it so as to gauge effects of trade liberalization or deliberalization on it. Toward establishing such processes, it will be useful to devote a separate subsection to the setup of the spillover matrix and another one to the process of abnormal returns.

Parameterizing the intensity of inter-stock spillovers
We will make use of an F × F identity matrix, I F , and H × 1 vectors as well as H × H matrices of ones, ι H and J H , respectively. 2 With regard to the latter H ∈ {S, F S} indicates vectors and matrices of dimension S and F S, respectively, where S indicates the number of sectors that the F firms together operate in, and F S is just the number of firms times the number of sectors. One useful matrix that will be used is 1 This is done to ensure that the end of the estimation window for the last event considered in this studynamely, the one on May 13, 2019-is December 31, 2017, which is prior to the first event of our analysis. Note that it is customary in the finance literature exploiting treatment effects at stock markets in event-type studies to use a number of days covering about one trading year to estimate the parameters used to determine abnormal returns at times after that year and a much shorter time window afterwards to estimate effects of certain events on those abnormal returns afterwards. The second window is kept short to ensure that the effects of events can be measured cleanly without much influence of other shocks than the interesting one. 2 This makes use of the notation introduced in Baltagi (2008).
where ⊗ is the Kronecker product. This matrix serves the purpose of summing all elements of postmultiplied vectors except for ones pertaining to the same firm. 3 Moreover, we will make use of an F S × 1 vector of output shares, h F S , where a typical element states how much firm i makes operating income with sector-s-type output relative to the sector's total operating income. Based on the latter, we define the matrix After defining the Chinese sector-S-to-sector-S input-output matrix, O S , we can define the matrix J F ⊗ O S to arrive at the weighted input-output matrix where • is the Hadamard (element-wise) product, and the firm sector-firm sector input-output matrix After defining an F S × F matrix which computes sums of post-multiplied vectors across all firms for each firm and sector and assigns that sum in all rows of a firm and sector, we may define the F × F firm-to-firm input-output matrix Of the latter, we will extract matrices of format F e × F e , which we call O F e , where F e ≤ F is an event-e-specific number of firms which can at most be as large as the unique number of firms in the data across all events. O F e captures the weighted impact of all firms present around event e, which is the same for all t days pertaining to that event. We will refer to the counterpart to O F e which has normalized entries so that the elements in each row sum up to unity by W F e , which is also of format F e × F e . 4

The process of determining abnormal returns
Let us stack u i,e+t into u e+t for all stocks i at date e + t. Then, we postulate the model for u e+t in t ∈ [−1, +10] as: We can write this differently, after defining the F e × F e identity matrix I F e , as When allowing the parameters (λ, κ, β 1 , β 2 ) to be firm-specific, 5 we obtain, after defining for the set of firms around event e as N e and the matrices L e = diag i∈N e (λ i ), Define x e = β 1 Tariff e,U S + β 2 Tariff e,C N or x e = x = B 1e Tariff e,U S + B 2e Tariff e,C N for all e, where Tariff e,U S and Tariff e,C N are the tariffs set by the USA and China, respectively, on event date e. The latter will be sufficient, as we will consider only effects and experiments associated with Tariff e+t,U S and Tariff e+t,C N which are nonzero on day t = 0 and zero otherwise.
Footnote 4 continued the relative magnitude of the amount is similar. This is the consequence of row normalization as a vector normalization. Note that scalar normalizations of network matrices would avoid destroying the notion of absolute distance or neighborliness (see LeSage and Pace 2008; Kelejian and Prucha 2010, for further details on the normalization of network matrices). 5 Note that with firm-specific parameters on two tariff determinants, one lagged-own-dependent-variable parameter and one lagged-network-dependent-variable parameter apart from a firm-specific constant, the overall residual degrees of freedom of the proposed model will be 660,514 or about 187 for the average firm when using the data which will be described and used below. As only five parameters have to be estimated per firm there are many-enough residual degrees of freedom to principally estimate the parameters of interest at sufficiently high precision when pooling the data across events. Note that, irrespective of whether parameters are common to all firms or not, researchers who are interested in the standard error of these parameters would have to consider that the dependent variable of this second setup is estimated. Hence, for inference, one would have to pursue a weighted least-squares approach (see Saxonhouse 1977). The latter is typically ignored in work on the determinants of abnormal returns. Note that parameters such as (λ, κ, β 1 , β 2 ) are typically assumed to be common across firms in the finance literature (see Moser and Rose 2014). However, recent work in econometrics points to the potential importance of parameter heterogeneity across cross-sectional units when estimating panel-data models (see Christodoulou and Sarafidis 2017;Sarafidis and Wansbeek 2020). While it turned out feasible to estimate all parameters specific to the level of the firm, optimizing the degree of pooling along the lines of Christodoulou and Sarafidis (2017) turned out computationally infeasible within reasonable time due to the size of the cross section of firms at hand.
In order to gauge the transition and long-run effects of a particular shock x e , we will let this process run, assuming that x e is absent (zero) for any time period prior to and after event time e, where t = 0. Hence, the effects of the shock x e will fade with time, if the elements of R e are properly bounded.
As E(ε e+t ) = 0 for any e and t, what will be relevant is Hence, with appropriately bounded elements of R e , the long-run effect from that series is ∞ p=0 R p e x e = (I F e − R e ) −1 x e . We can then consider the deviations of the long-run predictions where x e is estimated using firm-specific parameters (β 1i , β 2i ) and R e is estimated using firm-specific parameters (λ i , κ i ) from four more restrictive, alternative models: one where network effects are absent with κ i = 0 but all other parameters are firmspecific, (λ i , β 1i , β 2i ); one where the parameters are estimated from a pooled model so that (λ i , κ i , β 1i , β 2i ) are not firm-specific but common to all firms, (λ, κ, β 1 , β 2 ); one where λ = 0 and κ = 0 for all firms and (β 1i , β 2i ) are firm-specific, and one where λ = 0 and κ = 0 and (β 1 , β 2 ) are assumed to be identical across all firms.

Data
We use daily Chinese stock-market data in a window of 1 day before up until 10 days after each US-China "trade-war" tariff-announcement or-implementation event from Datastream. 6 We consider 19 events in this estimation. For year 2018, we use: March 29, April 2, 3, and 4, June 15 and 16, July 6 and 10, August 1, 3, 7, 8, and 23, September 18 and 24, December 14. And for year 2019, we use: May 5, 10, and 13. Note that as there is a time difference between the trading hours in the United States and China, we use the treatment day as to be e + 1 in China whenever the treatment (a tariff change) was announced by the USA. However, whenever China announces a tariff change, the USA reacts on the same day, consistent with the time difference. And when the tariff change was announced on weekends, the corresponding event date in our analysis would be the first weekday after that day. So the event dates we consider in our analysis for Chinese tariff changes are March 29, April 2 and 4, June 18, July 6, August 3, 8, and 23, September 18 and 24, and December 14 in year 2018, and May 13 in year 2019. And for US tariff changes, the event dates are April 4, June 18, July 9 and 11, August 2, 8, and 24, and September 19 and 25 in year 2018 , and May 6 and 13 in year 2019.
We retrieve stock-return data on active companies listed on the Shanghai and Shenzhen Stock Exchanges and returns on the Morgan Stanley Capital International (MSCI) national equity index for China from Datastream. Data on nominal exchange rates are from the World Bank. The import-share-weighted nominal exchange rate of China is constructed using import shares of China with respective to 39 countries and territories in the World-Input-Output Database (WIOD). 7 These data underly the dependent variable r i,e+t as well as the explanatory variables m e+t (for the MSCI index) and ex e+t used in the estimation of the abnormal returns, u i,e+t .
We then obtain the lists of products on which "trade-war" tariffs were announced or imposed for each event from each country's official website (see Egger and Zhu 2019). The USA and China report these tariffs by using HTS codes, which we convert to HS2017 6-digit product lines. We use concordance tables from the United Nations to convert 6-digit HS2017 codes first to 5-digit SITC rev. 3 codes and subsequently to ISIC rev. 3 4-digit codes. This is done, because firms in Datastream have between one and ten sector codes, while tariffs are levied on products rather than sectors. Given that we can map the sector classification of all listed firms in the data to the same (2-digit ISIC) sectoral classification, the "trade-war" tariff-change measures can be matched to each firm i around each event date e and day t around it. We weight the tariff changes by the operating income as a share of total operating income of each firm reported in datastream in up to ten sectors, so that tariff changes for firms in the same main sector can still vary between firms. The same sector-level operating income of the firms are used to compute the sector-level total operating income and the firm-specific share in it, when constructing the elements of h F S .
We convert the tariff-change data from the 4-digit ISIC industry classification into the ISIC rev. 3 2-digit sector classification used by the WIOD table. The latter permits measuring the input-share-weighted abnormal returns W F e u e+t−1 . Specifically, for this we use the WIOD table for 2011 as released in 2013. The China-China block in this WIOD table is used as O S in the construction of the firm-sector-to-firm-sector input-output matrix O F S above.
The final sample includes 3540 stocks/firms across all events. In the estimation of the model parameters in Sect. 4, we winsorize the abnormal returns and the tariff-ratechange variables in the first permille and the 999th permille to avoid a disproportionate influence of extreme values in the data on the estimates. We summarize some descriptive statistics on stock-market returns and abnormal returns in Tables 1 and 2 and "trade-war" tariff changes in Table 3.
First of all, the values of firm-/stock-specific stock-market returns, r i,e , in Table 1 are small, because these values reflect day-to-day log changes in the stock prices. Clearly, we use more data than the ones reported in the table, as the summary statistics there only pertain to the very date when a tariff shock (by announcement or implementation) was realized. Recall that in order to estimate abnormal returns, u i,e+t , we utilize data over an interval of about 250 trading days prior to any event. However, the table for just one day per event is still informative about the fact that there is a lot of variation in these returns, as the standard deviation of the returns on many event dates is larger than the average of these returns. Also the maximum and the interquartile ranges are quite large.

Table 1
Summary statistics of day-today log changes of individual stock returns (r i,e ) on each US-Chinese "tariff-war" event date considered The number of firms/stocks listed in this table corresponds to F e in the text. P25, P50, and P75 refer to the 25th, the 50th, and the 75th percentile of the distribution within an event. SD, Min, and Max refer to the standard deviation, the minimum, and the maximum value in the data  The number of firms/stocks listed in this table corresponds to F e in the text. P25, P50, and P75 refer to the 25th, the 50th, and the 75th percentile of the distribution within an event. SD, Min, and Max refer to the standard deviation, the minimum, and the maximum value in the data  Table 2 suggests that when netting out the common factor associated with the market-return index ( ζ 1i m e ) and import-share-weighted nominal exchanges ( ζ 2i ex e ) after estimating Eq. (1) there is a tendency for the variation in abnormal returns ( u i,e ) to be larger relative to their mean than it was the case for stock-market returns (r i,e ) in Table 1. This is not surprising, as the regressions based on Eq. (1) use data up until the day e − 361 prior to event e, and we would expect the variation in u i,e+t to increase as we move further forward from e − 361 to time e. Table 3 summarizes the tariff changes in percent on each event date. Recall that there is variation across Chinese firms in the tariff exposure for two reasons: First, the tariffs applied varied to a large extent across sectors (e.g., not all sectors were exposed to tariff increases, and the imposed tariffs were not identical across sectors where tariffs had been implemented); second, firms are active in various industries at different operating-income intensity so that they were exposed heterogeneously to the same tariffs schedule. In any case, this table suggests that there is a big variation in the average tariffs set between events, and also the variation within events between firms is large. The latter again shows in the standard deviation of tariff changes exceeding the mean in several events.
The estimates in Table 4 indicate that there is a large degree of variation in the respective parameters between firms. Even though we do not report standard errors on these coefficients, models with restrictions on these parameters to be common or zero are rejected against the proposed model with firm-specific parameters. As all considered more restrictive models are nested in the proposed, flexible one with firmspecific parameters, we can straightforwardly compare them statistically by way of F-statistics.
In Panel A of Table 5 we report on the F-statistics and the respective p values using event-specific data and appropriate residual and restrictions degrees of freedom for the three null models relative to the most flexible one, where (λ i , κ i , β 1i , β 2i ) are all firm-specific. The table indicates that every one of the considered restricted models is rejected at a high level of statistical significance against the flexible alternative model. Hence, serial correlation within and network correlation between the firms should not be ignored. Note in particular that the null model at the bottom of Table 5, which only restricts network effects to be absent but allows (λ i , β i ) to be firm-specific is rejected against the model which also considers firm-specific network effects in abnormal returns (κ i = 0). Some of the individual parameters λ i and κ i are quite large in absolute value. However, we should not conclude from this that the model is not stable. What matters is that (I F e − R e ) −1 exists, as is the case, here.  The number of firms/stocks listed in this table corresponds to F e in the text. P25, P50, and P75 refer to the 25th, the 50th, and the 75th percentile of the distribution within an event. SD, Min, and Max refer to the standard deviation, the minimum, and the maximum value in the data 27.520 0.000  Table 5 we report results on the F-tests of the firm-specific parameters model against ones with parameters that are common across sectors and event windows. Clearly, the sector-specific-parameters models are rejected against the one where all parameters are firm-specific. The sector-specific parameter densities overlap with the firm-specific ones, but the variance in the firm-specific parameters is much bigger, indicating that there is a lot of heterogeneity in the responses across firms.
Tables 6 and 7 focus on a summary of the resulting effect estimates. In the proposed model which allows all parameters (λ i , κ i , β i ) to vary across firms, the effects materializing instantly on event date e are x e , where the latter is estimated based on firm-specific parameters on tariff changes, β i . The corresponding results are summarized in Panel A of Table 6. In the long run, after all dynamic and rippling effects from this shock have faded, the cumulative effect of the tariff change (in absence of any further shocks in subsequent days) would be (I F e − R e ) −1 x e . The corresponding results are summarized in Panel B of Table 6. The findings of Table 6 are contrasted with the ones of a customary model in Table 7, where (β i = β) is assumed to be common across all firms and both dynamic and network-rippling effects are assumed to be absent, (λ i = 0, κ i = 0).
A comparison of the reported moments of the distributions of the effects indicates that the long-run effects tend to be larger than their short-run counterparts. Hence, there is dynamic (individual plus network) amplification and accumulation of the effects of tariff shocks on abnormal returns, according to Table 6. Moreover, a consultation of the moments of the effects distribution in Table 7, where dynamic and network effects are assumed to be absent (λ i = 0, κ i = 0) and immediate tariff-shock effects are assumed to be common across firms (β i = β) indicates that the bias in the results is Table 6 Effects of "trade-war" tariff shocks when all parameters are firm-specific The number of firms/stocks listed in this table corresponds to F e in the text. P25, P50, and P75 refer to the 25th, the 50th, and the 75th percentile of the distribution within an event. SD, Min, and Max refer to the standard deviation, the minimum, and the maximum value in the data Table 7 Direct effects of shocks when parameters are identical between firms and both dynamic and network effects are absent (short-run=long-run; (I The number of firms/stocks listed in this table corresponds to F e in the text. P25, P50, and P75 refer to the 25th, the 50th, and the 75th percentile of the distribution within an event. SD, Min, and Max refer to the standard deviation, the minimum, and the maximum value in the data quite substantial, when ignoring response heterogeneity and dynamic individual and network adjustment. Towards a quantitative interpretation of the results, note that the effects, e.g., in Table 6 can be contrasted with the numbers on abnormal returns (in logs) in Table 2. This comparison suggests the following insights. First, the average effects in Table 6 are smaller by roughly one order of magnitude than the average abnormal returns in logs. However, these average responses are driven by effects on directly exposed firms in China. Note that the effects at the median (P50) in Table 6 are small. Hence, most firms receive shocks only through the input-output network, but this indirect exposure is modest for the median firm. Firms which are directly exposed to shocks or closely related to ones in targeted sectors receive bigger shocks (see the reported changes in the tails of the distribution in Table 6).
In Tables 8 and 9, we present the effect estimates and associated tests of a model which accounts for the small minus big market capitalization (SMB) and the high minus low book-to-market value ratio (HML) proposed by French (1992, 1993) on top of the factors included otherwise in the first-step stock-market-returns model. The results in Table 8 suggest that the magnitudes of tariff-war shock effects are very similar to the model which excludes the SMB and HML factors, and the F tests in Table 9 support the same conclusions as drawn from the comparable tests in the benchmark model of Table 5.
In Tables 10 and 11, we summarize the results based on a third-order-lag model rather than a first-order lag one as used in the benchmark analysis. 8 The magnitudes of the short-run responses tend to be somewhat smaller on average while those of the long-run responses are somewhat larger in Table 10 than in Table 6. However, the range of the estimates is similar in broad terms in Table 10 to the ones reported in Table 6. The tests in Table 11 suggest that a firm-specific parameters model is supported relative to its more restrictive alternatives as before.

Potential sources of endogeneity
Regarding the estimation the model in Eq. (12), we can think of two sources of endogeneity: the one of the network connections between firms through W F e , and the one through the presence of the lagged endogenous regressor u e+t−1 . However, the latter is a residual obtained from earlier firm-specific regressions, so that we ignore the latter source of endogeneity and focus on the former one in this section.
Note that the network matrix W F e depends on the operating-income-share distribution across firms referred to as h F S above (for brevity, we will refer to operating income by sales in what follows). The latter is up to each firm i's choice, which establishes an endogeneity concern, since the latter choice of firms, even though cast in the past, may be correlated with unobservables in the ith element of ε e+t in (8). Then, the term κ W F e u e+t−1 would be endogenous and κ might be biased.  The number of firms/stocks listed in this table corresponds to F e in the text. P25, P50, and P75 refer to the 25th, the 50th, and the 75th percentile of the distribution within an event. SD, Min, and Max refer to the standard deviation, the minimum, and the maximum value in the data The model under the null (null model) indicates which parameters are restricted relative to the benchmark model (the Alternative model), where the parameters (λ i , κ i , β i ) are unrestricted. The F-statistic is the normalized difference in residual sums of squares of the respective model under the null and the benchmark model normalized by the difference in residual degrees of freedom relative to the benchmark model's residual sum of squares normalized by its residual sum of squares In this section, we address this concern as follows. Let us introduce the following definitions: sales is are the sales of firm i in sector s; S AL E S s = N i=1 sales is are the aggregate sales of all firms in our data in sector s; N i is the set of firms, whose sales shares are the biggest in the same sector as that of i but excluding firm i; N i is the number of firms (elements) in N i ; S AL E S is = j∈N i sales js are the aggregate sales of all firms except i in the set N i in any sector s; S AL E S is = SALES is /N i are the average sales of firms in N i (which excludes i) in sector s. Then, we can define h is = SALES is SALES s for each firm i and sector s instead of h is . This allows us to define Doing so suggests that the instrument is valid (suggested by the associated weak-instrument test statistics). However, a comparison of the OLS and two-stage least-squares (2SLS) parameters suggests that the bias associated with the addressed endogeneity of W F e is relatively minor. This can be seen from an inspection of Fig. 1, where we plot estimates of the parameters {λ, κ, β 1 , β 2 } based on 2SLS against ones based on OLS. By and large and with a few exceptions, the two sets of parameters are well aligned around the 45 • line.

Conclusions
This paper contributes to the literature which aims at estimating the effects of trade liberalization, trade deliberalization or other domestic or foreign shocks on the returns to individual stocks at stock markets. Related work tends to invoke three assumptions: (i) that the parameters on shocks which determine abnormal stock-market returns can be pooled across stocks/firms; (ii) that either adjustment costs are absent or at least common to all firms and small enough so that long-run effects of shocks on abnormal   The number of firms/stocks listed in this table corresponds to F e in the text. P25, P50, and P75 refer to the 25th, the 50th, and the 75th percentile of the distribution within an event. SD, Min, and Max refer to the standard deviation, the minimum, and the maximum value in the data The model under the null (null model) indicates which parameters are restricted relative to the benchmark model (the alternative model), where the parameters (λ i , κ i , β i ) are unrestricted. The F-statistic is the normalized difference in residual sums of squares of the respective model under the null and the benchmark model normalized by the difference in residual degrees of freedom relative to the benchmark model's residual sum of squares normalized by its residual sum of squares returns can be estimated from relatively small windows of time; and (iii) that shocks feature idiosyncratic effects on abnormal stock-market returns so that network effects among stocks are absent. The present paper abandons these assumptions by allowing the process of abnormal stock-market returns to be governed by firm-specific parameters and by an autoregressive process which accounts for individual autocorrelation as well as dynamic adjustment through network effects. The paper then employs a large data set of daily stock-market returns in 2018 and 2019 to estimate abnormal returns and then determine these abnormal returns as a function of three ingredients, all of which may carry stock-specific parameters: shocks as reflected in trade-war tariff announcements or implementations by the USA and China, own lagged abnormal returns of a stock, and network-weighted abnormal returns by a stock. The paper does so by postulating network effects to emanate from input-output relationships between stock-market listed firms about whom the operating revenue structure is known. The latter is done by establishing a listed-firm-listed-firm input-output matrix based on the revenue structure of firms across sectors as well as the input-output table between sectors in China.
The results suggest that all three customary assumptions are rejected by the data at hand: the process of abnormal returns appears to call for firm-specific parameters; there is evidence of idiosyncratic adjustment costs; own lagged abnormal returns display a large variance in their importance for current stocks' abnormal returns; and network effects among stocks contribute another important element in the variance of abnormal stock returns whose importance again varies largely between stocks.
We believe that the explicit consideration of adjustment costs and, particularly, of network effects could be an important future avenue of work. Network effects lead to nonlinear interactions among stocks, and shocks can have interesting and nontrivial effects on stock markets, an issue which is of potentially great importance when considering the vulnerability or resilience of stock markets and their dependence on core players in the network. In particular, it might be interesting to consider more flexible, multiplex networks in affecting stock markets than considered here.