Spatial GARCH models for unknown spatial locations – an application to financial stock returns

ABSTRACT Spatial GARCH models, like all other spatial econometric models, require the definition of a suitable weight matrix. This matrix implies a certain structure for spatial interactions. GARCH-type models are often applied to financial data because the conditional variance, which can be translated as financial risks, is easy to interpret. However, when it comes to instantaneous/spatial interactions, the proximity between observations has to be determined. Thus, we introduce an estimation procedure for spatial GARCH models under unknown locations employing the proximity in a covariate space. We use one-year stock returns of companies listed in the Dow Jones Global Titans 50 index as an empirical illustration. Financial stability is most relevant for determining similar firms concerning stock return volatility.


INTRODUCTION
Generalised autoregressive conditional heteroscedasticity (GARCH) processes are widely applied in time series modelling, particularly in financial econometrics, because the conditional volatility of stock returns typically depends on their past values (see, e.g., Bollerslev et al., 1992;Engle, 1982).In spatial statistics and econometrics, autoregressive dependence in conditional variances has received less attention.Thus, we focus on instantaneous GARCH-type dependence across the cross-sectional units or spatial locations.In particular, we want to empirically prove if such dependence exists for financial returns (i.e., if the volatility spills over in a GARCH-type sense).There are mainly two approachesthe spatial GARCH (spGARCH) model introduced by Otto et al. (2018) and Otto and Schmid (2019) and the spatial Log-GARCH model of Sato andMatsuda (2017, 2021) and Otto et al. (2022).While the first-mentioned models are direct extensions of time-series ARCH and GARCH models of Engle (1982); Bollerslev (1986), the latter class comprises elements of GARCH and exponential GARCH (E-GARCH) models that are particularly useful in spatial statistics because it can be transformed to a spatial autoregressive model of the logsquared observations.The spatial interaction is modelled in both approaches via a so-called spatial weight matrix.Each weight specifies the potential degree of spatial dependence between two locations up to an unknown parameter, which has to be estimated.Typically, this weight matrix is sparse; only nearby locations have non-zero weights.This is motivated by Tobler's first law of geography, which can be summarised by the fact that geographical proximity usually leads to statistical dependence between the observations (see Tobler, 1970).Commonly applied weight matrices are inverse-distance matrices (e.g., Maddison, 2006;Zhao et al., 2020), k-nearest neighbours matrices (e.g., Credit, 2019;Vroege et al., 2020), or contiguity-based matrices (e.g., Hoang et al., 2022;Wimpy et al., 2021).However, if the locations of the observations are unknown, the weights cannot be determined in this way.Moreover, Arbia et al. (2016) discuss the influence of missing locations on the estimation outcome of spatial econometric models.More specifically, they find assuming slightly misplaced locations leads to considerably biased estimation results.In this paper, we concentrate on case 3 described in Arbia et al. (2016), i.e., 'the observations […] are available, but their location is missing or not known with certainty'.In contrast to the methods presented by Bennett et al. (1984) for partially missing locations, we consider completely missing or uncertain spatial locations.This is particularly relevant for financial applications.Ever since the evolving reduction of transportation costs and time, the geographic locations of firms do matter with a decreasing magnitude.The same applies to the stock market, where assets can be traded worldwide without additional costs due to geographical distances.Only a few studies found geographical effects in asset prices, e.g., Shkilko and Sokolov (2020) showed geographical influences in high-frequency prices due to weather conditions.In general, however, asset prices are not expected to be spatially dependent in a geographical sense, but the correlation is rather determined by other factors such as fundamental values or business sectors.Similarly, Pirinsky and Wang (2006) studied the influence of the location of the headquarters of firms on their stock returns, whereby they clustered the stocks which are located in the same Metropolitan Statistical Area (critically discussed by Moon & LeSage, 2011).
For spatial autoregressive models, Santi et al. (2021) considered the locations unknown and suggested determining the weights as distance in a new space spanned by covariates.This paper extends this approach to spatial GARCH models motivated by financial applications.As outlined above, the geographical distances are less relevant when estimating volatility dependencies and spillovers between firms, but we expect that other quantities are rather relevant for (spatial) interactions.For such financial data, the interactions should be interpreted like processes on (financial) networks.That is, weighted edges connect the nodes (i.e., stocks), and we observe the financial returns of each stock as nodal attributes.However, the inherent network structure and the distance between the nodes is unknown and has to be estimated.This approach aims to exploit the correlation structure in a covariate space, which should be correlated with the true underlying network, to proxy the dependence structure.In the same way, the covariate space could be used to estimate geographical proximity in spatial applications.
Yet, there has been a variety of research on determinants for stock returns and their predictability using firm data.Early research of Collins (1957) finds that US banks' net profit and book value is highly correlated with their stock price in 1954 and 1955.Indicating that stock returns may be driven, to a specific end, by those covariates.Moreover, Gharbi et al. (2014) and Abarbanell and Bushee (1997) find that balance sheet data are essential for firms' stock returns.Moreover, the CAPM model supports the findings that fundamental data and risks are related to firms' stock returns, which may be related to firms' balance sheet data (Black et al., 1972).To this end, there might be a clustering of firms' returns according to their risks, which is not generally correlated with the actual geographical location of the firm.There might be geographical drivers in cases of political crises, wars or other catastrophic events which are regionally constrained.However, we exclude such cases from our analysis since we consider the 50 largest companies in, to some extent, a calm market phase.In our empirical application, we show that these interactions have a GARCH-type structure, and we can interpret the instantaneous interactions in the logvolatilities similarly to temporal GARCH dependence.
The remainder of the paper is structured as follows.In section 2, we summarise the considered spatial GARCH model, suggest using a quasi-maximum-likelihood (QML) estimator, and discuss how the weight matrix is constructed when the locations are unknown.In section 3, the results of a Monte Carlo simulation study are reported.In particular, we analyse the effect of an increasing distance between the true (data-generating) locations and the proxy locations in the covariate space.Then, the approach is applied to financial data, namely log-returns of the 50 worldwide-largest companies.By selecting the best-fitting covariate space, we could show which quantities of the firms' balance-sheet data most influence the autoregressive dependence in conditional volatility.The results of this study are reported in section 4. Finally, section 5 concludes the paper.

SPATIAL LOG-GARCH MODEL
We consider a spatial process in the (true) spatial domain S , R q with q . 1 and a finite set of distinct locations {s 1 , . . ., s n } in this space S. The observed process {Y (s i ) [ R:i = 1, . . ., n} follows a spatial Log-GARCH model (Sato & Matsuda, 2021).We suppose that the process is generated at the locations s 1 , . . ., s n , but these locations are not or only partially observable.Thus, we call S the true or data-generating spatial domain.In general, for spatial GARCH models, the observed process is given by: where 1(s i ) is an independent and identically distributed random error with mean zero and constant variance (Otto & Schmid, 2019).Furthermore, let h = ( log h(s 1 ), . . ., log h(s n )) ′ be an n-dimensional vector of all log h(s i ), and let Y = (log Y (s 1 ) 2 , . . ., log Y (s n ) 2 ) ′ be the vector of log-squared observations.For spatial GARCH models, h(s i ) be considered as a proxy to the lo conditional variance at location s i (Otto et al., 2019).Then, the spatial conditional heteroscedasticity model of a Log-GARCH process is given by (2) The n-dimensional spatial weight matrices W 1 = (w 1,ij ) i,j=1,...,n and W 2 = (w 2,ij ) i,j=1,...,n specify the spatial dependence structure.That is, w .,ijdetermines the influence of region s i to s j .If the weight is zero, s i is supposed to have no direct influence on s j .Note that s i may still indirectly influence s j via other linkages.Conditionally independent pairs of locations are those that have zeros in the precision matrix.Typically, this dependence decays with an increasing distance between the observations.Thus, W is based on a proximity criterion between the observations in the space S. In spatial econometrics, these weights are supposed to be non-stochastic and known quantities, which is often criticised in practice (LeSage & Pace, 2014).In some applications, prior knowledge about the structure of the spatial weights can be obtained from the underlying physical system (e.g., processes in river networks) or from visual inspection of descriptive plots (e.g., spatial autocorrelation functions or variograms).In the other cases, the best-fitting weight matrix can be selected from set of candidate schemes by maximising certain goodness-of-fit measures (e.g., Zhang & Yu, 2018) or they can be estimated on a fully data-driven basis (e.g., Ahrens & Bhattacharjee, 2015;Otto & Steinert, 2023).Moreover, if the distances between units in geographical space S can be approximated using another distance measure in another space, W can be accurately estimated.We will focus on this proxy covariate space in the following section.Finally, the two parameters a, r and l are unknown parameters, which have to be estimated.Thus, the spatial autoregressive dependence is always a multiple of the pre-specified weight matrices and r and l, so that the results have to be interpreted depending on the choice of W 1 and W 2 .When the weight matrices are row-standardised (i.e., W 1 1 = 1 and W 2 1 = 1), we can guarantee a solution of (2) if |r + l| , 1 (see Otto et al., 2022).Moreover, higher-order spatial lags could be considered, which, however, should not be in the focus of this paper.It is worth noting that the parameter space is more difficult to obtain (without too strong restrictions) in this case (see Elhorst et al., 2012).

Proxy covariate space
Now, suppose that the geographical locations s 1 , . . ., s n are not observable (i.e., unknown) or only partially known, so that no meaningful relation/distances between them can be computed.As a consequence, the spatial weight matrices W 1 and W 2 cannot be derived in a meaningful way.Like Santi et al. (2021), we further assume that there are a set of covariates, which are varying in S.More precisely, they are not first-order stationary in S but do not have to be related to S in geographical meaning.Thus, the distances in this proxy space are used to estimate distances for constructing the weight matrix W.Moreover, notice that geo-referenced data is not necessarily needed to apply this method.
In particular, this is interesting for financial applications, where GARCH models are widely applied.
In financial economics, characteristics beyond geographic distances are usually the driver of the interdependence between the firms or stock prices.Thus, let X be this space spanned by the covariates and W (X ) 1 and W (X ) 2 are weight matrices derived from this covariate space X .The covariates associated with the ith observation are denoted by X i .The accuracy of this approximation can be determined in the same way as shown by Santi et al. (2021).For this paper, we consider the k-nearest-neighbour (KNN) weights throughout the following sections.KNN approaches are often used in spatial econometrics and other settings such as discriminatory analysis (Fix & Hodges, 1989).More specifically, where k defines the threshold number of closest neighbours taken into account.The proximity is measured by the Euclidean distance, i.e., (3) Then, we select for each row i of W (X ) the k closest neighbours of i and set their value to w (X ) ij = 1/k and to 0 for the remaining n − k neighbours.Accordingly, the rows of the final matrix W (X ) sum up to 1.

Maximum likelihood estimator
To estimate the parameters, we follow a maximum likelihood (ML) approach assuming independent and identically distributed standard normal errors e(s i ) for all location s 1 , . . ., s 2 and replace W 1 and W 2 by W (X ) 1 and W (X ) 2 , respectively.From (2), we obtain which is a proxy of the true h, because of the approximated weight matrices. 1In general, the likelihood function is given by f 1 is the error distribution, j(a, r, l|y) is a function to map the observed values y = (y(s 1 , . . ., y(s n )) ′ to the residuals e = (e(s 1 ), . . ., e(s n )) ′ given a certain set of parameters (a, r, l) ′ , and J(j(a, r, l|y)) denotes the Jacobian matrix at e = j(a, r, l|y).Note that we used e(s) to denote the observed residuals (given any choice of the parameters) and 1(s) to denote the true data-generating errors.For the considered spatial Log-GARCH model, j is simply e(s i ) = y(s i )/ h (X ) (s i ) for all i, while the Jacobian matrix is given by J = (J ij ) i,j=1,...,n with The partial derivatives can be obtained by where ∂ log (h (X ) (s i ))/∂e(s j ) = 2d ij /e(s j ) and 1 (Otto & Schmid, 2019).Eventually, the ML estimator is given by ( â, r, l) ′ = arg max a,r,l f e (j(a, r, l|y)) 1 det(J(j(a, r, l|y))) .
In practice, usually the logarithmic likelihood is maximised for computational stability.

SIMULATION STUDY
In this section, we discuss the finite-sample performance of the estimators under different scenarios.We verify this using the spatial log-GARCH model introduced in section 2. Furthermore, different levels of correlation between the (generated) covariates and the true locations are used.Notice that we also consider the case with the true weight matrices (i.e., all locations are perfectly known) to assess the best possible precision of the estimates for the given parameter set-up and sample size n.

Simulation design
To compare the estimation performance of weight matrices obtained by the KNN method in different covariate proxy spaces, we assume that the covariates are linearly correlated with the true locations by a decreasing magnitude.Without loss of generality, we bound the simulation design to weight matrices taking only k = 10 nearest neighbours into account.More specifically, we sample n true locations from a two-dimensional uniform distribution in 0, n/50 2 to keep the density of locations stable with increasing dimension n.We hereinafter call the true locations S.Then, the simulated covariates X are obtained as where E N (0, I 2 ) and 1 − v defines the correlation between the true locations and the (generated) covariate.
To evaluate the estimation performance, the following three descriptive statistics are used: (1) mean( where u 0 = (a 0 , r 0 , l 0 ) ′ stands for the true parameter and { ûr = ( âr , rr , lr ) ′ :r = 1, . . ., m} is the sequence of estimates from m Monte Carlo experiments.Here, we set m = 1000 and the number of locations varies from n = 50, 100 to n = 200.Moreover, we simulated the process for different choices of the true parameters of the log-GARCH model given in (2) to cover different strengths of spatial dependence.Specifically, r 0 [ {0.1, 0.2, . . ., 0.7}, l 0 [ {0.2, 0.4} and a 0 = 0.5.Then, the true locations in S are simulated from an uniform distribution, i.e., s i U 0, n/50 2 for all i = 1, . . ., n.These locations are used to compute the KNN spatial weight matrix W of the data-generating process.Next, a log-spatial GARCH process is simulated as in (2), i.e., (y i ) i=1,...,n , with selected parameters a 0 , r 0 , l 0 , and We now turn to the estimation step of the simulation.Therefore, we generate covariates which are linearly correlated to the true locations in S. The magnitude of linear correlation v between the covariates and the true locations in S increases in four steps, v = 0 (i.e., the data-generating locations are identical to the locations in the covariate space), v = 0.1, v = 0.2, and v = 0.5 representing the strongest deviation between true locations in S and the locations in the covariate space X .The case with v = 0 can be seen as the best result, which can be obtained by the QML approach, i.e., there is no error due to the approximation of the geographical space.The observations in the covariate space are simulated as in (9).Using these simulated covariates, the inverse-distance matrix W (X ) is computed.In the last step, we estimate the parameters a, r, l of the process in (2) for the simulated series (y i ) i=1,...,n as described above.

Simulation results
The results shown in Table 1 allow for several insights regarding the suggested method to select the weight matrix using (linearly) correlated covariates.We focus on two interesting cases, namely {r 0 = 0.1, l 0 = 0.2} and {r 0 = 0.3, l 0 = 0.4}, while the results of other settings are reported in Appendix A in the online supplemental data, that is, in Tables A.3 (large spatial correlations r 0 and l 0 = 0.2), A.4 (weak spatial correlations r 0 and l 0 = 0.4), and A.5 (medium spatial correlations r 0 and l 0 = 0.4).We can summarise the results as follows.
The results of the simulation study show that as the sample size increases, the precision of estimating the parameters improves, as indicated by decreasing values of root mean square errors (RMSE), provided that the covariate space sufficiently mimics the true geographical coordinates (i.e., v is not too large).The simulations also reveal that a lower correlation between the covariates and true locations results in higher RMSE values, indicating lower parameter estimation accuracy.However, the difference in estimation outcomes between different correlation values is small, especially for low values of v.For example, when the correlation is 0.9, the RMSE values for r 0 = 0.3, l 0 = 0.4 and a 0 = 0.5 at n = 200 are 0.144, 0.288 and 0.045, respectively, compared to 0.115, 0.273 and 0.035 for the true weight matrix.This suggests that the proposed method using covariates to estimate the weight matrix performs well asymptotically, but a slight change in the ordering of proximity of the n locations may considerably alter the shape of the weight matrix.
In the first simulation set-up with r 0 = 0.1, l 0 = 0.2 and a 0 = 0.5, the estimation leads to lower RMSE values, especially with increasing divergence between the true observation locations and the proxy space.The remaining settings reported in Appendix A in the online supplemental data demonstrate that, in the majority of cases, the method with correlated covariates leads to slightly slower RMSE decays with higher standard deviations and mildly biased mean values of the series of parameter estimates.

APPLICATION
To demonstrate the proposed method for obtaining the weight matrix in practice, we investigate the spatial dependence between the 50 largest companies.For this purpose, we use the Dow Jones Global Titans 50 stock index.Figure A.2 in the online supplemental data displays the firms included in this index.We compute the yearly log-returns of the 50 companies between February 2019 and 2020.These long-term returns are depicted in Figure 1.Evidently, there appears to be a clustering of oil and gas firms with predominantly negative returns, juxtaposed with technology-related firms mainly exhibiting positive returns.However, it is important to note that the data have no clear separation by sector.
As outlined in the introduction, firm data may have an influence on its stock returns.Accordingly, we use the balance sheet data of these companies for the year 2019.In total, we investigate 66 balance sheet positions.All data are collected from Yahoo!-Finance.An overview and descriptive statistics can be found in Table A1 in the Appendix (online supplemental data).Since different balance sheet positions have, in general, different magnitudes, all covariates are standardised across the firms to allow for comparability of distances among the covariates.
For comparing estimation results, information criteria can be used based on the maximum value of the likelihood function (5), which additionally penalise the number of parameters and length of the time series.However, in our case, the number of model parameters remains the same since we compare different weight matrices.Using the AIC (Akaike, 1973) and BIC (Schwarz, 1978) criteria does not provide any additional informational output beyond the maximum of the log-likelihood value (LLV).Accordingly, we concentrate on the LLV to compare the model fit from now on.In other words, we chose the covariate space to achieve the best model for financial risk spillovers.

Results
The 66 balance sheet positions allow for an extensive amount of covariate combinations to be used for the weight matrix.To be precise, 2 66 combinations.To reduce the complexity of the model selection procedure, we apply a forward selection to find the best-fitting covariate combination.First, we use only one covariate to obtain 66 different weight matrices in total and compute the QML estimator for each weight matrix.Then, we iteratively added the remaining covariates to the best-fitting covariate(s) and computed the LLV.
The estimation results are depicted in Table 2.We find that the balance sheet position 'Free Cash Flow' is the most important value to unveil spatial dependencies among the largest 50 companies' stock returns since it is the covariate which weight matrix provides the best estimation results with an LLV of −98.119 after the first step.Free cash flow is a measure of a company's cash available for distribution to its shareholders after accounting for capital expenditures required to maintain or expand the business.Accordingly, it reflects, to some extent, a firm's Table 2. QML estimation results.
The columns represent the LLVs from each step of the step-wise selection procedure, when always one covariate is included in addition to the best-fitting model of the previous column (indicated by the braces).The highest LLV in each column is in bold face.
profitability.More specifically, a higher free cash flow indicates the amount of money available, for instance, for investments or debt reduction.A higher free cash flow may indicate more financial stability and vice versa.Consequently, the free cash flow might reflect the financial risk of a firm when used effectively.For a discussion of free cash flow and its interpretability concerning profitability, we refer the reader to Jensen (1986); Opler and Titman (1994).
When proceeding with the forward selection, we see an improvement by adding six further covariates to 'Free Cash Flow'.The highest likelihood (LLV = −88.045) is reached by combining 'Free Cash Flow' with 'Other Operating Expenses', 'Effect of Exchange Rate', 'Total Other Income/Expense Net', 'Total Stockholder Equity', 'Discontinued Operations' and 'Common Stock'.All of these underline the existing scientific results that a firm's profitability and financial stability is key for its stock returns (Abarbanell & Bushee, 1997;Blanchard et al., 1993;Chan et al., 1991).As pointed out by, for instance, Gordon (1959), current stock returns are determined by the current earnings of a firm being in line with our finding that present net profits determine the risks of stock returns to some extent.This outcome does not contradict the results of Chan et al. (2001) finding that R&D expenses and stock volatility are positively correlated.More specifically, firms with high expenses for R&D tend to have higher stock market uncertainty.R&D expenses may still be influencing the volatility.Nevertheless, the best volatility estimation is governed by balance sheet positions reflecting a firm's current and future profitability.It is worth noting that we found evidence for instantaneous spillovers in the conditional variances because of the spatial GARCH representation.Conventional (time-series) GARCH only allow for temporally lagged spillovers in the conditional variances.
The results of the empirical study are presented in Table 3, which shows the estimated parameters in each step of the forward-selection procedure.Overall, we find that the constant term in the volatility equation, i.e., a, is significantly different from zero.In addition to this constant term, the spatial interactions add to the overall variance levels given by ( 2).With an increasing For most cases, except for the best-fitting models with weight matrices based on five or seven covariates, r is not significantly different from zero.For our final model, we observe significant spatial interactions with r = 0.034.The spatial interactions are mainly driven by the effect of l, i.e., the spatial autoregressive effect in h.For the best-fitting model, l equals 0.869.

Detecting volatility relations across firms and sectors
The weight matrix associated with the highest LLV is displayed in Figure 2.For better visualisation, the firms are reordered such that companies cross-influencing each other are now close to each other.Based on the covariate space, our method unleashes similar companies and associated stock return volatility (risks).More specifically, companies related to similar industries are estimated to have closely related volatility magnitudes.For instance, BP, ENI, ConocoPhillips, Chevron and Gazprom are found to have closely related risks.However, firms from different Weight matrix in the covariate space W (X ) using the covariates obtaining the highest LLV.For further information see Table 2.The colour scheme from light yellow (indicating largest distance between two firms) to dark red (indicating closest distance).The density matrix visualisation is generated by the ComplexHeatmap package of Gu et al. (2016).
sectors tend to have related volatilities to oil and gas firms such as Merck & Co. and Hewlett Packard.This finding strengthens the results of Comin and Philippon (2005) and Hassan and Malik (2007) that firms of similar sectors have correlated volatilities to some extent.However, there are also spillovers from different sectors and firms associated with them.Accordingly, the idea of Santi et al. (2021) can also be applied for spatial GARCH models, and we demonstrate how to unleash such cross-sector spillovers since distances within balance sheet data may detect firms of similar sectors.Accordingly, the suggested method might bring together financial theory relating the profitability of firms with their stock returns and risks as well as theory stating that similar sectors may share risks.

CONCLUSION
In conclusion, real-world scenarios often involve the partial or complete inaccessibility of spatial locations.Moreover, spatial dependence in specific applications can extend beyond geographical proximity, influenced by various factors.Specifically, in financial contexts, the significance of firm and balance sheet data surpasses that of physical headquarters' locations.Therefore, we have expanded upon the ideas presented by Santi et al. (2021) for spatial autoregressive models, enabling their applicability to spatial GARCH models.To achieve this, we treat the spatial locations as unknown and approximate them by utilising locations in a secondary covariate space.
In a simulation study, we observed that the proposed procedure yields asymptotically unbiased estimates of the spGARCH model parameters when a large linear correlation is imposed between the actual spatial locations and the covariates.Furthermore, our method supports existing findings regarding the importance of balance sheet data in clustering firms based on return volatility.Specifically, research and development expenses largely explain firms' stock volatility, with firms having similar expense magnitudes exhibiting correlated stock volatilities in a GARCH-type manner.Additionally, costs and profits influence stock returns, highlighting the relationship between firm profitability and stock prices.Firms with close net profits are also correlated in terms of stock volatility.
A possible topic for future research is to what extent and which current balance sheet data helps identify companies' long-term risks.In addition, a spatiotemporal analysis might reveal further interesting results.In addition, different covariates can be considered to determine W 1 and W 2 so that one could reveal specific factors for the observed volatilities of the companies and the second-order effects.Moreover, given the computationally expensive nature of stepwise variable selection in the proxy covariate space, data-driven procedures such as regularised estimation methods could be considered to estimate and select the model simultaneously.

DISCLOSURE STATEMENT
No potential conflict of interest was reported by the author(s).

NOTE
1 For a discussion of identifiability, we refer the reader to Sato and Matsuda (2021).

Figure 1 .
Figure 1.Long-term logarithmic returns of the 50 largest companies.

Figure 2 .
Figure2.Weight matrix in the covariate space W (X ) using the covariates obtaining the highest LLV.For further information see Table2.The colour scheme from light yellow (indicating largest distance between two firms) to dark red (indicating closest distance).The density matrix visualisation is generated by the ComplexHeatmap package ofGu et al. (2016).

Table 1 .
Monte Carlo simulation results of the QML estimates for 1,000 replications.

Table 3 .
Parameter estimates and information criteria for spGARCH model using the best covariates according to the forward selection displayed in Table2.
log-likelihood value, we observe a tendency of increasing values of r and decreasing values of l.