Model Specification. One of the most important credit risk parameters is the probability of de- fault (PD) which is defined to measure the likelihood of the occurrence of a default event for a certain obligor over a one year horizon. Modern credit risk management is crucially based on the risk-adjusted pricing of loans and other credit-risk contingent claims which again heavily relies on a valid and accurate PD estimation methodology. The accuracy of PD estimates is of particular importance for virtually all pricing models for structured credit derivatives. While some pricing models need accurate measures of the aver- age PD of a bond or loan portfolio as an input, some more advanced models require the distribution of individual PDs of such a portfolio which imposes even higher challenges on the validity of the estimation models. Here, we use our general model framework (Equation 2.1) to assess the ac- curacy of PD estimates. Therefore, the “true” PD is taken as the latent variable. Rating outcomes from different sources (e.g., banks, rating tools, or rating agencies) are treated as noisy observations of this latent variable, because we assume that the raters cannot observe the “true” PD of the obligor due to informational asymmetry between firm owners and debt hold- ers which constitutes the cornerstone of modern corporate finance (e.g., Le- land and Pyle, 1977; Xxxx and DeMarzo, 2007). Possible reasons for this asymmetry are limited access to the existing information, such as incomplete accounting information (Duffie and Lando, 2001), and delayed observations of the driving risk factors (Xxx et al., 2009). The motivation for the specification used in this context builds on the main concept of structural or firm value models (e.g., Merton, 1974; Lando, 2004). The standard models assume the firm value to be the only driving factor of credit risk and to follow a Geometric Brownian Motion. As a consequence, an important stylized property of these models is that the probit of the PD is linear in the natural log of the firm value. Let Vi be the log asset value of firm i and PDi its probability of default. The basic model of Xxxxxx (1974) can be written as PDi = Φ(−DDi), DDi = ai + biVi, where Φ is the distribution function of the standard normal distribution, ai and bi are constants independent of Vi, and DDi is the so-called distance to default of firm i (Crosbie and Xxxx, 2003; Bharath and Xxxxxxx, 2008). Along the lines of Duffie and Lando (2001), we assume that the error in ...
Model Specification. In this section we develop a model framework to derive a consensus rating for raters providing ordinal rating information, e.g., external agency ratings. Our model is designed for a dynamic framework capturing a time dependent rating process. Despite the fact that the raters publish ordinal ratings, we assume that they estimate a numerical variable—representing the creditworthiness of the firm—in an internal rating process. Each firm is then assigned to a particular rating class if this variable lies within a certain interval (e.g., XxXxxx and Xxxxxx, 2007; Xxxxxxxxxx et al., 2009). In general, the specific rating process including both the estimation as well as the scale of the variable (representing the creditworthiness) is unknown. In the literature, modeling the creditworthiness, was first discussed by Xxxxxx (1968) who introduces the Z-score. Z-scores are used to predict corporate defaults and are an easy-to- calculate control measure for the financial distress status of companies. The Z-score uses multiple corporate income and balance sheet values to measure the financial health of a company. Furthermore, Xxxxxx (1974) assumes that the creditworthiness can be reflected by the distance-to-default (DD) capturing the distance of the firm’s asset value to its default threshold on the real line. Alternatively, the creditworthiness variable can also be the result of a ordered probit or logit regression model (e.g., Xxxxxx and Rijken, 2004). To obtain ordinal ratings, the estimated DD, the Z-score, or any other numerical variable representing the creditworthiness—which is in the following referred to as “rating score”—is mapped onto an ordinal rating scale by the raters. Let {1, . . . , Kj} be the set of possible non-default rating classes of rater j in descending creditworthiness. That is, 1 denotes the best credit quality and Kj the worst non-default rating class of rater j. Further, Sij(t) denotes the estimated rating score (e.g., negative DD, Z-score) and vij(t) the associated observed ordinal rating of firm i by rater j at time t. The relationship between vij(t) and Sij(t) is given by vij(t) = k ⇔ Sij(t) ∈ [λk−1,j, λk,j), (3.2) for a monotonically increasing sequence λk,j with k = 1, . . . , Kj. The class boundaries are assumed to be constant over time. The data consists of ob- servations for J raters and T time points. Observing rating k for a firm by rater j means that its rating score lies somewhere in the interval [λk−1,j, λk,j). In general, the ...
Model Specification. Following the work of Etale (2019) who examined the insurance sector development and economic growth in Nigeria with the following: GDP = ƒ (INV, PRE, CLA),… (3.1) The present study thus adapted the above model in line with the theoretical proposition in Finance growth Nexus theory thus: RGDP = f (GDPIS, INSTA & INSTC) (3.2) In econometric form, Eq(3.2) can be expressed as:
Model Specification. Initially we considered developing a mode choice model for only one type of commodity that is characteristic for the market for intermodal transport solutions. General cargo is the dominant ETIS manifestation type of the shipper survey typical shipments that are representative for this market. However, because general cargo is a diverse commodity type, because there are small shares of other commodity groups within the segment, and because of consistency with the network models, we decided to estimate a mode choice model that takes into account the type of commodity by NST/R 11 classification. The utility per mode alternative in the network model is decomposed into the following systematic components: V = Vn + Vn + V'n where, Vn is the portion of the utility associated with characteristics of the shipment, Vni is the portion of the utility associated with alternative i , V'ni is the portion of the utility that results from the interaction between alternative i and shipment n . It is assumed there are inherent perceptual differences in the decision-maker’s choice process when considering the land-based alternatives truck-only and rail-based versus sea- and air-based. To take this into account, we specify a nested logit model that considers a two-dimensional choice of geographical context (e.g., land and sea/air) and mode of transport (Figure 6.2). Upper Nest
Model Specification. The arena is spatially explicit, in a sense that it simulates the dynamics of carbon-related patches in space, but it is not aimed at simulating geographically verifiable outputs at pixel level. Thus, the model outputs should be evaluated at an aggregate level of pixels. In order to incorporate various possible patterns of Chomitzian landscape at an initial state as shown in Figure 2, the landscape is stratified in a vertical arrangement from top to bottom into three main sub arenas as described in Table 1, i.e.: forest core, forest margin and mosaic (Figure 1). At a fixed width of the landscape (i.e. 100 pixels), the height of each sub arena is defined based on an area fraction as follows: Where:
Model Specification. Initially we considered developing a mode choice model for only one type of commodity that is characteristic for the market for intermodal transport solutions. General cargo is the dominant ETIS manifestation type of the shipper survey typical shipments that are representative for this market. However, because general cargo is a diverse commodity type, because there are small shares of other commodity groups within the segment, and because of consistency with the network models, we decided to estimate a mode choice model that takes into account the type of commodity by NST/R 11 classification. Organization Code: Code European Commission Classification: Classification Sixth Framework Programme Version: Date: Number 03/05/2007 Contract : 513567 The utility per mode alternative in the network model is decomposed into the following systematic components: V = Vn + Vn + V'n where, Vn is the portion of the utility associated with characteristics of the shipment, Vni is the portion of the utility associated with alternative i , V'ni is the portion of the utility that results from the interaction between alternative i and shipment n . It is assumed there are inherent perceptual differences in the decision-maker’s choice process when considering the land-based alternatives truck-only and rail-based versus sea- and air-based. To take this into account, we specify a nested logit model that considers a two-dimensional choice of geographical context (e.g., land and sea/air) and mode of transport (Figure 6.2). Upper Nest
Model Specification. The model used in this study is adapted from the work of Etale (2019) who examined the insurance sector development and economic growth in Nigeria using the model: GDP = ƒ (INV, PRE, CLA),… (3.1) The present study thus adapted the above model in line with the theoretical proposition in Finance growth Nexus theory. Thus, the mathematical form of the model employed in this study is stated as follow: RGDP = f (GDPIS, INSTA & INSTC) (3.2) The econometric form of the model in equation (3.2) is expressed as: RGDP = 𝛽0 + 𝛽1RGDPIS + 𝛽2INSTA + 𝛽3INSTC + 𝑈𝑡 (3.3) By taking the log of Eq(3.3), we have: lnRGDP = 𝛽0 + 𝛽1lnRGDPIS + 𝛽2lnINSTA + 𝛽3lnINSTC + 𝑈𝑡 (3.4) The Autoregressive Distributed Lags form of the equation (3.4) is specified thus: 𝑝 𝛥𝑙𝑛𝑅𝐺𝐷𝑃𝑡 = 𝛽0𝑖 + 𝛽1𝑙𝑛𝑅𝐺𝐷𝑃𝐼𝑆𝑡−1 + 𝛽2𝑙𝑛𝐼𝑁𝑆𝑇𝐴𝑡−1 + 𝛽3𝑙𝑛𝐼𝑁𝑆𝑇𝐶𝑡−1 + ∑ 𝜃𝑖 𝛥𝑙𝑛𝑅𝐺𝐷𝑃𝑡−1 + 1 𝑖=1 𝑝 𝑝 𝑝 + ∑ 𝛾𝑖 𝛥𝑙𝑛𝑅𝐺𝐷𝑃𝐼𝑆𝑡−1 + ∑ 𝜆𝑖 𝛥𝑙𝑛𝐼𝑁𝑆𝑇𝐴𝑡−1 + ∑ 𝜑𝑖 𝛥𝑙𝑛𝐼𝑁𝑆𝑇𝐶𝑡−1 + 𝛹𝐸𝐶𝑀𝑡−1 𝑖=1 𝑖=1 𝑖=1 + 𝑈𝑡 … … … … … … … … … … … . … . (3.5)
