Modeling strategic group dynamics: A hidden Markov approach

Abstract

With competition playing a critical role in market-based strategic planning and implementation, identifying and understanding competiton and competitive dynamics has become critical. In this vein, the strategic groups perspective has emerged as a powerful means to understand such competitive phenomena. Empirical approaches to model competitive dynamics within the strategic groups framework, however, have been piece meal as researchers typically resort to distinct sequential analysis by time period. To overcome the limitations of these simplistic approaches, we develop a hidden Markov model to study strategic group (competitive) dynamics. In this approach, we explicitly account for competitive dynamics over time by modeling strategic group memberships as latent states that follow a first-order Markov process. Thus, we explictly model the notion that firms adopt their strategy for the next time period based on their current strategy and respective outcomes. We illustrate the model with longitudinal data from COMPUSTAT on 63 public banks from the tri-state region of NY-OH-PA. The results show the proposed model to be superior to a number of viable alternative approaches that have been suggested in the literature. We find the existence of three strategic groups: the leveraged group has low current assets compared to current liabilities, high debt to equity, and high total borrowing to assets. The lending group consists of the largest banks that focus on lending with high ratios of gross loans to securities and gross loans to deposits. The balanced group has the largest number of banks where the values of the financial and product ratios are intermediate compared to the leveraged and lending groups. The asymmetries in the switching probabilites are also evident as there seems to be a higher probability of switching into the balanced group than switching out of this group. The switching probabilites are symmetric between the the leveraged and lending groups.

Appendix I

1.1 Derivation of the full conditional distributions for the HMM dynamic model

The technical model specifications and derivations for the MCMC algorithm for the hidden Markov model are presented here. For each bank i = 1,...,N, we observe p strategic variables for t = 1,...,T time periods, denoted by the p × 1 vector Y _it . We assume that each firm belongs to one of K strategic groups in time period t, denoted by the (latent) variable S _it . Given S _it , we assume the strategic variables are independent and follow a p-variate normal distribution with mean ${\mu_{{S_{it}}}}$ and variance ${\Sigma_{{S_{it}}}}$. Furthermore, we assume that the group memberships change over time following a first-order Markov process, i.e., $p\left( {\left. {{S_{it}}} \right|{S_{i\left( {t - 1} \right)}}} \right) = \prod\nolimits_{j = 1}^K {\prod\nolimits_{k = 1}^K {\xi_{jk}^{I\left\{ {{S_{it}} = k,{S_{i\left( {t - 1} \right)}} = j} \right\}}} }$. Then,

$$\begin{array}{*{20}c} {p{\left( {Y_{{i1}} ,Y_{{i2}} , \ldots ,Y_{{iT}} {\text{ $|$ }}S_{{i1}} ,S_{{i2}} , \ldots ,S_{{iT}} } \right)} = {\mathop \prod \limits_{t = 1}^T }p{\left( {Y_{{it}} |S_{{it}} } \right)}} \\ { = {\mathop \prod \limits_{t = 1}^T }{\left\{ {{\mathop \prod \limits_{k = 1}^K }N_{p} {\left( {Y_{{it}} {\text{ $|$ }}\mu _{k} ,{\text{ $ \Sigma $ }}_{k} } \right)}^{{I\{ S_{{it}} = k\} }} } \right\}}} \\ { = {\mathop \prod \limits_{k = 1}^K }{\left\{ {{\mathop \prod \limits_{t:S_{{it}} = k} }N_{p} {\left( {Y_{{it}} |\mu _{k} ,{\text{ $ \Sigma $ }}_{k} } \right)}} \right\}},} \\ \end{array} $$

(1)

where the last product is only taken across those time periods t for which S _it  = k. Here $N_{l} {\left( {.|\mu ,{\text{ $ \Sigma $ }}} \right)}$ denotes the density function of a p-variate normal distribution with mean μ and variance-covariance matrix Σ. The joint distribution of the group memberships is given by:

$$ \begin{array}{*{20}{c}} {p\left( {{S_{i0}},{S_{i1}},\ldots,{S_{iT}}} \right) = \left( {\prod\nolimits_{t = 1}^T {p\left( {\left. {{S_{it}}} \right|{S_{i\left( {t - 1} \right)}}} \right)} } \right) \times p\left( {{S_{i0}}} \right)} \hfill \\ { = \left( {\prod\nolimits_{t = 1}^T {\prod\nolimits_{j = 1}^K {\prod\nolimits_{k = 1}^K {\xi_{jk}^{I\left\{ {{S_{it}} = k,{S_{i\left( {t - 1} \right)}} = j} \right\}}} } } } \right) \times p\left( {{S_{i0}}} \right)} \hfill \\ { = \left( {\prod\nolimits_{j = 1}^K {\prod\nolimits_{k = 1}^K {\xi_{jk}^{{N_{jk}}\left( {{S_{i1,}}{S_{i2, \cdots, }}{S_{iT}} = j} \right)}} } } \right) \times p\left( {{S_{i0}}} \right)} \hfill \\ \end{array}, $$

(2)

where N _jk (S _i1,S _i2,..., S _iT) counts the number of times that firm i switches from strategic group j to k across all T time periods. The complete data likelihood contribution:

$$ p\left( {{Y_i},{S_i}} \right) = p\left( {\left. {{Y_i}} \right|{S_i}} \right) \times p\left( {{S_i}} \right) $$

(3)

for firm t is given by the product of expressions (1) and (2), where Y _i  = (Y _i1, Y _i2,..., Y _iT ) and S _i  = (S _i1, S _i2,..., S _iT ). The full-sample, complete-data likelihood function is computed as the product of expression (3) across i = 1,...,N.

In the following, we denote the vector of all unknown parameters by θ. We specify independent normal priors for the groups mean μ _k , k = 1,...,K, with mean $\bar{\mu } = 0$ and variance $\Sigma = 25 \times {I_p}$. For each Σ _k , k = 1,...,K, we specify an inverse Wishart distribution as the prior distribution with parameters ${r_0} = p + 2$ and S ₀ = I _p . Each row of the matrix of transition probabilities $\xi = \left\{ {{\xi_{jk}}} \right\}$ is assumed to be independent a priori following a Dirichlet distribution with parameters $\left( {{M_{j1}},{M_{j2}}, \ldots, {M_{jk}}} \right) = \left( {1,1,...,1} \right)$, for j = 1,...K. We assume the process S _it starts at t = 0 from an arbitrary discrete distribution with probabilities (η ₁,...,η _K ). We treat the parameters η _K as unknown and specify a Dirichlet prior distribution for it with parameters $\left( {{N_{01}},{N_{02}}, \ldots {N_{0K}}} \right) = \left( {1,1, \ldots, 1} \right)$. Hence, the Markov chain is not assumed to be stationary a priori. Furthermore, we are not left to specify an ad-hoc distribution for the initial conditions, which becomes more important when T is relatively small. Hence, for each firm i, we have T + 1 latent variables S _it .

The posterior distribution is proportional to the product of the full-sample, complete-data likelihood function and the priors. This posterior is not available in closed form and we use MCMC sampling through the full conditional distributions to simulate the marginal posterior distributions of the model parameters. The full conditional distributions are obtained from the product of the likelihood and priors (e.g., Gilks et al. 1998).

Full conditional distribution for μ _k

The full conditional distribution for μ _k is obtained from a product of two normal densities (for notational simplicity, we only present the kernels here), and is proportional to:

$$ \exp \left\{ { - \frac{1}{2}\sum\nolimits_{\left( {i,t:{S_{it}} = k} \right)} {\left( {{Y_{it}} - {\mu_k}} \right)\prime \Sigma_k^{ - 1}\left( {{Y_{it}} - {\mu_k}} \right)} } \right\} \times \exp \left\{ { - \frac{1}{2}\left( {{\mu_k} - \bar{\mu }} \right)\prime \Sigma_k^{ - 1}\left( {{\mu_k} - \bar{\mu }} \right)} \right\}. $$

(4)

Both quadratic forms in the exponent terms can be completed and combined as follows:

$$ \mu_k^\prime \left( {{\Sigma^{ - 1}} + {n_k}\Sigma_k^{ - 1}} \right){\mu_k} - \mu_k^\prime \left( {{\Sigma^{ - 1}}\bar{\mu } + \Sigma_k^{ - 1}{\Sigma_{\left\{ {i,t:{S_{it}} = k} \right\}}}{Y_{it}}} \right) - \left( {{\Sigma_{\left\{ {i,t:{S_{it}} = k} \right\}}}{Y_{it}}\prime \Sigma_k^{ - 1} + \bar{\mu }\prime {\Sigma^{ - 1}}} \right){\mu_k}, $$

(5)

where n _k is the number of observations across i and t that are in strategic group k. From expression (5), it follows that the full conditional for μ _k is a p-variate normal distribution with mean ${{\text{V}}^{ - {1}}}\left( {{\Sigma^{ - {1}}}\bar{\mu } + \Sigma_k^{ - 1}{\Sigma_{\left\{ {i,t:{S_{it}} = k} \right\}}}{Y_{it}}} \right)$ and variance V ⁻¹, with $V = {\Sigma^{ - 1}} + {n_k}\Sigma_k^{^{ - 1}}$.

Full conditional distribution for Σ _k

The full conditional for Σ _k is derived from the product of a normal density from the likelihood and the inverted Wishart prior, i.e.,:

$$ \left( {{{\prod\nolimits_{i,t:{S_{it}} = k} {\left| {{\Sigma_k}} \right|} }^{ - \frac{1}{2}}}\exp \left\{ { - \frac{1}{2}\left( {{Y_{it}} - {\mu_k}} \right)\prime \sum\nolimits_k^{ - 1} {\left( {{Y_{it}} - {\mu_k}} \right)} } \right\}} \right) \times {\left| {{\Sigma_k}} \right|^{ - \frac{{{r_0} + p + 1}}{2}}}\exp \left\{ { - \frac{1}{2}tr\left( {{S_0}\Sigma_k^{ - 1}} \right)} \right\} = {\left| {{\Sigma_k}} \right|^{ - \frac{{{n_k}}}{2}}}\exp \left\{ { - \frac{1}{2}tr\left( {\sum\nolimits_{\left( {i,t:{S_{it}} = k} \right)} {\left( {{Y_{it}} - {\mu_k}} \right)\left( {{Y_{it}} - {\mu_k}} \right)\prime \Sigma_k^{ - 1}} } \right)} \right\} \times {\left| {{\Sigma_k}} \right|^{ - \frac{{{r_0} + p + 1}}{2}}}\exp \left\{ { - \frac{1}{2}tr\left( {{S_0}\Sigma_k^{ - 1}} \right)} \right\}. $$

(6)

From the previous equation, it follows that the full conditional distribution for Σ _k is an inverted Wishart distribution with parameters $\sum\nolimits_{\left( {i,t:{S_{it}} = k} \right)} {\left( {{Y_{it}} - {\mu_k}} \right)\left( {{Y_{it}} - {\mu_k}} \right)} \prime + {S_0}$ and r ₀ + n _k .

Full conditional distribution for ξ

Each row of the matrix ξ may be sampled independently. The full conditional for each row $\left( {{\xi_{j1}},{\xi_{j2}}, \ldots, {\xi_{jK}}} \right)$, for j = 1,...,K, is derived from the product of the full-sample likelihood contribution of the latent group memberships ${S_1},{S_2}, \ldots, {S_N}$, where ${S_i} = \left( {{S_{i1}},{S_{i2}}, \ldots, {S_{iT}}} \right)$, and the Dirichlet prior for ${\xi_{j1}},{\xi_{j2}}, \ldots, {\xi_{jK}}$. Let ${S_i} = \left( {{S_{i1}},{S_{i2}}, \ldots, {S_{iT}}} \right)$ and $S = \left( {{S_1},{S_2}, \ldots, {S_N}} \right)$, it follows that the full conditional is proportional to:

$$ \left\{ {\prod\nolimits_{t = 1}^N {\left( {\prod\nolimits_{k = 1}^K {\xi_{jk}^{{N_{jk}}\left( {{S_i}} \right)}} } \right)} } \right\} \times \left( {\prod\nolimits_{k = 1}^K {\xi_{jk}^{{M_{jk}}}} } \right) = \left( {\prod\nolimits_{k = 1}^K {\xi_{jk}^{{N_{jk}}(S)}} } \right) \times \left( {\prod\nolimits_{k = 1}^K {\xi_{jk}^{{M_{jk}}}} } \right), $$

(7)

where N _jk (S) counts the number of transitions from states j to k across all firms i and time periods t. Hence, it follows that the full conditional distribution for the j-th row of ξ is a Dirichlet distribution with parameters ${N_{jk}}(S) + {M_{jk}}$, k=1,...,K.

Full conditional distribution for (η ₁,η ₂,...,η _K )

Using similar logic as for deriving the full conditional distribution for the j-th row of ξ, it follows that the full conditional distribution for the group sizes at t = 0 is also Dirichlet with parameters (N _0k  + M _0k ), k=1,...,K, where N _0k is the number of banks i in group k at time t = 0.

Data augmentation for the hidden states

The paths of the hidden states (S ₁,S ₂,...,S _N ) are updated using a multi-move sampler (Scott 2002; Frühwirth-Schnatter 2006). For each bank, we sample S _i from the full conditional distribution:

$$ p\left( {\left. {{S_t}} \right|{Y_i},\theta } \right) = \left( {\prod\nolimits_{t = 0}^{T - 1} {p\left( {\left. {{S_{it}}} \right|{S_{i\left( {t + 1} \right)}};{Y_i},\theta } \right)} } \right)p\left( {\left. {{S_{iT}}} \right|{Y_i},\theta } \right), $$

(8)

where:

$$ p\left( {\left. {S_{it} } \right|S_{i\left( {t + 1} \right)} ;Y_i ,\theta } \right) = \frac{{p\left( {\left. {S_{i\left( {t + 1} \right)} } \right|S_{it} ;Y_i ,\theta } \right)p\left( {\left. {S_{it} } \right|Y_i ,\theta } \right)}}{{p\left( {\left. {S_{i\left( {t + 1} \right)} } \right|Y_i ,\theta } \right)}} \propto \xi _{S_{it} ,S_{i\left( {t + 1} \right)} } p\left( {\left. {S_{it} } \right|Y_{it} ,Y_{i2} , \ldots Y_{it} ,\theta } \right). $$

(9)

The probabilities $p\left( {\left. {{S_{it}}} \right|{Y_{i1}},{Y_{i2}}, \ldots, {Y_{it}},\theta } \right)$ are the filtered state probabilities. Hence, it follows from (8) that a path S _i may be obtained recursively by first sampling a value ${\tilde{S}_{iT}}$ from the filtered state probability distribution $p\left( {\left. {{S_{iT}}} \right|{Y_i},\theta } \right)$, and subsequently, for $t = T - 1,T - 2, \ldots, 0$, a value for S _it is sampled from the conditional distribution $p\left( {\left. {{S_{it}}} \right|{{\tilde{S}}_{i\left( {t + 1} \right)}};{Y_i},\theta } \right)$ in (9). The filtered state probabilities are sampled using a forward sampling algorithm (e.g., Frühwirth-Schnatter 2006) from:

$$ p\left( {\left. {{S_{it}}} \right|{Y_{i1}},{Y_{i2}}, \ldots, {Y_{it}},\theta } \right) \propto p\left( {\left. {{Y_{it}}} \right|{S_{it}},{Y_{i1}},{Y_{i2}}, \ldots, {Y_{t\left( {t - 1} \right)}},\theta } \right)p\left( {\left. {{S_{it}}} \right|{Y_{i1}},{Y_{i2}}, \ldots, {Y_{t\left( {t - 1} \right)}},\theta } \right), $$

(10)

where $p\left( {\left. {{Y_{it}}} \right|{S_{it}},{Y_{i1}},{Y_{i2}}, \ldots, {Y_{i\left( {t - 1} \right)}},\theta } \right) = p\left( {\left. {{Y_{it}}} \right|{S_{it}},\theta } \right)$ is the p-variate normal density with mean ${\mu_{{S_{it}}}}$ and variance ${\Sigma_{{S_{it}}}}$. The one-step ahead probabilities $p\left( {\left. {{S_{it}}} \right|{Y_{i1}},{Y_{i2}}, \ldots, {Y_{i\left( {t - 1} \right)}},\theta } \right)$ are obtained from:

$$ \begin{array}{*{20}{c}} {p\left( {\left. {{S_{it}}} \right|{Y_{i1}},{Y_{i2}}, \ldots, {Y_{i\left( {t - 1} \right)}},\theta } \right) = \sum\nolimits_{k = 1}^K {p\left( {\left. {{S_{it}},{S_{it\left( {t - 1} \right)}} = k} \right|{Y_{i1}},{Y_{i2}}, \ldots, {Y_{i\left( {t - 1} \right)}},\theta } \right)} } \\ { = \sum\nolimits_{k = 1}^K {\xi k{S_{it}}p\left( {\left. {{S_{i\left( {t - 1} \right)}} = k} \right|{Y_{i1}},{Y_{i2}}, \ldots, {Y_{i\left( {t - 1} \right)}},\theta } \right).} } \\ \end{array} $$

(11)

Hence, the filtered state probabilities can be computed recursively for t = 1,...,T from expressions (10) and (11). For t = 1 we set $p\left( {{S_{i\left( {t - 1} \right)}} = \left. k \right|{Y_{i1}},{Y_{i2}}, \ldots, {Y_{i\left( {t - 1} \right)}},\theta } \right) = {\eta_k}$, for k = 1,...,K, in expression (11).