logLikStratSel function

Log-Likelihood Function of an Agent Error Model with Correlated Errors (strategic selection model)

This function calculates the log-likelihood value for an agent error model (belongs to the general class of quantal response models) with correlated errors. The underlying formal structure is

1 
 /\  
/  \  
    /    \ 2  
   u11   /\  
   /  \  
  /    \  
 0     u14  
 0     u24

and shows a game where there are two players which move sequentially. Player 1 decides to move left or right and if she does move right player 2 gets to move. The final outcome in this case depends on the move of player 2.

logLikStratSel(x11, x14, x24, y, beta)

Arguments

• x11: A vector or a matrix containing the explanatory variables used to parametrize u11.
• x14: A vector or a matrix containing the explanatory variables used to parametrize u14.
• x24: A vector or a matrix containing the explanatory variables used to parametrize u24.
• y: Vector. Outcome variable which can take values 1, 3, and 4 depending on which outcome occurred.
• beta: Vector. Coefficients of the model whereas the last element is the correlation coefficient $\rho$. Note, that this parameter has been re-paramterized (see details).

Details

This function provides the likelihood of an agent error model (Signorino, 2003) but in addition allows the random components to be correlated and hence can take selection into account. The correlation parameter is re-paramaterized (see Note). Further, as with probit and logit models, one needs to assume an error variance to achieve identification, here 1 is chosen as with a regular probit model. Finally, u13 and u23 are set to 0 to achieve identification.

Returns

Returns a numeric value for the log-likelihood function evaluated for $\beta$.

References

Lucas Leemann. 2014. "Strategy and Sample Selection - A Strategic Selection Estimator", Political Analysis 22: 374-397.

Author(s)

Lucas Leemann lleemann@gmail.com

Note

The notation $\boldsymbol{\Phi_2}(a;b;c)$ indicates a bivariate standard normal cumulative distribution evaluated at the values a,b whereas the two random variables have a correlation of c.

$$\ell\ell = \sum_{i=1}^n \log\left(\boldsymbol{\Phi_2}(p_{i4}(\mathbf{x}_{i14} \boldsymbol{\beta}_{14})-\mathbf{x}_{i11}\boldsymbol{\beta}_{11}; \mathbf{x}_{i24} \boldsymbol{\beta}_{24}; -\rho)^{(1-I(y_{i}=1))(1-I(y_{i}=4))} \right)ll= \sum log(\Phi_2(p_i4(x_i14*\beta_14)-x_i11*\beta_11; x_i24*\beta_24; -\rho )^(1-I(y_1=1))(1-I(y_1=4))$$ $$+ \sum_{i=1}^n \log\left(\boldsymbol{\Phi_2}(p_{i4}(\mathbf{x}_{i14} \boldsymbol{\beta}_{14})-\mathbf{x}_{i11}\boldsymbol{\beta}_{11}; \mathbf{x}_{i24} \boldsymbol{\beta}_{24}; \rho)^{(1-I(y_{i}=1))I(y_{i}=4)} \right)+ \sum log(\Phi_2(p_i4(x_i14*\beta_14)-x_i11*\beta_11; x_i24*\beta_24; \rho )^(1-I(y_1=1))I(y_1=4)$$ $$+ \sum_{i=1}^n \log\left(1-\Phi(p_{i4} \mathbf{x}_{i14} \boldsymbol{\beta}_{14} -\mathbf{x}_{i1} \boldsymbol{\beta}_{11})\right)\sum log(1-\Phi(p_i4*x_i14*\beta_14 - x_i11*\beta_11))$$

whereas

$$p_{i24} = \Phi(x_{24}\cdot\beta_{24})p_i4 = \Phi(x_24 \beta_24)$$

and

$$p_{i1} = \Phi(x_{11}\cdot\beta_{11}-p_{24}(x_{14}\cdot\beta_{14}))p_i1 = \Phi(x_11* \beta_11-p_24* (x_14 * \beta_14))$$

The re-parametrization is as follows:

$$\rho = \frac{2}{1-exp(-\theta)}- 1\rho = 2/(1-exp(-\theta))- 1$$

See Also

StratSel