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# Maximum Likelihood Models
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Full information maximum likelihood estimation for spatial lag and error models with analytical derivatives and concentrated log-likelihood functions.
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## Capabilities
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### Spatial Error ML Models
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```python { .api }
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class ML_Error:
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    def __init__(self, y, x, w, epsilon=0.0000001, hard_bound=False, vm=False,
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                 name_y=None, name_x=None, name_w=None, name_ds=None, latex=False):
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        """
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        Maximum likelihood spatial error model.
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        Parameters:
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        - y (array): nx1 dependent variable
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        - x (array): nxk independent variables
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        - w (sparse matrix): Spatial weights matrix
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        - epsilon (float): Convergence criterion
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        - hard_bound (bool): Constrain lambda to [-1,1]
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        - vm (bool): Include variance-covariance matrix
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        """
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class ML_Error_Regimes:
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    def __init__(self, y, x, regimes, w, constant_regi='many', cols2regi='all',
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                 regime_err_sep=False, cores=False, epsilon=0.0000001,
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                 hard_bound=False, vm=False, name_y=None, name_x=None, 
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                 name_regimes=None, name_w=None, name_ds=None, latex=False):
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        """ML spatial error model with regimes."""
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```
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### Spatial Lag ML Models
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```python { .api }  
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class ML_Lag:
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    def __init__(self, y, x, w, epsilon=0.0000001, hard_bound=False, vm=False,
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                 name_y=None, name_x=None, name_w=None, name_ds=None, latex=False):
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        """
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        Maximum likelihood spatial lag model.
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        Parameters:
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        - y (array): nx1 dependent variable  
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        - x (array): nxk independent variables
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        - w (sparse matrix): Spatial weights matrix
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        - epsilon (float): Convergence criterion
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        - hard_bound (bool): Constrain rho to [-1,1]
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        - vm (bool): Include variance-covariance matrix
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        """
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class ML_Lag_Regimes:
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    def __init__(self, y, x, regimes, w, constant_regi='many', cols2regi='all',
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                 regime_lag_sep=False, cores=False, epsilon=0.0000001,
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                 hard_bound=False, vm=False, name_y=None, name_x=None,
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                 name_regimes=None, name_w=None, name_ds=None, latex=False):
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        """ML spatial lag model with regimes."""
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```
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## Usage Examples
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### Basic ML Spatial Error
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```python
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import numpy as np
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import spreg
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from libpysal import weights
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# Data setup
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n = 49
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x = np.random.randn(n, 2)
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y = np.random.randn(n, 1)
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w = weights.lat2W(7, 7)
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# ML spatial error estimation
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ml_error = spreg.ML_Error(y, x, w.sparse, name_y='y', name_x=['x1', 'x2'])
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print(ml_error.summary)
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```
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### ML Spatial Lag Model
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```python
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# ML spatial lag estimation  
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ml_lag = spreg.ML_Lag(y, x, w.sparse, name_y='y', name_x=['x1', 'x2'])
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print(ml_lag.summary)
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print(f"Spatial lag parameter (rho): {ml_lag.rho}")
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```