internal partial class EntitySelector
{
    public EntitySelector()
    {
        InitializeComponent();

        var viewModel = new EntitySelectorViewModel();
        DataContext = viewModel;

        var binding = new Binding
        {
            Path = new PropertyPath("SelectedEntityCode"),
            Source = viewModel,
            Mode = BindingMode.TwoWay,
            UpdateSourceTrigger = UpdateSourceTrigger.PropertyChanged
        };
        BindingOperations.SetBinding(this, SelectedEntityProperty, binding);
    }

    public static readonly DependencyProperty SelectedEntityProperty =
        DependencyProperty.Register
        (
            "SelectedEntity",
            typeof(string),
            typeof(EntitySelector),
            new FrameworkPropertyMetadata(default(string), 
                FrameworkPropertyMetadataOptions.BindsTwoWayByDefault,
                SelectedEntityProperty_Changed)
        );

    public string SelectedEntity
    {
        get { return Convert.ToString(GetValue(SelectedEntityProperty)); }
        set { SetValue(SelectedEntityProperty, value); }

    }

    private static void SelectedEntityProperty_Changed(DependencyObject dobj, DependencyPropertyChangedEventArgs e)
    {
       //Some code     
    }
}

public string SelectedEntityCode
{
    get { return _selectedEntityCode; }
    set
    {
        if (_selectedEntityCode != value)
        {
            _selectedEntityCode = value;
            OnPropertyChanged(() => SelectedEntityCode);
        }
    }
}