(Requires Matrix Algebra)The population multiple regression model can be written in matrix form as
Y = Xβ + U
where
Y = ![(Requires Matrix Algebra)The population multiple regression model can be written in matrix form as Y = Xβ + U where Y = ,U = ,X = ,and β = Note that the X matrix contains both k endogenous regressors and (r +1)included exogenous regressors (the constant is obviously exogenous). The instrumental variable estimator for the overidentified case is where Z is a matrix,which contains two types of variables: first the r included exogenous regressors plus the constant,and second,m instrumental variables. Z = It is of order n × (m+r+1). For this estimator to exist,both ( Z)and [ Z( Z)-1 X] must be invertible.State the conditions under which this will be the case and relate them to the degree of overidentification.](https://d2lvgg3v3hfg70.cloudfront.net/TB2833/11eab146_bd89_99e5_896a_7f0c3dda6078_TB2833_11.jpg)
,U = ![(Requires Matrix Algebra)The population multiple regression model can be written in matrix form as Y = Xβ + U where Y = ,U = ,X = ,and β = Note that the X matrix contains both k endogenous regressors and (r +1)included exogenous regressors (the constant is obviously exogenous). The instrumental variable estimator for the overidentified case is where Z is a matrix,which contains two types of variables: first the r included exogenous regressors plus the constant,and second,m instrumental variables. Z = It is of order n × (m+r+1). For this estimator to exist,both ( Z)and [ Z( Z)-1 X] must be invertible.State the conditions under which this will be the case and relate them to the degree of overidentification.](https://d2lvgg3v3hfg70.cloudfront.net/TB2833/11eab146_bd89_99e6_896a_6d2e94f2d092_TB2833_11.jpg)
,X = ![(Requires Matrix Algebra)The population multiple regression model can be written in matrix form as Y = Xβ + U where Y = ,U = ,X = ,and β = Note that the X matrix contains both k endogenous regressors and (r +1)included exogenous regressors (the constant is obviously exogenous). The instrumental variable estimator for the overidentified case is where Z is a matrix,which contains two types of variables: first the r included exogenous regressors plus the constant,and second,m instrumental variables. Z = It is of order n × (m+r+1). For this estimator to exist,both ( Z)and [ Z( Z)-1 X] must be invertible.State the conditions under which this will be the case and relate them to the degree of overidentification.](https://d2lvgg3v3hfg70.cloudfront.net/TB2833/11eab146_bd89_99e7_896a_6bdda3b19972_TB2833_11.jpg)
,and β = ![(Requires Matrix Algebra)The population multiple regression model can be written in matrix form as Y = Xβ + U where Y = ,U = ,X = ,and β = Note that the X matrix contains both k endogenous regressors and (r +1)included exogenous regressors (the constant is obviously exogenous). The instrumental variable estimator for the overidentified case is where Z is a matrix,which contains two types of variables: first the r included exogenous regressors plus the constant,and second,m instrumental variables. Z = It is of order n × (m+r+1). For this estimator to exist,both ( Z)and [ Z( Z)-1 X] must be invertible.State the conditions under which this will be the case and relate them to the degree of overidentification.](https://d2lvgg3v3hfg70.cloudfront.net/TB2833/11eab146_bd89_c0f8_896a_57c8d097b8c9_TB2833_11.jpg)
Note that the X matrix contains both k endogenous regressors and (r +1)included exogenous regressors (the constant is obviously exogenous).
The instrumental variable estimator for the overidentified case is ![(Requires Matrix Algebra)The population multiple regression model can be written in matrix form as Y = Xβ + U where Y = ,U = ,X = ,and β = Note that the X matrix contains both k endogenous regressors and (r +1)included exogenous regressors (the constant is obviously exogenous). The instrumental variable estimator for the overidentified case is where Z is a matrix,which contains two types of variables: first the r included exogenous regressors plus the constant,and second,m instrumental variables. Z = It is of order n × (m+r+1). For this estimator to exist,both ( Z)and [ Z( Z)-1 X] must be invertible.State the conditions under which this will be the case and relate them to the degree of overidentification.](https://d2lvgg3v3hfg70.cloudfront.net/TB2833/11eab146_bd89_c0f9_896a_53308949d5e9_TB2833_11.jpg)
where Z is a matrix,which contains two types of variables: first the r included exogenous regressors plus the constant,and second,m instrumental variables.
Z = ![(Requires Matrix Algebra)The population multiple regression model can be written in matrix form as Y = Xβ + U where Y = ,U = ,X = ,and β = Note that the X matrix contains both k endogenous regressors and (r +1)included exogenous regressors (the constant is obviously exogenous). The instrumental variable estimator for the overidentified case is where Z is a matrix,which contains two types of variables: first the r included exogenous regressors plus the constant,and second,m instrumental variables. Z = It is of order n × (m+r+1). For this estimator to exist,both ( Z)and [ Z( Z)-1 X] must be invertible.State the conditions under which this will be the case and relate them to the degree of overidentification.](https://d2lvgg3v3hfg70.cloudfront.net/TB2833/11eab146_bd89_c0fa_896a_3b05b78dda27_TB2833_11.jpg)
It is of order n × (m+r+1).
For this estimator to exist,both ( ![(Requires Matrix Algebra)The population multiple regression model can be written in matrix form as Y = Xβ + U where Y = ,U = ,X = ,and β = Note that the X matrix contains both k endogenous regressors and (r +1)included exogenous regressors (the constant is obviously exogenous). The instrumental variable estimator for the overidentified case is where Z is a matrix,which contains two types of variables: first the r included exogenous regressors plus the constant,and second,m instrumental variables. Z = It is of order n × (m+r+1). For this estimator to exist,both ( Z)and [ Z( Z)-1 X] must be invertible.State the conditions under which this will be the case and relate them to the degree of overidentification.](https://d2lvgg3v3hfg70.cloudfront.net/TB2833/11eab146_bd89_c0fb_896a_71b246d1f668_TB2833_11.jpg)
Z)and [ ![(Requires Matrix Algebra)The population multiple regression model can be written in matrix form as Y = Xβ + U where Y = ,U = ,X = ,and β = Note that the X matrix contains both k endogenous regressors and (r +1)included exogenous regressors (the constant is obviously exogenous). The instrumental variable estimator for the overidentified case is where Z is a matrix,which contains two types of variables: first the r included exogenous regressors plus the constant,and second,m instrumental variables. Z = It is of order n × (m+r+1). For this estimator to exist,both ( Z)and [ Z( Z)-1 X] must be invertible.State the conditions under which this will be the case and relate them to the degree of overidentification.](https://d2lvgg3v3hfg70.cloudfront.net/TB2833/11eab146_bd89_c0fc_896a_43d802456eb0_TB2833_11.jpg)
Z( ![(Requires Matrix Algebra)The population multiple regression model can be written in matrix form as Y = Xβ + U where Y = ,U = ,X = ,and β = Note that the X matrix contains both k endogenous regressors and (r +1)included exogenous regressors (the constant is obviously exogenous). The instrumental variable estimator for the overidentified case is where Z is a matrix,which contains two types of variables: first the r included exogenous regressors plus the constant,and second,m instrumental variables. Z = It is of order n × (m+r+1). For this estimator to exist,both ( Z)and [ Z( Z)-1 X] must be invertible.State the conditions under which this will be the case and relate them to the degree of overidentification.](https://d2lvgg3v3hfg70.cloudfront.net/TB2833/11eab146_bd89_c0fd_896a_2dc3f5bef957_TB2833_11.jpg)
Z)-1 ![(Requires Matrix Algebra)The population multiple regression model can be written in matrix form as Y = Xβ + U where Y = ,U = ,X = ,and β = Note that the X matrix contains both k endogenous regressors and (r +1)included exogenous regressors (the constant is obviously exogenous). The instrumental variable estimator for the overidentified case is where Z is a matrix,which contains two types of variables: first the r included exogenous regressors plus the constant,and second,m instrumental variables. Z = It is of order n × (m+r+1). For this estimator to exist,both ( Z)and [ Z( Z)-1 X] must be invertible.State the conditions under which this will be the case and relate them to the degree of overidentification.](https://d2lvgg3v3hfg70.cloudfront.net/TB2833/11eab146_bd89_c0fe_896a_25f5de80a00a_TB2833_11.jpg)
X] must be invertible.State the conditions under which this will be the case and relate them to the degree of overidentification.
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