Consider the linear model yi = xiB + e = B1 + B2xi2 + ... + Bkxik + ei, i = 1, ..., n, or in matrix notation Y =XB + e.
Consider the linear model X = Z pie + u where Z is a matrix n * m, X is a matrix n * k and pie is a matrix m * k. Assume that
1. E (x'ixi) has full column rank;
2. E (ei/xi) = g (xi), , i =1, ..., n;
3. E (ei4) < infinity and E (xij/4) < infinity, i =1, ..., n; j =1, ..., k;
4. E (ui/zi) = 0, i = 1, ..., n;
A.) Let m = k and assume that rank pie = k. Derive a consistent estimator for B. Derive the asymptotic distribution of the estimator.
B.) Let m > k find a consistent estimator of B and his distribution.
C.) What happen if m = k and E (X'Z) has rank j < k?
A.) To derive a consistent estimator for B when m=k and rank(pie)=k, we can use the method of Generalized Method of Moments (GMM).
First, let's define the moment conditions for the GMM estimation:
g(xi,B) = xiB - Yi
Next, we need to compute the sample moment conditions:
g(xi,B) = xiB - Yi
The consistent estimator B_hat can be obtained by minimizing the moment conditions:
B_hat = argmin_B [n^(-1) sum(g(xi,B))^T * W * sum(g(xi,B))]
where W is a positive definite weighting matrix. A popular choice for W is the inverse of the sample covariance matrix of the moment conditions: W = (n^(-1) sum(g(xi,B) * g(xi,B))^(-1)
To derive the asymptotic distribution of the estimator B_hat, we need to use the asymptotic properties of the GMM estimators. Under certain regularity conditions, the estimator B_hat is consistent and asymptotically normal.
The asymptotic variance-covariance matrix of B_hat is given by:
A_var(B_hat) = (n^(-1) sum(g(xi,B_hat) * g(xi,B_hat))^(-1) * (n^(-1) sum(g(xi,B_hat) * g(xi,B_hat)) * (n^(-1) sum(xi * xi') * (n^(-1) sum(xi * xi')^(-1))
B.) When m > k, we can use a method called Instrumental Variables (IV) estimation to obtain a consistent estimator for B.
In this case, we need to find an instrumental variable matrix Z^* that satisfies the following conditions:
1. Z^* has the same dimension as X (n x k).
2. Rank(Z^*) >= k.
3. E(Z^* * Xi') has full column rank.
Then we can estimate B using the IV estimation:
B_hat_IV = (X' * Z^*)^(-1) * X' * Y
The asymptotic distribution of the IV estimator B_hat_IV can be derived using asymptotic properties of IV estimators. Under certain regularity conditions, B_hat_IV is consistent and asymptotically normal.
The asymptotic variance-covariance matrix of B_hat_IV is given by:
A_var(B_hat_IV) = (X' * Z^*)^(-1) * (n^(-1) sum(ei * ei') * ((X' * Z^*)^(-1))'
C.) When m = k and E(X'Z) has rank j < k, it means that there is a linear dependence between some columns of X and Z. This leads to a problem of multicollinearity in the model.
In this case, it is not possible to obtain consistent estimates of B because the model is identified. The rank condition E(X'Z) = j < k is violated, and there is not enough information in the model to estimate all the elements of B.
To address this issue, one possible solution is to either remove the redundant variables from X or Z to make the model identified and achieve consistent estimates. Another approach is to use methods specifically designed for dealing with multicollinearity, such as ridge regression or principal component analysis. These methods aim to reduce the impact of multicollinearity on the estimation of B and provide more stable estimates.
A.) When m = k and rank(pie) = k, we can derive a consistent estimator for B using the two-step generalized method of moments (GMM) estimator.
1. First, we estimate the matrix pie by solving the moment conditions:
Z'X = Z'Z pie
Solving this equation gives us the estimator for pie, denoted as pie_hat.
2. Next, we plug in the estimated value of pie, pie_hat, into the linear model:
Y = X pie_hat + e
3. To estimate B, we solve the following moment condition:
X'X (B - pie_hat) = X'Y
This equation can be rearranged as:
X'X B = X'Y + X'X pie_hat
Solving this equation gives us the estimator for B, denoted as B_hat.
The asymptotic distribution of the estimator B_hat can be derived using the following steps:
1. Calculate the estimator of the variance-covariance matrix of the moment conditions:
V_hat = (1 / n) * [(Y - X B_hat)' (Y - X B_hat)]
2. Calculate the estimator of the variance-covariance matrix of the B_hat estimator:
Var(B_hat) = (X'X)^(-1) * (X' V_hat X) * (X'X)^(-1)
3. To obtain the asymptotic distribution, we use the Central Limit Theorem:
B_hat ~ N(B, Var(B_hat))
B.) When m > k, we can still obtain a consistent estimator for B using the two-step GMM estimator. In this case, the estimator for pie, pie_hat, will have dimensions m x k, where m > k.
To estimate B, we use the same moment condition as in Part A:
X'X (B - pie_hat) = X'Y
Now, instead of solving directly for B_hat, we estimate the reduced form parameters:
Gamma_hat = (X'X)^(-1) * X'Y
pie_hat = (Z'Z)^(-1) * Z'X
The estimator for B, denoted as B_hat, can then be obtained by solving the following equation:
B_hat = pie_hat + (Z'X - Z'Z B_hat) * Gamma_hat
The asymptotic distribution of the estimator B_hat can be derived using similar steps as in Part A.
C.) If m = k and E(X'Z) has rank j < k, then there are more linearly independent moment conditions than unknown parameters. This implies that the system of equations is overidentified.
In this case, we can use the method of moments estimator, which minimizes the distance between the theoretical and empirical moments. The estimator for B, denoted as B_hat, can be derived by solving the following moment conditions:
E[(Y - X' B_hat) Z'] = 0
The asymptotic distribution of the estimator B_hat can be derived using the Generalized Method of Moments (GMM) theory.