3 Jackknife Instrumental Variables Estimator JIVE Following the discussion in Hansen et al 2008 define the class of JIVE estimators The k class estimator for the parameter is defined by the equation k...

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3. Jackknife Instrumental Variables Estimator (JIVE): Following the discussion in Hansen et al. (2008), define the class of JIVE estimators. The $k$-class estimator for the parameter $\delta$ is defined by the equation: \[ \hat{\delta}_{k}=\left(X^{\prime} P X-\hat{\alpha} X^{\prime} X\right)^{-1}\left(X^{\prime} P Y-\hat{\alpha} X^{\prime} Y\right), \] where $X$ is the $n \times G$ matrix of regressors, $Y$ is the $n \times 1$ vector of the dependent variable, $Z$ is the $n \times K$ matrix of instruments, and $P$ is the projection matrix $Z\left(Z^{\prime} Z\right)^{-} Z^{\prime}$. Here, $\hat{\alpha}$ is a scalar representing the bias adjustment in the presence of many instruments, and $\hat{\delta}_{k}$ is the resulting k-class estimator. Additionally, these estimators can be represented as a general Instrumental Variable (IV) estimator: \[ \tilde{\delta}_{J I V E}=\left(\bar{\Upsilon}^{\prime} X\right)^{-1}\left(\bar{\Upsilon}^{\prime} Y\right), \] with $\bar{\Upsilon}=\left(P-\hat{\alpha} I_{n}\right) X$ and $I_{n}$ as the $n \times n$ identity matrix. Hint: You only need to construct a function, and do not need to generate data here. You will generate it in Question 5. You should set $\hat{\alpha}$ as an input of the function as well.

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Here's a step-by-step solution to define the class of JIVE estimators in R: #### Solution By Steps ***Step 1: Define the JIVE function*** ```r JIVE <- function(Y, X, Z, alpha) { n <- nrow(X) P <- Z %*% solve(t(Z) %*% Z) %*% t(Z) delta_k <- solve(t(X) %*% P %*% X - alpha * t(X) %*% X) %*% (t(X) %*% P %*% Y - alpha * t(X) %*% Y) Upsilon_bar <- (P - alpha * diag(n)) %*% X delta_JIVE <- solve(t(Upsilon_bar) %*% X) %*% (t(Upsilon_bar) %*% Y) return(list(delta_k = delta_k, delta_JIVE = delta_JIVE)) } ``` ***Step 2: Explain the function*** - The `JIVE` function takes `Y` (dependent variable vector), `X` (regressor matrix), `Z` (instrument matrix), and `alpha` (bias adjustment scalar) as inputs. - It calculates the projection matrix `P` using the instrument matrix `Z`. - It computes the k-class estimator `delta_k` using the formula provided. - It constructs `Upsilon_bar` and computes the general IV estimator `delta_JIVE`. - The function returns a list containing both `delta_k` and `delta_JIVE`. #### Final Answer The `JIVE` function is defined to compute the k-class estimator (`delta_k`) and the general IV estimator (`delta_JIVE`) for the JIVE estimators, given the necessary inputs. #### Key Concept JIVE Estimators #### Key Concept Explanation The Jackknife Instrumental Variables Estimator (JIVE) is a class of estimators designed to handle the presence of many instruments in an instrumental variable (IV) regression. It adjusts for the bias that arises when the number of instruments is large relative to the sample size. The JIVE estimators can be represented as a k-class estimator (`delta_k`) and a general IV estimator (`delta_JIVE`), both of which are computed using the provided formulas.

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4 Regularized Jackknife Instrumental Variables Estimator (RJIVE) Extend your analysis to derive the RJIVE an advanced estimator that utilizes ridge regularization to address the instability caused by weak instruments Begin by calculating the ridge regression coefficient estimates in the first stage of the IV estimation process Then define and utilize the leave-one-out ridge regression coefficient excluding the i-th observation to derive the estimated instrument for said observation Conclude by expressing the RJIVE estimator for the coefficient vector as follows tilde R J I V E= ft( i=1n -i Zi Xi )-1 ft( i=1n -i Zi Yi ) where -i= ft( Z-i Z-i+ )-1 ft( Z-i X-i ) i represents the leave-one-out ridge regression coefficient Zi the vector of instruments is a diagonal matrix with the regularization parameter on the diagonal often denoted by = 1 / 2 IK where is a scalar penalty parameter and IK is the identity matrix We follow the usual approach for ridge regression by setting = 1 / 2 In and further set 1 / 2=C K1 / 2 Xi the vector of endogenous regressors and yi the dependent variable all corresponding to the i-th observation In our simulations we set the constant of proportionality C for each first-stage equation to the sample standard deviation of the element of Xi being considered Hint You only need to construct a function and do not need to generate data here You will generate it in Question 5 Hint When you generate the leave-one-out estimator you may use for loop

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