2
$\begingroup$

My model of interest is given by

$Y_1 = X_1\beta + \epsilon_1$

with $Y_1\in\mathbb{R}^{n_1}$ , $X_1\in\mathbb{R}^{n_1}$ , $\beta\in\mathbb{R}$ and $\epsilon_1\in\mathbb{R}^{n_1}$. However, $X_1$ can not be observed in the same sample as $Y_1$, and therefore has to be replaced by a predicted regressor $\widehat{X}_1$.

Therefore, the following two models are used:

$X_1 = Z_1\Pi + \eta_1$

where $Z_1\in\mathbb{R}^{n_1\times q}$ is a matrix of instruments that can be observed together with $Y_1$ in sample 1, $\Pi\in\mathbb{R}^{q}$ is a vector of coefficients and $\eta_1\in\mathbb{R}^{n_1}$ is a vector of disturbance terms. Furthermore, it is possible to observe variable $X_1$ in a second independent sample, denoted by $X_2$. Again, it holds that

$X_2 = Z_2\Pi + \eta_2$

where $Z_2\in\mathbb{R}^{n_2\times q}$ is a matrix of instruments, $\Pi\in\mathbb{R}^{q}$ is a vector of coefficients and $\eta_2\in\mathbb{R}^{n_2}$ is a vector of disturbance terms.

Since $X_2$ and $Z_2$ are observed jointly in one sample, it is possible to estimate the model in a first-stage regression via OLS, and to find an estimate of $X_1$ via

$\widehat{X}_1 = Z_1\cdot \widehat{\Pi}_{OLS} = Z_1\cdot (Z_2'Z_2)^{-1}Z_2'X_2$

Thus, the model of interest can be estimated in a second stage via $\widehat{X}_1$ and the resulting two-sample TSLS estimator is given as

$\widehat{\beta}_{TSTSLS} = (\widehat{X}_1'\widehat{X}_1)^{-1}\widehat{X}_1Y_1$

This is the framework described in the paper of Solon & Inoue (2010). They also give advice on how to obtain the repsective standard error of the estimator. They recommend to estimate the asymptotic covariance matrix of the estimator as follows:

$\widehat{\Sigma}_{TSTSLS} = (\widehat{\sigma}_{11} + \frac{n_1}{n_2}\widehat{\beta}_{TSTSLS}'\widehat{\Sigma}_{\eta}\widehat{\beta}_{TSTSLS})\cdot (\widehat{X}_1'\widehat{X}_1)^{-1}$

where $\widehat{\Sigma}_{\eta}$ is a consistent estimate of the covariance matrix for the first-stage disturbances and $\widehat{\sigma}_{11}$ is the sample mean squared residual from the second-stage regression.

Now, I would like to know how to calculate this estimated covariance matrix in case of an additional constant term in the model of interest, i.e. what happens if I consider the model

$Y_1 = \alpha + X_1\beta + \epsilon_1$

I am only interested in the standard error of $\widehat{\beta}_{TSTSLS}$. Can I simply ignore the constant term and calculate the covariance as mentioned above, or should I include a vector of ones in the regressor matrix and then proceed as proposed by Solon & Inoue?

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.