Calculating implied values of a model

Question

How are the implied values and their standard errors calculated here? Thank you in advance.

http://imgur.com/a/3dMdx

Edit

We have this production function with $\alpha+\beta<1$.

$Y=K^{\alpha}H^{\beta}(AL)^{1-\alpha-\beta}$

$Y$ output, $K$ physical capital, $H$ human capital, $A$ technology.

There are cross-section data for a number of countries, found in the appendix here: http://eml.berkeley.edu/~dromer/papers/MRW_QJE1992.pdf

The model is:

$ln(\frac{Y}{L})=ln(A_0)+gt+\frac{\alpha}{1-\alpha-\beta}ln(s_k)+\frac{\beta}{1-\alpha-\beta}ln(s_h)-\frac{\alpha+\beta}{1-\alpha-\beta}ln(n+g+\delta)$

$I/GDP$ is $s_k$ and $SCHOOL$ is a proxy for $s_h$.

I'm unsure about how to find the implied values of $\alpha$ and $\beta$ and their standard errors.

Answer 1

I found this article which goes a little bit into the computation of alpha:

http://www.portfolioprobe.com/2013/05/20/implied-alpha-and-minimum-variance/

So it looks like it would just be:

Alpha = 2cVw

Where c is a scaling factor, w your weight vector, and V your variance matrix.

That same article is also cross referenced by Harvard website here:

http://tagteam.harvard.edu/hub_feeds/1981/feed_items/189302

There is also a link there that holds more info about implied alpha.

The article "Measuring Systematic Risk Using Implied Beta in Option Prices" by Lin and Paxson seems to imply that the beta value is linked to the slope of your regression result and the alpha to the intercept.

Answer 2

I know it's a bit late but for anybody else who needs this, it's actually quite straightforward.

This is from the seminal economics paper Mankiw, Romer and Weil 1992.

As stated in the question, the equation we are trying to estimate here is:

\begin{align*} log(Y(t)/L(t)) = log(A(0)) + gt + \frac{\alpha}{1 - \alpha - \beta}log(s_k) - \frac{\alpha + \beta}{1 - \alpha - \beta}log(n + g + \delta) + \frac{\beta}{1 - \alpha -\beta}log(s_h) \end{align*}

What we get we we perform a standard OLS regression on this is an equation of this type:

\begin{align*} log(Y(t)/L(t)) = \theta_0 + \theta_1 log(s_k) - \theta_2 log(n + g + \delta) + \theta_3 log(s_h) + \epsilon \end{align*}

So we end up getting the values of the parameters: $\theta_1$, $\theta_2$ and $\theta_3$

To get the point estimate of $\alpha$ and $\beta$ we simply solve the system of equations pertaining to the coefficients of $s_k$ and $s_h$ \begin{align} \theta_1 = \frac{\alpha}{1 - \alpha - \beta} \theta_3 = \frac{\beta}{1 - \alpha - \beta} \end{align}

You should solve to get these solutions:

\begin{align} \alpha = \frac{\theta_1}{1 + \theta_1 + \theta_3} \beta = \frac{\theta_3}{1 + \theta_1 + \theta_3} \end{align}

Sadly not I'm not sure how to do the standard errors bit