Is there empirical evidence on usability of Cobb-Douglas production function?

Question

This is my first question at Stack Exchange. I am trying to estimate simple Cobb-Douglas production function $Y=AL^{\alpha}K^{\beta}$ on US data and receive an unrealistic estimate of $\alpha$ and a high p-value for $\beta$. Structure of regression is as follows:

$\frac 1 Y \frac{\mathrm dY}{\mathrm dt} = \frac 1 A \frac{\mathrm dA}{\mathrm dt} + \alpha \frac 1 L \frac{\mathrm dL}{\mathrm dt} + \beta \frac 1 K \frac{\mathrm dK}{\mathrm dt}$

and assuming technological growth constant:

$growth_Y = a_0 + \alpha\cdot growth_L + \beta\cdot growth_K$

I have the following output of linear regression in R:

\begin{array}{lllll} &\text{Estimate}\quad & \text{Std. error}\quad & \text{t value} & \mathrm{Pr}(>|t|) \\ \text{(Intercept)}\ & 0.008256 & 0.006070 & 1.360 & 0.186 \\ growth_L & 1.388552 & 0.283867 & 4.892 & 5.47\mathrm e^{-05}\ ^{***} \\ growth_K & 0.194165 & 0.255572 & 0.760 & 0.455 \\ \\ R^2 = 58\% \end{array}

Estimate for the coefficient in front of $growth_L$ ($\alpha$ in Cobb-Douglas function) seems unrealistically high (1.388552) and estimate for coefficient for $growth_K$ ($\beta$ from Cobb-Douglas function) has high p-value (0.455).

Before running the regression I excluded outliers (observations outside 2.5% and 97.5% for growth_gdp, growth_L and growth_K). I also checked correlation between growth_L and growth_K and it is 42% which is not that high to impact stability of coefficients of the regression that much.

Unfortunately I could not find in the internet a simple and reproducible analogue of the above analysis.

My questions are the following:

1. Is there an empirical analysis of simple Cobb-Douglas function producing realistic estimates of coefficients? So that I would understand my key mistakes
2. If not, what would be the simplest form of production function which could be supported by empirical analysis?

Thank you!

P.S. For the full reproducibility of the analysis below is more info on data and R code used in estimation of the coefficients. I take data from OECD Economic outlook database

Variables used: GDPV – states for Real GDP, HRS states for hours worked per worker, LF – states for number of workers in Economy, KTPV states for productive capital stock. I calculated L as $LF*HRS$

Full R code and the input data set can be found at pastebin.

Answer 1

The simple answer to why the Cobb-Douglas functional form is used is because it is at least a log-linear approximation to some higher-order production function. That is, suppose you take a functional form that looks like this: $\log Y_t = f(A, K, L)$. Then a linear approximation would look like the Cobb-Douglas production function. (For a small $1\%$ increase in $K$, we get approximately an $\alpha \%$ increase in $Y$.)

One potential source of the strange estimation results you're getting is likely a bias due to the endogeneity of the regressors. Total factor productivity (TFP) in your notation is $A$. It is changing over time. In your regression, it shows up in the error term. Changes in productivity affect the composition of the inputs $L$ and $K$. Thus, $L$ and $K$ are correlated with error term. To see this, solve any typical cost minimization for a firm. $L$ and $K$ will depend on $A$. Also note that the behavior will be different in the short-run vs the long-run.

Hope this helps!

EDIT:

If you're interested, here is a problem set (no solutions available) that walks you through a demonstration of the bias described above. In the end, one potential improvement to the estimation procedure is explored via Monte Carlo simulation. This improvement relies on the availability of panel data of firm input/output data.