I am trying to estimate a simple Cobb-Douglas production function $Y=AL^{\alpha}K^{\beta}$ on US data and get 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:

$\text{growth}_Y = a_0 + \alpha\cdot \text{growth}_L + \beta\cdot \text{growth}_K$

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 $\text{growth}_L$ ($\hat\alpha$) seems unrealistically high (1.388552) and estimate for coefficient for $\text{growth}_K$ ($\hat \beta$) 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 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?

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.

  • $\begingroup$ Thank you for this interesting question, Petr! Here is advice for new users. For your information, you can share long code samples with pastebin.com . $\endgroup$ – ahorn Sep 29 '19 at 16:37
  • $\begingroup$ Thanks for warm welcome ahorn! I transferred my code to pastebin and edited question accordingly. $\endgroup$ – Petr Sep 29 '19 at 18:08

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!


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.

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  • $\begingroup$ thank you so much for your answer! When I look how TFP is calculated, it looks like it is calculated based on some production function (e.g. Cobb-Douglas) as A=Y/(L^α*K^β). Therefore it looks to me as a circular link, as we can either estimate α and β and then use them to calculate A, or we have to somehow know α and β without regression and then calculate A as a residual. What I am trying to understand is how to estimate/calibrate simple production function (e.g. Cobb-Douglas) without indirectly using α and β estimates (which are indirectly embedded in A) of other people. Thanks! $\endgroup$ – Petr Sep 28 '19 at 14:43
  • $\begingroup$ @Petr The problem you are describing is a form of the identification problem (or see the identification article). The problem isn't the functional form, it's that you don't have the data to estimate it. Unfortunately, there isn't an easy solution for you. You could add more assumption about the functional form of production, but this isn't likely a good solution. There is a whole literature devoted finding solutions to this problem, so you're not alone. Good luck! $\endgroup$ – jmbejara Sep 28 '19 at 22:18
  • $\begingroup$ thanks for the clarification! I will further go through the links provided by you! And will reply later on if the question could be closed or if there would still be significant missing pieces of this picture in my head :) $\endgroup$ – Petr Sep 29 '19 at 9:21
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    $\begingroup$ @ahorn, I will shortly accept the answer as soon as I understand that the answer solved my problem and before doing so I need to go through the links provided by jmbejara $\endgroup$ – Petr Sep 29 '19 at 18:17
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    $\begingroup$ @emeryville Fixed! $\endgroup$ – jmbejara Apr 18 at 20:03

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