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I am estimating Cobb-Douglas production function for US using time series framework:

$$Y_t=A_t \times L_t^\alpha K_t^\beta $$ where $Y$ is output, $A$ is $TFP$, $L$ is labor and $K$ is capital. After log-transformation I estimate the following linear model:

$$lnY_t=\alpha_0+\alpha_1lnL+\alpha_2lnK+\epsilon_t$$ The estimation results are: $\alpha_0=16.8$, $\alpha_1=-0.72$ and $\alpha_2=0.64$. I am concerned about negative coefficient of labor. Is there any theoretical/empirical explanation for it?

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    $\begingroup$ This may happen because of endogenous inputs, typically correlated with $A_t$ whose variations are unobserved and catched up by $\epsilon$. It is common to use IV or GMM in these circumstances (as well as panel data). $\endgroup$
    – Bertrand
    Nov 3 '20 at 12:31
  • $\begingroup$ @Bertrand Thank you for your comment. I am using time series framework (rather than panel), do you have in mind how I can solve endogeneity problem (using IV, which instruments is common to consider?), or could you please suggest any relevant paper? Thanks! $\endgroup$
    – Duo
    Nov 4 '20 at 7:27
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    $\begingroup$ I do not know many recent contributions with aggregate times series. For IV, I think that Olley and Pakes (1996), and Blundell and Bond (2000) are good starting points. Lagged wages and input prices and $L_{t-s}$ could be used as intruments. With few observations, however, IV estimates often have poor properties. See also the references given by @1mouflon1. $\endgroup$
    – Bertrand
    Nov 4 '20 at 8:34
  • $\begingroup$ @Bertrand Thank you! $\endgroup$
    – Duo
    Nov 4 '20 at 10:18
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The negative coefficient could imply decreasing returns to scale under some specifications. For example, Miller & Upadhyay (2002) show that under certain assumptions (see the paper for details) the Cobb-Douglass functions of a form $Y =AK^{\alpha}L^{\beta}$ could be rewritten as:

$Y =AK^{\alpha}L^{\alpha +\beta-1}\implies \ln Y = \ln A + \alpha \ln K + (\alpha+\beta-1)\ln L$

The authors even find some negative estimates for coefficient on labor, although most of them not statistically significant. However, the study above is done on a panel with short $T$ and might not necessarily be applicable to time series setting that you seem to imply you are having by only including $t$ subscripts.

Moreover, if you just directly applied the specification you are showing above most likely your coefficients are biased so that could be an explanation for this result.

As pointed out by @Bertrand in his +1 comment there is likely to be an endogeneity present, and hence it is common to use methods that address that. For example, the Wooldridge method that relies on GMM is becoming very popular in the literature (see . Wooldridge, 2009), and some older but still popular approaches include the Levinsohn and Petrin (2003) or Olley and Pakes (1996) approaches. Also, Van Beveren, I. (2012). Total factor productivity estimation: A practical review and sources cited therein provide good overview of different estimation methods.

Furthermore, if you are estimating the production function on datasets with long $T$ you need to take into account possible unit-roots (or use some cointegration models) and in addition to the stochastic trends also deterministic ones. If you run this on a time series data I would find it very likely that the result is just due to some stochastic or deterministic trend that was not properly taken into account.

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  • $\begingroup$ Thank you for your answer. I am using time series framewrok. As suggest ed by @Bertrand and you, the reason for negative cofficient for labor could be emdogeneity problem. So do you know which instrumnets are common to use to address this problem (under IV method) or could you propose any relevant literature? Thanks in advance. $\endgroup$
    – Duo
    Nov 4 '20 at 7:37
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    $\begingroup$ @Duo yes I think all relevant literature is already cited in this answer you will find there also examples of instruments... it’s I think most often people use lags of variables themselves as instruments but I seen various other things being used - like some measures of exogenous shocks on labor/capital markets, i think I even seen weather in some old paper), mostly I think people nowadays use lags but it’s always extremely case sensitive, people often also try multiple different instruments as robustness checks $\endgroup$
    – 1muflon1
    Nov 4 '20 at 14:17
  • $\begingroup$ Got it, thanks! $\endgroup$
    – Duo
    Nov 5 '20 at 11:40

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