# Estimating $TFP$ using Cobb-Douglas production function

Suppose we want to estimate total factor productivity (TFP) under time series framework. Let assume that the production function is given in the Cobb-Douglas form, i.e. $$Y_t=A_tK_t^\alpha L_t^\beta,$$ where $$A_t$$ is the total factor productivity (TFP), $$K_t$$ is capital stock and $$L_t$$ is labor. After log-linearization the empirical model is given (time series are $$I(1)$$, therefore we take in the first-difference) $$\Delta ln Y_t=\mu+\alpha \Delta lnK_t+\beta \Delta ln L_t+\epsilon_t,$$ where $$TFP_t=\widehat \mu+\widehat \epsilon_t$$;$$\widehat \mu$$ is average factor productivity and $$\widehat \epsilon$$ is the deviation from average over time.

Questions

1. In the above specification, by construction, $$\mathbb{E}\widehat \epsilon_t=0$$ (moment condition). It means that by estimating the above specification yields to $$TFP_t=\widehat \mu+\widehat \epsilon_t$$, which hasn't any trend (upward- or dawnward-sloping). However, it is natural to assume that $$TFP_t$$ time series can have a time trend. If no, what is the economic interpretation of no-trend in the $$TFP_t$$? Is it possible to derive time-trend in the $$TFP_t$$?
2. What is the interpretation of the estimated $$TFP_t$$?
3. Suppose the estimated $$TFP_{2020}=2$$ or $$TFP_{2019}=3$$, how these numbers can be interpreted?

If you want to control for some deterministic trend you could add a trend term to your equation. For example, if you think there is a linear trend you can add $$\gamma t$$:

$$\Delta \ln Y_t= \mu+ \gamma t + \alpha \Delta \ln K_t+ \beta \Delta \ln L_t+\epsilon_t.$$

However, this being said since the equation is already estimated in first differences you have to assume there is a trend in the growth rate of TFP not just in TFP.

Regarding the interpretation, the TFP itself is unitless and there is no agreed upon measurement but higher is better. What studies usually do is not to look at TFP itself but at its growth. So in this case if TFP in 2020 is 2 and TFP in 2019 is 3 the growth rate would be $$\frac{2-3}{3}=\approx -0.3$$. That would mean that economy technologically regressed somewhat between the two years.

This being said one has to be very careful when interpreting TFP as its estimates can also partially reflect changes in returns to scale, markups due to imperfect competition, or gains from sectoral reallocations (see this World Bank brief). This is especially problem when you look at an aggregate data rather than at firm level panel data.

• Thanks for very useful answer. Just for clasrification serevral questions: 1) If the estimated specification is in the first-differences, than $\mu+\epsilon_t$ is not the $TFP$, but the growth rate of the $TFP$,right? And if I want to derive $TFP$ rather than growth rate of $TFP$ I should estimate the equation in levels? 2) If I assume that the series of $TFP$ growth rate has a trend then I define it as $TFP_t=\mu+\gamma t +\epsilon_t$. But if I don't use cointegration property (if exists), than how I deal with the non-stationarity of time trend ($t$) in the specification?
– Duo
Nov 23 '20 at 19:47
• @Duo 1. yes it will be the average growth rate. Well if you want just TFP then you could do it on a level provided that there are no unit root issues (and there likely will be some) or other problems that would bias the results - but generally people go for growth rate of TFP in studies on aggregate data dont see any reason why would you want TFP itself given that it is unitless. 2. non-stationarity is issue with stochastic trends $\gamma t$ is deterministic trend - it wont lead to any issues (aside of sacrificing 1 degree of freedom - could be issue in small dataset)
– 1muflon1
Nov 23 '20 at 19:54
• Thanks for the comment. Is it legit to add $\gamma t$ as an independent variable in the equation? If yes, how it is legit given that $\gamma t$ is non-stationary (non-constant mean)? Could you please give reference to read about this procedure, if any? Thanks!
– Duo
Nov 23 '20 at 20:09
• @Duo yes it is completely legit, in fact it is so trivial that most people wont even discuss that much but you can see discussions of this in Verbeek a guide to modern econometrics in chapter when verbeek deals with ADF unit root test. Also, as I mentioned in my previous comment the reason why this is legit is that $\gamma t$ is deterministic trend not a stochastic trend so it will not cause unit root. In fact one reason why you usually add such trend to ADF is to control for deterministic trends which can bias unit root tests
– 1muflon1
Nov 23 '20 at 20:13
• Got it, many thanks!
– Duo
Nov 23 '20 at 20:17