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
- 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 $TFP_t$?
- What is the interpretation of the estimated $TFP_t$?
- Suppose the estimated $TFP_{2020}=2$ or $TFP_{2019}=3$, how these numbers can be interpreted?
Thanks in advance!