Empirically estimating TFP Suppose we assume that production function has a Cobb-Douglass form:
$$Y=A\times K^\alpha\times L^\beta,$$ where $Y$ is output (GDP), $A$ is Total Factor Productivity and $L$ is labor. By log-linearizing the production function we have:
$$y=a+\alpha k+\beta l,$$ where $y=log(Y)$, $k=log(K)$ and $l=log(L)$. Hence, the model that we empirically estimate can be written as:
$$y_t=a+\alpha k_t+\beta l_t+\epsilon_t,$$
where $\epsilon$ is the error term. Suppose applying OLS we have the estimated parameters, i.e. $\widehat{a}$, $\widehat{\alpha}$ and $\widehat{\beta}$.
Question Does only $\widehat{a}$ refer to $TFP$, or the $TFP=\widehat{a}+\epsilon$? As far as I know, $\epsilon$ is also called Solow residual. Please elaborate.
Thanks!
The total factor productivity (TFP) would be $a+\epsilon_t$ where $a$ is the average TFP and $\epsilon_t$ (where Solow residual is technically actually $\Delta \epsilon$) tells us how TFP varies across time. Let me explain:
First, the $A$ should also be function of time in time series model as technology can change (I doubt you want to impose restriction that technology has to be constant and if so then having time varying residual would not make sense) so actually the production function should look like this:
$$Y_t = A_t K_t^{\alpha} L_t^{\beta}$$
Hence log linearizing would give us:
$$y_t = a_t + \alpha k_t + \beta l_t,$$
where lower case letter denote logs $ \ln X =x$. Now when you make mistake is in specifying your OLS. The $a_t$ actually is the residual. Since we can only observe $k_t$ and $l_t$ we cannot include $a_t$ in regression and it will be the residual because it can be calculated as:
$$y_t - \alpha k_t - \beta l_t = a_t, a_t \equiv TFP$$
So actually $a_t$ is the residual $\epsilon_t$. So the specification would be:
$$y_t = \alpha k_t + \beta l_t + \epsilon_t.$$
However, the above specification is unnecessarily restrictive as it forces TFP to have 0 mean (although we  can always rescale any variable to have zero mean this could bias $\hat{\alpha}$ and $\hat{\beta}$). As a result we can add a constant term $\beta_0$ to the above regression.
$$y_t = \beta_0+ \alpha k_t + \beta l_t + \epsilon_t.$$
In this case TFP ($\ln A_t$) would be $\ln A_t = \beta_0+ \epsilon_t$ where $\beta_0$ represents the average factor productivity and $\epsilon_t$ would be the deviation from the average over time (see Van Beveren, I. (2012). Total factor productivity estimation: A practical review and sources cited therein - the source talks about panel data applications but I think the basic explanation holds up even in pure time series even if time series has its own issues that require attention). Also as mentioned at the beginning if you want to assume TFP is constant $A_t=A$ then $\epsilon_t=0, \forall t$.
Lastly, Solow residual is actually defined in growth terms so actually it is $\Delta \ln A_t = \beta_0 +\epsilon_t - (\beta_0 + \epsilon_{t-1}) = \Delta \epsilon_t$, since Solow residual is defined as productivity growth (see Barro &amp; Sala-i-Martin Economic Growth 2nd ed. pp 434-435).
PS: if you actually are going to perform the estimations on a time series you should take into account that all series will most likely be $I(1)$ and estimate the whole model in first differences where the interpretation of constant would be average rate of growth of TFP. In the above I did not explored this issue in order to avoid unnecessarily adding more confusion.