<p>The total factor productivity (TFP) would be <span class="math-container">$a+\epsilon_t$</span> where <span class="math-container">$a$</span> is the average TFP and <span class="math-container">$\epsilon_t$</span> (where Solow residual is technically actually <span class="math-container">$\Delta \epsilon$</span>) tells us how TFP varies across time. Let me explain:</p>
<p>First, the <span class="math-container">$A$</span> 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:</p>
<p><span class="math-container">$$Y_t = A_t K_t^{\alpha} L_t^{\beta}$$</span></p>
<p>Hence log linearizing would give us:</p>
<p><span class="math-container">$$y_t = a_t + \alpha k_t + \beta l_t,$$</span></p>
<p>where lower case letter denote logs <span class="math-container">$ \ln X =x$</span>. Now when you make mistake is in specifying your OLS. The <span class="math-container">$a_t$</span> actually is the residual. Since we can only observe <span class="math-container">$k_t$</span> and <span class="math-container">$l_t$</span> we cannot include <span class="math-container">$a_t$</span> in regression and it will be the residual because it can be calculated as:</p>
<p><span class="math-container">$$y_t - \alpha k_t - \beta l_t = a_t, a_t \equiv TFP$$</span></p>
<p>So actually <span class="math-container">$a_t$</span> is the residual <span class="math-container">$\epsilon_t$</span>. So the specification would be:</p>
<p><span class="math-container">$$y_t = \alpha k_t + \beta l_t + \epsilon_t.$$</span></p>
<p>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 <span class="math-container">$\hat{\alpha}$</span> and <span class="math-container">$\hat{\beta}$</span>). As a result we can add a constant term <span class="math-container">$\beta_0$</span> to the above regression.</p>
<p><span class="math-container">$$y_t = \beta_0+ \alpha k_t + \beta l_t + \epsilon_t.$$</span></p>
<p>In this case TFP (<span class="math-container">$\ln A_t$</span>) would be <span class="math-container">$\ln A_t = \beta_0+ \epsilon_t$</span> where <span class="math-container">$\beta_0$</span> represents the average factor productivity and <span class="math-container">$\epsilon_t$</span> would be the deviation from the average over time (see <a href="https://onlinelibrary.wiley.com/doi/full/10.1111/j.1467-6419.2010.00631.x?casa_token=P2GBG5wKAqQAAAAA%3A6zBROGdCkmn-4yFWAM3xHa5YLSuon-kHuij_zjcFY_Vb7MKcOel8dNRGoJkeBjpRIsx1hBL0SXEutuk" rel="nofollow noreferrer">Van Beveren, I. (2012). Total factor productivity estimation: A practical review</a> 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 <span class="math-container">$A_t=A$</span> then <span class="math-container">$\epsilon_t=0, \forall t$</span>.</p>
<p>Lastly, Solow residual is actually defined in growth terms so actually it is <span class="math-container">$\Delta \ln A_t = \beta_0 +\epsilon_t - (\beta_0 + \epsilon_{t-1}) = \Delta \epsilon_t$</span>, since Solow residual is defined as productivity growth (see Barro &amp; Sala-i-Martin Economic Growth 2nd ed. pp 434-435).</p>
<p>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 <span class="math-container">$I(1)$</span> 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.</p>

CC BY-SA 4.0
2020-10-09T20:54:52.420Z
I got it, thanks! Though a small question: you say that "the above specification is unnecessarily restrictive as it forces TFP to have a 0 mean. As a result we can add a constant term ..." So if TFP in the model is represented by $\epsilon_t$, which is error-term, it is natural to assume that $E \epsilon_t =0$, so why it is restrictive? In other words, why we add intercept? I did not get this.Thanks!
Duo
CC BY-SA 4.0
2020-10-09T21:26:25.647Z
@Duo this is good question. On one hand you are right that it should not matter. The constant is anyway just some level and we care about variation. An equivalent in physics would be to change degrees Celsius in a way that freezing point is no longer 0 but lets say 30 - it is a transformation that does not change information that thermometer would provide. However, setting constant to be zero has some serious implications for estimation of $\alpha$ and $\beta$. To be specific it forces both $\alpha$ and  $\beta$ to pass through the origin even if some other line would provide better fit.
1muflon1
CC BY-SA 4.0
2020-10-09T21:28:06.973Z
that is quite restrictive and even as a matter of fact often in research where even theory says there should not be any constant (for example when testing with ADF test unit root in some series where we know for a fact that the series varies around zero and so there should be no constant) people still add the constant in to allow model to estimate $\beta$s without any restrictions.
1muflon1
CC BY-SA 4.0
2021-02-22T23:41:44.167Z
Nice explanation! +1 for that and for the reference.
Thomas Winckelman
CC BY-SA 4.0
2020-10-09T19:01:25.723Z
2020-10-09T20:28:44.687Z
2020-10-09T20:28:44.687Z
1muflon1
<p>Suppose we assume that production function has a Cobb-Douglass form:
<span class="math-container">$$Y=A\times K^\alpha\times L^\beta,$$</span> where <span class="math-container">$Y$</span> is output (GDP), <span class="math-container">$A$</span> is Total Factor Productivity and <span class="math-container">$L$</span> is labor. By log-linearizing the production function we have:
<span class="math-container">$$y=a+\alpha k+\beta l,$$</span> where <span class="math-container">$y=log(Y)$</span>, <span class="math-container">$k=log(K)$</span> and <span class="math-container">$l=log(L)$</span>. Hence, the model that we empirically estimate can be written as:
<span class="math-container">$$y_t=a+\alpha k_t+\beta l_t+\epsilon_t,$$</span>
where <span class="math-container">$\epsilon$</span> is the error term. Suppose applying OLS we have the estimated parameters, i.e. <span class="math-container">$\widehat{a}$</span>, <span class="math-container">$\widehat{\alpha}$</span> and <span class="math-container">$\widehat{\beta}$</span>.</p>
<p><strong>Question</strong> Does only <span class="math-container">$\widehat{a}$</span> refer to <span class="math-container">$TFP$</span>, or the <span class="math-container">$TFP=\widehat{a}+\epsilon$</span>? As far as I know, <span class="math-container">$\epsilon$</span> is also called Solow residual. Please elaborate.</p>
<p>Thanks!</p>

CC BY-SA 4.0
2020-10-09T18:15:42.080Z
Did you mean $y=\log(Y)$ (switching the letter capitalization)?
Herr K.
<production-function>
Empirically estimating TFP
Duo
# Empirically estimating TFP

(Asked by: username_0 on 2020-10-09)

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!

**Tags:** production-function
> **Comments:**
> **username_1**: Did you mean $y=\log(Y)$ (switching the letter capitalization)?  

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## 1 answers

### Answer by username_2 on 2020-10-09. Score: 4 | Accepted answer

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](https://onlinelibrary.wiley.com/doi/full/10.1111/j.1467-6419.2010.00631.x?casa_token=P2GBG5wKAqQAAAAA%3A6zBROGdCkmn-4yFWAM3xHa5YLSuon-kHuij_zjcFY_Vb7MKcOel8dNRGoJkeBjpRIsx1hBL0SXEutuk) 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 & 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.

> **Comments:**
> **username_0**: I got it, thanks! Though a small question: you say that "the above specification is unnecessarily restrictive as it forces TFP to have a 0 mean. As a result we can add a constant term ..." So if TFP in the model is represented by $\epsilon_t$, which is error-term, it is natural to assume that $E \epsilon_t =0$, so why it is restrictive? In other words, why we add intercept? I did not get this.Thanks!  
> **username_2**: @Duo this is good question. On one hand you are right that it should not matter. The constant is anyway just some level and we care about variation. An equivalent in physics would be to change degrees Celsius in a way that freezing point is no longer 0 but lets say 30 - it is a transformation that does not change information that thermometer would provide. However, setting constant to be zero has some serious implications for estimation of $\alpha$ and $\beta$. To be specific it forces both $\alpha$ and  $\beta$ to pass through the origin even if some other line would provide better fit.  
> **username_2**: that is quite restrictive and even as a matter of fact often in research where even theory says there should not be any constant (for example when testing with ADF test unit root in some series where we know for a fact that the series varies around zero and so there should be no constant) people still add the constant in to allow model to estimate $\beta$s without any restrictions.  
> **username_3**: Nice explanation! +1 for that and for the reference.  

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