University of Cambridge
Question
Asked 22 March 2018
Production function estimation and TFP as a residual?
Hello. I was reading a literature on the estimation of a (firm-level) production function.
y = a0 + a1x + e
where e is an error
To avoid the endogeneity issue, I found that the econometricians divide the error term into one which is observable by the firm, but not by the econometrician (u) and the other i.i.d error term not observable by both (v) as below
y = a0 + a1x + u + v
Then, they control for 'u' using a proxy (because it is the source of endogeneity). My questions is that, after obtaining consistent estimates for a0 and a1, they use the residual `u_hat' as the measure of the level of TFP as
TFP = u_hat = y - a0_hat - a1_hat * x
However, isn't it true that the above TFP also contains v_hat such that
TFP = u_hat + v_hat = a0_hat - a1_hat *x?
Is there any particular reason that the current literature use the former and dismiss v_hat?
Thank you so much in advance!
Most recent answer
Oh. Thank you so much! :D
All Answers (18)
University of Copenhagen
Hi Myungun
I would guess because the u_hat equation you refer to is written in terms of estimates, that is expectations (the E-operator), and as v is i.i.d. (0, sigma) in your writing, E(v) = 0.
University of Cambridge
Thank you for your kind answer!
I may be confused, so please correct me if I'm wrong.
I think the equation above is not written in expectations in the literature.
Also, I don't understand why v_hat in the above equation should be removed only because it is i.i.d.
When we estimate a very simple regression as below
y = a0 + a1*x + e
where e is an i.i.d error term.
Suppose we obtain consistent estimates of a0 and a1, which are a0_hat and a1_hat, then isn't the residual e_hat is expressed as
e_hat = y - a0_hat - a1_hat x?
which is not zero?
Thank you very much indeed!
I suggest you take a look at Henningsen, A. (2018). Introduction to Econometric Production Analysis with R.( https://leanpub.com/ProdEconR). It is a very good guide to do production estimates. Hope it helps!
Centre for Economic and Social Studies
see if it is useful.....in addressing your problem...https://poseidon01.ssrn.com/delivery.php?ID=405072103003094092087086082093113098018009066012037092123120124089029089073112101108110101057026123124054117026103118008105103058005037041060010006020111075083101027042010007109118126093085031028095120115007005116008085064065022012008119068024001025&EXT=pdf
University of Cambridge
Thank you very much for your kind suggestions!
I've read the literature you suggested, but unfortunately, they do not seem to directly answer my question. Thank you though! :D
Centre for Economic and Social Studies
My understanding is as follows:
As we understand the Solow Residual method in Growth Accounting framework, the TFP consists of all that residual part other than that accounted for labour, capital or any other observable factor (i.e. in case of KLEMS). It is a non-parametric approach.
In that case it is true TFP = u_hat + v_hat = a0_hat - a1_hat *x
But in parametric approach through econometric method residual is further bifurcated (e= u + v) and here the estimated TFP = u_hat = a0_hat - a1_hat *x and that still leaves residual part i.e. v_hat , out.
Here TFP represents the technological change with an assumption of Hicksian-neutral technical change (i.e. technical change affects all the input factors proportionately and it is output augmenting).
This v_hat part consists many other things including the measurement error.
When we understand productivity growth, the Solow residual in fact consists of technical changes, efficiency changes, scale economies, measurement errors.
Even if TFP variable (by a proxy) capture all other things including technical, efficiency change and scale economies depending on the proxy one constructs, it has to leave some space for measurement error. Hence a part of the residual still remains as residual (v-hat).
So, v-hat part is not TFP.
If u-hat part could represent and capture most of the factors associated with TFP and if it is done with a minimal measurement error of all the input factors included, the v-hat part remains very small.
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University of Cambridge
Thank you so much, Venkatanarayana!
I absolutely agree with your view and I also think that v-hat is not part of TFP. It is true that given that there is a minimum level of measurement errors, v-hat will remain small. However, what I argue is that it will be small, but not zero.
I guess we are looking at the same page?
Centre for Economic and Social Studies
Yes...definitely it is not zero...as we are not perfect in measuring whatever the input factors (K&L or KLEMS, any additional ones) we identify to include in the function.
So the production function in terms of econometric is a stochastic not a deterministic.
University of Cambridge
Thank you.
Do you think they leave v_hat out in the estimation of TFP, because they assume that they are , as you pointed out, minimal?
Centre for Economic and Social Studies
No....it represent the residual error term....after accounting for TFP...so it is part of estimating equation....
University of Cambridge
Exactly..
Back to the original question, I don't understand why they use
TFP = u_hat = y - a0_hat - a1_hat * x
leaving v_hat out.
Centre for Economic and Social Studies
Sorry...v-hat is there in production function equation but not TFP estimation formula. Yes, TFP is after accounting for other input factors (k,l) and leaving out a error/residual representing measurement error.
University of Cambridge
Oh I see what you mean...
Please correct me if I'm wrong.
The following formula
y - a0_hat - a1_hat * x
allows us to find the residual from the regression, which will be 'u_hat + v_hat'.
However, we also agree that TFP = u_hat , not TFP = u_hat + v_hat.
So can TFP = y - a0_hat - a1_hat * x ?
1 Recommendation
Centre for Economic and Social Studies
We have to subtract v-hat as well.
TFP=y-aohat-a1hat*x-vhat
The estimating production function equation is
Y=a0+a1*x+u+v
Hope I am not wrong...
1 Recommendation
Centre for Economic and Social Studies
Further, suggest you to see the following paper:
J. RODRIGO FUENTES and MARCO MORALES, "ON THE MEASUREMENT OF TOTAL FACTOR PRODUCTIVITY: A LATENT VARIABLE APPROACH", Macroeconomic Dynamics, 15, 2011, 145–159. doi:10.1017/S1365100509991040
In this paper, authors have produced in a table their estimated percentages of residuals (say part of v-hat) along with TFP (i.e. u-hat) and main input factors (K&L).
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