Most of us cannot get what you want to do. Be clear so that we can give you the act helip. Based on my understanding money values if indexed in 2005. Then, re-index the 2015 to 2005 then divide.by the re-index.First, u can use any  base period since you will take a natural log of the final value.. You can rebate the following made-up data by dividing all values by 2337 and multipying by 100 so that 100 appears in the row associated with 2005 or you can rebase by using 2015 value 4973 to divide every value so that 2004 44.3; 2005 is 447.0 and  2015 is 100 . I hope this helps:

 2004        $  2203

2005           2337  

2006           2146

2007           3568

2008           2473

2009           2917

2010           3546

2011            3721

2012            3642

2013             4352

2014             4726

2015             4973

You may choose any base year, but for practical reasons I recommand to take the same year (period) for all variables.

Anton's point is excellent if you are working with other variables that require indexing! If only one requires indexing then there is no need to index them. Another suggestion that I have is the natural log all variables (if possible) for ease of interpretation. 

I support Chuck's proposal.

1. O f course, no deflation or indexing is necessary, if you already have a real variable. E.g. if energy input is measured in Joule or Wh it is already real, measured in energy cost it has to be deflated by an energy price index.

2.  Only if there are good (theoretica)l reasons, that there is a linear relation between y and the xi's, you cannot change to a log-approach. For such a linear function, I would take the indices with one common base year But for the realtion of real variables, this assumption is not realistic, because for real variables percentage changes are more informative than absolute values.

3. Therefore, an estimation in ln is peferalble to one in absolute valures. The coefficients show the elasticities and are easy to interpret. I personally would prefer a regression in percentage changes, because I think, in practice nobody calculates in logs or in differences of logs (although dln x is very near to %x, as long as %x is not too high).

4. Before estimating, one should put much effort, to formulate the "correct" function (specificatioon). Then one can think over which transformations are necessary to get a form which is tractable by an econometric method (e.g. linearisation for OLS).

Anton has already answered your basic question. If you are using official price indices as deflators rebase all your price indices to the same base before deflating. You will have much more problems deciding to use a particular functional form i.e. Cob-Douglas, CES, one of the various functional forms or using a cost function approach. You may also encounter various problems arising from non-stationarity.

@ Hi all,

Thanks for all replies. What I really want to know is shall I use an earlier or and older year as a basis/constant year ?. For instance, I have data 1970-2000, shall I use 1985 as a basis year or use 1999 as a basis year ?

It doesn't matter since it is a variable in your equation. Also, use natural log for ease of interpretation. I am very sure of what I am telling you Muhamed!

Muhamad you can get firstly a general price  index (GPI) basic 2004,  by mean accumulative annual inflation rate (p) 2004-2015 period, for instance if anual inflation rate was 3% all term:

years    GPI

2004 = 100

2005 = 103, if  p = 3%

2006 = 103 x 1.03 = 106.09,  if p = 3%

2007 = 106.9 x 1.03 = 110.107

...

2015

Then performance (Production factors values/GPI) x 100

In fact your data are different; however proceeding is same, known in monetary theory and macroeconimic as nominal variable conversion in real variables, by discount price effect.

If me suppousse Chuk A ejemple values 

2004        $  (2203/100)100 = 2203 value no change

2005           (2337/103) x 100 = 2268.9

2006           (2146/106.09) X 100 = 2007.48

2007           (3568/110.07) x 100 = 3241

2008           2473

2009           2917

2010           3546

2011            3721

2012            3642

2013             4352

2014             4726

2015             4973

In fact your data are different; however proceeding is same, known in monetary theory and macroeconimic as nominal variable conversion in real variables, by discont price effect. I hope helpe you. Mario

Are there any gauge to set the constant price ? - ResearchGate. Available from: https://www.researchgate.net/post/Are_there_any_gauge_to_set_the_constant_price [accessed Jul 8, 2016].

@ Hi Mario, Good elucidations!!

Really? Thank Chuk

Changing the base year involves multiplying every observation in a series by the same number. If you are working with the logs of a series changing the base adds a constant to the log of each observation. If you are doing regressions with logs as explanatory variables (e.g. Cobb-Douglas production functions) changing the base does not change the coefficient of the variable. If you are working in levels of the variable (unlikely) changing the base will change the value of the coefficient of the variable but not its significance. Review the result of linear transformations to variables in regression analysis.

A minor warning - Some official statistics may involve changes in the weights used to calculate price indices and you should be aware of any quirks that this may involve. Look at the documentation.

Also be aware that the functional form that you use to analyse your data may impose certain restrictions on expansion paths and elasticities of substitution between factors. A  Cobb-Douglas function imposes unit elasticity, a CES constant elasticity, a flexible function such as the translog allows for different elasticies at different inputs and outputs. There are also various other flexible functional forms.  

David F. Heathfield (Author), Soren Wibe (1987), An Introduction to Cost and Production Functions, Macmillan. is a good introduction to these matters.   It does require some calculus but with this topic this can not be avoided. You might also review the work of Christensen, Jorgensen and Lau 

I think, Muhamad knows how to deflate nominal data to get real ones. His question was which base year to choose. And here Chuck' proposal comes in: to use logs. In this case, the coefficients of the variables (elasticities) will be independent from the base year. Implicitly that means to estimate a Cobb-Douglas function. The mostly used CD function is:  production Q = A^t*X1^a*X2^b*X3^c etc. with Xi...Inputs, t...time index and A...total factor productivity (with technical progress). Taking the logs, it is easy to estimate the coefficients by OLS. I prefer regressions in differences:

%Q = a*%X1+b*%X2+c*%X3   ......(.+ constant) with % as percentage change to preceding year.  The constant shows you the average annual change of TFP. The statistical tests for differences are, in general, much "sharper". Another reason for using percentage changes is that I am not a believer of the CD function, but that there are likely similar relations between inputs and output for other functional forms, too.

@Anton A regression in % changes is almost the same as a regression in first differences of logs. (The first difference of logs is marginally better). Thus if the Cobb-Douglas form is acceptable the regression in first differences of logs is inefficient (MA introduced to disturbance in regression. The regression in first differences is valid if there is no loang term relationship betwen the factors and output. (Omitted vatiable bias arising from omission of adjustment to equilibrium. Any thing I am saying about regressions in first differences of logs applies also to regressions in % changes

Dear John,

I know that for not too high percentage changes %x these are near dln x. That is easy to see from the definition of ln(1+%x) =%x-%x^2/2+%x°3/3-%x^4/4+....etc. The main problem is that, I think, one cannot find one production process in the world which can really be discribed by a Cobb-Douglas function. Most production prosesses seem to be near limitational, i.e. there is limited scope of substitution between inputs. If one combines different limitational functions, total production may seem to follow a CD, CES or VES function. The main reason why most economists love these functions is that it is relatively easy to derive derivatives and optima. Another reason may be to make "proofs"  non-understandable for normal people, because maybe 90% do not really know about logs (sometimes even the economists applying logs do not), but most know what a percentage change is.  Even if one does not believe in CD functions, one can assume, that, in general, an increase of inputs will bring a somewhat higher output. And the best possibility of showing such a relation is elasticities, i.e. the percentage increase of production for a one percent increase of an input. Of course, one has to acknowledge, that this relation is not as clearcut as the theoretical production functions suggest. Therefore, one may have to accept raher high variances of the estimated coefficients.

Enclosed you find a simple simulation model which shows how biased (OLS) estimates can be even with unrealistically small random errors. The inputs follow a random trend (annual changes between -2% and +6%. In the yellow cells you find the coefficients of the Cobb-Douglas function which you can be change according to your assumptions. For the output, I included an equally distributed random error between -1% and +1%. I know that an equal distribution is not very realistic, but for these small margins it does not matter. Any time you press the enter or del key you get a new simulation. On the right, you see the results of the OLS regressions. Although the statistics for the ln-regression look impressive (high R2, low variances), the results are worse than for the regressions with differences (dln and %). In most cases, the true coefficients are far outside the 99%-significance interval. That is because the TFP coefficient "eats" a part of the technological coefficients (collinearity). The difference equation (which show similar results - as expected) show, in general, good estimates for the coefficients, mostly they seem to underestimate the annual increase of TFP (= constant).

Anton

Can you have look at your simulations. At the momet, I pnly have accessto an Android tablet. It would not be easy to replicate your analysis on it. I will look at it in more detail later.

In your your simulation there is no disturbance term. If you do your regression with the forst four observations. (or indeed any four ob6servations) you will get a much better fit.

John

There is a disturbance term in my simulation, because the output calculated with the CD-function is multiplied by (,99+random()*,02). Of course, this is a very small disturbance with a maximum deviation of 1%. But I wanted to show that even with such a small error (i.e. optimal conditions for statistics) one has to be cautious with the results. With no or a very low disturbance, one can expect that regressions (with logs) for a shorter period will bring better results, because the trends of inputs and output play a minor role and cannot (statistically) "eat" as much from the technological coefficients as in the long run.

Yes

Anton is correct!

Anton

Sorry I misread the relevant formula on my tablet. 

@Anton and @Chuck I have now got access to a PC and have had the opportunity to look again at the simulations and estimates. 

In log terms the model simulated is

lnO = 0.5*lnI1 + 0.3*lnI2 + 0.2*lnI3 + log(1.01)*tt + e

where

Input1_t = Input1_{t-1}*(0.98+RAND()*0.08), Input2 and Input3 are defined similarly. tt is a time trend running from 1 to the number of obs being simulated. lnIi is the log of Inputi , i =1, 2, 3.

As far as I can make out in the log model Chuck omits the tt variable in his regression specification. I have programmed the same simulation in R (reg2.r) and the regression results are

Call:

lm(formula = lnO ~ lnI1 + lnI2 + lnI3 + tt)

Residuals:

Min 1Q Median 3Q Max

-0.010110 -0.003904 -0.001243 0.005114 0.009688

Coefficients:

                       Estimate   Std. Error    t value     Pr(>|t|)

(Intercept)   -0.447194  0.268046    -1.668     0.104

lnI1                0.483713  0.035342   13.687     1.29e-15 ***

lnI2                0.339844  0.034702     9.793     1.46e-11 ***

lnI3                0.273326  0.039813     6.865      5.71e-08 ***

tt                    0.008343  0.001130      7.380      1.24e-08 ***

---

Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.006071 on 35 degrees of freedom

Multiple R-squared: 0.9998, Adjusted R-squared: 0.9997

F-statistic: 3.515e+04 on 4 and 35 DF, p-value: < 2.2e-16

As you will see all the estimated coefficients are well within two standard deviations of the known model values. If I omit the trend variable I get the same kind of silly results as Chuck. Inference is on the basis that the input variables are held constant over repeated simulations and that only the disturbance term is simulated in each replication.

I have not looked at estimation using the log differences (or percentage differences). It should be noted that first differencing will induce an MA term in the disturbances. Standard OLS estimates of the standard errors of the coefficient estimates are not valid.

If in the simulations you allow new input variables to be simulated for each simulation.the log input variables become I(1). While there is cointegration the standard errors are again probably not valid.

The use of %changes on occasions such as this can lead to problems. First note that % changes are not linearly additive. a 10% change following a 10% aggregates to a 21% change. If you have a 50% fall you the need a 100% rise to compensate. Year on Year these effects may not be significant but aggregated of a number of years they may cause problems. Log differences are additive,

I have attached two files (1) reg2.r which contains the R script which produces the regression results above and (2) reg2-output.txt which contains the complete output of that program.

Thank you, John.

It was my fault, I mixed-up the numbers of the estimated coefficients. After correction, I get similar results as you. Nevertheless, I prefer estimating in differences, because from a practical point of view a Cobb-Douglas function looks really strange. Economists mostly use it, because it is so comfortable, not because it is realistic. Of course, there are, in general, correlations between inputs and outputs, but these are not as strict as a production function implies. If one estimates a relation in absolute values or logs of these, then you will often have seemingly good statistical results, although the correltation may be weak or non-existing.

Thank you John for your silly calculations, with implied assumptions. One thing is clear as along as the measurement errors are stationary your argument becomes uninteresting to me. Why would I want a time trend included? Even when different variables do not use th same base periiod, it is still correct to log them for elasticity interpretation purposes , The key point is that base periods are irrelevant in cointegration analysis so long as the variable is logged and measurement errors are stationary. You seem focused on one variable when you should be focused on the intended usage. Most macrovariable  become stationary after differencing and differencing makes them satationry. Actually if I remember correctly an article my M.P Taylor and Ron Macdonald addressed this issue in ReStat many years agon and I know Hendry touched on it but I cannot remember in which paper anymore. Anyway, I will be continue to use logs and rebasing because this is what is done in the Fedaeral Resaeve office. Take care and good luck ! 

Perhaps I have moved away from the original topic here. My first interventions about (16 and 18) days ago agrees with a lot of what Chuck has said about logs, elasticities and re-basing. My later comments were about   Anton's simulation and estimation exercise with Excel. I felt that his estimation results were too different from the simulated model. I proceeded to examine aspects of his exercise using R.  My intention was to clarify some aspects of Anton's exercise. 

I included a time trend in the estimation because a time trend was included in the simulation.  This is specific to this simulated model.  In other models a time trend may or may not be necessary.  In output models a time trend is often included as a proxy for technical progress.

If I recall, correctly, the Hendry type results on including (stationary) variables included a requirement that the residuals be "clean".  If you first difference here there is an MA term in the disturbances and the residuals will not be clean. In David Hendry's terms the model will not be data congruent.   If you proceed with OLS in the presence of MA (or AR) disturbances the OLS estimates of the standard errors of your coefficients will not be correct. If your residuals are "clean" there is no problem.

I don't understand what you mean by  "focused on one variable when you should be focused on the intended usage" as my intended usage was the estimation of a particular model and the effect of various assumptions on, in particular the estimates of standard errors of the estimated coefficients. Perhaps this should have been clearer.

I regard the Cobb-Douglas form as a first order linearisation of a function where the input variables have been transformed. A translog is a second order approximation of the actual function. In using these one should be aware of their limitations. (e.g. constant returns to scale and unit elasticity of substitution in the case of Cobb-Douglas form). One often accepts linear approximations  when doing regressions. 

It is important in econometrics to know if your variables are (1) pre-determined or (2) stochastic and stationary or (3) stochastic and non-stationary.  When you have non-stationary variables and cointegration you must take this into account.

Hi John,

Please accept my apologies! No harm was intended  & Sorry!

Regards,

Prof. Chuck

Hi John and Chuck,

We seem to agree about the estimation of the usual Cobb-Douglas function with technical progress or total factor productivity. My main point is that I do not believe the function. I think that most production processes have only very limited space for factor substitution, i.e. they are very near to a limitational (Leontieff) function.

Two remarks: If the errors are small, I think, one need not care much about stationarity and cointegration, if they are big, then one gets poor results with any method. With time series data, it is, in general, not possible to estimate the form of isoquants, because the data are to near to a ray, i.e. the relation between factors does not change much.

Best wishes

Anton

Anton and Chuck,

I don't think that there was that much difference between us - perhaps a question of different emphasis  dependent on our different experience. I think it was an interesting debate.

Regards

John

Thank you, John, you know the common saying academic exercise  is one thing that is better than s-thing (I leave this your imagination). The reason is probably because we spend most of our life time studying. I truly enjoyed the interaction but we need to pay attention to other aspects of our lives, so that  when our time nears to say good bye, we could say we had a full one. Thanks John and lots of love!