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### Topic 5.4

1. 1. ECON 377/477<br />
2. 2. Topic 5.4<br />Measuring TFP change<br />
3. 3. Outline<br />Introduction<br />Malmquist TFP index and panel data<br />Calculating TFP change using DEA frontiers<br />Calculating TFP change using SFA frontiers<br />Measuring TFP change: conclusions<br />Summary of methods<br />3<br />ECON377/477 Topic 5.4<br />
4. 4. Introduction<br />In this part, we consider the case where we have access to better data<br />In particular, we assume we have data on a sample of firms in periods s and t that are sufficient to obtain an estimate of the production technology in these two periods<br />We can then calculate the required distances directly and relax a number of the assumptions made to construct the TFP index numbers<br />In particular, we need no longer assume all firms are operating on the surface of the production technology (are technically efficient)<br />ECON377/477 Topic 5.4<br />4<br />
5. 5. Introduction<br />We no longer have the situation where the ratio of the distance functions provides a measure of TFP change that is identical to technical change (TC) (i.e. frontier shift), as was the case in earlier parts of this topic<br />Thus, when panel data are available, we can obtain a measure of TFP change that has two components:<br />a TC component<br />a TE change component<br />ECON377/477 Topic 5.4<br />5<br />
6. 6. Introduction<br />The remainder of this part is organised as follows<br />The Malmquist TFP index is briefly described<br />Then, we describe how to calculate these indices using DEA-like methods<br />Next, we describe the calculation of these indices using the SFA methods<br />We then refer to detailed application of some of these methods in CROB<br />Finally, we make some brief concluding comments<br />ECON377/477 Topic 5.4<br />6<br />
7. 7. Malmquist TFP index and panel data<br />Malmquist TFP change measures can be decomposed into various components, including TC and TE change<br />These measures could be calculated using distances measured relative to DEA frontiers<br />The Malmquist TFP index measures the TFP change between two data points by calculating the ratio of the distances of each data point relative to a common technology<br />ECON377/477 Topic 5.4<br />7<br />
8. 8. Malmquist TFP index and panel data<br />If the period t technology is used as the reference technology, the Malmquist (output-orientated) TFP change index between period s and period t is defined as:<br />Alternatively, if the period s reference technology is used, it is defined as:<br />ECON377/477 Topic 5.4<br />8<br />
9. 9. Malmquist TFP index and panel data<br />The notation represents the distance from the period t observation to the period s technology<br />A value of mo greater than one indicates positive TFP growth from period s to period t while a value less than one indicates a TFP decline<br />These period s and period t indices are only equivalent if the technology is Hicks output-neutral<br />It is Hicks output-neutral if the output distance functions may be represented as <br /> for all t<br />ECON377/477 Topic 5.4<br />9<br />
10. 10. Malmquist TFP index and panel data<br />The Malmquist TFP index is often defined as the geometric mean of the two indices with period s and period t technology:<br />The distance functions can be rearranged to show that it equates to the product of a technical efficiency change index and an index of TC:<br />TE change<br />TC<br />ECON377/477 Topic 5.4<br />10<br />
11. 11. Malmquist TFP index and panel data<br />The ratio outside the square brackets in the second equation on the previous slide measures the change in the output-oriented measure of Farrell technical efficiency between periods s and t<br />The remaining part of the index in this equation is a measure of TC: the geometric mean of the shift in technology between the two periods, evaluated at xt and xs<br />ECON377/477 Topic 5.4<br />11<br />
12. 12. Malmquist TFP index and panel data<br />A number of additional possible decompositions of these technical efficiency change and TC components have been proposed, notably:<br /><ul><li>decomposing TC into input bias, output bias and ‘magnitude’ components
13. 13. decomposing TE change into scale efficiency and ‘pure’ technical efficiency components (this can only be done when the distance functions are estimated relative to a CRS technology</li></ul>ECON377/477 Topic 5.4<br />12<br />
14. 14. Malmquist TFP index and panel data<br />The decomposition into scale efficiency involves taking the efficiency change measure and decomposing it into a pure efficiency change component (measured relative to the arguably true VRS frontier),<br /> and a scale efficiency component,<br />ECON377/477 Topic 5.4<br />13<br />
15. 15. Malmquist TFP index and panel data<br />The scale efficiency change component is the geometric mean of two scale efficiency change measures<br />The first is relative to the period t technology and the second is relative to the period s technology<br />Note that the extra subscripts, v and c, relate to the VRS and CRS technologies, respectively<br />Refer to CROB for an evaluation of this method and an alternative approach<br />The returns to scale properties of the technology are very important in TFP measurement<br />ECON377/477 Topic 5.4<br />14<br />
16. 16. Calculating TFP change using DEA frontiers<br />A number of different methods could be used to estimate a production technology and, hence, measure the distance functions that make up the Malmquist TFP index<br />The most popular method has been the DEA-like LP method, discussed in this section<br />The other main alternative approach is the use of stochastic frontier methods, described in the next section<br />ECON377/477 Topic 5.4<br />15<br />
17. 17. Calculating TFP change using DEA frontiers<br />Given suitable panel data, we can calculate the distance measures using DEA-like LPs<br />For the i-th firm, four distance functions are calculated to measure the TFP change between two periods<br />This requires solving four LP problems using a CRS technology in the TFP calculations<br />This ensures that TFP change measures satisfy the property that if all inputs are multiplied by the (positive) scalar  and all outputs are multiplied by the (non-negative) scalar , the resulting TFP change index will equal /<br />ECON377/477 Topic 5.4<br />16<br />
18. 18. Calculating TFP change using DEA frontiers<br />The required four LPs are:<br /> [dot(qt, xt)] - 1 = max,<br /> st -qit + Qt 0<br /> xit - Xt0<br />0<br /> [dos(qs, xs)] - 1 = max,<br /> st -qis+ Qs0<br /> xis – Xs0<br /> 0<br /> …<br />ECON377/477 Topic 5.4<br />17<br />
19. 19. Calculating TFP change using DEA frontiers<br /> [dot(qs, xt)] - 1 = max,<br /> st -qis + Qt 0<br /> xis - Xt 0<br />  0<br /> [dos(qt, xt)] - 1 = max,<br /> st -qit + Qs 0<br /> xit – Xs 0<br />  0<br />ECON377/477 Topic 5.4<br />18<br />
20. 20. Calculating TFP change using DEA frontiers<br />In the third and fourth LPs, where production points are compared with technologies from different time periods, the  parameter need not be greater than or equal to one, as it must be when calculating Farrell output-orientated TEs<br />The s and s are likely to take different values in the above four LPs, which must be solved for each firm in the sample<br />As extra time periods are added, it is necessary to solve an extra three LPs for each firm (to construct a chained index)<br />ECON377/477 Topic 5.4<br />19<br />
21. 21. Calculating TFP change using DEA frontiers<br />Decomposing technical efficiency change into scale efficiency and ‘pure’ technical efficiency measure requires the solution of two additional LPs (when comparing two production points)<br />These LPs would involve repeating the first two with the convexity restriction (I1 = 1) added to each<br />This provides estimates of distance functions relative to a VRS technology<br />ECON377/477 Topic 5.4<br />20<br />
22. 22. Calculating TFP change using DEA frontiers<br />CROB (pages 295-300) provide a numerical example of the application of the DEA-like method to construct Malmquist TFP indices, using the DEAP computer program<br />Four distances are calculated for each firm in each year, relative to:<br />the previous period’s CRS DEA frontier<br />the current period’s CRS DEA frontier<br />the next period’s CRS DEA frontier<br />the current period’s VRS frontier<br />ECON377/477 Topic 5.4<br />21<br />
23. 23. Calculating TFP change using DEA frontiers<br />All indices are calculated relative to the previous year<br />Hence, the output begins with year 2<br />Five indices are presented for each firm in each year:<br />TE change relative to a CRS technology<br />technological change<br />pure TE change relative to a VRS technology<br />scale efficiency change<br />TFP change<br />ECON377/477 Topic 5.4<br />22<br />
24. 24. Calculating TFP change using DEA frontiers<br />Summary tables of these indices follow for the different time periods (over all firms) and for the different firms (over all time periods)<br />Note that all indices are equal to one for time period 3 because the data for year 3 are identical to the year 2 data in the example data set used<br />ECON377/477 Topic 5.4<br />23<br />
25. 25. Calculating TFP change using SFA frontiers<br />The distance measures required for the Malmquist TFP index calculations can also be measured relative to a parametrically estimated technology<br />We focus our attention on the production frontier case, which is a single-output special case of the more general (multi-output) output distance function<br />ECON377/477 Topic 5.4<br />24<br />
26. 26. Calculating TFP change using SFA frontiers<br />Consider a translog stochastic production frontier defined as:<br /> i = 1,2,...,I , t = 1,2,...,T<br /> where qit is the output of the i-th firm in the t-th year; xnit denotes the n-th input variable; t is a time trend representing TE; the s are unknown parameters to be estimated; the vits are random errors, and the uits are the technical inefficiency effects<br />ECON377/477 Topic 5.4<br />25<br />
27. 27. Calculating TFP change using SFA frontiers<br />The above model has the time trend, t, interacted with the input variables to allow for non-neutral TC<br />The technical efficiencies of each firm in each year can be predicted using the approach outlined in Topic 4<br />We obtain the conditional expectation of exp(-uit), given the value of eit= vit - uit<br />These TE predictions are between zero and one, with a value of one indicating full TE<br />ECON377/477 Topic 5.4<br />26<br />
28. 28. Calculating TFP change using SFA frontiers<br />We can use measures of TE and TC to calculate the Malmquist TFP index via equations (11.4-11.6) in CROB (see next slide)<br />TE change = TEit/TEis<br />The TC index between period s and t for the i-th firm can be calculated directly from the estimated parameters<br />First, the partial derivatives of the production function are evaluated with respect to time using the data for the i-th firm in periods s and t<br />ECON377/477 Topic 5.4<br />27<br />
29. 29. Malmquist TFP index and panel data<br />From an earlier slide (slide 9), the Malmquist output index equates to the product of a technical efficiency change index and an index of TC:<br />TE change<br />TC<br />CROB, equation (11.4)<br />ECON377/477 Topic 5.4<br />28<br />
30. 30. Calculating TFP change using SFA frontiers<br />The TC index between the adjacent periods s and t is calculated as the geometric mean of these two partial derivatives<br />When a translog function is involved, this is equivalent to the exponential of the arithmetic mean of the log derivatives, and TC equals:<br />The indices of TE change and TC can then be multiplied together to obtain a Malmquist TFP index<br />ECON377/477 Topic 5.4<br />29<br />
31. 31. Calculating TFP change using SFA frontiers<br />The above technical change measure involves derivative calculations, which appears to contradict the earlier comments that these indices are derived from distance measures<br />It can be easily shown (for the translog case in which a time trend is used to represent technical change) that the geometric means of the distance ratios in equation (11.6) in CROB are equivalent to the geometric means of the derivative measures<br />ECON377/477 Topic 5.4<br />30<br />
32. 32. Calculating TFP change using SFA frontiers<br />A criticism of this method is that the TFP index may produce biased measures because the productivity changes due to scale changes are not captured<br />One possible solution to this problem is to impose CRS upon the estimated production technology<br />Another option is to derive a Malmquist TFP decomposition identical to that proposed above, and address the scale issue by including a scale change component in the TFP measure<br />ECON377/477 Topic 5.4<br />31<br />
33. 33. Calculating TFP change using SFA frontiers<br />Scale change =<br /> where , and<br />This scale change index is equal to one if the production technology is CRS, when the scale elasticity ( ) equals one<br />An empirical application to rice production data is provided by CROB (pages 302-309)<br />ECON377/477 Topic 5.4<br />32<br />
34. 34. Measuring TFP change: conclusions<br />Some of the advantages of the frontier approach are:<br /><ul><li>It does not require price information
35. 35. It does not assume all firms are fully efficient
36. 36. It does not need to assume a behavioural objective such as cost minimisation or revenue maximisation
37. 37. It permits TFP change to be decomposed into components such as TC, TE change and scale change</li></ul>ECON377/477 Topic 5.4<br />33<br />
38. 38. Measuring TFP change: conclusions<br />An important advantage of the Tornqvist approach is that it can be calculated using only two data points<br />In contrast, the frontier approach needs a number of firms to be observed in each time period so that the frontier technology in each year can be estimated<br />If one has suitable panel data, the frontier approach provides richer information and makes fewer assumptions<br />ECON377/477 Topic 5.4<br />34<br />
39. 39. Measuring TFP change: conclusions<br />But if only aggregate time-series data are available, the Tornqvist approach provides useful estimates of TFP change, given that the assumptions specified earlier in the topic notes are reasonable<br />ECON377/477 Topic 5.4<br />35<br />
40. 40. Summary of methods<br />We have considered four principal methods:<br />Least-squares econometric production models <br />TFP indices (Tornqvist/Fisher)<br />DEA<br />SFA<br />They differ in various ways, as demonstrated in the table on the next slide<br />ECON377/477 Topic 5.4<br />36<br />
41. 41. Summary of methods<br />ECON377/477 Topic 5.4<br />37<br />
42. 42. Summary of methods<br />Efficiency is generally measured using either DEA or SFA<br />Advantages of SFA over DEA include:<br /><ul><li>it accounts for noise
43. 43. it can be used to conduct conventional tests of hypotheses</li></ul>Disadvantages include:<br /><ul><li>the need to specify a distributional form for the inefficiency term
44. 44. the need to specify a functional form for the production function (or cost function)</li></ul>ECON377/477 Topic 5.4<br />38<br />
45. 45. Summary of methods<br />TC (or TFP) is usually measured using either least squares econometric methods or Tornqvist/Fisher index numbers<br />Some of the advantages of index numbers over least-squares econometric methods are:<br /><ul><li>only two observations are needed
46. 46. they are easy to calculate
47. 47. the method does not assume a smooth pattern of TC</li></ul>The principal disadvantage is that it requires both price and quantity information<br />ECON377/477 Topic 5.4<br />39<br />
48. 48. Summary of methods<br />Both approaches assume that firms are technically efficient, which is unlikely to be true<br />To relax this assumption, frontier methods can be used, assuming panel data are available, to calculate TFP change<br />ECON377/477 Topic 5.4<br />40<br />
49. 49. Summary of methods<br />Some of the advantages of the SFA approach over the Tornqvist/Fisher index numbers approach are that it:<br /><ul><li>does not require price information
50. 50. does not assume all firms are fully efficient
51. 51. does not require the assumption of cost minimisation and revenue maximisation
52. 52. permits TFP to be decomposed into TC and TE change</li></ul>ECON377/477 Topic 5.4<br />41<br />
53. 53. Summary of methods<br />But an important advantage of the index-number approach is that it only requires two data points, say observations on two firms in one time period or observations on one firm in two time periods<br />The frontier approaches need a number of firms to be observed in each time period so that the frontier technology in each year can be calculated<br />ECON377/477 Topic 5.4<br />42<br />