The Canadian Productivity Review

15-206-XIE

Volume 2007
Number 5

Depreciation Rates for the Productivity Accounts

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Depreciation Rates for the Productivity Accounts

Micro-economic Analysis Division

Executive summary

The productivity program at Statistics Canada requires estimates of both outputs and inputs to the production process. Inputs are classified as intermediate materials, labour and capital. Intermediate materials are products that are essentially completely consumed over the course of one year in the production process. Capital, on the other hand, is provided by assets whose life extends beyond one period and whose use therefore extends over several years.

Measures of capital that are applied to the production process in any particular year require information on investments that have been made over a period of time, and some method of weighting investments of different vintages. Estimates of depreciation are used for the latter task. For example, the net value of capital today from an investment made last year is just the gross investment made last year minus the value by which it has declined because of use — the amount it has depreciated. Net values of investments from different years are then summed to provide an aggregate value of capital that is employed in the production process today.

Estimates of depreciation are therefore central to attempts to provide summary measures of the amount of capital that is being applied to the production process. But obtaining estimates of the rate of depreciation (the amount of depreciation in a particular year divided by its initial value) provides numerous difficulties. While depreciation is a concept that is applied directly to the accounts of companies and is used in the calculation of taxes owed to the government, the commonly used concepts are not always perceived as being those required by the productivity program. This can occur for a number of reasons — not the least of which is that depreciation allowances used for taxation purposes may differ from the 'real' rate, either because the tax system lags changes in the world or because the tax system may deliberately choose a rate that is different because it is attempting to stimulate or decelerate investment.

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The statistical community has, therefore, long wrestled with alternate methods of estimating depreciation rates. Originally, estimates derived from tax codes were generally chosen in North America. These rates were then arbitrarily adjusted in order to try to accommodate what were widely perceived to be outdated estimates in the tax code. More recently, the United States made use of the prices of used assets to estimate depreciation. And the Canadian productivity accounts made use of ex ante estimates of the length of life derived from a survey of what was expected in the way of life upon initial investment and several arbitrary assumptions about the rate of decline of an asset (what has been referred to here as the DBR or declining-balance rate).

In 2003, the Productivity Accounts at Statistics Canada moved to make use of used-asset prices in estimating the rate of depreciation for calculating the growth of capital stock and capital services (see Harchaoui and Tarkhani, 2003). A background paper (Gellatly, Tanguay and Yan, 2002) describes how depreciation rates for a range of assets were estimated by employing used-asset prices. It also compared differences that arose from the ex post estimates and an alternate method that used ex ante estimates of expected length of life — finding that the differences between the two were not large across most asset classes.

Gellatly, Tanguay and Yan (2002) use Weibull survival models to estimate patterns of economic depreciation based on rich samples of used-asset prices and discards. Two variants of the estimation framework were proposed: a simple linear model estimated via average prices, and models that generate depreciation estimates directly from the entire sample of micro-data. The second used a maximum likelihood formulation of the price survival function that adjusted for patterns of digit preference.

The depreciation profiles generated by the econometric techniques were, on balance, accelerated, producing convex age-price curves. Declines in value early in life were apparent for many assets in the machinery and equipment class, as well as for certain structures. Evidence that rates of depreciation are constant over the service life was, on balance, mixed.

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This paper extends the earlier work. It does so in several ways. First, it enlarges the database on used-asset prices and makes use of additional editing techniques on that database. This enlarges the number of observations to around 30,000. The size of this database is unique.

Second, it revisits the issue of the choice of the estimation technique. In our original version (Gellatly, Tanguay and Yan, 2002), we compared a very simple ordinary least squares model to what was referred to as "a maximum likelihood survival model" and chose the latter. In this paper, we extend our econometric techniques. We explore several other econometric techniques than were used in the original study and we investigate the differences between the different estimates. We ask if there are clear advantages of one technique over the latter, both in terms of making use of background theory, and in terms of their ability to handle different datasets. In the latter case, we use Monte Carlo simulation techniques to examine the ability of each to provide accurate estimates in the face of both misspecification of functional forms and imperfect data.

We discover that differences in the econometric estimates stemmed not so much from differences in techniques but rather from the nature of the sample that was being used. The data are not generated by a process that is necessarily random and this may have an influence on the different econometric techniques. The paper examines the nature of the data and finds evidence of non-randomness in the distributions of the used-asset prices that are generated by the survey source and corrects for this.

Differences in the econometric formulations can give rise to discordant impressions about how rapidly asset values erode over the course of service life. Therefore, we briefly discuss the advantages and disadvantages of each technique — based on theoretical and practical considerations. Some of the techniques require less precision in specifying underlying functional forms; others are more consistent but require knowledge about the functional forms. Since these considerations do not yield strong preferences for one technique over another, we also use Monte Carlo simulation techniques to discriminate among the various estimates. The simulation results suggest a slight preference for a technique that simultaneously estimates both the discard process and the price-age profile.

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But when we compare the differences in the depreciation rates produced by the three different techniques, we find that, after we account for the non-random nature of the process that generated the data and reweight our sample, the differences in the estimated mean depreciation rates across the three methods are small. And the estimates of individual assets were generally not significantly different from one another. More importantly, the rates of growth of capital services associated with each estimate of the depreciation rate are quite similar. Since our purpose is to estimate the growth in capital stock and capital services as part of the process by which productivity estimates are produced, we conclude that for our purposes there is little to choose between our estimates — at least for those assets where we have an adequate number of observations.

We also compared the estimates derived from our econometric ex post approach to ex ante methods using estimates of the expected length of life of assets. We do so for two reasons. First, it is inherently interesting to know whether the two estimates yield approximately the same results. Do managers predict the length of life of their assets correctly? If accounting records are based on these ex ante predictions, we would like to know how accurate they are. Second, knowing whether ex post and ex ante estimates are approximately the same is important if we are to produce estimates of depreciation of those assets where we cannot do so via the ex post technique but can do so via the ex ante approach. There are a large number of fixed assets that fall in the building and engineering construction categories where we have an ex ante prediction of the length of life but where we do not have enough used-asset transactions to employ the ex post technique.

We find that the ex ante and ex post approaches are approximately the same for those assets where we have enough observations to provide estimates of both. The ex ante approach suffers a number of problems. Managers have to correctly forecast length of life in a changing world. They need to have in mind an optimal maintenance schedule when they provide expectations on length of life. The ex post approach in turn suffers from other difficulties. Discard data can suffer from a number of imperfections — not the least of which is accuracy of recall of the original purchase price, all relevant upgrades, and the asset's age. Despite these problems, the two techniques provide remarkably similar results.

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We therefore combine information from both approaches to generate depreciation rates across our asset classes. We propose a set of depreciation rates that make use of both the ex ante and ex post approaches. The ex ante information that is provided in Statistics Canada's surveys only pertains to the expected length of life of the asset. Derivation of a (geometric) depreciation rate from the expected life of the asset also requires a shape parameter of the rate — what is referred to as the DBR (the declining-balance rate). It is this parameter that determines how much of total life-time depreciation occurs early in life. And here we make use of information on similar assets where we have been able to estimate the ex post approach to infer what the DBR is likely to be.

Despite the progress that has been made in updating the database, and modifying the estimation techniques, the new growth rates in capital stock and capital services are not very different than those previously used.

Finally, it must be stressed that the adequacy of any set of depreciation estimates depends on the use to which they are being put. Statistics Canada's standard for quality is that its estimates pass a "fitness for use" test. The motivation for this paper is to produce depreciation estimates that have a degree of accuracy that is appropriate for the productivity program. Throughout this exercise, we have asked how robust our estimates of productivity growth are to the various econometric techniques employed once the data are reweighted to take into account non-randomness. In the end, we find little difference across the alternate estimates of depreciation derived from the different estimation techniques. But that is for the construction of the productivity accounts. We are not necessarily advocating their use for determining capital consumption allowances asset by asset in the tax code. For that purpose, we believe the estimates reported here might provide useful starting information — but they would need to be bolstered by case studies and other information.

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