Forecasting income shares: are mean-reversion assumptions appropriate?

Abstract

The article examines some statistical evidence that supports the view that US labour and capital shares of income return to some long-run historical values. We estimate the long-run share values and the length of time it takes to converge to them. We account for the interdependence of the shares by using a vector error–correction model, and this specification is tested against a VAR alternative using Johansen's method to characterize the properties of the cointegrating vector. We find support for the idea that labour and capital shares have historically been mean reverting, in spite of the fairly restrictive assumptions implied when invoking the Cobb–Douglas production function as the rationale. The cumulative impulse response functions indicate that for capital and labour shares, the time required to revert back to long run levels is in the order of thirty quarters.

Acknowledgements

I thank Ralph Monaco, Danny Bachman and Michael Falkenheim for comments. The editor and a referee also provided helpful comments. Views expressed are the author's and not those of the Department of the Treasury.

Notes

¹ A useful recent summary of the history and progression of multi-equation modelling techniques and issues in cointegration is provided in this journal (Harvey and Mills, 2005).

² BCC is a monthly survey of fifty-four private-sector professional forecasters.

³ Income taxes and payroll taxes for Social Security affect revenues more than taxes on corporate profits. In fiscal year 2005, CBO estimates a shift of 1 percentage point of GDP out of profits into wages and salaries leads to a gain in revenue of $11 billion–almost 3% of that year's current law deficit–rising to $25 billion by 2015. This excludes the reduction in outlays that would be attributable to reduced debt service. Furthermore, CBO states that ‘Uncertainty about the path of the shares of income that affect revenue projections has been a source of error in CBO's budget projections in the past and is a major risk to the accuracy of the current forecast.’ Source, The Budget and Economic Outlook: Fiscal years 2006 to 2015, January 2005 pp. 47.

⁴ BCC does contain consensus projections of corporate profits and disposable personal income for one calendar year ahead. However, consensus estimates of the pre-tax components of labor's share–wages and salaries, worker compensation, and proprietor income are not provided.

⁵ Highly regarded macro-econometric models used in these forecasts calculate profits as a residual after other components of national income are calculated. The equations say nothing about whether shares return to their long run historical values.

⁶ This note touches on a few of the measurement issues pertaining to the National Income and Product Accounts, particularly the difficulty in allocating proprietor income to capital and labor shares, and the role of interest income in distorting profit's share as a proxy for returns to capital. The NIPA framework divides GDI into specific payment types. Rental income–which includes labor income and proprietors’ income–which includes noncorporate capital income do not neatly fit into a working definition of labor and capital shares, because the capital/compensation split of these items is unobservable. So the way the data are reported means that labor and capital shares cannot be precisely measured. Moreover, corporate profits as measured in NIPA can be affected by the way corporations choose to leverage their debt. Consider a decline in nominal interest rates––(or the debt to equity ratio, which would raise the profit share of GDI while the capital share is really unchanged. So the shift between equity and bond financing would be undetected using corporate profit alone to measure capitals’ share. While the share of capital in GDI should include profits, it also should also include interest payments to corporate debt holders. Our measure of capital ‘share’ includes interest payments but does not separate out interest payments to corporate debt holder.–Interest payments played a significant role in explaining the divergence of profits and capital shares of GDI. During the 1980's interest payments rose relative to profits because firms issued bonds rather than equity. Later, in the early 90's the reverse occurred as interest rates fell.

⁷ Email correspondence with James MacKinnon, Queens University, and Kingston, Ontario.

⁸ We also examined differing sub-periods of the data because structural shifts have been known to result in the test being unable to reject unit roots.

⁹ To avoid confusion, the ‘levels’ transformation refers to the absence of logit or tobit transformation. This nomenclature should not be taken to suggest that the data are always in ‘levels’ during the VAR and VEC estimation procedure.

¹⁰ This procedure is described in Griffiths, 2004. ‘p’ denotes the order of lag on the Δx _t determined by the Akaike information criterion. MacKinnon (1996) provides the critical values of these statistics based on Monte Carlo simulation.

¹¹ Proprietors’ income and taxes on imports and business were excluded from further analysis. As Table 1 shows, the stationary nature of proprietors’ income indicates that it is not a part of a cointegrated system, since it must be part of a group of nonstationary variables which in combination must be stationary for this to apply. Also, an unreported squared trend term was found to be statistically significant which suggests nonstandard behavior of this series. Second, as noted earlier the allocation of proprietors’ income into ‘capital’ and ‘labor’ shares is unobservable because it contains both elements. Proprietors’ income comprises a relatively small share of total shares with a mean of 8.3% and a SD of 2.2%. In studying many different combinations of variables, the inclusion of taxes on imports and business resulted in the impulse response functions on wages and profits not perfectly converging to zero, a necessary stability condition for a valid model. While this aberrant behavior is a puzzle, a possible clue may be that the factors determining this share are in part policy-determined, which may interfere with the normal interaction across the shares, rather than purely the result of economic forces. Taxes on imports and businesses comprise a relatively small share of the total, and add virtually no explanatory power to any of the equations. It has a mean of 7.8% and a SD of 0.5%.

¹² A linear combination of the variables in long-run equilibrium implies πX = 0, so the previous period's deviation from long run equilibrium has the representation πX _t−1 = θ, where θ is the previous period's deviation from equilibrium.

¹³ Johansen (1988) provides the details to this procedure.

¹⁴ If the rank of π is equal to r there are r linearly independent combinations of the X sequences that are stationary. If π is of full rank equal to n then each of the n variables in the vector X must be stationary.

¹⁵ The inverse is applied to the logit and tobit transformed data so that direct comparison with the untransformed data can be made.

¹⁶ The causal ordering runs from profits to wages and the remaining variables. This was determined from unreported Granger causality testing and determines the ordering of the variables in the model.

¹⁷ Our experience with data revisions suggests that there is some modest variability in the projected terminating value when different vintages of GDI data release are employed. The data revisions can result in significant changes to the recent history of income shares. Hence the model's share input variables, and to a much lesser extent, the coefficients of the re-estimated model can be affected. The average terminating compensation share over several vintages of data is 58.3, which is the same as the sample mean of the compensation share between 1970Q1 and 2004Q4.