Gauging the performance of the linear-quadratic inventory model

Abstract

The explanatory and predictive abilities of the infinite-horizon linear-quadratic inventory model are gauged using the flexible-accelerator model as a baseline. Tests of explanatory power for six nondurables industries indicate that the flexible accelerator has superior explanatory power for inventories and output in the majority of industries. Tests of predictive ability during the late 1990s also support the superiority of the flexible-accelerator model.

Acknowledgements

The initial draft of this paper was written while the author was Senior Fulbright Teaching Scholar at Tallinn Technical University, Tallinn Estonia. I am grateful to the Council for the International Exchange of Scholars and Indiana University-Purdue University Indianapolis for their support during this period. I am also grateful to Robert Rossana, Peter Pedroni, Chris Plantier, Ellis Tallman, and several anonymous referees for their helpful comments and insights.

Notes

This feature is appealing because the models are relatively easy to estimate, unlike lot-size models in which production takes place only when stocks on hand reach a critical point. Iturriaga (2000) rejects the lot-size model in his analysis of Spanish firms in favour of a target-stock model.

Sensier (2003) examines the symmetry assumption implicit in certainty equivalence models using aggregate British data. Her model allows for asymmetric responses when inventories are above or below their target value and when production is rising or falling. She finds that allowing for asymmetries improves explanatory and predictive power in some cases.

For the infinite horizon model, λ is set to 0.995 implying an annual real rate of interest of about 6%.

The marginal cost function is the second derivative of E(V) with respect to X_t : (1 + λ)a _{Δ X}  + a_X . Since a _{Δ X} is non-negative, this expression can only be negative if a_X is negative. The evidence on the sign of a_X is mixed but Bairam (1996) concludes that the sign is positive using annual firm-level data for major corporations in the USA.

Typically, these terms have not performed well in empirical models of inventory behavior. An exception is Durlauf and Maccini (1995). They employ specially constructed cost series for each industry. Unfortunately, the series have not been updated. The series employed here is less industry-specific as described below but still seems to perform moderately well.

Nevertheless, the simple flexible accelerator model remains a popular vehicle for analysing inventory movements. See Iturriaga (2000) and Nguyen and Andrews (1989) for two example. No doubt, much of its appeal arises from its simplicity. But another advantage of the model is that it allows for complicated definitions of the target stock. Carpenter et al. (1998) rely on the model to test for financing constraints among smaller firms. They argue in favour of the myopic specification on the grounds that inventories are more liquid than, say, fixed capital. Therefore the firm can make optimal decisions with a shorter forecasting horizon. Nguyen and Andrews (1989) compare the flexible-accelerator model with an alternative developed by Feldstein and Auerbach (1976) in which inventories adjust very rapidly toward a slowly-changing target. Their results suggest that a compromise model incorporating characteristics of the two alternatives performs best.

The assumption of cost minimization in Equation 6 rather than profit maximization in Equation 3 is not relevant for the decision rules because sales are exogenous.

Blinder (1986) first uncovered evidence that output is less stable than sales among manufacturers. Iturriaga (2000) uncovers the same result for Spanish firms.

As Equation 5a indicates, a_F /a _{Δ X} is also a function of π₁ and π₂. The reduced-form model estimates a_F δ/a _{Δ X} as the coefficient on S2t and Equation 5a is used to define a_F and δ from this coefficient.

In each iteration, Equation 4 is first estimated with a serial correlation correction. If the estimated serial correlation coefficient is insignificant at the 5% level, then the model is re-estimated under the assumption that the errors are serially independent.

Details are available from the author.

Unfortunately, the estimates for the chemicals industry do not satisfy the Legendre–Clebsch second-order condition for a maximum. This condition requires the second-derivative of the first-order condition with respect to finished goods to be positive. Specifically, the requirement is that (West, 1995)

The point estimates satisfy this condition for every industry except chemicals. For that industry, the value is negative and significant at the 5% level. Naturally, this calls into question the restrictions imposed by the linear-quadratic model for this industry. For our purposes, however, the theoretical consistency of the estimates is secondary to the model's explanatory power and predictive ability.

The nondurables sector is another obvious candidate, but it is not pictured because the estimate of δ is negative and this makes it difficult to determine if the differences in the impulse responses are due to differences in the underlying models or to the negative sign on δ in the linear-quadratic model.

There is no evidence of serial correlation in 5 industries for either inventories and output. In the remaining industry, the coefficient was significant at the 5% level but not at the 1% level. Also, a Jarque–Bera test for normality rejected the null hypothesis of a normally distributed residual in only two industries for output and one industry for inventories at the 5% level. Thus the standard t- and F-tests are appropriate. For modifications to the test in the event of non-normally distributed errors, see Harvey et al. (1998). West (2001) has recently provided an alternate test for the case in which one of the forecasts is based upon estimated regression parameters. Judging from his results, the sample size of 156 observations together with a forecast period of 31 observations should results in relatively little size distortion. Thus the reported results are based upon a standard t-test.