We begin from the simple accounting identity relating the profit rate r(t) to the net operating surplus s(t) and the net capital stock k(t) scaled by revenue (or GDP):
(1) r(t) = s(t) / k(t) → ln r(t) = ln s(t) − ln k(t)
The net capital stock k(t) is measured at mid-year (Z(t) = ½ [K(t) + K(t – 1)], where K(t) is the end-of-year capital stock). The index (t) denotes the time period (quarter or annual data). This log-linear identity provides the starting point for tractable econometric specifications.
1. Baseline ARDL(1,0,0) Specification
A minimal autoregressive distributed lag representation consistent with the profit rate identity is:
(2) ln r(t) = a + ρ ln r(t−1) + b ln s(t) + c ln k(t) + u(t),
where u(t) is the random error with mean zero and finite variance..
This regression form (2) involves four scalar parameters (a, ρ, b, c), and is a direct application of the ARDL(1,0,0) structure discussed in Pesaran & Shin (1999, eq. 2.2) and in Asteriou & Hall (2016, Chapter 17). ARDL denotes the autoregressive distributed lag model.
2. Long-Run Multipliers
From the ARDL(1,0,0), the implied long-run multipliers are the indirect least squares (ILS) of the regression:
(3) ln r(t) = α + β ln s(t) + γ ln k(t) + v(t),
where α = a / (1 − ρ) is the reduced-form intercept. The partial slope coefficients are β = b / (1 − ρ) and γ = c / (1 − ρ). The transformed residual error is v(t).
These indirect least-squares coefficients are the parameters of ln s(t) and ln k(t) in the long-run equilibrium condition in which ln r(t) = ln r(t−1). Under the profit rate identity, the null hypothesis is H0: β = 1, γ = −1. This parallels the bounds-testing approach of Pesaran, Shin & Smith (2001, Section 3).
3. Equivalent ECM Form
Reparameterizing, we obtain the equivalent error-correction model (ECM):
(4) Δ ln r(t) = a0 + π [ ln r(t−1) − β ln s(t−1) + γ ln k(t−1) ] + λs Δ ln s(t) + λk Δ ln k(t) + u(t).
The bracketed expression multiplied by pi is the equilibrium condition, or error correction elements (ECM). Here π = ρ − 1 is the speed of adjustment, expected to be negative if deviations from the long-run relationship are corrected over time. This representation (4) is emphasized in Asteriou & Hall (2016, Chapter 17) as a direct reparameterization of the ARDL model.
4. Practical Implementation
The ARDL(1,0,0) model can be estimated using Python’s Statsmodels (tsa.ardl). The researcher estimates (a, ρ, b, c), then computes the long-run multipliers (β, γ) and the error-correction coefficient (π). The beta and gamma (β, γ) multipliers can also be estimated with the regression equation (3), without apparent dynamics.
Diagnostics (residual autocorrelation, stability tests) guide whether higher-order lags are necessary (ARDL(1,1,1), for example). Like model (3), model (4) can be estimated by ordinary least squares (OLS) with Newey-West corrected errors.
5. Conclusion
By tying the simple profit rate identity r(t) = s(t)/k(t) to an ARDL(1,0,0) framework, we gain an empirically tractable model with interpretable long-run coefficients. This provides a rigorous yet simple bridge between accounting identities and econometric dynamics. In principle, the ARDL model applies to any profit indicator defined as a ratio (such as return on assets, profit margin, or profit markup) in transfer pricing.
References
Asteriou, D., & Hall, G. (2016). Applied Econometrics, 3rd Edition. Palgrave Macmillan. See Chapter 17 for the reparameterization of ARDL into ECM.
Kmenta, J. (1986). Elements of Econometrics, 2nd Edition. Macmillan. See Section 14.5 on distributed lag models.
Maddala, G. S. (1992). Introduction to Econometrics, 2nd Edition. Macmillan. See Chapter 16 on dynamic models.
Pesaran, M. H., & Shin, Y. (1999). “An Autoregressive Distributed-Lag Modelling Approach to Cointegration Analysis,” in S. Strom (ed.), Econometrics and Economic Theory in the 20th Century. Cambridge University Press. See equation (2.2).
Pesaran, M. H., Shin, Y., & Smith, R. J. (2001). “Bounds Testing Approaches to the Analysis of Level Relationships,” Journal of Applied Econometrics, 16(3): 289–326. See Section 3 on the bounds test.