Risk shocks with time-varying higher moments
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Victor Dorofeenko
Abstract
Within the context of a financial accelerator model, we model time-varying uncertainty (i.e. risk shocks) through the use of a mixture normal model with time variation in the weights applied to the underlying distributions characterizing entrepreneur productivity. Specifically, we model capital producers (i.e. the entrepreneurs) as either low-risk (a relatively small second moment of productivity) or high-risk (a relatively large second moment of productivity) and the fraction of both types is time-varying. We show that this modeling feature implies that the aggregate distribution of productivity shocks is non-normal and has time varying kurtosis and skewness; both of these features have important effects on equilibrium characteristics. In particular, after estimating the steady-state share and the change in the fraction of risky entrepreneurs, we show that a small change in the fraction of risky types can result in a large quantitative effect of a risk shock relative to standard models for both financial and real variables. Moreover, the bankruptcy rate and the risk premium in the economy are very sensitive to a change in the composition of entrepreneurs.
Acknowledgements
We thank the seminar and conference participants at the 2015 Business Cycle Conference, LAEF, Bundesbank Research Seminar, University of Regensburg and 2016 Asian Econometric Society Meeting. Gabriel Lee and Johannes Strobel gratefully acknowledge financial support from the German Research Foundation (DFG) LE 1545/1-1.
A Appendix
A.1 Data
The productivity data are from NBER-CES Manufacturing Industry Database, see also https://www.nber.org/data/nberces5809.html, that uses Annual Survey of Manufactures (ASM).data. There are two versions of TFP in the NBER-CES Manufacturing Database: 4-factor and 5-factor. We use the 5-factor version, that separates out energy from non-energy materials. We broadly define the industries displayed in Table 5 to be part of the capital-good producing sector.
Industries in the capital-good producing sector.
| SIC4 | Industry | Benchmark | Robustness |
|---|---|---|---|
| 2200–2299 | Textile and Mill Products | x | x |
| 2800–2899 | Chemicals and Allied Products | x | x |
| 3000–3099 | Rubber and Misc. Plastic Products | x | − |
| 3200–3299 | Stone, Clay, Glass, and Concrete Products | x | − |
| 3300–3399 | Primary Metal Industries | x | − |
| 3400–3499 | Fabricated Metal Prdcts, Except Machinery & Transport Eqpmnt | x | x |
| 3500–3599 | Industrial and Commercial Machinery and Computer Equipment | x | x |
| 3600–3699 | Electronic, Elctrcl Eqpmnt & Cmpnts, Excpt Computer Eqpmnt | x | x |
| 3700–3799 | Transportation Equipment | x | − |
| 3800–3899 | Mesr/Anlyz/Cntrl Instrmnts; Photo/Med/Opt Gds; Watchs/Clocks | x | x |
An “x” means that the industry is included for the computation. The column “Benchmark” (“Robustness”) refers to the benchmark computation (robustness check).
The summary statistics for the benchmark data as well as for the robustness check are displayed in Table 6.
Descriptives of productivity data.
| Obs. | Mean | St. Dev. | Skewness | Kurtosis | p-value SK-Test | |
|---|---|---|---|---|---|---|
| Benchmark | 9269 | −0.0017 | 0.258 | 4.406 | 85.562 | 0.0000 |
| Robustness | 6587 | −0.004 | 0.292 | 4.278 | 73.177 | 0.0000 |
SK-Test refers to Skewness and Kurtosis Test. The null hypothesis is that the data are normally distributed.
A.1.1 Variables used for Bayesian estimation
Desciption of variables used for Bayesian estimation.
| Time series | Source | Code | Obs. | Note |
|---|---|---|---|---|
| GDP | Bureau of Economic Analyses | BEA_1.3.3RealVA_QI | 1947–2016 | BEA_1.3.3RealVA_QI |
| Investment | Bureau of Economic Analyses | Gross private domestic investment | 1947–2016 | Table 1.1.3. Real Gross Domestic Product |
| Consumption | Bureau of Economic Analyses | Personal Consumption Expenditures | 1947–2016 | Table 1.1.3. Real Gross Domestic Product |
| Hours worked | Bureau of Economic Analyses | BEA_6.9BHours | 1947–2016 | BEA_6.9BHours |
| Bankruptcy Rate | FRED | DRBLACBS | 1947–2014 | Delinq. rate on commercial loans are a proxy |
| Real interest rate | Gomme, Ravikumar, and Rupert (2011) | Real return on business capital | 1988–2014 |
A.2 Competitive equilibrium
A competitive equilibrium is defined by the decision rules for (aggregate capital, entrepreneurs capital, household labor, entrepreneur’s labor, the price of capital, entrepreneur’s net worth, investment, the cutoff productivity level, household consumption, and entrepreneur’s consumption) given by the vector:
The first equation represents the labor-leisure choice for households given the functional forms specified above, while the second equation is the necessary condition associated with household’s savings decision. For each unit of investment it that a household wishes to purchase, it gives qt consumption goods to the CMF. The additional capital resulting from time t investment becomes productive with a one-period delay. The third and fourth equation are from the optimal lending contract while the fifth equation is the necessary condition associated with entrepreneur’s savings decision. The sixth equation is the determination of net worth while the seventh gives the evolution of entrepreneur’s capital. The final two equations represent the laws of motion for the aggregate technology and fraction of risky entrepreneurs, respectively.
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Article Note
The previous version of this paper was circulated under the title “On Modeling Risk hocks”.
Supplementary Material
The online version of this article offers supplementary material (DOI: https://doi.org/10.1515/snde-2018-0028).
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Articles in the same Issue
- Research Articles
- Constrained interest rates and changing dynamics at the zero lower bound
- A threshold mixed count time series model: estimation and application
- Temporal aggregation of random walk processes and implications for economic analysis
- Forecasting the unemployment rate over districts with the use of distinct methods
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