Home P-star model for India: a nonlinear approach
Article
Licensed
Unlicensed Requires Authentication

P-star model for India: a nonlinear approach

  • Aditi Chaubal ORCID logo EMAIL logo
Published/Copyright: August 18, 2018
Become an author with De Gruyter Brill
Submit Manuscript Author Information
Studies in Nonlinear Dynamics & Econometrics
From the journal Studies in Nonlinear Dynamics & Econometrics Volume 22 Issue 5

Abstract

Inflation in India has been a major cause for concern in the recent past (2008–2012). This study examines the Indian wholesale price index inflation from 1951 to 2012 using P-star (or P*) models after accounting for the nonlinearities in the data by establishing the presence of a nonlinear long-run equilibrium. The paper establishes the presence of a threshold vector error correction model (TVECM) between prices and their long-run equilibrium with three optimal regimes to explain the short-run and long-run dynamics based on an error correcting transition term. Based on these results, the study classifies the various regimes that Indian inflation goes through based on historical economic events. The P* models (price gap, output gap and velocity gap models) were implemented regime-wise. The price gap models (output gap and income velocity gap determine inflation) were found to be optimal in the first and second regimes and consistent with theory. The velocity gap model (which has monetarist foundations) was found to be optimal in the third regime.

Keywords: Inflation; price gap model; P-star models; threshold vector error correction; velocity gap and output gap models
JEL Classification: E31; C24; C32; C34

Article Note

This paper was presented at the Financial and Nonlinear Dynamics 2017 conference held in Paris (June 1–2, 2017). The views expressed here do not reflect the views of the Australian Consulate General, Mumbai (my affiliation when this paper was presented and accepted). This paper was part of my PhD thesis submitted and defended at the Indira Gandhi Institute of Development Research, Mumbai.

Acknowledgement

The author is extremely grateful for the valuable comments received from the anonymous referee which led to considerable improvement in the paper. Responsibility for any remaining shortcomings and errors rests solely with the author.

A Appendix

A.1 Data transformations

A.1.1 GDP interpolation method

The quarterly estimates of GDP are available in Khetan, Handa, and Waghmare (1976) and Das (1993) for the periods 1951–1966 and 1970–1990, respectively. Thus, there are gaps in the GDP data from 1967 to 1969 and from 1991 to 1995 (post–1995, quarterly data is available from 1996 onwards).

Suppose the quarterly GDP data points are given as Y(i, t) [e.g. Y(1, 1954), Y(2, 1954), Y(3, 1960), etc.] where i denotes the quarter and t denotes the year.

Then, let P(i, t) denote the proportions of GDP for each quarter (i = 1, …,4) in year t.

(23)P(i,t)=Y(i,t)∑t=14⁡Y(i,t),∀t

Collate all the different proportions quarter-wise (set 1, set 2, …, set 4). For each set of quarters (set 1–3), run the following regression:

(24)P(i,t)=αi+βit,t=1,2,3

for the period 1951–1966. The estimates for quarters P(1, t), P(2, t) and P(3, t) for the missing years (1967–1969) are calculated by simple extrapolation of the above equation [equation 24].

The estimates of the fourth quarter proportions (P(4, t)) for the period 1967–1969 are then calculated as given below:

(25)P(4,t)=1−P(1,t)−P(2,t)−P(3,t)

These estimated proportions are then multiplied with the annual GDP (at constant 2004–2005 prices) to get the GDP at constant 2004–2005 prices.

This process is repeated for the 1970–1990 data to obtain the estimates of GDP (at constant 2004–2005 prices) for 1991–1995.

Splicing of GDP

a) The GDP data that has been taken from various data sources needs to be spliced to the same base year prior to using it for further analysis: For e.g.:

YearGDP at new base (2004–2005)QuarterGDP at old base (e.g. 1970–1971)SumProportionSplicing factor (SF)New base (2004–2005) GDP
1970

X1

1975Q1

1975Q2

1975Q3

1975Q4		Y1

Y2

Y3

Y4		S1 = ΣYiZ1 = Y1/S1

Z2 = Y2/S1

Z3 = Y3/S1

Z4 = Y4/S1		SF = X1/S1Z1*SF

(where S is the annual GDP figure at 2004–2005 prices)

Steps: e.g. to splice 1980–1981 to 2004–2005 (no overlapping data exists)

  1. Take sum of each year in quarterly data.

  2. Find splicing factor by using data from (i) and the annual data (at base 2004–2005).

  3. Splice data.

A.2 WPI

1) Monthly data for WPI is available from the DBIE Handbook of Statistics on the Indian economy (with overlapping data given for the different base years).

a) Overlapping data is not available for the WPI at 1980–1981 and 1993–1994 base years.

  1. But, the Economic Affairs Ministry has released the linking factor to convert 1993–1994 base to 1980–1981 base. Thus, the inverse of the linking factor gives (equals) the splicing factor to convert 1993–1994 base to 1980–1981 base.

  2. Thus, the 1980–1981 base data can be converted to 1993–1994 base data using the splicing factor derived in i).

  3. Since overlapping data exists for 1993–1994 and 2003–2004, the splicing factor can be found for converting the former to the latter base.

This in turn can now be used to splice the “new spliced” data (from 1980–1981 to 1993–1994) from ii) to 2003–2004 base.

A.3 Income velocity of money, V

1) Estimation of quarterly income velocity:

The quarterly income velocity of money is calculated as per the RBI definition:

Income velocity of broad money = [GDP (at current market prices)/M3] where M3 is the average broad money supply.

  1. The GDP at current market prices that is used is calculated in the same manner as given above (Appendix A.1.1) for the GDP at constant prices. These figures are also given at a base year of 2004–2005 and hence have been spliced to the same year.

  2. These quarterly estimates are used along with the aggregated M3 figures (which are the simple averages of the monthly broad money M3, for each quarter).

2) Aggregation from monthly to quarterly:

Once all the data is spliced to the same base (2003–2004), it needs to be aggregated to quarterly data. Compute the simple arithmetic mean across quarters for each year.[29] The quarterly WPI dataset is thus, the requisite price data for the P* model.

B Technical appendix: Why BK filter vs. HP filter?

The Baxter-King (BK) filter is chosen to calculate the potential output and potential velocity of money. The advantage of using this filter over the Hodrick-Prescott (HP) filter is that it minimizes the possibility of identifying a spurious cyclical structure in the series. The BK filter filters low frequency and high frequency cycles which enables retaining periodic fluctuations (cycles) between the low and high frequencies (Woitek, 1998; Baum, 2006).

The BK filter can be implemented on raw data, xt, to isolate the component of xt with period of oscillation between pl and pu (where 2 ≤ pl ≤ pu ≤ ∞) as follows (Christiano & Fitzgerald, 1999):

y^t=B0xt+B1xt+1+…+BT−t−ixT−1+B~T−txT+B1xt−1+…+Bt−2x2+B~t−1x1,

where t = 3, 4, …, T−2 where y^ approximates yt (where yt is the data generated by applying the ideal band pass filter to xt); Bj=sin⁡(jb)−sin⁡(ja)πj,j≥1,B0=b−aπ,b=2πpu,a=2πpl and B~T−t,B~t−1 are simple linear functions of Bj s.

The HP filter computes the smoothed series st of the raw data xt by minimizing the variance of xt around st subject to a constraint on the second difference of st. The HP filter minimizes:

∑t=1T⁡(xt−st)2+λ∑t=2T−1⁡((st+1−st)−(st−st−1))2

where λ is the smoothing parameter.

C Model Evaluation

Optimal SARIMA Models across regimes (Model E)

RegimesRegime 1Regime 2Regime 3
CoefficientsConstant0.0733**Constant0.0835**Constant0.0428**
AR(1)0.5201**MA(1)−0.1604MA(1)0.5994**
MA(4)−0.2142**SMA(4)−0.1918SMA(4)−0.2080
Modelsy(t) = 0.07 + 0.5y(t–1) + e(t) – 0.2e(t–4)y(t) = 0.08 + (1 – 0.16L)(1 – 0.19L4)e(t)y(t) = 0.04 + (1 + 0.599L)(1 – 0.21L4)e(t)

References

Aparicio, F., A. Escribano, and A. García. 2003. “Range Unit Root (RUR) Tests.” Working Paper 03-11 Statistics and Econometrics Series 26,” Universidad Carlos III de Madrid.Search in Google Scholar

Aparicio, F., A. Escribano, and A. Siplos. 2006. “Range Unit Root (RUR) Tests: Robust Against Non-Linearities, Error Distributions, Structural Breaks and Outliers.” Journal of Time Series Analysis 27: 545–576.10.1111/j.1467-9892.2006.00474.xSearch in Google Scholar

Balke, N. S., and T. B. Fomby. 1997. “Threshold Cointegration.” International Economic Review 38: 627–645.10.2307/2527284Search in Google Scholar

Baum, C. 2006. “Time-Series Filtering Techniques in STATA.” North American STATA Users’ Group Meetings.Search in Google Scholar

Boskin, Michael J., E. Dulberger, R. Gordon, Z. Griliches, and D. Jorgenson. 1996. “Toward a More Accurate Measure of the Cost of Living.” Final Report to the Senate Finance Committee: Boskin Commission.Search in Google Scholar

Chong, Y. Y., and D. F. Hendry. 1986. “Econometric Evaluation of Linear Macro-Economic Models.” The Review of Economic Studies 53(4): 671–671. DOI: 10.2307/2297611.Search in Google Scholar

Christiano, L., and T. Fitzgerald. 1999. “The Band Pass Filter.” NBER Working Paper Series, No. 7257.10.26509/frbc-wp-199906Search in Google Scholar

Czudaj, R. 2011. “P-star in Times of Crisis – Forecasting Inflation for the Euro Area.” Economic Systems 35: 390–407.10.1016/j.ecosys.2010.09.006Search in Google Scholar

Das, R. K. 1993. “Quarterly Estimates of GDP: 1970–1971 to 1990–1991.” RBI Staff Studies, Department of Economic Analysis and Policy, Reserve Bank of India, Bombay.Search in Google Scholar

Dua, P., and U. Gaur. 2010. “Determination of Inflation in an Open-Economy Phillips Curve Framework: The Case of Developed and Developing Asian Countries.” Macroeconomics and Finance in Emerging Market Economies 3: 33–51.10.1080/17520840903498107Search in Google Scholar

Dufrénot, G., and V. Mignon. 2002. Recent Developments in Nonlinear Cointegration with Applications to Macroeconomics and Finance. Netherlands: Kluwer Academic Press, Dordrecht.10.1007/978-1-4757-3615-1Search in Google Scholar

Granger, C. W., and T. Teräsvirta. 1993. Modelling Nonlinear Economic Relationships. Oxford: Oxford University Press.10.1093/oso/9780198773191.001.0001Search in Google Scholar

Habibullah, M. S. 1998. “The Applicability of the P-Star approach of Modelling Inflation in a Developing Country: The Case of Malaysia.” Analisis 5: 33–45.Search in Google Scholar

Hallman, J., R. Porter, and D. Small. 1989. M2 per unit of Potential GNP as an Anchor for the Price Level, Staff Study No. 157 (April), Board of Governors of the Federal Reserve System, Washington, DC.Search in Google Scholar

Hallman, J., R. Porter, and D. Small. 1991. “Is the Price Level Tied to the M2 Monetary Aggregate in the Long Run?.” American Economic Review 81: 841–858.Search in Google Scholar

Hansen, B. E., and B. Seo. 2002. “Testing for Two-Regime Threshold Cointegration in Vector Error-Correction Models.” Journal of Econometrics 110: 293–318.10.1016/S0304-4076(02)00097-0Search in Google Scholar

Hoeller, P., and P. Poret. 1991. “Is P-Star a Good Indicator of Inflationary Pressure in OECD Countries?.” OECD Economic Studies No. 17.Search in Google Scholar

Keenan, D. M. 1985. “A Tukey Nonadditivity-Type Test for Time Series Nonlinearity.” Biometrika 72: 39–44.10.1093/biomet/72.1.39Search in Google Scholar

Khetan, C. P., J. Handa, and R. R. Waghmare. 1976. The Monetary Structure of the Indian Economy: A Quarterly Econometric Model. Macmillan Co. of India, Delhi.Search in Google Scholar

Kool, C., and J. Tatom. 1994. “The P-Star Model in Five Small Economies.” Federal Reserve Bank of St. Louis Review No. 3.10.20955/r.76.11-30Search in Google Scholar

Mujeri, M., M. Shahiduzzaman, and M. Islam. 2009 Measuring Inflationary Pressure in Bangladesh: The P-Star ApproachWorking Paper Series: WP 0901Policy Analysis Unit (PAU), Bangladesh Bank.Search in Google Scholar

Nachane, D., and R. Lakshmi. 2002. “Dynamics of inflation in India – a P-Star Approach.” Applied Economics 34: 101–110.10.1080/00036840010020385Search in Google Scholar

Nau, R. 2014. Seasonal ARIMA Models [Online], Available at http://people.duke.edu/-rnau/seasarim.htm, Accessed on August, 13, 2014.Search in Google Scholar

Pallardo, V., and V. Esteve. 1999. “The P-Star Model and its Performance for the Spanish Economy.” Departamento de Economía Aplicada II Universidad de Valencia, Working Paper 99/11.Search in Google Scholar

Raj, J., and S. Misra. 2011. “Measures of Core Inflation in India: An Empirical Evaluation.” RBI Working Paper Series, Department of Economic and Policy Research, Reserve Bank of India.Search in Google Scholar

Reserve Bank of India. 2013. Handbook of Statistics of the Indian Economy. Mumbai: Reserve Bank of India.Search in Google Scholar

Reserve Bank of India. 2014. “The Report of the Expert Committee to Revise and Strengthen the Monetary Policy Framework (Urjit Patel Committee Report).” Mumbai: Reserve Bank of India.Search in Google Scholar

Samuelson, P., and W. Nordhaus. 1998. Economics. 16th ed. New York: Tata McGraw-Hill.Search in Google Scholar

Seo, M. 2006. “Bootstrap Testing for the Null of no Cointegration in a Threshold Vector Error Correction Model.” Journal of Econometrics 127: 129–150.10.1016/j.jeconom.2005.06.018Search in Google Scholar

Singh, B. K. An Assessment of Inflation Modelling in India. ICRIER Working Paper No. 259 Indian Council for Research on International Economic Relations, 2012.Search in Google Scholar

Srinivasan, N., V. Mhambre, and M. Ramachandran. 2008. “Modelling Inflation in India: A Critique of the Structuralist Approach.” Working Paper Series, Indira Gandhi Institute of Development Research, Mumbai.10.1007/BF03546447Search in Google Scholar

Stigler, M. 2011. “Threshold Cointegration: Overview and Implementation.” Working Paper, Available at http://cran.r-project.org/web/packages/tsDyn/vignettes/ThCointOverview.pdf.Search in Google Scholar

Tatom, J. 1990. “The P-Star Approach to the Link Between Money and Prices.” Working Paper 1990-008A, Federal Reserve Bank of St. Louis, Research Division, http://research.stlouisfed.org/wp/1990/90-008.pdf.10.20955/wp.1990.008Search in Google Scholar

Teräsvirta, T. 1994. “Specification, Estimation, and Evaluation of Smooth Transition Autoregressive Models.” Journal of the American Statistical Association 89: 208–218.10.1080/01621459.1994.10476462Search in Google Scholar

Teräsvirta, T., D. Tjøstheim, and C. W. J. Granger. 1994. “Aspects of Modelling Nonlinear Time Series.” In Handbook of Econometrics, edited by R. F. Engle and D. McFadden, 4. Amsterdam: Elsevier.10.1016/S1573-4412(05)80017-0Search in Google Scholar

Tsay, R. S. 1986. “Nonlinearity Tests for Time Series.” Biometrika 73(2): 461–466. DOI: 10.1093/biomet/73.2.461.Search in Google Scholar

Tsionas, E. G. 2001. “P-STAR Analysis in a Converging Economy: The Case of Greece.” Economic Modelling 18: 49–60.10.1016/S0264-9993(00)00027-4Search in Google Scholar

Woitek, U. 1998. “A Note on the Baxter-King Filter.” Discussion Paper, University of Glasgow.Search in Google Scholar

Supplementary Material

The online version of this article offers supplementary material (DOI: https://doi.org/10.1515/snde-2017-0067).

Published Online: 2018-08-18

©2018 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Interview
  2. An Interview with Timo Teräsvirta
  3. Research Articles
  4. Nonlinear and asymmetric pricing behaviour in the Spanish gasoline market
  5. Testing for misspecification in the short-run component of GARCH-type models
  6. Closed-form estimators for finite-order ARCH models as simple and competitive alternatives to QMLE
  7. Time-varying asymmetry and tail thickness in long series of daily financial returns
  8. Modeling changes in US monetary policy with a time-varying nonlinear Taylor rule
  9. Financial fragmentation and the monetary transmission mechanism in the euro area: a smooth transition VAR approach
  10. P-star model for India: a nonlinear approach
  11. Can a Taylor rule better explain the Fed’s monetary policy through the 1920s and 1930s? A nonlinear cliometric analysis
  12. Modeling time-variation over the business cycle (1960–2017): an international perspective
https://doi.org/10.1515/snde-2017-0067
Keywords for this article
Inflation; price gap model; P-star models; threshold vector error correction; velocity gap and output gap models

Articles in the same Issue

  1. Interview
  2. An Interview with Timo Teräsvirta
  3. Research Articles
  4. Nonlinear and asymmetric pricing behaviour in the Spanish gasoline market
  5. Testing for misspecification in the short-run component of GARCH-type models
  6. Closed-form estimators for finite-order ARCH models as simple and competitive alternatives to QMLE
  7. Time-varying asymmetry and tail thickness in long series of daily financial returns
  8. Modeling changes in US monetary policy with a time-varying nonlinear Taylor rule
  9. Financial fragmentation and the monetary transmission mechanism in the euro area: a smooth transition VAR approach
  10. P-star model for India: a nonlinear approach
  11. Can a Taylor rule better explain the Fed’s monetary policy through the 1920s and 1930s? A nonlinear cliometric analysis
  12. Modeling time-variation over the business cycle (1960–2017): an international perspective
Sign up now to receive a 20% welcome discount
Institutional Access
How does access work?
Have an idea on how to improve our website?
Please write us.
© 2025 De Gruyter Brill
Downloaded on 7.11.2025 from https://www.degruyterbrill.com/document/doi/10.1515/snde-2017-0067/html
Scroll to top button