University of Dundee. ARCH and structural breaks in United States inflation Russell, William. Published in: Applied Economics Letters

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1 University o Dundee ARCH and structural breaks in United States inlation Russell, William Published in: Applied Economics Letters DOI: / Publication date: 2014 Document Version Peer reviewed version Link to publication in Discovery Research Portal Citation or published version (APA): Russell, B. (2014). ARCH and structural breaks in United States inlation. Applied Economics Letters, 21(14), DOI: / General rights Copyright and moral rights or the publications made accessible in Discovery Research Portal are retained by the authors and/or other copyright owners and it is a condition o accessing publications that users recognise and abide by the legal requirements associated with these rights. Users may download and print one copy o any publication rom Discovery Research Portal or the purpose o private study or research. You may not urther distribute the material or use it or any proit-making activity or commercial gain. You may reely distribute the URL identiying the publication in the public portal. Take down policy I you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim. Download date: 17. Feb. 2017

2 ARCH AND STRUCTURAL BREAKS IN UNITED STATES INFLATION Bill Russell* 3 December 2013 ABSTRACT United States Phillips curves are routinely estimated without accounting or the shits in mean inlation. As a result we may expect the standard estimates o Phillips curves to be biased and suer rom ARCH. We demonstrate this is indeed the case. We also demonstrate that once the shits in mean inlation are accounted or the ARCH is largely eliminated in the estimated model and the model deining expected rate o inlation in the New Keynesian model plays no signiicant role in the dynamics o inlation. Keywords: Philips curve, ARCH, structural breaks, inlation, markup. JEL Classiication: C22, E INTRODUCTION This paper examines the hypothesis that the observed ARCH in estimated models o United States inlation is the result o unaccounted shits in the mean rate o inlation in the data. This hypothesis is based on the ollowing understanding o the data. Russell (2006, 2011) and Russell and Chowdhury (2013) argue inlation in the developed world cannot be a truly integrated process as it appears to have a lower boundary around zero and an upper boundary at some moderate rate o inlation. They also argue that inlation cannot be a stationary process with a constant mean over the past ity years as this would imply only one long-run rate o inlation, one expected rate o inlation, and one short-run Phillips curve. Furthermore, this implies that Phillips (1958) original curve did not break down with changes in expected inlation towards the end o the 1960s. They conclude that while inlation is a non-stationary process it is most likely to be a stationary process around a shiting mean. The latter is due to * Economic Studies, School o Business, University o Dundee, Dundee DD1 4HN, United Kingdom (work phone), (ax), brussell@brolga.net. I wish to thank Genaro Sucarrat who counselled me on the estimation o ARCH models and Tom Doan or generously making available the Bai-Perron programmes on the Estima web site. All data are available at 1

3 discrete changes in monetary policy and allows or numerous long-run and associated expected rates o inlation that are central tenants o modern theories o the Phillips curve. 1 Graph 1 shows United States quarterly inlation or the period March 1960 to March 2013 measured as the quarterly change in the natural logarithm o the gross domestic product (GDP) implicit price delator at actor cost multiplied by 400 to provide an annualised rate o inlation. 2 The graph reveals two characteristics o United States inlation. First, the shiting mean rate o inlation discussed above is evident. These shits can be identiied ormally by applying the Bai and Perron (1998) technique to identiy multiple breaks in the mean rate o inlation. 3 Nine breaks in mean are identiied in the inlation data implying there are ten inlation regimes within which we believe statistically the mean rate o inlation is constant. The identiied mean rates o inlation in each regime are shown on Graph 1 as solid thin horizontal lines. The second characteristic is the variance in inlation increases during the turbulent high inlation years o the 1970s beore declining to lower levels ollowing the Volker delation in the early 1980s. This clustering o high and low variance into discrete periods suggests that the variance in inlation may be serially correlated. Since Engle s (1982) seminal Nobel Prize winning paper, a popular way to model this clustering in the variance in the inlation data is to estimate some orm o auto-regressive conditional heteroscedastic (ARCH) type models o inlation. Much o the ARCH literature on inlation has ocused on (i) the unpredictability o inlation when the variance increases; and (ii) the relationship between the mean and the variance o inlation. 4 Interest in the ormer is due to the welare costs associated with agents holding mistaken inlation expectations and the later due to Friedman s (1977) conjecture in his Nobel Prize lecture that the mean rate o inlation and the variance o inlation are positively related. Importantly this literature interprets the ARCH as due to some real economic interaction between central banks, irms and agents. An alternative explanation o the ARCH in inlation is that it is simply a statistical arteact due to the miss-speciication o the estimated model. In particular, our visual impression rom Graph 1 that the variance o inlation is not constant and our ormal tests o heteroscedasticity proceed under the assumption that inlation is a stationary process with a constant mean. However, as argued above, inlation may contain a shiting mean and i these shits are accounted or then the conditional variance may be constant and the data homoscedastic. We 1 The term non-stationary in this paper encompasses all statistical processes other than stationary with a constant mean. It thereore includes stationary around a shiting mean. Modern theories include the Friedman-Phelps expectations augmented, New Keynesian and hybrid theories o the Phillips curve. 2 The Data Appendix provides details o the data used in this paper. 3 Details o how the breaks are estimated are provided in Section 3.1 and Table 1 below. 4 For example, see Engle (1983, 1988), Cosimano and Jansen (1988), Baillie, Chung and Tieslau (1996), Grier and Perry (2000) and Boero, Smith and Wallis (2008). 2

4 might then conclude that the observed heteroscedastic nature o inlation has no economic or behavioural relevance. In the next section we set out and then estimate in Section 3 a general hybrid Phillips curve. In support o our hypothesis we ind that once we account or the shits in mean inlation there is little statistical evidence o ARCH in our estimated models o inlation. We also ind that expected inlation as commonly measured in the standard New Keynesian literature plays no signiicant role in the dynamics o inlation ater allowing or the shits in mean inlation. 2. THE HYBRID PHILLIPS CURVE To examine our hypothesis we estimate a hybrid Phillips curve o the orm: 5 e π t = δ + δ π t+1 + δ b π t 1 + δ μ μ t + 10 m=1 φ m D m + j=2 ω j I j + ε t (1) e where inlation, t, depends on expected inlation, π t+1, lagged inlation, t 1 4, a orcing variable, t, shit dummies representing the 10 inlation regimes, D m, identiied earlier by the Bai-Perron technique, and an error term, ε t. The dates o the shit dummies are those estimated in the inlation data using the Bai-Perron technique and reported in Graph 1. With the shit dummies included we are estimating 10 short-run Phillips curves associated with the 10 inlation regimes where we believe statistically that the mean (or long-run) rate o inlation is constant. The shits in mean are rare events and diicult or agents and economists to estimate. Following Russell and Chowdhury (2013) the shits in mean inlation are assumed to be independent o the mean reversion process in the inlation data. On a practical level this allows us to estimate the dates o the shits in mean independently o estimating equation (1). This independence assumption is commonly applied when estimating Gaussian mean reversion processes with structural breaks and irst applied to shits in mean inlation by Russell and Chowdhury. They deend the independence assumption with the ollowing thought experiment. Assume an inlation regime o n periods is stationary with a constant mean. Now assume that j periods beore the end o the regime agents can predict the next shit in mean. This implies that inlation will begin to adjust towards the new mean which implies the mean rate o inlation in the last j periods o the regime is dierent rom the irst n-j periods. This contradicts the initial assumption that the inlation regime o n periods has a constant mean. 6 Thereore, we 5 For the sake o space the our Phillips curve theories nested in the hybrid Phillips curve are not reproduced here but can be ound in the reerences cited below. 6 In the transition between two long inlation regimes there may be a number o short inlation regimes where the mean is constant but these regimes cannot be identiied with current statistical techniques. However, the 3

5 can conclude that i the data is stationary around a shiting mean as argued above then (i) agents cannot predict the next shit in mean, (ii) the process driving the breaks and the mean reversion process in the data are independent and uncorrelated, and (iii) estimating the dates o the structural breaks in mean independently o estimating equation (1) is valid. Note that the meanshit-mean-reversion independence assumption conorms to our general understand that predicting, or orecasting, structural breaks in the mean rate o inlation is not possible, or at the very least, very diicult. The three modern theories o the Phillips curve are nested within the hybrid Phillips curve. In the standard unrestricted hybrid models o Galí and Gertler (1999) and Galí, Gertler and Lopez-Salido (2001) agents are both backward and orward looking and 1 d where 1 d is the discount rate. I instead we restrict 0 and 1 then the hybrid curve collapses to the Friedman (1968) and Phelps (1967) (F-P) expectations augmented Phillips curve where agents are purely backward looking. Conversely, i we restrict 1 d and 0 then the hybrid curve becomes the New Keynesian (NK) Phillips Curve o Clarida, b Galí and Gertler (1999) and Svensson (2000) where agents are purely orward-looking. Finally, the statistical process consistent (SPC) Phillips curve o Russell and Chowdhury (2013) where 0 and 0 δ b < 1 provides a ourth model o the Phillips curve. b b I inlation is stationary around a shiting mean then estimating equation (1) without incorporating the regime shit dummies we should expect; (i) the estimates o both δ and δ b will be biased upwards; 7 and (ii) an over-rejection o the null hypothesis o conditional homoscedasticity. 8 Thereore we might expect that i the shit dummies in equation (1) are signiicant then the bias on the estimates o the dynamic inlation terms will be reduced along with any evidence o ARCH in the estimated model. 3. ESTIMATING UNITED STATES SHORT-RUN PHILLIPS CURVES 3.1 The Data The model is estimated with quarterly seasonally adjusted United States data or the period March 1960 to March Inlation is measured as the quarterly change in the natural logarithm o the gross domestic product (GDP) implicit price delator at actor cost. In keeping logic o the thought experiment remains that within each short inlation regime agents cannot predict the next shit in mean. 7 This is a generalisation o Perron (1989). Russell et al. (2011) provides an extensive analysis o the Perron eect due to not accounting or the shits in mean when estimating United States Phillips curves and ind the bias is substantial. 8 See Lumsdaine and Ng (1999). 4

6 with much o the recent NK literatures the orcing variable is the markup and measured as the natural logarithm o the price series divided by unit labour costs which is equivalent to the inverse o labour s share o national income measured at actor cost. The inlation regimes are identiied with the Bai and Perron (1998) algorithm that minimises the sum o the squared residuals in an estimated shiting means model: k+1 π t = i=1 γ i + τ t (2) where γ i is a series o k 1 constants that estimate the mean rate o inlation in each o n = k + 1 inlation regimes and τ t is a random error. The inal model was chosen using the Bayesian Inormation Criterion. The estimated dates and mean rates o inlation or the ten identiied inlation regimes are reported in Table 1. Also reported in Table 1 is some historical economic context associated with each inlation regime. While the Bai-Perron technique is a purely statistical process the identiied inlation regimes are broadly consistent with economic history in the United States. e Expected inlation, π t+1, is commonly measured in the standard NK literature as its orecasted value based on inormation publicly available at time t and published in t 1. 9 The orecasted value o inlation is obtained by regressing inlation on lags o inlation and the markup or periods t 2 to t 5. The regression is estimated using ordinary least squares (OLS) and the static orecast o inlation is included in the estimation o the model with a lead o one period. To overcome the possible simultaneous nature o inlation and the markup the contemporaneous markup in equation (1) is also replaced with its orecast value based on the data published in period t 1. The static orecast, μ t, is obtained rom regressing the markup on lags o itsel and inlation or periods t 1 to t 4. Finally, the models based on equation (1) are estimated with OLS with White heteroscedastic consistent standard errors and covariances The Estimated Models Model 1 reported in Table 2 retrieves the standard results o the hybrid Phillips curve literature by estimating equation (1) assuming the inlation data is stationary with a constant mean. This is achieved by restricting the shit dummies to be the same across all the inlation regimes. We see the sum o the estimated coeicients on the dynamic inlation terms δ + δ b is insigniicantly dierent rom 1 with the estimated value or δ considerably larger than δ b. The residuals o the estimated model are non-normal, serially correlated and heteroscedastic. Introducing our impulse dummies in Model 2 that correspond to the largest our residuals in the estimated model reduces but does not eliminate evidence o ARCH in the standard model. The dates o the impulse dummies are March 1972, September and December 1974 and 9 e This is one conceptual interpretation o using instrumental variables or π t This is equivalent to estimating equation (1) with a two-stage-least-squares estimator. 5

7 December 1977 which coincide with the turbulent period during and ater the irst OPEC oil price shock and represent the idiosyncratic nature o the inlation process at that time. These results are consistent with the standard literatures on the hybrid Phillips curve and ARCH inlation. Model 3 in the same table reports the estimates o equation (1) with the unrestricted shit dummies and the impulse dummies. We now see that the sum o the dynamic coeicients are signiicantly less than 1, the residuals are normally distributed, serially uncorrelated and conditionally homoscedastic. Furthermore, the estimated coeicient on expected inlation is now insigniicantly dierent rom zero. Finally Model 4 in Table 2 reports the estimated Phillips curve with the insigniicant expected inlation term restricted to zero. The estimated value or δ b remains signiicantly less than one by a wide margin and signiicantly greater than zero. The residuals remain normally distributed and serially uncorrelated and there is no signiicant evidence that the residuals are heteroscedastic. With the shits in mean accounted or in the estimation o the Phillips curve we ind that (i) there is little evidence o ARCH in the estimated models; and (ii) the data is inconsistent with the hybrid, New Keynesian and Friedman-Phelps theories o the Phillips curve and consistent with the SPC Phillips curve. 4. CONCLUSION The estimates above suggest that the apparent ARCH process in United States inlation data can be explained as a statistical arteact due to unaccounted breaks in the mean rate o inlation implying the observed ARCH may have no behavioural signiicance. Furthermore, the estimates indicate that i we account or the shits in mean inlation then the estimates suggest that there is no signiicant role or the model deining New Keynesian expected rate o inlation in the dynamics o inlation. The standard interpretation o the F-P Phillips curve is also not supported by the data. In contrast, once the shits in mean inlation are accounted or, the statistical process consistent Phillips curve is supported by the data. 6

8 5. REFERENCES Bai, J. and P. Perron (1998). Estimating and testing linear models with multiple structural changes. Econometrica 66: Baillie, R.T., Chung, C.F. and Tieslau, M.A. (1996). Analysing inlation by the ractionally integrated ARFIMA-GARCH model, Journal o Applied Econometrics, 11, Boero, G., J. Smith and K.F. Wallis (2008). Modelling UK Inlation Uncertainty, , Robert F. Engle Festschrit Conerence, San Diego, 21 June Clarida, R., Galí, J. and M. Gertler (1999). The Science o Monetary Policy: a New Keynesian Perspective, Journal o Economic Literature, vol. 37, pp Cosimano, T.F., and D.W. Jansen (1988). Estimates o the variance o U.S. Inlation Based upon the ARCH Model: Comment, Journal o Money, Credit and Banking, vol. 20, no. 3, August, pp Engle, R.F. (1982). Autoregressive conditional heteroscedasticity with estimates o the variance o United Kingdom inlation, Econometrica, 50, Engle, R.F. (1983). Estimates o the Variance o U.S. inlation Based on the ARCH Model, Journal o Money, Credit and Banking, vol. 15, no. 3, August, pp Engle, R.F. (1988). Reply to Cosimano and Jansen, Journal o Money, Credit and Banking, vol. 20, no. 3, August, pp Friedman, M. (1968). The role o monetary policy, American Economic Review, 58, 1 (March), pp Friedman, M. (1977). Nobel Lecture: Inlation and Unemployment, The Journal o Political Economy, vol. 85, no. 3, pp Galí, J., and M. Gertler (1999). Inlation Dynamics: A Structural Econometric Analysis, Journal o Monetary Economics, vol. 44, pp Galí, J., Gertler M., and J.D. Lopez-Salido (2001). European Inlation Dynamics, European Economic Review, vol. 45, pp Grier, K.B. and Perry, M.J. (2000). The eects o real and nominal uncertainty on inlation and output growth: some GARCH-M evidence, Journal o Applied Econometrics, 15, Lumsdaine, R.L. and S. Ng (1999). Testing or ARCH in the presence o a possibly misspeciied conditional mean, Journal o Econometrics, vol. 93, pp

9 Perron, P., (1989). The Great Crash, the Oil Price Shock, and the Unit Root Hypothesis, Econometrica, vol. 57, no. 6, November, pp Phelps, E.S. (1967). Phillips curves, expectations o inlation, and optimal unemployment over time, Economica, 34, 3 (August), pp Phillips, A. W. (1958). The Relation Between Unemployment and the Rate o Change o Money Wage Rates in the United Kingdom, , Economica, 25, pp Russell, B. (2006). Non-Stationary Inlation and the Markup: an Overview o the Research and some Implications or Policy, Dundee Discussion Papers, Department o Economic Studies, University o Dundee, August, No Russell, B. (2011). Non-stationary Inlation and Panel Estimates o United States Short and Long-run Phillips Curves, Journal o Macroeconomics, vol. 33, pp Russell, B., A. Banerjee, I. Malki and N. Ponomareva (2011). Dundee Discussion Papers, Economic Studies, University o Dundee, April, No Russell, B. and R.A. Chowdhury (2013). Estimating United States Phillips Curves with Expectations Consistent with the Statistical Process o Inlation, Journal o Macroeconomics, vol. 35, pp Svennson, L.E.O. (2000). Open Economy Inlation Targeting, Journal o International Economics, vol. 50, pp

10 APPENDIX 1 DATA APPENDIX The United States data are seasonally adjusted and quarterly or the period March 1960 to March The United States national accounts data are rom the National Income and Product Account tables rom the United States o America, Bureau o Economic Analysis. The aggregate data were downloaded via the internet on 31 May The data are available at Variable Details Inlation: Nominal gross domestic product (GDP) at actor cost is nominal GDP (Table 1.1.5, line 1) plus subsidies (NIPA Table 1.10, line 10) less taxes (NIPA Table 1.10, line 11). The price series is the GDP implicit price delator at actor cost calculated as nominal GDP at actor cost divided by constant price GDP at 2005 prices (NIPA Table 1.1.6, line 1). Inlation is the irst dierence o the natural logarithm o the price series. Note that Graph 1 shows the estimated inlation regimes multiplied by 400 to provide an annualised rate o inlation. The Markup: Calculated as the natural logarithm o nominal GDP at actor cost divided by compensation o employees paid (NIPA Table 1.10, line 2). Table 1: Estimated Inlation Regimes using the Bai-Perron Technique Regime Dates o the Inlation Regimes Mean Historical Context 1 March 1960 to September Low inlation ollowing recession. 2 December 1964 to September December 1967 to December Loose monetary & iscal policy associated with unding the Vietnam War. World commodity price boom and breakdown o the Bretton-Woods exchange rate system. 4 March 1973 to March First OPEC oil price shock. 5 June 1978 to September Second OPEC oil price shock. 6 December 1981 to December Volker delation. 7 March 1985 to June Depreciation o the $US ollowing the Plaza Accord. 8 September 1991 to September Decline to low inlation ollowing recession. 9 December 2003 to September Failure o the Federal Reserve to unwind the loose monetary policy ollowing the 9/11 terrorist attach and the end o the technology share price bubble. 10 December 2007 to March Financial/debt crisis recession. The minimum regime size is assumed to be 12 quarters (three years). The model was estimated using baiperron.src and multiplebreaks.src programmes written by Tom Doan in RATS 7.2. Note that in Graph 1 the mean rate o inlation in each regime is multiplied by 400 and shown as solid thin horizontal lines. 9

11 Table 2: United States Phillips Curves - Two Stage Least Squares Hybrid Phillips Curve SPC Phillips Curve No Spike Dummies All Dummies Only Dummies All Dummies π t+1 (6.5) (7.4) (1.6) π t (2.9) (3.2) (2.5) (3.0) μ t (-0.8) (0.0) (-1.9) (-2.8) Constant (0.8) (0.0) (2.0) (3.0) Dummies (see Table 2b) No Yes Yes Yes Coeicient Restriction Tests b [0.0000] [0.0000] [0.0039] [0.0035] b 1 [0.5125] [0.3010] [0.0001] [0.0000] b φ 2.. φ 10 = 0 [0.0246] [0.0000] Inormation Criteria 2 R Akaike Schwarz Residual Diagnostics LM(1) LM(1 to 4) [0.0069] [0.0008] [0.4141] [0.1690] [0.7242] [0.8201] [0.5385] [0.8802] DW J-B [0.0000] [0.0167] [0.2981] [0.2567] ARCH [0.0000] [0.7024] [0.3693] [0.3797] White [0.0000] [0.0021] [0.0751] [0.0793] B-P-G [0.0000] [0.0021] [0.0718] [0.0737] 10

12 Table 2b: United States Phillips Curves Estimated Shit and Impulse Dummies Hybrid Phillips Curve SPC Phillips Curve Spike Dummies Only All Dummies All Dummies D (2.4) (3.5) D (1.8) (3.1) D (2.8) (5.4) D (4.0) (7.6) D (2.6) (3.9) D (2.7) (4.0) D (1.2) (1.5) D (2.9) (4.9) D (1.6) (2.1) I Mar (17.1) (12.4) (12.1) I Sept (14.4) (10.7) (10.9) I Dec (10.7) (9.3) (9.3) I Dec (17.0) (11.2) (11.2) Notes to Tables 2 and 2b The dummy coeicient estimates reported in the table correspond to the same numbered models in Table 2. The models are estimated using quarterly seasonally adjusted data or the period March 1960 to March 2013 using 208 observations. Reported as ( ), ( ) and [ ] are t-statistics, -statistics and probability values respectively. See Appendix 1 or details concerning the data and Section 3.1 or the estimation o the inlation regimes. Models are estimated with ordinary least squares with the lead in inlation and the markup replaced by their static orecast values (see Section 3.1) which is equivalent to estimating the model with two-stage-least-squares. Models are estimated with White heteroscedastic standard errors and covariances. Coeicient restrictions are Wald F tests. LM(1) and LM(1 to 4) reports the Breusch-Godrey LM test o serially correlated residuals or one lag and one to our lags respectively. J-P reports the test that the skewness and kurtosis o the residuals are rom a normal distribution. The ARCH test is the Engle Lagrange multiplier test or ARCH in the residuals. White tests the null hypothesis o no heteroskedasticity against heteroskedasticity o unknown general orm in the residuals. B-P-G is the Breusch-Pagan-Godrey test which is a Lagrange multiplier test o the null hypothesis o no heteroskedasticity against heteroskedasticity o the orm σ t 2 = σ 2 h(z t α), where z t is a vector o the independent variables rom the mean equation. The null hypotheses o the LM, J-P and heteroskedasticity tests are no serial correlation, normally distributed residuals and no heteroskedasticity in the residuals respectively. Models estimated with Stata/SE 8.2, Eviews 7.1 and RATS

13 Annualised Quarterly Inlation Graph 1: United States Annualised Quarterly Inlation, Seasonally Adjusted, March 1960 March Mar-60 Mar-65 Mar-70 Mar-75 Mar-80 Mar-85 Mar-90 Mar-95 Mar-00 Mar-05 Mar-10 Notes: Horizontal dashed lines indicate the ten inlation regimes identiied by the Bai-Perron technique. Annualised quarterly inlation is measured as the change in the natural logarithm o the price index multiplied by