Inflation Forecasts: An Empirical Re-examination. Swarna B. Dutt University of West Georgia. Dipak Ghosh Emporia State University

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1 Southwest Business and Econoics Journal/ Inflation Forecasts: An Epirical Re-exaination Swarna B. Dutt University of West Georgia Dipak Ghosh Eporia State University Abstract Inflation forecasts are an iportant part of odern day activist stabilization policy, and hence their accuracy and credibility is of vital iportance. We exaine the accuracy of these forecasts by testing for rationality in the expectation foration process using the Survey of Professional Forecasters data surveys of annual inflation forecasts. he non-stationarity of the actual and forecasted series allows for the application of cointegration techniques. We sequentially apply the Johansen Maxiu Likelihood and the (recently available) Bieren Non-Paraetric Cointegration tests. hese techniques are copleentary, and thus help strengthen our results. We find evidence indicating the presence of a cointegrating vector between the actual and forecasted series, which is evidence of the presence of rationality in the expectations foration process. I. Introduction Inflation forecasts are an iportant part of acroeconoic odels, especially in the context of proactive onetary and fiscal policy. In the corporate world, inflation forecasts are used with the prevailing rate of interest to deterine the expected real rate of interest, and these forecasts have a significant ipact on firs future investent on factories, equipent, and inventories. Inflation expectations are also used in household expenditure calculations of consuers, especially for the purchase of consuer durables. Governent uses inflation forecasts in calculating the long ter solvency of the Social Security syste. Research efforts have been aied at studying the efficiency, unbiasedness and rationality of forecasts such as the Livingston data, the Decision Makers Poll, Michigan Household Survey, Money Market Survey, etc. hese forecasts are of questionable relevance, and their usefulness in acro odels is doubtful in the absence of these properties. hus, a necessary condition for using inflation forecasts in the forulation of policy is the credibility of these forecasts. Results are far fro conclusive. Soe surveys are conducted qualitatively, and then subsequently converted to quantitative data, aking their use questionable. Other studies, such as Evans and Gulaani (1984), Batchelor (1986) and Kanoh and Li (1990), have exained the credibility of qualitative versus quantitative data and the errors in the conversion fro one for to the other. A second line of research exaines short horizon and long horizon forecasts and the statistical characteristics of the forecasts as the tie frae changes. Studies along these lines are Stenius (1986), hopson and Ottosen (1993) and Gagnon (1996). But research in this area has revolved around easuring the accuracy and credibility of these forecasts through an epirical exaination of econoic hypotheses such as unbiasedness, efficiency and rationality. Such studies in this field are 9

2 Inflation Forecasts: An Epirical Re-exaination Batchelor and Dua (1987), Hvidding (1987), Engsted (1991) and Marini and Scaraozzino (1993). hoas and Grant (2000) copares the relative accuracy of three standard inflation forecasting techniques, naely surveys (Livingston and Michigan Surveys), tie series ethods (ARIMA processes), and structural econoetric odels (as developed by the Federal Reserve Bank of San Francisco). hey use saples fro each forecast and copare these with the actual inflation over the 1980s and 1990s. Findings indicate that the surveys are unbiased over the two decades of forecasts, and clearly superior to backward looking ARIMA, as well as structural odels like the Fisher equation. Expert forulation of survey forecasts includes all relevant inforation in the arket regarding onetary and fiscal policy, and therefore indirectly iplies the incorporation of bandwagon effects by the forecasters in their predictions. A bandwagon effect would result when arket participants use the forecasts of other participants in forulating their forecasts, instead of doing so independently (i.e., participants jup on to the bandwagon.) his study exaines the rationality of inflation forecasts using the Survey of Professional Forecasters data set, and applies a new econoetric procedure. he paper is divided into 5 sections. Section 2, discusses the odel and the data set used, followed by the non-stationarity tests in section 3. Section 4 sequentially applies the Johansen Multivariate Cointegration test followed by the recently available Bierens Non- Paraetric Cointegration technique. hese are copleentary procedures, and hence enhance the credibility of our results. Section 5 contains soe concluding rearks. II. Model and Data A standard representation of a rational expectations odel is: S t+1 = t S e t+1 + t (1) where S t+1 is the actual inflation rate one period ahead and t S e t+1 is the expected inflation for the next period forecasted (in survey for) in the current period with inforation set I t. hus, the odel regresses actual inflation on its forecasted value. aking expectations of both sides of the equation yields: E(S t+1 ) = t S e t+1 + E( t ) (2) he error ter t has an assued ean of 0, and a test of the Rational Expectations Hypothesis (REH) involves first estiating equation (1), and then testing the error ter for stationarity. A stationary error structure would iply that the actual and the forecasted values are cointegrated (ove together over tie). Cointegration of actual and forecasted values is a necessary condition for the REH since if future inflation and expected inflation rates do not ove together over tie, we cannot find any relation (including REH) between the. his paper uses the Survey of Professional Forecasters data set, which is currently aintained by the Philadelphia Federal Reserve Bank (forerly known as the ASA- NBER data set). his survey is an iportant data source (for policy akers, both public and private) covering a large nuber of acroeconoic variables over an extended period of tie. he survey also includes professional forecasters fro business, finance, governent and acadeia, who state their forecasts over ultiple tie horizons. Data is quantitative, and therefore avoids the controversy associated with any conversion fro qualitative data. he odel uses quarterly data for actual inflation one quarter ahead (annualized, S t+1 ) and one quarter ahead (annualized, S e t+1) forecasts of expected inflation fro 1981 (3 rd quarter) to 2003 (4 th quarter). (1) Suitability of the data set for the tests of rationality (since the data is the direct forecast of arket participants) along with the 10

3 Southwest Business and Econoics Journal/ copleentary cointegration tests used in this paper (which ake the results independent of any particular econoetric procedure, and therefore ore general) ake our results about arket rationality particularly relevant to the literature in the field. A graph of the data is given in Figure 1. Figure 1 A c tu a l a n d F o r e c a s te d In fla tio n Quarterly Data for Actual and Forecasted Inflation INFLA FORE he graph indicates soe differences between the two series. Specifically, fro approxiately , , , the two series see to ove in opposite directions, and fro even though both series see to ove together, a large difference in their values is present. An econoetric test of cointegration (long run cooveent) is necessary to gather evidence for or against this phenoenon, and the results showing evidence for or against the presence of cointegration between actual and forecasted inflation will have iportant iplications for the use of survey data for research. III. ests of Stationarity Standard tests for stationarity such as the augented Dickey-Fuller (henceforth ADF, 1979, 1981) and the Phillips-Perron (henceforth PP,1988), are based on the conventional null of no cointegration hypothesis, which is unable to distinguish between unit root and near unit root stationary processes. In addition to these tests, this paper also uses the Kwiatkowski et. al. (henceforth KPSS, 1992) test, where the null is of stationarity and the alternative is the presence of a unit root. he KPSS procedure ensures that the null hypothesis will be rejected only when the results indicate there is strong evidence against it. 2 11

4 Inflation Forecasts: An Epirical Re-exaination able 1: Unit Root ests ADF est (Null Hypothesis H 0 : Unit root without a drift or tie trend) t-test Statistic Critical Value Actual Inflation Forecasted Inflation Phillips-Perron est (Z t test statistic) (Null Hypothesis H 0 : Unit root without a drift or tie trend) est Statistic Critical Value Actual Inflation Forecasted Inflation KPSS est Actual Inflation Forecasted Inflation Critical Value at 5 % level for the KPSS test: and Notes: All three tests were run using RAS 6.02b. he critical values are provided by the software. ADF test indicates that there is a unit root in the actual inflation series, but not in the forecasted series. he Phillips-Perron series indicates that there is no unit root in either series. Due to the drawbacks of these two tests, as described in section 3, we use the results of the KPSS test. he KPSS test and are test statistics that correspond to the null of stationarity with and without a tie trend respectively. Since the estiated test statistics for with tie trend) is not significant, and without tie trend) is significant (greater than the critical value), as explained in note 3, the appropriate tie series odel for our data is without a drift or a trend, is the appropriate statistic and it indicates that both the current inflation and the one year ahead forecast series have a unit root. Results for the ADF, PP and KPSS tests are given in able 1. he ADF test cannot reject the null of a unit root for the actual inflation series; whereas, it does reject the null for the forecasted inflation series. he PP test rejects the null of a unit root for both the actual and forecasted series. he KPSS test statistics and are the null of stationarity with and without a tie trend respectively. he test statistic for the null hypothesis of stationarity (when the odel includes a tie trend) is insignificant, whereas, the test statistic without the tie trend is significant. As explained in endnote 3, evidence suggests that the data can be accurately described by a odel that does not include a tie trend. herefore, the statistic is appropriate, and indicates rejection of the null hypothesis of stationarity in favor of the existence of a unit root. Based on all three tests, the reasonable observer ay therefore conclude that both the actual and forecasted inflation series have a unit root. 3 V. Syste Cointegration ests Once the non-stationarity of the relevant series is confired, the study runs two systes cointegration tests, naely the Johansen-Juselius (henceforth JJ, 1990) ultivariate test, followed by the Bierens non-paraetric cointegration (NPC) test. hese two tests are copleentary procedures and help strengthen our results by aking the independent of any specific cointegration test. 12

5 Southwest Business and Econoics Journal/ In applying the JJ test, first the appropriate lag length was selected using the likelihood-ratio test (Greene, 1993). In case of the trace test (), Johansen and Juselius recoend starting with the null hypothesis of the nuber of cointegrating vectors r = 0 and then oving upwards. he idea is that accepting r n iplies that one should also accept the r n+i, where I =1,2,3..., one stops the first tie he/she is unable to reject the null. According to JJ (1990) for conclusive evidence regarding the exact nuber of cointegrating vectors in the syste, one has to check the axiu eigenvalue test (ME) results. Here the null hypothesis is specific about the nuber of cointegrating vectors in the syste as r = I, where I = 0, 1, 2,... for the coplete odel and therefore iplies that we are testing a null hypothesis of r =0 against an alternate hypothesis that r =1, r =1 against r =2, etc. able 2: Johansen-Juselius Syste est NH Critical Value NH ME Critical Value r = r = r r = Notes: NH: Null Hypothesis, : race est, ME: Maxiu Eigenvalue est. Since none of the estiated test statistics for either the trace test or the axiu eigenvalue test exceeds the critical values, we are unable to reject the null hypothesis that there are no cointegrating vectors. Critical values are taken fro the Johansen-Juselius (1990) paper. Both the trace () and the axiu eigenvalue test (ME) yield significant statistics at the 5 percent level in able 2. But the null hypothesis of r 1 (trace test) and r=1(axiu eigenvalue test) would not be rejected at significance level less than 5% (for exaple at 1%), indicating that one cointegrating vector exists in the syste. hus, the actual and forecasted series ay be cointegrated, that is, they ove together over the long run, and this result can be confired by estiating the Bierens nonparaetric test. Next, the application of the Bierens nonparaetric cointegration (henceforth NPC, 1997) test exaines for cointegration between the relevant series. 4 he Bierens procedure is not only a consistent estiator of the nuber of cointegrating vectors in the syste under consideration, it is also a nonparaetric test, and hence does not need any specification of the data generating process, aking it highly flexible in application and avoiding the estiation of structural (and soeties nuisance) paraeters. Bierens (1997) states that our approach is capable of giving the sae answers regarding the nuber of cointegrating vectors and the cointegrating vectors theselves as Johansen s ML ethod, with less effort. He also includes soe Monte Carlo results which indicate that his test perfors a little better than Johansen s Labda-ax test. He goes on to conclude that. Our approach cannot copletely replace Johansen s approach, because the latter provides additional inforation, in particular regarding possible cointegrating restrictions on the drift paraeters, and the presence of linear trends in the cointegrating relations hus, rather than being substitutes, the two approaches are copleents. Consequently, this study looks at the results of the Johansen and the Bierens procedures together, and not independent of each other. A brief description of the Bierens procedure is given in Appendix A. 13

6 Inflation Forecasts: An Epirical Re-exaination able 3A: Bierens Non-paraetric Cointegration est Labda-in 5 percent critical region Result (0, 0.017) Reject the H 0 We are unable to reject the null hypothesis of zero cointegrating vectors as the calculated value of labda-in does not lie in the critical region. he critical region is calculated by the Easyreg international software, which was used to estiate the Bierens procedure. Here the paper tests the null hypothesis (H 0 ): r = 0, iplying there are no cointegrating vectors in the syste against the alternative hypothesis (H 1 ): r = 1, iplying one cointegrating vector. he null hypothesis will be rejected if the calculated value of the labda-in statistic lies inside the critical region (is less than the critical value). he null hypothesis is rejected in favor of the alternate hypothesis of cointegration, iplying that there is one cointegrating vector in the syste. he presence of one cointegrating vector is confired by estiating the function gˆ ( r) (as outlined in section 4.4 of Bierens (1997), which will converge in probability to infinity if the true nuber of cointegrating vectors is not equal to r, and will converge in probability to zero if the true nuber of cointegrating vectors is equal to r. Results are given in able 3B. able 3B R ˆ ( r) g Since r=1 has the lowest value of gˆ ( r) this is further proof that there is one cointegrating vector. Since r = 1 has the lowest value of ˆ ( r) (and this is uch lower than the value g of gˆ ( r) for the other two values of r), this just confirs the conclusion that the nuber of cointegrating vectors is one. hus, the one-quarter ahead (annualized) inflation rate series and its forecasts are cointegrated. Since (as discussed above) Bierens (1997) suggests that the JJ(1990) test and his test are copleentary procedures, the JJ results given above and the results fro the Bierens tests together indicate that inflation forecasts and actual inflation are cointegrated. VI. Conclusion his study exaines the experts inflation expectations foration process for unbiasedness over a one quarter ahead (annualized) forecast horizon. he epirical results support the presence of cointegration between the actual and the forecasted inflation series, thus indicating rational foration. Over a tie horizon of one year (and longer) it is ore than likely that fundaental deterinants of inflation will be relevant to forecasting, rather than just historical data. herefore, the arket is possibly unable to accurately forecast inflation over a long tie horizon because of a lack of consensus on the fundaental deterinants of inflation and how certain econoic policies will have an ipact on inflation. Englander and Stone (1989) get a siilar result, as they conclude that inflation expectations, in spite of having a significant forward looking coponent, are not 14

7 Southwest Business and Econoics Journal/ rational in the theoretical sense. Engsted (1991), using cointegration (error correction) techniques, also finds evidence against rational expectations in that sense. Dutt and Ghosh (1999) exained unbiasedness of the CPI expectations process at ultiple tie horizons (fro one to four quarters ahead forecasts). hey report the actual and forecasted series to be cointegrated at the short (one quarter horizon) and not cointegrated as the forecast horizon grows longer and longer, which is consistent with the results reported in this paper. hus cointegration between actual and forecasted inflation supports the bandwagon effect hypothesis in the expectations foration process, at least over the short run (as exained here) i.e., the actual and forecasted series ove together in the statistical sense. In the long run, econoic fundaentals becoe ore iportant in forecasting inflation than just the historical values of inflation, leading to a rejection of cointegration between actual and expected inflation, since fundaental deterinants of inflation are not included in cointegration studies. References Batchelor, R. A. (1986). Qualitative versus quantitative easures of inflation expectations. Oxford Bulletin of Econoics and Statistics, 48, Batchelor, R. A., & P. Dua (1987). he accuracy and rationality of UK inflation expectations: Soe quantitative evidence. Applied Econoics, 19, Bierens, H. J. (1997). Nonparaetric cointegration analysis. Journal of Econoetrics, 77, Caner, M., & Kilian, L. (2001). Size distortions of tests of the null hypothesis of stationarity: Evidence and iplications for the PPP debate. Journal of International Money and Finance, 20(5), Dickey, D. A., & Fuller, W. A. (1979). Distribution of the estiators for autoregressive tie series with a unit root. Journal of Aerican Statistical Association, 74, Dickey, D. A., & Fuller, W. A. (1981). Likelihood ratio statistics for autoregressive tie series with a unit root. Econoetrica, 49, Dutt, S. D., & Ghosh, D. (1999). Are price forecasts rational? An epirical exaination using survey data. Kentucky Journal of Econoics and Business, 18, Elder, J., & Kennedy, P.E. (2001). esting for unit roots: What should students be taught. Journal of Econoic Education, 32(2), Elliot, G., Rothenberg,.J., & Stock, J.H. (1996). Efficient tests for an autoregressive unit root. Econoetrica, 64 (4), Englander, A. S., & Stone, G. (1989). Inflation expectations surveys as predictors of inflation and behavior in financial and labor arkets. Federal Reserve Bank of New York, Research paper # Engsted,. (1991). A note on the rationality of survey inflation expectations in the United Kingdo. Applied Econoics, 23, Evans, G., & Gulaani, R. (1984). ests of rationality of the Carlson-Parkin Inflation Expectations Data. Oxford Bulletin of Econoics and Statistics, 46, Gagnon, J. E. (1996). Long eory in inflation expectations : Evidence fro international financial arkets. International Finance Discussion Papers, No.538. Greene, W. H. (1993). Econoetric Analysis, 2 nd edition, New York: Macillan Publishing Copany. Hvidding, J. M. (1987). Measureents errors and tests for expectational rationality using survey data. Southern Econoic Journal,

8 Inflation Forecasts: An Epirical Re-exaination Johansen, S., & Juselius, K. (1990). Maxiu likelihood estiation and inference on cointegration-with applications to the deand for oney. Oxford Bulletin of Econoics and Statistics, 52, Kwiatkowski, D., Phillips, P.C.B., Schidt, P., & Shin, Y. (1992). esting the null hypothesis of stationarity against the alternative of a unit root. Journal of Econoetrics, 54, Kanoh, S., & Li, Z. D. (1990). A ethod for exploring the echanis of inflationary expectations based on qualitative survey data. Journal of Business and Econoic Statistics, 8, Marini, G., & Scaraozzino, P. (1993). Inflation expectations, price flexibility and output variability in acro odels. Structural Change and Econoics Dynaics, 4, Perron, P. (1988). rend and rando walks in acroeconoic tie series. Journal of Econoic Dynaics and Control, 12(2), Phillips, P. C. B., & Perron, P. (1988). esting for a unit root in tie series regression. Bioetrica, 75, Steinus, M. (1986). Separating short ter and long ter inflation expectations using observations fro financial arkets. Epec, 11, hoas, L. B., & Grant, A.P. (2000). Forecasting inflation-surveys versus other forecasts. Business Econoics, hopson, D. N., & Ottosen, G.K. (1993). Long ter inflation expectations, the personal savings rate and credit standards. Business Econoics, Appendix A Bierens Non-paraetric Cointegration Procedure 5 he data are assued to be generated by the following expression: z z u (1) t t 1 t z t is a q-variate unit root process with drift, u t is a zero ean stationary process and is a vector of drift paraeters. Bierens akes the following assuptions: Assuption 1: he process u t can be written as given above with v t i.i.d N q (0,I q ) and C(L) = C 1 (L) -1 C 2 (L), where C 1 (L) and C 2 (L) are finite-order lag polynoials, with all the roots of det(c 1 (L)) lying outside the coplex unit circle. Assuption 2: Let R r be the atrix of eigenvectors of C(1)C(1) corresponding to the r zero eigenvalues. hen the atrix R D( 1) D(1) R is nonsingular. It is assued that the cointegrating relations R trend. Assuption 3: R 0 r r zt he Bierens test is based on the atrices n, k n, k k 1 r r are stationary about a possible intercept but not a tie ˆ, ˆ A a a B b b, q, and a n,k, b n,k are defined in Bierens k 1 (1997). n, k n, k 16

9 Southwest Business and Econoics Journal/ heore 1 of Bierens (1997) states: Let ˆ... ˆ be the ordered solutions of the generalized eigenvalue proble 1, q, ˆ ˆ 2 det[ ( ˆ 1 A B n A )] 0, and let ˆ 1,... ˆ q r, be the ordered solution of the generalized eigenvalue proble * * * * det X k X k Yk Yk 0 (2) k 1 k 1 * * where the X ' s and Y ' s are i. i. d. N (0, I ). If z t is cointegrated with r linear i j independent cointegrating vectors, then under assuptions 1-3 ( ˆ q r q r... ˆ ) 1, q, ˆ q r, converges in distribution to ( ˆ... ˆ 1, q r,,0,... 0 ). he test statistic can be used for testing the null hypothesis H r that there are r cointegating vectors against the alternative H r+1 (that there are r+1 cointegrating vectors). his is called the labda-in test. he critical values for the test are given in an appendix to Bierens (1997). his is a left sided test, i.e., if the test statistic is less than the critical value, then the null hypothesis will be rejected. he choice of the paraeter is iportant to this procedure, as it has an ipact on the liiting distribution of the labda-in statistic and the critical values and the power function. he power of the null hypothesis against the alternative is given by ˆ * P( q r, K, q r, ) P( 1, n K, q r, ) where K, q r, are the 100% critical values. We can choose such that the right-hand side of the above expression is axial, subject to the condition q. Notes 1. hough ost forecasts of acroeconoic variables are available fro 1968 onwards, the inflation series was collected fro he forecasts were obtained fro the Survey of Professional Forecasters data set, available on the web site of the Federal Reserve Bank of Philadelphia. he actual inflation rate was obtained fro the web site of the Bureau of Econoic Analysis. 2. he DF and PP tests are aong the first, and ost widely used, unit root tests. he KPSS test is an alternate approach to testing for unit roots. We ran all the tests and found siilar results. he results of the DF, PP and KPSS results are reported in able 1. It was pointed out by a referee that a discussion of testing for unit roots with or without a trend should be included. As pointed out by Perron (1988), and Elder and Kennedy (2001), a general to specific strategy (starting with a odel with a drift and a trend and working our way down) would be appropriate. Perron (1988) does describe how such a strategy should be ipleented. However, Elder and Kennedy (2001) point out that this procedure is needlessly coplicated. hey suggest that it is straightforward to deterine whether a tie series odel should include a tie trend, based on theoretical arguents and graphs, and it is not necessary to have an econoetric test for this. hey also show that in such cases the ADF t-test is equivalent to testing for the presence of a tie trend, and ore coplicated F-tests, or coplicated stepwise procedures like those recoended by Perron (1988) are not necessary. Based on 17

10 Inflation Forecasts: An Epirical Re-exaination this strategy, it would see that the appropriate null hypothesis is a unit root with no tie trend and no drift, and we are unable to reject this null hypothesis. 3. It has been pointed out to us that the KPSS test suffers fro severe size distortions, and therefore ultiple unit root tests should be included. herefore, we have included results fro ADF and PP tests in addition to the KPSS tests, all of which support the conclusion of unit roots in the data. In addition, Caner and Killian (2001), who discuss this issue of size distortions in soe detail, also suggest that the power of the asyptotically efficient DF-GLS test (as discussed in Elliot et. al. (1996)) copares favorably to the ADF test. We ran the DF-GLS test on our data and the results supported the presence of unit roots in both the actual and forecasted inflation series. 4. he coputations for the Bierens procedure were done using the EasyReg International software ade available by Heran Bierens on the web-site: 5. We give a brief description about the Bierens NPC procedure since the details of the procedure can be obtained fro that paper. 18