Forecasting Demand for Various Denominations of Notes and Coins Using Error Correction Models

Vale, Bent Research Report Forecasting Demand for Various Denominations of Notes and Coins Using Error Correction Models Staff Memo, No. 1/2015 Provided in Cooperation with: Norges Bank, Oslo Suggested Citation: Vale, Bent (2015) : Forecasting Demand for Various Denominations of Notes and Coins Using Error Correction Models, Staff Memo, No. 1/2015, ISBN 978-82-7553-858-9, Norges Bank, Oslo, https://hdl.handle.net/11250/2506559 This Version is available at: https://hdl.handle.net/10419/210309 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. http://creativecommons.org/licenses/by-nc-nd/4.0/deed.no STAFF MEMO Forecasting demand for various denominations of notes and coins using error correction models NO. 1 | 2015 BENT VALE MARKETS AND BANKING SERVICES Staff Memos present reports and documentation written by staff members and affiliates of Norges Bank, the central bank of Norway. Views and conclusions expressed in Staff Memos should not be taken to represent the views of Norges Bank. © 2015 Norges Bank The text may be quoted or referred to, provided that due acknowledgement is given to source. Staff Memo inneholder utredninger og dokumentasjon skrevet av Norges Banks ansatte og andre forfattere tilknyttet Norges Bank. Synspunkter og konklusjoner i arbeidene er ikke nødvendigvis representative for Norges Bank. © 2015 Norges Bank Det kan siteres fra eller henvises til dette arbeid, gitt at forfatter og Norges Bank oppgis som kilde. ISSN 1504-2596 (online) ISBN 978-82-7553-858-9 (online) Forecasting demand for various denominations of notes and coins using error correction models Bent Valey Norges Bank (The central bank of Norway) March 23, 2015 Abstract In this paper we present a set of error correction models in order to forecast separately the change in demand for each of the notes and coins issued by Norges Bank. Such forecasts can play a role in planning how many new banknotes and coins Norges Bank should order from its producers. The explanatory variables upon which we condition the forecasts inculde: households’ point of sales consumption, the number of EFTPOS terminals, money market interest rate, as well as some dummies for particular events. The estimated effects seem overall to correpsond with findings by other studies. Interstingly, we find that as consumers to a larger extent use EFTPOS cards for their point of sales consumption, holdings of the lowest value denomination coin increase, something we attribute to consumers leaving stocks of low value coins at home since they are less frequently used for paying. Thanks to Trond Eklund who initiated this project and together with Karianne Sæther provided the description of the system for production and distribution of cash in Norway. The latter also provided the data. Both colleagues have in addition given valuable feedback. Furthermore, many thanks to Farooq Akram and Øyvind Eitrheim for helpful discussions during the work with this paper. Also thanks to André Anundsen and Claudia Foroni for comments to a recent version of this paper. Nevertheless, views and conclusions presented here are the responsibility of the author and cannot necessarily be attibuted to the persons mentioned above. yAddress: Norges Bank, Box 1179, Sentrum, N-0107 Oslo, Norway. E-mail: bent.v[email protected] 1. Introduction This paper presents models for forecasting changes in the demand for the various denominations of banknotes and coins in Norway. The model is used in order to determine the necessarry production of new coins and notes. The paper is organized as follows: In Section 2 we present a brief overview of the system for production and distribution of cash in Norway, whereas section 3 sets out the system for determining the production of new notes and coins and how the forecasting model fits into that framework. In Section 4 we then present the general model to be used, discuss some econometric issues like identification and potential endogeneity problems, and not the least we discuss how to use the models estimated for forecasts. Data are presented in Section 5. In Section 6 the estimated models and their ability to forecast are presented. Finally, we provide some concluding remarks in Section 7. 2. Norges Banks role and responsibility in cash distribution Under the Norges Bank Act, Norges Bank (NB) has the exclusive right to issue notes and coins. This implies an obligation to issue in order to meet the needs of the economy and to ensure that an adequate supply of the notes and coins issued is available to the public. Notes and coins in circulation must be of a certain quality in order to function as an efficient means of payment. Norges Bank has an overriding responsibility for maintaining this quality as well as an obligation to accept worn and damaged notes and coins and replace them with notes and coins of acceptable quality. Norges Bank acts as a wholesaler in the cash distribution. Notes and coins are produced by external suppliers: Oberthur and Giesecke & Devrient produce notes, and Mint of Norway produces coins. All these producers have been chosen in a competitive process after Norges Bank invited tenders internationally. Given full information concerning costs, prices and quality, the market participants are best qualified to find good solutions for cash distribution. Norges Bank wishes to encourage them to constantly seek the best solutions based on cost/benefit analyses, and to ensure that distribution sites and processing solutions change in line with the assessments of market participants. This would 2 Figure 2.1: The system for distibution of notes and coins in Norway initially imply that Norges Bank supplies banks from just one business site. Security and logistics considerations may however necessitate having emergency stocks at more than one site. In addition, Norges Bank appears better equipped than banks to transport large amountsoverlongdistances. ThisimpliesthatNorgesBankshouldhaveadditionaldepots and business sites and handle the transport between them in a system that can constitute the “central nerve” of the supply and distribution of cash. Since 2005, Norges Bank has had 5 regional depots in addition to a central distribution vault, as depicted in Figure 2.1. Within the five regions, banks are responsible for supplying cash to their customers and for redistributing cash among different banks and bank branches. Moreover the banks are responsible for quality sorting. Banknotes have to be sorted into fit and unfit according to the central bank guideline prior to deposits in the central bank. To facilitate an efficient redistribution within the region, Norges Bank offers a measure referred to as private depot where Norges Bank on certain conditions pays interest rate compensation on banks’ holdings of coins and notes. Norges Bank issues five denominations of notes: 1000 kroner, 500 kroner, 200 kroner, 100 kroner and 50 kroner. Four denominations of coins are also issued: 20 kroner, 10 kroner, 5 kroner and finally the smallest denomination, 1 krone. Until 1 May 2012 a 50 øre 3 coin, worth 0.5 kroner was also legal tender. Although it circulated in the period covered by our data, we do not present any demand model for this denomination. Furthermore, we do not estimate a model for the 20-krone coin as Norges Bank has a huge stock of this coin. The reason being that gambling machines that used 20-krone coins were outlawed in 2006 and 2007, resulting in a huge return of 20-krone coins to Norges Bank. 3. Why estimate the need for cash? Norges Bank is responsible for the availability of a sufficient quantity of cash in order to meet the demand, and thus will have to estimate the future demand for cash. Forecasts of demand for individual denominations are necessary both for ordering and for keeping optimal stock levels of banknotes and coins. Basically, the need to order new banknotes and coins depends on the need to replace the amount of destroyed denominations of banknotes and coins and the change in external demand. The total need for orders of new banknotes and coins to keep Norges Bank’s stock level unchanged , NS, may be expressed as: NS =WD+S. (3.1) Wis total withdrawals of banknotes and coins by banks or private depots from Norges Bank’s vaults. Dis their total deposits into Norges Bank’s vaults. Sis deposited unfit notes that will be shredded and therefore need to be replaced.1Thus, WDis the net increase in circulation of notes and coins, which will be addressed per denomination in the forecasting model. Our existing forecasting model provides information on aggregate developments in cash circulation (see Aastveit (2005)). Such models increase the understanding of factors of importance for the total amount of cash circulation and contribute to the information needed for future procurements. In the case of actual orders, there is, however, need for a model which splits the denominations. In Norway the total amount of cash in circulation has been stable for a period of more than five years while the denomination mix has changed. To meet its mandate of supplying sufficient coins and banknotes to the general public, the Norges Bank has to develop a well defined currency stock policy and a comprehensive 1The need for replacement of destroyed denominations depends to a certain degree on the Norges Bank’s clean note policy. 4 database for monitoring and planning purposes. A well-founded stock policy is a key measure, it secures the ability to meet variations in demand and defined extraordinary situations in a cost effective way. A policy for holding stocks of coins and banknotes within the Norges Bank covers: transaction stocks decided by ordering policy; buffer stocks, or safety stocks, to buffer against uncertainty regarding demand for cash and uncertainty regarding supplier lead-time; and contingency stocks to meet extraordinary circumstances. Transaction stocks serve to cover the expected requirements during the period between one delivery and the next from the producer. When a new delivery arrives, the transaction stock level should be close to zero. The average level of the transaction stock is decided by the order policy — the size of the order and the ordering frequency — which is set to minimize the sum of order, transport and storage costs, see Figure 3.1. The graph illustrates the framework that is used to decide on the timing of an order. The decline in stocks is estimated on the basis of the forecasts for future change in demand per denomination, as well as destruction of obsolete banknotes and coins, see equation (3.1). Forecasts should be updated quarterly from an updated dataset. The initial forecast together with the anticipated initial lead time is illustrated with the dotted lines. An updated forecast is illustrated with the solid lines. The updated forecast indicates a need to order earlier. In addition, the graph explains another measure, a renegotiated shorter lead time. These graphs give the necessary information for ordering (time and volume) from suppliers of notes and coins. By minimizing total costs for ordering and for stockholding, and at the same time applying an acceptable risk for obsolete banknotes, one may determine the optimal order volume and subsequent magnitude of transaction stocks. Buffer stocks or safety stocks serve as a buffer, primarily against uncertainty regarding demand for cash during the lead time. This is referred to as service level or choice of supply capability, i.e., the probability of being able to meet demand for a denomination 5 Figure 3.1: Actual and forecasted level of transaction stock 6 5. Data and special events relating to demand for cash The data consist of quarterly observations on the numbers of outstanding notes and coins of each denomination issued by Norges Bank from 1998q1 until 2010q3, i.e., for 51 periods. However we only use data from 1998q1 until 2008q3 for estimation, leaving the eight quarters 2008q4 – 2010q3 for post-sample predictions. Adjusting for the various lags used in the eight models the number of observations in estimation ranges from 35 to 39. Notes and coins outstanding include all notes and coins held by households, financial8 and non-financial firms, local and central government institutions other than Norges Bank, as well as residents, private and government institutions in other countries. Data for the quarterly POS consumption is collected from the quarterly national accounts.9 The annual numbers of EFTPOS terminals are from Norges Bank’s Annual Report on Payment Systems. The number of terminals increases from 52,235 in 1998 to 131,079 in 2009.10 The stock numbers of outstanding notes and coins are shown i Figures 5.1 and 5.2. As is apparent from Figure 5.1, for the 200-krone notes and the 500-krone notes there was a steady increase during our period of almost 11 years. During this period, the former has continuously been a standard ATM denomination whereas the latter has become a widely used ATM denomination. There was also a fairly steady but weaker growth in the use of 50-krone notes (a non-ATM denomination), whereas the 100-krone note had a steady and fairly strong decline during the first half of the period. This is the time when it largely was phased out as an ATM denomination. The 1000-krone note is only rarely used as an ATM denomination. Over the whole period the number of 1000-krone notes outstanding fell, but there was a slight increasing trend between 2003 and 2007. This 8That includes the bank owned cash handling firm NOKAS 9For POS consumption actual data from the quarterly national accounts are used until 2010q2. For 2010q3 we assume the same 12 months growth as in overall household consumption from 2009q3 till 2010q3. 10See http://www.norges-bank.no/en/about/published/publications/annual-report-on-payment-systems/ . Numbers are recorded from the report for 2009 and previous. For 2010 we assume the same growth rate in EFTPOS terminals as from 2008 to 2009. 13 10000 20000 30000 40000 1998q1 2000q1 2002q1 2004q1 2006q1 2008q1 2010q1 50-krone notes 100-krone notes 200-krone notes 500-krone notes 1000-krone notes Numbers in 1000s Stock of notes in circulation Figure 5.1: Number of notes outstanding in 1000s from 1998q1 till 2010q3 14 note is probably mostly used as a store of value and in some large POS transcations like second hand car purchases. It may also be used in illegal transactions more so than other notes, simply because it is the largest value denomination. For all denomiations there is evidence of cyclical patterns. 500000 600000 700000 800000 900000 1000000 1-krone coins 80000 100000 120000 140000 1998q1 2000q1 2002q1 2004q1 2006q1 2008q1 2010q1 5-krone coins 10-krone coins 1-krone coins Numbers in 1000s Stock of coins in circulation Figure 5.2: Number of coins outstanding in 1000s from 1998q1 till 2010q3. Figure 5.2 shows that there was a steady rise in the stock of both the 1-krone coin and the 5-krone coin. A notable feature is that the 1-krone note is by far the most widely used coin in terms of the number of coins in circulation. By the end of the sample there were almost 1 billion 1-krone coins in circultaion, whereas the number of 5-krone coins in circulation was "only" a little less than 140 millions. Unlike for the notes, there is little evidence of any cycliacl patterns in the number of 1-krone coins in circulation. There is a fall in the number of 10-krone coins in circulation between 2002q1 and 2004q1 This is related to intoduction of restrictions on slot machines for gambling in this period. 15 6. Results of model estimation and forecasting First we report the estimated models and forecasts for the 5 notes in increasing order of denomination value. Then follows the 3 coin debominations in the same order. We report the values of the estimated parameters with corresponding t-values, as well as misspecification tests and tests of stationarity for the residiuals of the long run model. Economic interpretations of the models and their abillity to forecast the change in demand for the respective notes and coins 8 quarters ahead out of sample, are discussed. Estimation sample for all models is 1998q1 to 2008q3, with adjustments of the starting period as required by the number of lags. The out-of-sample forecasting period thus stretches from 2008q4 to 2010q3. Ending the estimation sample as early as 2008q3 enables us to compare the models’ forecast to the actual developmement in the circulation of each denomination over a horizon corresponding to the lead time between orders and deliveries of notes, cf. Figure 3.1. 6.1. Forecasting models for the notes All five models for the notes are estimated in one step. 6.1.1. 50-krone note For the 50-krone note we have the following preferred model 4M50t= 4925:5 (6:13) + 0:353 (4:29) 4M50t20:894 (8:17) M50t4+ 0:102 (7:19) PCt4(6.1) +774:5 (4:53) rkt1+ 0:148 (3:62) 4PCt875:1 (6:05) q3t898:3 (5:28) q4t R20.8397 AR 1–2 test F(2;27) 0.53 ARCH 1–1-test F(1;27) 0.59 Normality test 2(2) 0.33 Hetero test F(11;17) 0.37 RESET test F(1;28) 0.16 Number of obs. 37 For the misspecification tests, p-values are reported. The estimated forecasting model passes the misspecification tests. From (6.1) we can derive the following long run equilibrium model M50t4= 5508:9+0:114PCt4+ 774:5rkt1+t. (6.2) 16 The Dickey-Fuller test for stationarity of tconfirms that this is a cointegrating relationship. The test statistic is 5:307 wheras the 1% critical value for this test is 3:668. As one might expect, the demand for 50-krone notes depends positively on the POS consumption. As intended, the demand also reacts positively to introduction ofthe interest compensation for holding notes and coins. Being a small denomination note, the demand for it does, however, not react to the level of interest rate. Neither does it depend on the availability of EFTPOS terminals. This is in accordance with findings by ten Raa and Shestalova (2002) and Bergman et al. (2007) who calculate the maximum size of a consumer transaction where cash is preferrable to cards, to be somewhat above the value of the 50-krone note. To check the ability of (6.1) to forecast we present a plot of the actual values of 4M50t(in red) and the predicted values (in blue). The residuals or forecast errors are marked as green vertical spikes. Observations to the left of and including at the grey dotted vertical line represent in-sample forecasting, whereas the out of sample forecasting is to the right of the vertical line. 17 -1000 01000 2000 1999q1 2002q1 2005q1 2008q1 2011q1 Linear prediction Actual values Residuals in 1000s estimated to 2008q3 Changes in circulation of 50-krone notes over 4 quarters Actual values in red and the predicted values in blue. The residuals or forecast errors are marked as green vertical spikes. Observations to the left of and including at the grey dotted vertical line represent in-sample forecasting, whereas the out of sample forecasting is to the right of the vertical line. The model predicts fairly well in-sample, whereas the out-of-sample forecasts have somewhat larger residuals. But, the model seems to be able to capture a sightly increasing trend in 4M50tafter 2009q1. The forecast does anyhow seem to be somewhat more volatile than the actual development of 4M50t. In order to smooth the forecast of the change in circulation of the 50-krone notes we calcualte the average of 4M50tfor all the 8 post-sample quarters. In that way we avoid the risk of putting too much weight on one of the outlying forecasts. Using this smoothing method we arrive at a forecast of growth in the circulation of 50-krone notes from 2008q3 to 2010q3 of 757,333 notes compared to the actual growth of 687,471 notes. If instead we had added the two fourquarters forecasts at 2009q3 and 20010q3 we would have got a forecast of 1367,115, the double of the actual value. In this case the smoothing method performs far better than simple adding up of two ensuing four-quarters forecasts. 18 6.1.2. 100-krone note When estimating a forecast for the 100-krone notes we need to take into account the gradual phasing out of the 100-krone note as an ATM denomination during the first part of our estimating sample. Unfortunately we lack data on how many ATMs that offer the various denominations. Instead we approach this phasing out by adding a simple linear trend variable as follows atm100t=176 tif t2003q4 0if t > 2003q4. The numerical value of 2004q1 is 176.11 Thus, atm100ttakes the value of 18 in the first quarter of the sample over which (6.3 ) below is estimated (1999q3), it becomes 1 in 2003q4 and 0 thereafter. The preferred model for forecasting 4M100t, the four quarter change in the circulation of 100-krone notes is 4M100t= 8211:0 (3:16) + 0:300 (2:57) 4M100t20:773 (4:14) M100t4(6.3) +0:077 (4:25) PCt4+ 395:3 (3:42) atm100t4+ 0:194 (5:29) 4PCt 0:107 (10:59) 4etrmt10:071 (2:88) 4etrmt4+ 727:9 (1:95) q2t771:6 (2:72) q3t R20.9079 AR 1–2 test F(2;25) 0.76 ARCH 1–1-test F(1;25) 0.92 Normality test 2(2) 0.00 Hetero test F(16;10) 0.81 RESET test F(1;26) 0.20 Number of obs. 37 For the misspecification tests, p-values are reported. The estimated model passes misspecification tests except the one for normality of the residuals, this calls for some caution in the interpretation of t-values assigned to the coefficient estimates. From (6.3) we can derive the following long run equilibrium model M100t= 10622:3+0:1055PCt+ 511:4atm100t+t. (6.4) From the Dickey-Fuller test of twe get clear support of (6.4) being a cointegrating relationship. The test statistic is 6:594 versus a 1% critical value of 3:668. 11The numerical representation of tis set equal to the number of quarters since 1960q1 in the statistsics package we are using for organizing the data, STATA. 19 The transaction variable PCtalso shows up in this model, as do the trend variable for phasing out the 100-krone note as an ATM denomination. Note that the trend variable has a higher value the more ATMs that offer 100-krone notes, hence its positive sign. As in the long run equilibrium model for the 50-krone note (6.2), the number of EFTPOS terminals does not enter (6.4). Nevertheless, it is part of the short run dynamics in (6.3). To assess the ability of (6.3) to forecast, below we present a plot of actual values and predicted values of 4M100t. -6000 -4000 -2000 02000 1999q1 2002q1 2005q1 2008q1 2011q1 Linear prediction Actual values Residuals in 1000s estimated to 2008q3 Changes in circulation of 100-krone notes over 4 quarters Actual values in red and the predicted values in blue. The residuals or forecast errors are marked as green vertical spikes. Observations to the left of and including at the grey dotted vertical line represent in-sample forecasting, whereas the out of sample forecasting is to the right of the vertical line. The model predicts rather well in-sample. Out-of-sample it is able to capture the rising trend in the actual values, but it shows higher volatility than can be found in the actual values. Using the same smoothing technique as described in Section 6.1.1 for the 50-krone note in order to forecast the change in circulation of 100-krone notes during the eight quarters following 2008q3, we get a forecast of 683;453 notes as opposed to an actual increase of 110,643 notes. By instead adding the two ensuing four quarter forecasts at 2009q3 and 2010q3, we get a forecast of 66,156, closer to the actual value. I.e., in this case simply adding the two four quarter forecasts would produce a better 20 forecast. Nevertheless, it is evident from the plot that the model has a slight tendency of underpredicting in the post-sample period. 6.1.3. 200-krone note For the 200-krone notes we have the following preferred model 4M200t= +10961:7 (4:96) 0:796 4:36 M200t4+ 0:038 (1:18) PCt4(6.5) + 0:049 (3:67) etrmt4+ 1443:1 (3:28) rkt4+ 1236:7 (2:83) 4rkt +0:180 (7:31) 4etrmt30:176 (7:95) 4etrmt4+ 1189:7 (2:40) q2t+ 2006:0 (3:88) q4t R20.6749 AR 1–2 test F(2;27) 0.33 ARCH 1–1-test F(1;27) 0.43 Normality test 2(2) 0.47 Hetero test F(14;14) 0.93 RESET test F(1;28) 0.31 Number of obs. 39 For the misspecification tests, p-values are reported. The estimated model passes all the misspecification tests. From (6.5) we can derive the following long run equilibrium model M200t= 13771:0+0:048PCt+ 0:062etrmt+ 1813:4rkt+t. (6.6) The Dickey-Fuller test for tclearly indicates that (6.6) is a cointegrating relationship. The test statistic is 10:063 versus a 1% critical value of 3:655. Like with the 50-krone note, but unlike the 100-krone note, there is a verystrong effect from the introduction of interest compensation. POS consumption in levels is forced to be part of the dynamic model although it has a t-value of only 1.18. In the long run though it has a t-value of 1.36 and corresponding p-value of approximately 20%. The number of EFTPOS-terminals has a positive influence on the demand for 200-krone notes. That may seem at odds with cash and debit cards being substitutes when consumers buy POS goods or services. However, we need to take into account the avaialbility of cash-back at most EFTPOS terminals. Consumers paying with debit cards have strong incentives to witdraw cash from the check-out registries in stores rather than using ATMs. There is no fee on getting cash-back when paying with an EFTPOS card as opposed to getting cash 21 from an ATM. In addition, the density of ATMS has decreased in the large cities. When withdrawing cash the 200 kroner note is a popular currency, as it is covenient to use for many cash transactions. This effect seems to outweigh the effect of substitution between cash and debit cards. Below, we plottheactualvalues and theforecasted values ofthe four quarterlychanges in the circultion of the 200-krone note. -2000 02000 4000 1999q1 2002q1 2005q1 2008q1 2011q1 Linear prediction Actual values Residuals in 1000s estimated to 2008q3 Changes in circulation of 200-krone notes over 4 quarters Actual values in red and the predicted values in blue. The residuals or forecast errors are marked as green vertical spikes. Observations to the left of and including at the grey dotted vertical line represent in-sample forecasting, whereas the out of sample forecasting is to the right of the vertical line. The model predicts fairly well in-sample. Out-of-sample, it overprdicts for the first five quarters then underpredicts for the next three. Using the smoothing technique (see Section 6.1.1) we get a forecast of the eight quarter change in the circulation of 200- krone notes after 2008q3 of 1,785,372 compared to an actual change of 2,280,188. By instead adding the tow four-quarters forecasts at 2009q3 and 20010q3 , we get a forecast of 2,369,776, even closer to the actual value. I.e., in this case, as with the 100-krone note simply adding the two four quarter forecasts gives a better prediction. 22 -20000 020000 40000 60000 1999q1 2002q1 2005q1 2008q1 2011q1 Linear prediction Actual values Residuals in 1000s estimated to 2008q3 Changes in circulation of 1-krone coins over 4 quarters Actual values in red and the predicted values in blue. The residuals or forecast errors are marked as green vertical spikes. Observations to the left of and including at the grey dotted vertical line represent in-sample forecasting, whereas the out of sample forecasting is to the right of the vertical line. The model predicts well in-sample. In the first part of the out-of-sample period the model overpredicts a litlle, whereas it underpredicts in the last part. The actual change in the number of 1-krone coins in circulation during the 8 quarters following 2008q3 was 55,155,570. The forecast using the smoothing method, explained in Section 6.1.1, is 56,085,280. By just adding the two 3rd quarter forecasts following 2008q3 we get 55,177,540. Both forecasts are close the actual value, but by just adding the two 3rd quarter forecasts we get closest to the actual number. 6.2.2. 5-krone coin For the 5-krone coin we first estimate the long run equilibrium model for the sample 1998q1 to 2008q3 M5t= 26967:5 (6:00) + 0:821 (16:95)PCt+ 3513:4 (3:07) rkt(6.13) +9686:4 (9:33) q1t+ 3838:5 (5:63) q2t+ 1349:7 (2:17) q3t 29 R2of this model is 0.9832. The Dickey-Fuller stationarity test of the residual of (6.13) ECM5t, gives a test statistic of 4:729 with a 1% critical value of 3:634. Hence, (6.13) is cointegrating. As with other denominations, the equilibrium demand for 5-krone coins depends positively on POS consumption and it also reacts positively to the intoduction of interest rate compensation to banks and their cash depositories for storing cash. Unlike the 1-krone coin there is no effect on the demand for 5-krone coins from the number of EFTPOS terminals. The preferred short run dynamic model for the 5-krone note is 4M5t= 0:569 (6:18) 4M5t10:546 (3:67) ECM5t4(6.14) +0:341 (3:83) 4PCt+ 1508:4 (2:98) rkt R20.9609 AR 1–2 test F(2;32) 0.03 ARCH 1–1-test F(1;32) 0.51 Normality test 2(2) 0.46 Hetero test F(7;26) 0.19 Hetero-X test F(13;20) 0.26 RESET test F(1;33) 0.79 Number of obs. 38 For the misspecification tests, p-values are reported. The model passes the misspecification tests except for the AR 1–2 test. Below, we plotthe actualvaluesand theforecasted values of thefour quarterlychanges in the circulation of the 5-krone coin. 30 -5000 05000 10000 1999q1 2002q1 2005q1 2008q1 2011q1 Linear prediction Actual numbers Residuals in 1000s estimated to 2008q3 Changes in circulation of 5-krone coins over 4 quarters Actual values in red and the predicted values in blue. The residuals or forecast errors are marked as green vertical spikes. Observations to the left of and including at the grey dotted vertical line represent in-sample forecasting, whereas the out of sample forecasting is to the right of the vertical line. The model predicts relatively well in-sample. Out-of-sample it seems to overpredict somewhat more than it underpredicts. The actual 8 quarter change in the circulation of 5-krone coins after 2008q3 was 3,384,044 coins. The forecast for the same period using the smoothing method described in Section 6.1.1 is 5,413,705. By instead just adding the two four quarter forecasts for 2009q3 and 2010q3 we get 6,333,825. Hence in this case the smoothing method gives a better forecast than just adding the two four quarter forecasts. 6.2.3. 10-krone coin For the 10-krone coin we did not succeed in finding a cointegrating long run eequilibrium relationship. Hence, we are left to rely on a short run dynamic model only containing 31 variables as first-differences and dummies. The preferred model is as follow 4M10t= 0:593 (5:59) 4Mt10:219 (3:34) 4Mt4(6.15) +0:249 (2:98) 4PCt1+ 0:218 (2:75) 4PCt2 +2028:5 (7:12) 4rkt31353:7 (2:52) DM10t1996:6 (2:99) DM10t3 R20.9767 AR 1–2 test F(2;26) 0.58 ARCH 1–1-test F(1;26) 0.78 Normality test 2(2) 0.38 Hetero test F(11;16) 0.99 RESET test F(1;27) 0.63 Number of obs. 35 For the misspecification tests, p-values are reported. The model passes all the misspecification test. We also check that its error term is stationary. The Dickey-Fuller test statistic is 7:558 versus a critical 1% value of 3.689. Hence, the error term can be considered stationary. The four quarter change in POS consumption contributes to explain the four quarter growth in the circulation of 10-krone coins. There was also a boost to the demand for 10-krone coins from the introduction of interest compensation to banks and their cash deposits for holding notes and coins. In addition, the period between early 2002 and early 2004 when slot machines machines became more strictly regulated had a clear negative impact on the demand for 10-krone coins. Below, we plotthe actualvaluesand theforecasted values of thefour quarterlychanges in the circulation of the 10-krone coin. 32 -5000 05000 10000 2000q1 2002q1 2004q1 2006q1 2008q1 2010q1 Linear prediction Actual numbers Residuals in 1000s estimated to 2008q3 Changes in circulation of 10-krone coins over 4 quarters Actual values in red and the predicted values in blue. The residuals or forecast errors are marked as green vertical spikes. Observations to the left of and including at the grey dotted vertical line represent in-sample forecasting, whereas the out of sample forecasting is to the right of the vertical line. The model predicts well in-sample. Out-of-sample, i.e., after 2008q3 it first has a tendency of overpredicting, then underpredicring somewhat the growth in the demand for 10-krone coins. The actual 8 quarter growth in the circulation of 10-krone coins after 2008q3 was 4,523,668. Using the smoothing method described in Section 6.1.1 we get a forecast of the 8 quarter growth from 2008q3 at 6,080,144. If instead we simply add the two four quarter forecasts ending in 2009q3 and 2010q3 we get a forecast of 5,135,694. I.e., in this case the simple adding of two four quarter foracsts gives a forecast closer to the actual number than does the smoothing method. 7. Concluding remarks In this paper, based on quarterly data, we have presented a set of error correction models in order to forecast the change in demand for notes and coins issued by Norges Bank. We have etsimated one model for each denomination. Such forecasts can play a role in planning how many new banknotes and coins Norges Bank should order from its producers. 33 The variables upon which we condition the forecasts: households’ point of sales consumption, the number of EFTPOS terminals, money market interest rate, as well as the ability of banks to receive interest payments for cash they keep in their vaults, have good foundation in economic theory on demand for cash. Furthermore, they are also found to influence demand for banknotes and coins in empirical papers using data from other countries. This holds to a large extent true for studies estimating demand for aggregates of all denominations and studies estimating demand for larger versus smaller denominations. Interestingly, we made one finding which we are not aware of in any other studies: as consumers to a larger extent use EFTPOS cards for their point of sales consumption holdings of the lowest value denomination coins increases, something we attribute to consumers leaving stocks of low value coins at home since they are less frequently used for paying. Our models produce four quarters forecast of the change in demand for the various denominations. The typical interval between orders of new supplies for a denomination is 8 quarters. The out-of-sample forecasts for the change in demand over this horizon (8 quarters) perform reasonably well. However, quarter by quarter our four quarters forecasts may be quite inaccurate. In order to arrive at a more relible number for the change in demand over an 8 quarters horizon we need to smooth the quarter by quarter forecasts by using an average. Instead of smoothing one could just add the two sequential nonoverlapping four quarter forecasts. In this paper, the method of adding the two four quarter forecasts till 2009q3 and 2010q3 gives predictions closer to the actual numbers than does the smoothing method in five of eight cases. Nevertheless, one should still regard the smoothing method as more reliable, in the sense that getting forecasts that are far away from the true numbers is less likely with the smoothing method. As can be seen from the plots, relying on just one or two observations in the post sample period may lead one to pick observations that deviate the most and in the same direction from the actual numbers. The simple adding method scoring 5 – 3 over the smoothing method in the results presented here, is most likely a coincidence. Alternatively, one could have tried using a vector error correction approach or a structural VAR approach in order to capture interdepencies between various denominations. However, we leave that for future research. 34 References Aastveit, K. A. (2005, February). 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