An Analysis of Tax Revenue Forecast Errors

An Analysis of Tax Revenue Forecas Errors Marin Keene and Peer Thomson N EW Z EALAND T REASURY W ORKING P APER 07/02 M ARCH 2007

NZ TREASURY WORKING PAPER 07/02 An Analysis of Tax Revenue Forecas Errors MONTH/ YEAR March 2007 AUTHORS Marin Keene The Treasury PO Box 3724 Wellingon 6140 New Zealand Email marin.keene@reasury.gov.nz Telephone +64 4 917 6037 Fax +64 4 499 0992 Peer Thomson Saisics Research Associaes Ld PO Box 12649 Wellingon 6144 New Zealand Email peer@sasresearch.co.nz Telephone +64 4 934 4753 Fax +64 4 475 4206 ACKNOWLEDGEMENTS We graefully acknowledge he assisance and suppor received from our colleagues wihin he New Zealand Treasury, paricularly Bob Buckle, John Carran, Kay Henderson, Dean Hyslop, John Janssen, Daniel Lawrence, Kam Szeo, Wayne Tan, and everyone else wihin he Treasury who has aken an ineres in his research. NZ TREASURY New Zealand Treasury PO Box 3724 Wellingon 6140 NEW ZEALAND Email informaion@reasury.gov.nz Telephone 64-4-472 2733 Websie www.reasury.gov.nz DISCLAIMER This documen was commissioned by he New Zealand Treasury. However, he views, opinions, findings and conclusions or recommendaions expressed in i are sricly hose of he auhors, do no necessarily represen and should no be repored as hose of he New Zealand Treasury. The New Zealand Treasury akes no responsibiliy for any errors, omissions in, or for he correcness of, he informaion conained in his Paper. ISSN 1176-9505 Treasury:931300v2

Absrac The New Zealand Treasury forecass ax revenue for he wice-yearly Economic and Fiscal Updaes. The accuracy of hese forecass is imporan for he governmen s annual budge decisions as hey affec key fiscal aggregaes such as he operaing balance and deb levels. Good decision-making in his area is imporan for macroeconomic sabiliy and susainabiliy, one of he Treasury s oucomes. Over he pas six years, Treasury ax forecass, and he macroeconomic forecass on which hey are based, have underesimaed he acual ouurns. This repor presens an analysis of he Treasury s ax revenue forecas errors, boh in aggregae and disaggregaed by individual ax ype. The analysis focuses primarily on he annual one-year-ahead Budge forecass ha are ypically based on raing up pas ax revenues by growh raes in relaed macroeconomic variables such as GDP. The obecive of he analysis is o beer deermine he maor sources of ax revenue forecas error and o idenify he poenial for mehodological improvemens. A review of he Treasury's ax forecasing mehods is given and a general class of models proposed ha encompasses hese mehods. Adoping one of he simples of hese as a benchmark, he individual ax revenue forecas errors are firs disaggregaed ino componen errors due o forecasing he macroeconomic drivers used as a proxy for he ax base, and a componen due o forecasing he ax raio, or raio of ax revenue o proxy ax base. The ax raio is furher disaggregaed ino a componen error due o forecasing he ax raio rend and random error. The laer provides a measure of he bes accuracy ha can be achieved using he benchmark models adoped. Among oher findings, he repor shows ha he main source of ax revenue underforecasing is he underforecasing of he macroeconomic variables used as axbase proxies. The ax raio forecass were generally unbiased, bu less precisely deermined han he macroeconomic forecass. This and oher evidence indicae ha beer ax raio forecass are likely o be achieved, even wih he simple benchmark model used here. The benchmark models have meri as compeing models ha could be invesigaed furher alongside oher simple srucural ime series models in a sysemaic evaluaion using hisorical daa. JEL CLASSIFICATION KEYWORDS C53 Forecasing and Oher Model Applicaions E17 Forecasing and Simulaion H68 Forecass of Budges, Deficis, and Deb Tax revenue forecasing; forecas error decomposiions; disaggregaion; benchmark models WP 07/02 AN ANALYSIS OF TAX REVENUE FORECAST ERRORS i

Table of Conens Absrac...i Table of Conens...ii Lis of Tables...ii Lis of Figures...ii 1 Background...1 2 The Treasury's ax forecasing process...2 2.1 Curren forecasing mehods...3 3 Towards a model framework...6 3.1 Alernaive models...7 4 Forecas error decomposiions...8 4.1 Benchmark model...9 4.2 Decomposiion of oal ax revenue by ax ype...9 4.3 Separaing ou he macroeconomic forecas errors...11 4.4 A model-based decomposiion of ax forecas errors...12 5 Daa...12 5.1 Daa issues...13 6 Analysis...15 6.1 Toal ax revenue decomposiion...17 6.2 Individual ax revenue decomposiions...19 6.3 Oher forecas horizons...31 7 Conclusions...31 Appendix: Plos for wo-year-ahead forecas errors...34 References...42 Lis of Tables Table 1 Means and sandard deviaions of individual ax revenues...16 Table 2 Summary saisics for he weighed percenage forecas errors...18 Table 3 Conemporaneous correlaions for he weighed percenage forecas errors...19 Table 4 Summary saisics for oal ax revenue percenage forecas errors...20 Table 5 Summary saisics for PAYE revenue percenage forecas errors...22 Table 6 Summary saisics for GST revenue percenage forecas errors...25 Table 7 Summary saisics for corporae ax revenue percenage forecas errors...27 Table 8 Summary saisics for ne oher persons ax revenue percenage forecas errors...29 Table 9 Summary saisics for oher ax revenue percenage forecas errors...31 Lis of Figures Figure 1 Tax revenues as a percenage of oal ax revenue...15 Figure 2 Forecas errors for oal ax revenue and maor ax ypes...17 Figure 3 Toal ax revenue and nominal GDP...20 Figure 4 Source deducions (PAYE) and compensaion of employees (COE)...22 Figure 5 Goods and services ax (GST) and nominal consumpion...24 Figure 6 Corporae ax and operaing surplus...26 WP 07/02 AN ANALYSIS OF TAX REVENUE FORECAST ERRORS ii

Figure 7 Ne oher persons ax and enrepreneurial income...28 Figure 8 Oher ax revenue and nominal GDP...30 Figure 9 Tax-share weighed percenage forecas errors...32 Figure 10 Two-year-ahead forecas errors for oal ax revenue and maor ax ypes...34 Figure 11 Toal ax revenue and nominal GDP: wo-year-ahead forecass...35 Figure 12 PAYE and COE: wo-year-ahead forecass...36 Figure 13 GST and nominal consumpion: wo-year-ahead forecass...37 Figure 14 Corporae ax and operaing surplus: wo-year-ahead forecass...38 Figure 15 Ne oher persons ax and enrepreneurial income: wo-year-ahead forecass...39 Figure 16 Oher ax revenue and nominal GDP: wo-year-ahead forecass...40 Figure 17 Tax-share weighed, wo-year-ahead, percenage forecas errors...41 WP 07/02 AN ANALYSIS OF TAX REVENUE FORECAST ERRORS iii

An Analysis of Tax Revenue Forecas Errors 1 Background In each of he las hree years, he New Zealand Treasury has published a repor analysing he performance of is macroeconomic, ax and fiscal forecass. These repors can be found a hp://www.reasury.gov.nz/forecass/performance/. In paricular, Treasury forecass have persisenly underesimaed acual ax flows over he pas six years. Why his should be and wha he sources of error are, form he background for his sudy where he primary obecives are o beer deermine he maor sources of ax revenue forecas errors and o idenify he poenial for mehodological improvemens. The sudy builds on and complemens Schoefisch (2005) which focussed on he Treasury's general ax forecasing mehods and processes raher han specific forecasing models and mehodology. Schoefisch (2005) noed ha a sudy (Mühleisen, Danninger, Hauner, Kranyák, and Suon, 2005) published by he Inernaional Moneary Fund (IMF) showed ha he Treasury's ax forecasing performance compared well wih he performances of governmen agencies in oher counries over he period 1995 o 2003. I would seem ha New Zealand is no unique in erms of persisen underesimaion of ax, and we are aware of similar reviews underaken in Ausralia, Canada and he Unied Kingdom ha also address hese issues. Some of hese reviews have no ye been published, bu ohers have, including O'Neill (2005) which reviews Canadian federal fiscal forecasing. In general, he lieraure on ax forecasing seems o be sparse and largely he preserve of official governmen agencies or organisaions such as he IMF. In addiion o hose already cied, some relaively recen examples of his lieraure are Basu, Emmerson and Frayne (2003), who examine Unied Kingdom corporae ax forecass by he Insiue for Fiscal Sudies, London, and Rich, Bram, Haughwou, Orr, Rosen and Sela (2005) who use regional economic indices o forecas ax revenues for New York. Furher publicaions are lised in he references of he publicaions already menioned. The Treasury's ax forecass are based on raing up pas ax revenues by growh raes in relaed macroeconomic variables such as gross domesic produc (GDP) which also need o be forecas. Included in he 2003 and 2004 forecas performance repors was an aemp o disaggregae each one-year-ahead Budge oal ax forecasing error ino a macroeconomic componen and a ax componen. Separaing ou he wo sources of forecas error allows one o es he proposiion ha he ax forecass migh sill be oo low even in he case of a perfec macroeconomic forecas. For oal ax, par of he forecas error was aribued o errors in he forecass of nominal GDP, wih he remaining porion WP 07/02 AN ANALYSIS OF TAX REVENUE FORECAST ERRORS 1

being aribued o he ax forecasing process. In boh he 2003 and 2004 analyses, here was insufficien evidence o deermine wheher he proposiion was rue. Neverheless, hese analyses suggesed ha errors in nominal GDP forecass were making a considerable conribuion o errors in he ax forecass. However, he Treasury does no explicily use nominal GDP o forecas oal ax. Raher, forecass of various macroeconomic variables are used o forecas each of he componen ax ypes and he forecass of he ax ypes are hen aggregaed ino a oal ax forecas. Schoefisch (2005) noed hese earlier aemps o spli he ax forecas errors ino macroeconomic and ax componens and recommended ha resources should be direced owards analysing he relaive conribuions of macroeconomic and ax errors o he oal ax error for he various ax ypes (PAYE, GST, company ax, ec). This is a primary obecive of he analysis presened in his repor. A review of he Treasury's ax forecasing processes and mehods is given in Secion 2 and a general class of models is proposed in Secion 3 ha encompasses hese mehods. In Secion 4, one of he simples of hese models is adoped as a benchmark model where ax revenue is expressed as a produc of a ax raio and a suiable macroeconomic variable ha can be regarded as a proxy for he ax base. Using his model, expressions for suiable forecas error decomposiions are also derived. Individual ax revenue forecas errors are firs decomposed ino componen errors due o forecasing he underlying macroeconomic driver used as he ax-base proxy, and a componen due o forecasing he ax raio. The ax raio is hen furher disaggregaed ino a componen error due o forecasing he ax raio rend and random error. The laer provides a measure of he bes accuracy ha can be achieved using he benchmark models adoped. Using hese decomposiions, an analysis was underaken of hisorical daa from 1995 o 2005 where a discussion of he daa and adusmens applied is given in Secion 5. Resuls and discussion of he analysis are given in Secion 6 wih conclusions presened in Secion 7. As noed by Schoefisch (2005), he overriding purpose of his analysis is o enhance he undersanding of key deficiencies in forecas performance and provide a base for oher research proecs designed o improve forecas qualiy. 2 The Treasury's ax forecasing process Twice each year, he Treasury produces economic and fiscal forecass. The firs forecas each year is usually prepared for he governmen's annual Budge and published as he Budge Economic and Fiscal Updae (BEFU) around May or June. The second forecas is usually released in December, a week or wo before Chrismas, and published as he Half-Year Economic and Fiscal Updae (HYEFU). Prior o 2005, his was known as he December Economic and Fiscal Updae (DEFU). In an elecion year, here may be anoher forecas published four o six weeks before he general elecion, called he Pre- Elecion Economic and Fiscal Updae. The Treasury's economic forecass are produced by he macroeconomic forecasing eam, which devoes four o six weeks o he ask a each forecasing round. The macroeconomic forecasers examine recen economic daa, he forecass produced by oher New Zealand economic forecasers and discuss he sae of he New Zealand economy wih many business people around he counry. They hen run a variey of forecasing models o produce forecass of many macroeconomic variables, such as GDP WP 07/02 AN ANALYSIS OF TAX REVENUE FORECAST ERRORS 2

and he Consumer Price Index (CPI) of inflaion. These forecass ypically cover he curren year and he nex four years. The ax forecass are prepared more-or-less concurrenly wih he economic forecass. In he period under examinaion, a eam of hree forecasers and a manager prepared forecass for each of he 20 or so ax ypes. A ax ype is a caegory of oal ax such as PAYE, which is he income ax paid by salary and wage earners, or GST or company ax. Like he economic forecass, he ax forecass cover he curren year and he following four years. Unlike he economic forecass, he ax forecass are for June years. The macroeconomic variables are forecas eiher quarerly or in March years. As each par of he economic forecas is prepared, he ax forecasers prepare forecass of he relevan ax ypes. For example, as he labour marke forecas is prepared, he ax forecasers prepare PAYE forecass using variables from he labour marke forecas. The economic and ax forecasers hen examine he labour marke and PAYE forecass ogeher o ensure ha hey are consisen wih each oher and wih he oal economic and ax forecas. When forecass have been prepared for each ax ype, hey are aggregaed ino a oal ax forecas. Three axes are no forecas by he Treasury. The Minisry of Transpor supplies forecass for Road User Charges (RUC) and moor vehicle licensing fees (MVF), and he Minisry of Economic Developmen supplies forecass of Exhausible Resource Levies (ERL). Collecively, hese axes accoun for less han 2% of oal ax. The ax forecass are prepared in erms of boh receips, he ax acually paid o he collecing agency (usually Inland Revenue), and revenue, he ax ha is acually due, regardless of wheher or no i has been paid. The remainder of his documen focuses on ax revenue, alhough he reasoning is idenical for receips and he resuls of he analysis for boh revenue and receips are similar. 2.1 Curren forecasing mehods In common wih many oher official agencies around he world, he New Zealand Treasury uses mainly spreadshee-based ax forecasing models and procedures comprising he following phases. Phase 1: Deermine he nominal ax revenue for he las available year which is he base year. Phase 2: Adus he nominal ax revenue for he base year by removing any known anomalies o esablish he rue underlying ax posiion for ha year. Phase 3: Apply he forecas growh raes of relevan macroeconomic variable(s) o forecas ax for 1 o 5 years ahead, applying elasiciies if required. Phase 4: Adus he ax forecass for anomalies such as ax policy changes, expeced shifs in paymen daes or axpayer behaviour, and include any udgemenal forecasing adusmens ha may be deemed appropriae. More deailed descripions of some of he maor ax ypes follow. These serve o illusrae he general naure of he forecasing mehods used and how hey are implemened. WP 07/02 AN ANALYSIS OF TAX REVENUE FORECAST ERRORS 3

2.1.1 Source deducions This is he larges single ax ype. I makes up abou a hird of he oal ax collecions, around NZ$18 billion ou of a oal of NZ$51 billion in he year o June 2005. Approximaely 97% of source deducions are pay-as-you-earn (PAYE) deducions on wages, salaries and social assisance benefis, wih he oher 3% being specified superannuaion conribuion wihholding ax (SSCWT). The forecasing model used during he 1995-2005 period under examinaion was a quarerly muliplicaive model ha sars wih a hisory of collecions up o he mos recenly complee quarer. I proecs forward by muliplying he collecions base by macroeconomic forecass of wage, salary and employmen growh, bu also makes adusmens for he progressiviy of he individuals' ax scale (higher raes a higher incomes) and payday weighings (mos people are paid on a fornighly cycle). Two componens of he source deducions forecas are prepared by oher agencies and combined wih he Treasury's forecas o produce he final forecas. These componens are he Minisry of Educaion forecass of PAYE on eachers' salaries, and he Minisry of Social Developmen forecass of PAYE on social assisance benefis, which are boh small in relaion o oal source deducions. More formally, oal source deducions SD q for quarer q are given by SD = G + T + B q q q q where G q denoes general source deducions excluding PAYE on eachers' salaries T q and PAYE on social assisance benefis B q. Forecass of G q are obained by raing up pas values by macroeconomic growh raes according o he recursion G q = G E 1+ E 1+ 1. q q 4 q 4 2 Eq 4 W q W W q 4 q 4 where E q denoes oal employmen and W q denoes oal, payday-weighed wages and salaries. Noe he elasiciies of 1 on employmen and 1.2 on wages and salaries, wih he wage and salary elasiciy being esimaed by empirical research. Using his recursion and associaed forecass of he macroeconomic growh raes, a forecas for G q is now calculaed for each quarer for he nex 20 quarers or so. These quarerly forecass are hen accumulaed ino annual forecass. Furher ad-hoc adusmens may be made o he annual forecass, such as adusmens for changes in ax policy or udgemenal adusmens. Throughou he remainder of his repor we use he simple abbreviaion PAYE o denoe oal source deducions (PAYE plus SSCWT). 2.1.2 Oher persons ax This is ax paid mainly by individuals and russ on income ha is no wihheld, or is underwihheld, a source. Typically, his is ax paid by small-business operaors and invesors. Terminal ax from wage and salary earners also falls ino his caegory. Ne oher persons ax, or oher persons ax less refunds o all individuals, makes up abou 8% of he oal ax ake. WP 07/02 AN ANALYSIS OF TAX REVENUE FORECAST ERRORS 4

This ax ype is nooriously difficul o forecas. Since i is an amalgam of ax on a variey of sources, i is difficul o find a reliable macroeconomic driver o use for he forecass. In he pas, he Treasury has ried various muliplicaive models and a micro-simulaion model, bu wih varying degrees of success. Currenly, he Treasury uses a composie approach wih some smaller anomalous componens forecas separaely using simple sraigh line exrapolaion. The laer include revenue from income summaries (end-of-ax-year reconciliaions for salary and wage earners), and rebaes for chariable donaions and child-minding/housekeeping expenses. Wih hese componens removed, he adused oher persons ax is monhly revenue ha has resuled from personal income. This is allocaed ino pas ax years based on sampled daa from he Inland Revenue Deparmen (IRD), New Zealand's main ax-gahering agency. Then he monhly daa are accumulaed o give ax revenue and ax receips oals for he respecive ax years. The resuling annual adused oher persons ax for ax year is denoed O. Alhough various macroeconomic variables have been used over he years o forecas O, he one ha has been a he hear of all of he models considered is a measure of household enrepreneurial income calculaed by Saisics New Zealand. In essence, he Treasury forecasing model assumes ha forecass of O follow he recursion I 1 O = O 1 1+ + I 1 I A where I denoes enrepreneurial income, and A denoes any udgmenal or policy adusmens made. The annual ax-year forecass of revenue and receips ha resul are hen spread across budge years using inerpolaion based on he monhly seasonal paerns observed. Finally, hese forecass are accumulaed ogeher wih he annual forecass for income summaries and chariable donaion rebaes o yield an overall forecas for ne oher persons ax. 2.1.3 Oher ax ypes Mos of he oher ax ypes use models similar o hese. Fringe benefi ax, company ax and GST forecasing models, for example, are based on growh raes of compensaion of employees, oal operaing surplus and nominal domesic consumpion respecively. Forecasing models for some of he smaller ax ypes are even simpler wih forecass of excise duies, for insance, based on rend growh esimaes. A differen model is adoped for forecasing wihholding ax on ineres. This uses a regression relaionship wih compensaion of employees, domesic consumpion, house prices and ineres raes as regressor variables. WP 07/02 AN ANALYSIS OF TAX REVENUE FORECAST ERRORS 5

3 Towards a model framework The ax revenue forecasing procedures used by he Treasury sugges a muliplicaive model for he monhly, quarerly or annual levels of axaion revenue considered. A simple example is Y β = αx e (1) where Y denoes a paricular ax revenue, X denoes a macroeconomic predicor such as GDP, and e denoes muliplicaive error which varies abou a mean of uniy. If he β ransformed macroeconomic predicor X can be hough of as a proxy for he relevan ax-base, hen α can be inerpreed as a mean ax rae. Oher muliplicaive variables can be included and parameers such as α and β may also be ime-dependen. Monhly or quarerly variables may have seasonal variaion and all are likely o be affeced, o some degree, by longer-erm economic cycles. In erms of coninuously-compounding growh raes, (1) becomes Δlog Y = β Δlog + ε (2) X where Δ denoes he difference operaor ( ΔZ = Z Z 1 ), he parameer β can be inerpreed as an elasiciy, and he ε now correspond o addiive errors (possibly saionary) wih zero mean. Noe ha he approximaion yields log ( 1+ x) x (x small) (3) ΔlogZ Z Z Z 1 1 o a good degree of approximaion provided he righ-hand side of he above (a simple growh rae) is small. Throughou his repor, we mainly consider he more-commonly used compound growh raes of he form Δlog Z raher han heir simple growh rae equivalens. The reasons for his are largely echnical convenience and a direc link o coninuous ime growh models, bu lile is los in adoping eiher definiion since hey differ very lile in pracice. Consider (2) and forecasing he ax revenue growh rae Δlog Y. If he predicor Δlog X is known and he ε are independen, in addiion o having zero mean, hen he bes predicor of Δlog Y is given by ΔlogYˆ = β Δlog (4) X or, using he above approximaions and simple growh raes, ˆ = ˆ 1 1 1 X X Y Y + β (5) X 1 provided he various growh raes are relaively small. Forecas funcions such as hese lie a he hear of he Treasury's curren ax forecasing mehods described in Secion 2.1. WP 07/02 AN ANALYSIS OF TAX REVENUE FORECAST ERRORS 6

This suggess ha he Treasury's ax forecasing mehods could be regarded as opimal predicors for simple models ha are suiable varians of (1). This linkage beween a model, such as (1), and is forecas funcion is no unique since oher models can be found ha will yield he same forecas funcion (5). However he simpliciy of (1) and is growh rae model (2) make i a suiable saring poin for a model framework wihin which he Treasury s ax forecasing mehods can be embedded. This is he sraegy ha has been adoped here. 3.1 Alernaive models Schoefisch (2005) noes ha a number of he Treasury s ax revenue forecasing mehods assume ha he elasiciy β in (1) is idenically uniy. He quesions his assumpion, noing ha β may well depend on he phase of he economic cycle which could impac differenially on he various componens of GDP. In addiion, he parameer α is also likely o change slowly over ime o accommodae srucural changes in New Zealand's economy. Such consideraions sugges a more general model of he form Y β = α X e (6) where Y, X and e are as in (1), bu now he mean rae α and elasiciy β are assumed o vary over ime. In his case, models for he evoluion of α and β are needed in order o use (6) for forecasing. I is also possible ha seasonal facors will need o be included in (6) in he case of monhly or quarerly daa. The analysis of such a model would direcly address many of he recommendaions made in Schoefisch (2005). In erms of growh raes, (6) becomes ( Δβ ) log X ε Δlog Y = Δlogα + β Δlog X + + (7) where, as in (2), he ε correspond o addiive errors wih zero mean. In pracice, he mean rae α and elasiciy β are likely o evolve smoohly over ime and so Δlog α, Δβ will ypically be small, or have small variance, relaive o he oher sources of variaion. Such consideraions lead o modelling log α as a sochasic rend wihin a suiable srucural ime series framework. See Harvey (1989) for a full discussion of his general class of models. Many oher variaions of (6) are possible using differen ax-base proxies where X is replaced by geomeric combinaions of one or more macroeconomic regressors and heir lagged values. An example is 1 Y β 0 β X 1 1 = αx KX β p p e which allows he ax-base proxy o be a moving geomeric combinaion of curren and pas values of he macroeconomic driver X. This model can also be framed in erms of growh raes and, in his case, a long-run co-inegraing relaionship beween log Y and log X may be needed. Alernaively, hese macroeconomic drivers could be replaced by lagged values of Y or some oher ax revenue series. A simple example is Y = β α Y 1 e WP 07/02 AN ANALYSIS OF TAX REVENUE FORECAST ERRORS 7

where he α are assumed o be evolving smoohly over ime and he log e are whie noise errors. If β < 1, hen log Y reduces o a convenional ime series rend plus addiive error where he laer is a firs-order auoregression. If β = 1, hen he ax revenue growh raes Δlog Y follow a rend plus error model. This simple model is readily generalised o include oher more complex ime series models. In shor, he general model (6) provides a flexible modelling framework for forecasing ax revenues and heir growh raes, eiher in erms of suiable macroeconomic drivers and heir lags ha are proxies for he ax-base, or ime series models involving us he ax revenue alone, or a combinaion of boh. 4 Forecas error decomposiions A beer undersanding of he source and naure of he Treasury's ax revenue forecasing errors is an imporan prerequisie o building more accurae and robus ax forecasing models. To his end, and as recommended in Schoefisch (2005), we now develop decomposiions of he Treasury's pas ax revenue forecasing errors ino suiable srucural componens. The decomposiions considered include: he disaggregaion of oal ax revenue forecas errors ino heir componen ax ypes (Secion 4.2); he decomposiion of individual ax revenue forecas errors ino a componen due o forecasing he macroeconomic variables ha are a proxy for he ax-base used, and a componen due o forecasing he raio of ax revenue o proxy ax-base, or ax raio (Secion 4.3); a furher decomposiion (Secion 4.4) of he ax raio ino a rend measuring an underlying mean ax rae and a random error componen. The former provides a benchmark agains which ax raio forecass can be benchmarked and he laer is ypically non-informaive noise ha is prediced only by is mean. As discussed in Secion 2.1, he Treasury's ax revenue forecass are ypically variaions of simple muliplicaive models ha proec ax revenue forward a he same rae as he growh forecas of he macroeconomic aggregae ha serves as a proxy ax-base. As migh be expeced, wihin each ax ype no one mehod has been used consisenly and, insead, he Treasury's mehods have been refined and modified over ime. In addiion, any forecass produced by he mehodology described in Secion 2.1 are furher modified by udgemenal facors, boh a he individual ax level by he Treasury s ax forecasing uni, and subsequenly a an aggregae level by an inernal review panel of senior Treasury saff. As a consequence, he Treasury's ax revenue forecasing models and processes canno be replicaed exacly. These consideraions have led us o consider a simple benchmark model for each ax ype ha faciliaes he decomposiions referred o above. Alhough based on similar axbase proxies, hese models are no he same as he Treasury models, bu do have he virues of ransparency, since hey have a simple srucural inerpreaion, and consisency over ime. The benchmark model provides a srucural decomposiion of he individual ax revenues agains which he Treasury's forecass can now be assessed. WP 07/02 AN ANALYSIS OF TAX REVENUE FORECAST ERRORS 8

4.1 Benchmark model The general srucure of axaion suggess he simple model Y = R X, R α e = (8) which is a special case of (6) wih uni elasiciy β = 1. Here he macroeconomic variable X is o be regarded as a proxy for he relevan ax-base of he ax concerned, he ax raio R = Y /X is he observed raio of ax revenue o proxy ax-base, and he muliplicaive errors e have uni mean so ha α can be inerpreed as an underlying mean ax rae. A simpler version of his model was used in O'Neill (2005) o decompose ax forecasing errors ino suiable componens wih X se a Canada's nominal GDP. The sysemaic componen α is assumed o evolve smoohly over ime o accommodae minor policy changes whose effecs are phased in gradually over ime, and also any discrepancies beween he proxy ax-base X and he underlying rue ax base. To mainain his assumpion, i is possible ha any abrup one-off changes will need o be accouned for by prior adusmens made o he daa. See Secion 5. Wih hese assumpions and caveas in mind, he muliplicaive model (8) can now be ransformed ino he addiive model log Y logr + log X =, = logα + log R ε (9) where log α is a rend and he ε = log e will be assumed o be saionary whie noise, independen of α. This simple model for log R belongs o he general class of srucural ime series models discussed in Harvey (1989). Decomposiions of he Treasury's ax forecasing errors can now be underaken using his simple srucural model and is componens as a benchmark. 4.2 Decomposiion of oal ax revenue by ax ype Consider he case where he oal ax revenue is denoed by Y() and he componen revenues by Y () (=1,,m) so ha Y m () = Y () = 1 and he Y () follow models of he form (8). Given forecass Ŷ () of he individual componens Y (), a forecas of he aggregae Y() is given by Yˆ m () = Yˆ () = 1 yielding he forecas error decomposiions Yˆ m ( ) () Y () = Yˆ () Y () = 1 WP 07/02 AN ANALYSIS OF TAX REVENUE FORECAST ERRORS 9

and, in erms of proporionae errors, Yˆ () Y () Y () = m = 1 where P () Y () Y () P () Yˆ () Y ( ) Y () = measures Y () as a proporion of he oal ax revenue Y(). Using he approximaion (3), he laer decomposiion can now be framed in erms of logarihms o yield m ( ) ( () ) = P () e Y () e Y = 1 (10) where e ( Y () ) logyˆ () logy () = (11) wih he e(y ()) defined similarly. As before, he qualiy of he approximaion is such ha hese errors can be inerpreed as simple proporionae errors. Noe ha he forecas errors defined have he opposie sign o hose more commonly adoped (log Y() log Ŷ() for proporionae errors and Y() Ŷ() for acual errors). However definiion (11) allows for more naural inerpreaions wih posiive errors implying over-forecasing and negaive errors implying under-forecasing. In Secion 6.1, i is shown ha he P () evolve slowly and vary lile over ime by comparison o he e(y ()). In his case, he proporionae forecas error for he oal ax revenue Y() has mean-squared error given by where [ e( Y () )] var[ e( Y () )] ( bias[ e( Y ( ) )]) 2 MSE = + (12) m 2 [ ( Y () )] = P () var e Y () = 1 [ ( )] + 2 P () Pk () cov[ e( Y () ) e( Yk () )] var e, bias m [ ( )] [ e( Y () )] = P () E e Y () = 1 wih var[.], cov[.], E[.] denoing variance, covariance and expecaion respecively. The decomposiions (10) and (12) provide an appropriae framework for evaluaing he relaive conribuions of he various ax forecasing errors o boh individual ax componens and heir aggregaes. Noe ha hese paricular decomposiions are no dependen on he benchmark model (8). < k WP 07/02 AN ANALYSIS OF TAX REVENUE FORECAST ERRORS 10

4.3 Separaing ou he macroeconomic forecas errors Consider he benchmark model (8) where, as before, Y denoes a paricular ax revenue, he macroeconomic variable X is a proxy for he ax-base, and R is he associaed ax raio. Given forecass Ŷ, Xˆ of Y, X respecively, a naural forecas of R is given by Yˆ ˆ = (13) Xˆ R so ha he hree forecass saisfy he simple relaionship Yˆ = Rˆ Xˆ. (14) If R, X are independen or, more generally, if hey are condiionally uncorrelaed given pas daa, hen he bes predicors of Y, R and X will saisfy (14). These and oher consideraions lead us o assume ha (14) holds for he forecass considered in his repor so ha R can be forecas by he simple predicor (13). From he muliplicaive relaionships (8) and (14) we obain logyˆ logy = logrˆ logr + log Xˆ log X or, using he noaion inroduced in (10), ( Y ) e( R ) e( X ) e = + (15) where hese quaniies are he proporionae forecas errors for each componen. Noe ha (15) addiively decomposes he oal ax revenue proporionae forecas error e(y ) ino wo componen proporionae errors, one due o forecasing he ax raio R and he oher he macroeconomic variable X used as he ax-base proxy. Muliplying (15) by Y and using he approximaion (3) yields he acual forecas error decomposiion Yˆ Y = X ( Rˆ R ) + R ( Xˆ X ) which shows he influence of he respecive errors in absolue erms. The mean-squared proporionae forecas error of Y is given by [ e( Y )] var[ e( Y )] ( bias[ e( Y )]) 2 MSE = + (16) where [ e ( Y )] = var[ e( R )] + var[ e( X )] 2cov[ e( R ), e( X )] var + [ e( Y )] E[ e( R )] E[ e( X )] bias = +. This decomposiion and (15) provide a suiable framework for separaing ou he forecas errors for he ax raio R from hose of he macroeconomic ax-base proxy X. They can also be used in conuncion wih decomposiions (10) and (12) o examine he relaive conribuions of he various ax-raio forecasing errors. WP 07/02 AN ANALYSIS OF TAX REVENUE FORECAST ERRORS 11

4.4 A model-based decomposiion of ax forecas errors Here we consider he ax raio R given by he benchmark model (8). Since α has been assumed o be independen of ε and e = exp(ε ) has uni mean, he bes forecas of R will always be he same as he bes forecas of α. In essence, he benchmark model decomposes R ino a srucural forecasable componen α and a non-informaive noise componen e. These consideraions lead us o assume, in addiion o (14), ha he forecass considered in his repor saisfy ˆ α = Rˆ (17) where Rˆ, αˆ are he forecass of R and α respecively. Now (6) and (17) yield he decomposiion log Rˆ log R = log ˆ α logα ε or ( R ) e( ) n e = α + (18) where hese proporionae error componens are defined in he same way as before and n = ε is non-sysemaic whie noise error. Here he mean-squared proporionae forecas error of R is given by [ e( R )] var[ e( R )] ( bias[ e( R )]) 2 MSE = + (19) where, o a good approximaion, [ e( R )] = var[ e( α )] var[ n ] var + bias [ e( R )] = E[ e( α )] provided E(ε ) is close o zero and he forecass forecass of R. Rˆ are closely correlaed o he opimal These decomposiions can now be used o deermine he relaive conribuions, wihin ax revenue ypes, of he proporionae forecas errors of he ax raio rend α and, us as imporanly, he naure and size of he non-informaive noise componens ε. They can also be used in conuncion wih he decomposiions given in he previous secions o beer undersand he iner-relaionships beween he various ax ypes and heir error componens. 5 Daa Tax forecas daa have been aken from he various Budge Updae and December/Half- Year Updae publicaions produced by he Treasury since he 1994 Budge. The daa were colleced from spreadshees sored on Treasury compuers and cover all of he ax ypes forecas by he Treasury, he Minisry of Transpor (road user charges and moor vehicle licensing fees) and he Minisry of Economic Developmen (energy/exhausible resource levies). WP 07/02 AN ANALYSIS OF TAX REVENUE FORECAST ERRORS 12

The Treasury has mainained a daabase of forecass of macroeconomic variables ever since he 2001 Budge and his daabase was used as a primary source. Forecass of macroeconomic variables prior o he 2001 Budge were colleced from macroeconomic and ax forecasing spreadshees sored on Treasury compuers. The daa colleced were resriced o he primary nominal macroeconomic variables used in he Treasury s ax forecasing models such as GDP, compensaion of employees, consumpion, operaing surplus and enrepreneurial income. Up unil he end of 1999, he Treasury produced ax and macroeconomic forecass for he curren year and he following hree years. From 2000 onwards, his was exended o include a fourh year. Alhough daa across all hese forecas horizons were colleced, he analysis underaken focuses on one-year-ahead forecass, since hese are he forecass agains which ax oucomes are measured by he New Zealand Parliamen. Acual ax oucomes are calculaed by he Treasury each monh and published on he Treasury's websie. The macroeconomic daa oucomes used in he analysis were he laes available from Saisics New Zealand. 5.1 Daa issues The hisorical ax and macroeconomic daa analysed were, for he mos par, final oucomes raher han unrevised esimaes. However, he corresponding forecass were ofen based on unrevised daa available a he ime or, in some cases, were prepared in advance of significan policy changes. To correc for hese and oher such effecs, a number of prior adusmens were made o he forecass. Deails are given below. Policy changes Tax forecas daa have been adused for policy changes ha affeced he final oucomes, bu which were no known abou a he ime he forecas was made. This is o ensure ha he forecass, and he acual ax oucomes hey are being compared wih, were prepared on he same policy basis. For example, PAYE forecass prepared for he 1995 Budge have been adused for he personal income ax rae reducions of July 1996 and July 1997 ha were announced in December 1995. In he 1997 Budge, he July 1997 personal income ax rae reducions were deferred unil July 1998, so forecass prior o ha have also been adused for his deferral. The adusmens used are he acual policy cosings ha were available a he ime he new policy was announced. We have decided o use hese adusmens, raher han recalculae he acual effec of he policy change, as hese adusmens are he closes we can ge o he acual adusmens ha he Treasury s ax forecasers would have made had he new policy been known abou a he ime of forecasing. Macroeconomic daa impuaion As previously menioned, some of he earlier forecass of macroeconomic variables were colleced from macroeconomic and ax forecasing spreadshees. Alhough we canno be absoluely sure ha hese were he final published macroeconomic forecass, in all cases he ax forecass in hese spreadshees mached he final ax forecass made and so i seems reasonable o assume ha hese spreadshees also conained he final macroeconomic forecass. WP 07/02 AN ANALYSIS OF TAX REVENUE FORECAST ERRORS 13

Some of he forecass of macroeconomic variables were sored as growh raes raher han as levels. We have convered hese forecas growh raes ino forecas levels using base level oucomes ha were known a he ime, bu have no way of knowing how close hese reconsruced forecass are o he acual final macroeconomic forecass of levels. Neverheless, any discrepancies inroduced are likely o be small. Daa revisions Macroeconomic daa available from Saisics New Zealand are subec o revision. For example, iniial esimaes of nominal GDP for he year o June 1999 were around $100 billion, whereas he laes esimae is some 5% higher han his a around $105 billion. For he mos par, revisions o nominal GDP in he period under examinaion have been upward, alhough his is no necessarily so for all of he componens of GDP. The Treasury's macroeconomic forecass ypically apply forecas growh raes o he hisorical macroeconomic daa available a he ime, he mos recen of which will ofen be an unrevised or parially-revised esimae. To ensure ha he analysis is no unduly influenced by such daa revisions, each macroeconomic forecas was muliplied by he raio of he mos recen macroeconomic esimae o he unrevised or parially-revised esimae available a he ime of making he forecas. This simple correcion facor is based on he assumpion ha he macroeconomic forecas is, a leas approximaely, he produc of he las available value of he macroeconomic variable and a forecas growh facor. Adusing he macroeconomic forecass by scaling in his way is less han perfec since he forecass may no be scale-invarian. For example, he composiion of nominal GDP has been revised over he inervening period, somehing ha scaling does no necessarily accoun for. Perhaps he bes way o adus he economic forecass for subsequen daa revisions would be o repea each forecas using he laes daa. This would mean using he same forecasing models, he same forecasers and replicaing he udgmenal processes ha were used a he ime. Such a ask would be ime-consuming and cosly and may no lead o beer adusmens. Scaling has he advanages of being quick, simple and ransparen. Tax revenues are no subec o revision and so heir forecass did no have o be adused. However, he forecass of he ax raios R were implicily adused since hey are based on (13) which is he raio of he ax revenue forecas o he adused forecas of is associaed macroeconomic variable or ax-base proxy. Daa mis-alignmen The Crown's accouns are prepared on a Budge year, ie June year, basis. Thus ax revenues are repored as June year oals and ax revenue forecass are prepared accordingly. However, some macroeconomic variables are forecas solely in March years. In hese cases, he March year forecass and acual oucomes have been used in he analysis and no aemp has been made o correc for any emporal mis-alignmen. WP 07/02 AN ANALYSIS OF TAX REVENUE FORECAST ERRORS 14