Applying Auto-Regressive Binomial Model to Forecast Economic Recession in U.S. and Sweden

Transcription

1 Applying Auo-Regressive Binomial Model o Forecas Economic Recession in U.S. and Sweden Submied by: Chunshu Zhao Yamei Song Supervisor: Md. Moudud Alam D-level essay in Saisics, June 200. School of Technology & Business Sudies, Dalarna Universiy

2 Conens ABSTRACT... Inroducion...2. Background Aim Daa and Mehods Daa descripion The recession indicaor The explanaory variables Mehods AR logi model Parameers esimaion Parameers es Forecasing Model selecion crieria Resuls Model selecion and in-sample resuls Empirical analysis o U.S Empirical analysis o Sweden Ou-of-sample resuls Ou-of-sample resuls of U.S Ou-of-sample resuls of Sweden Conclusion...23 References...26

3 ABSTRACT In his paper, we consruc an auo-regressive AR logi model which can be applied in he predicion of he U.S. and Sweden economic recessions. The predicive variables of his model vary beween U.S. and Sweden. The prediced power of his AR logi model is checked in erms of in-sample and ou-of-sample performance. We find ha he goodness-of-fi of his model o daa is remarkable and his model can give reasonable degree of accuracy in he predicing U.S. and Sweden business cycle. Keywords: Auo-regressive logi model, Business cycle, Recession

4 Inroducion. Background The laes recession in U.S. occurred ever since he bankrupcy of Lehman Brohers Holding Inc. due o he sub prime morgage crisis in 2008 and lased more han wo years. The consequence of his recession is quie significan which is viewed hrough several economic performance indicaors, e.g. 0% unemploymen rae and a deeply decline of real GDP. Alhough nowadays he governmen of U.S. declares ha he recession in U.S. has end, he economy in Europe is experiencing a new round of sharply decline, such as he 200 Euro crisis including Greece, Ireland, Spain, and Porugal. Thus, how o find an efficien ways o model and forecas economic recession is nowadays a popular opic among policy makers and researchers. A binary AR model has been considered as a useful model o forecas economic recession in U.S., hus we wan o invesigae wheher his kind of model is also applicable in European counries such as Sweden. Before going hrough he definiion of recession, i is necessary o inroduce a general idea abou business cycle, since recession is regarded as a par of business cycle. In he view of macroeconomics, business cycle is a well-known financial variable o measure he economy-wide flucuaions in producion over a long period. These flucuaions exi when he economic environmen changes from an expansion o a recession and vice versa. Generally, people are more concerned wih recession raher han expansion, since a majoriy of papers are focus on discussing recession. The definiion of recession varies beween differen research insiues and herefore no exac definiion of recession exiss. One commonly used definiion is provided by he Naional Bureau of Economic Research NBER in US. In heir words, a recession is a significan decline in economic aciviy spread across he economy, lasing more han a few monhs, normally visible in real GDP, real income, 2

5 employmen, indusrial producion, and wholesale-reail sales. From his definiion, we can see ha being able o describe and predic a recession is of grea ineres for he people. More specifically, policy makers who work on macroeconomic issues are more concern wih he rend of real GDP and real income, since a wrong predicion of he fuure of economic sae leads o he ineffecive of policy iself. And he employees in indusry are mosly ineresed in he sae of employmen, and so on. Thus, a reasonable assessmen of economic aciviies such as business cycle is of grea imporance o he whole sociey. In modeling recession, AR binomial model has been found very useful. Kauppi and Saikkonen 2005 predic U.S. recessions wih dynamic binary response models. They develop a unified model framework ha accommodaes mos previously analyzed dynamic binary ime series models as special cases Kauppi and Saikkonen, The previously analyzed works are mainly focus on how o improve he predic model from saic one o dynamic one. Sarz 2006 gives an empirical sudy on U.S. recession by using binomial AR moving average BARMA models. The disinc advanage of BARMA model is o eliminae he curse of dimensionaliy found in long lag Markov models. Alhough, he BARMA 2, 2 model fi well wih real daa, comparing o radiional Markov model and oher BARMA models wih differen lagged values, he auhor did no repor he predicive power of his model. Thus, we lack he evidence ha how well his model perform when applying o predic he process of recession. Nyberg 2009 exend he AR binomial model from univariae case o bivariae case. Unlike some ypical cases in dynamic models based on laen variables Chauve and Poer, 2005, he measures he sae of he economy in erms of recession periods defined by he business cycle and he growh rae cycle indicaors. Besides he dependen variable and is AR erm, he model also akes ino accoun oher financial variables as predicor. The empirical applicaion in U.S. suggess he bivariae model ouperform he univariae model. Since he mos empirical resuls are only available for he U.S. recessions, how he AR binomial model performance in oher homogeneous counries is of grea ineres. Yasuhiro Omori 2003 use discree hp:// 3

6 duraion model wih AR random effecs o analyse Japanese diffusion index. Ulf Hamberg and David Versändig 2009 apply saic logisic regression model o forecas business cycle under Swedish condiions. Unlike wha Hamberg and Versändig 2009 did in heir hesis, firs we apply he AR logi model in U.S. o examine wheher he model works well. Second we exend he saic logisic regression model o a binomial AR model and hen applying he model wih real daa se from Sweden. By using our model, we forecas he saus of economy in hree quarers ahead. To he bes of our knowledge, hese have no been done previously. This essay generally includes four secions. Secion one inroduces he background and basic definiions hen proposes he aim of his essay. Secion wo describes he daa we employed and he mehod we use. Secion hree presens he sudy resuls which we divided ino wo pars, in-sample forecasing resuls and ou-of-sample forecasing resuls. Secion four concludes..2 Aim The aim of his essay is o examine wheher he AR logi model which is found useful for forecasing U.S. recession is also applicable o forecas he business cycle of he European counries, like Sweden. If yes, how well he performance of his model does. 2 Daa and Mehods 2. Daa descripion All our models are based on binomial AR model, hus he recession indicaor has a binary form which means i can ake only one of wo values, or 0.Since 4

7 modeling recession is of ineres raher han modeling expansion, we le denoe economic recession and le 0 denoe economic expansion. All he daa applied in our analysis is quarerly daa. Esrella and Mishkin 998 sugges ha quarerly daa are more preferable since he ime series become less variable, hus reducing noise and usually leading o a beer goodness-of-fi in he model. 2.. The recession indicaor The recession indicaor applied in U.S. case is he sae of business cycle repored by NBER. According o he definiion given by NBER Moore, 967, business cycle peak daes mark he end of a period of expansion and he beginning of a period of conracion; rough daes mark he end of a period of conracion and he beginning of a period of expansion. Thus, we label he period from a peak o he following rough as which indicaes a process of recession and label he period from a rough o he following peak as 0 which means an expansion of economy. The recession indicaor applied in Sweden is he sae of business cycle repored by Economic Cycle Research Insiue ECRI 2. Since his insiue also repors he business cycle of oher counies besides U.S. and Sweden, he idenificaion of peak and rough beween differen counries is consisen when applying hem o models The explanaory variables Esrella and Mishkin 998 give a deailed discussion abou he financial variables as leading indicaors when predicing U.S. recessions. Series such as ineres raes and spreads, sock prices, and moneary aggregaes can play an imporan role in macroeconomic predicion. Among hem, he erm spread beween a shor-erm and a long-erm ineres rae and sock prices indexes emerge as he mos useful simple

8 financial indicaors. Hence, i is of ineres o examine wheher he erm spread and sock price indexes have predicive power in our models. Besides, we also employ he sock marke reurn and German erm spread, he same explanaory variables as Nyberg 2009, as he predicors in our models. The lagged values of all hese variables are he same as in Nyberg Specifically: Term spread a ime TS: TS R i where R is en-year reasury bond yield rae wih consan mauriy and i is hree-monh reasury bill rae on secondary marke 2. The expeced sign of his variable should be negaive. Since for a given long-erm yield, a large value of yield spread indicae a decreasing of shor-erm yield which associaed wih increased expecaion of srong economic aciviy. Sock marke reurn SMR: SMR logs & P500 where S&P 500 is he sandard and poor 500 index 3. To make sure he variable is saionary, which is he basic assumpion of ime series model, we ake he log-difference value of S&P500 index. The expeced sign of his variable should be negaive. Since an increasing sock marke reurn indicaes a high expecaion of invesmen reurn and herefore associaed wih an expansion of economic. German erm spread GTS: GTS R GE i GE German erm spread is consruced as he difference beween 0-year Federal securiy 4 and he hree-monh money marke rae 5. The expeced sign of his variable should be negaive as well. The reason for his assumpion is he same as erm spread. We generally apply he same variables employed in U.S. case when considering he Swedish case. Hamberg and Versändig 2009 ake ino accoun six differen explanaory variables as he independen variables; hey are changes in a sock price hp://research.slouisfed.org/fred2/daa/gs0.x 2 hp://research.slouisfed.org/fred2/daa/tb3ms.x 3 hp://finance.yahoo.com/q/hp?s%5egspc&a00&b3&c950&d&e23&f2009&gm 4 hp:// hp://sas.oecd.org/index.aspx?queryid86 97M-972M8 5 hp:// 6

9 index, oil price, a confidence indicaor, European GDP, residenial building as well as he spread beween a shor-erm and a long-erm ineres rae. However, some variables used by Hamberg and Versändig 2009 do no work in our model, we jus use he erm spread, sock marke reurn and erm spread in German as he leading indicaors. Specifically: We use OMX Sockholm All Share OMXS-PI including all he shares lised on OMX Nordic Exchange Sockholm as he indicaor of sock price. One advanage of OMX Sockholm PI index is o reduce he impac of firm-specific risk in accordance wih normal porfolio heory Bodie, Kane and Marcus, 2005, since oher sock marke index may jus include a small number of socks issued by specific companies. We ake is log-difference ransformaion o make i consisence wih wha we did in U.S case. The expeced sign of his variable should be negaive. SMR log OMX PI Swedish erm spread STS: STS R SW r SW Where SW R is he long-erm ineres rae measured by Swedish governmen bond wih 0 years mauriy 2 and SW r is shor-erm ineres rae measured by Swedish Treasury Bill wih 3 monhs mauriy 3. The expeced sign of his variable should be negaive. 2.2 Mehods 2.2. AR logi model Consider having a random process y, recession indicaor, which is binary value, ha is, 0- value and his can be expressed in indicaor funcion form, hp:// 2 hp:// 3 hp:// 7

10 y I recession a period So, condiional on he informaion se Ω { y-, x h h }, where y is he lagged value of y and x is he explanaory variable wih lag h, h y can be fied as a binary model wih canonical link, E y Ω p logi p η X -h β Where X -h is he design marix and simplifies he noaion of lagged explanaory variables because differen explanaory variables can have differen lags and p is he condiional probabiliy ha economy is in recession a period. Model has been used in former economic recession sudy Hamberg and Versändig, A raional exension o model is a dynamic specificaion by adding a lagged value of recession indicaor y logi p η µ y X -h β 2 Based on former sudy Kauppi and Saikkonen, 2005 and model 2, a firs order AR logi model can be proposed logi p η α η µ y X -h β Parameers esimaion Suppose, here are observed processes y TL, x i, i n, TL where T is he number of observaion and n is he number of explanaory variables. Define a corresponding parameer vecor α, µ, β where β β Lβ. Then model 3 can be wrien as n E y Ω p 4 8

11 9 n n,, 0 β β β µ α j - -i x x y η η p logi L Based on he binomial model 4, he condiional likelihood funcion has he following form by muliplying individual conribuion T y - y - T y y T η logi - η logi p - p l L. The esimaed parameers can be carried ou direcly by Newon-Raphson mehod. In order o simplify he calculaion, he log-likelihood funcion should be considered as T - - logi log y logi log y L log LL η η. The score funcion of log-likelihood funcion is,,,,, n 0 β β β µ α LL LL LL LL LL LL S L The large sample heory can be applied o he parameers esimaed and is pracical use is he calculaion of informaion marix, T ' T p S S T lim I According o Newon-Raphson mehod, he formal ieraive funcion is n n S I And assume here is an iniial value 0 of parameer vecor, he ieraive value can be calculaed ou by 0 S I When 0 n n as n, or 0 S as n, he process of ieraion can be over. And finally, he parameers are esimaed ou.

12 2.2.3 Parameers es In order o es he significance of correlaion coefficien under he null hypohesis H : 0, Wald es can be used in large sample, and is has form ˆ 0 z, SE ˆ 0 Where SE ˆ is he sandard error of parameer esimaes, ˆ, calculaed from he non-null model. The Sandard Error of ˆ can be esimaed by Fisher s Informaion marix, I, so he Wald es can be wrien as z ˆ 0 diag I, Where diag is he diagonal elemen of he parameer marix. If he p-value, calculaed by quanile z under normal disribuion, is larger han he significan level under wo side es, he null hypohesis equals o zero is acceped, oherwise, he esimaed value of is acceped Forecasing Kauppi and Saikkonen 2005 menioned ha one-period or muli-period forecas can be made based on he explici formula. Based on he given informaion se Ω { y, x h }, he forecas condiional recession probabiliy given as - h - - p logi η logi αη µ y X -h β 5 where h is he forecas horizon. A ime -, he forecasing funcion 5 can give one period ahead forecas direcly. Bu i will be a lile complicaed bu sill sraigh forward, if more han one 0

13 period ahead forecas is needed. For simpliciy, suppose he forecas horizon h equals o wo, he ieraive process of wo-period ahead forecas is given as following. The linear predicor based on informaion up o ime -2 is η αη µ y X -2β α αη 2 α η 2 2 µ y 2 i i α 2 X µ y i -3 β µ y α -i-i X X β The informaion of y can no be obained a forecasing ime -2, bu he - condiional probabiliy p equals logi αη 2 µ y 2 X -3β -2 β, which will be used o calculae he predicive recession probabiliy p p E logi η p logi η y p logi η y 0 So, he righ hand side of forecasing funcion 5 can make wo-period ahead forecas ieraively a forecasing ime -2. Thus, in order o make h-period ahead forecas, he AR erm should be ailored o mach informaion available a forecasing ime -h. The process of calculaion is lile complicaed bu can be solved by repeiively. The general formula is h h i i η α η h α µ y i α X -i- i p E logi y 0, η h i p β y - i y i i i p logi η y L y h In conclusion, he general form of forecasing funcion 5 can give one period ahead forecas direcly for he condiional probabiliy p P y a ime -, and h period ahead forecas ieraively a forecasing ime -h, using informaion up o ime -h. 6

14 2.2.5 Model selecion crieria Model comparison can be carried ou by evaluaing he goodness-of-fi measures, one of which is Schwarz-Bayesian informaion crierion, BIC Schwarz, 978, log T BIC -logl K 2 Where logl is he log-likelihood funcion, K is he number of esimaed parameers, and T is he number of observaions. The model is fiing beer when he value of his saisic is declining. A faciliaed measure of evaluaing he performance of predicive model is he percenage of correc predicion. The goodness-of-fi of model o daa can be checked visually by a cross able as able 4. Finally, sandard errors and p-values of esimaed parameers are also considered when checking he goodness-of-fi of specificaion models. 3 Resuls In his par, he performance of AR logi model 3 is checked by forecasing business cycle recession periods for U.S. and Sweden respecively. The predicive power of AR logi model can be examined by corresponding figures and cross ables. We are mainly ineresed in ou-of-sample performance of he model, bu i is also very helpful o check is in-sample forecasing firs in choosing he explanaory variables and corresponding lags. 3. Model selecion and in-sample resuls 3.. Empirical analysis o U.S. The in-sample performance of AR logi model can be evaluaed by he goodness- 2

15 of-fi o he daa and he accuracy of recession predicion. Based on daa descripion secion, explanaory variable candidaes are U.S. erm spread, TS, sock marke reurn, SMR, and German erm spread, GTS. So, he design marix in model 3 has he form, X -h, TS -i, SMR- j, GTS -k And he proposed model is logi p η αη µ y β β β β GTS η 0 TS i 2SMR j 3 -k 7 Afer he selecion of explanaory variables, he nex sep is o fix he lags of he variables of model 7 ha fi he daa bes. Par of he in-sample resuls of model 7 for direcly forecasing U.S. economy saus one quarer ahead and corresponding selecion crieria values are lised in able. Table he summaries of model 7 wih differen lags Predicor Lag i, j, k,,,,2 2,, 2,2,2 inercep η y TS i SMR j GTS k Log-likelihood BIC From able, he coefficiens of SMR - j in he four models are posiive which does no make economic sense, and he posiive sign of he parameer goes hrough all he models wih differen lags no jus limied o he models refer o able. So, he explanaory variable, SMR, should be removed from he final model. Using quarerly daa from he firs quarer of 97 o he fourh quarer of

16 The modified model of 7 is logi p η αη µ y β β β GTS η 0 TS i 2 - j 8 And a par of summaries of differen models, explanaory variables wih differen lags, is lised in able 2. Table 2 he summaries of model 8 wih differen lags Lag i, j Log-likelihood BIC Lag i, j Log-likelihood BIC, , , , , , , , , Based on he summary, explanaory variables wih lags 2, 3 have he bigges prediced power in recession because of he smalles value of BIC. And a summary of he bes model is in able 3. Table 3 he summary of model 8, explanaory variables wih lags 2, 3 Predicor Lag 2, 3 Crieria inercep Log-likelihood η BIC y 5.8 TS i GTS j And he sign of he wo explanaory variables are negaive, ha is, when U.S. erm spread or German erm spread is going down, he probabiliy of U.S. economy will be in recession goes up. So, he beer AR logi model used o forecas he U.S. economy is logi p η η η 5.8y TS GTS 3 9 4

17 In order o illusrae he in-sample performance of model 9, figure shows he esimaed probabiliy, he solid line, ha he U.S. economy will be in recession in one quarer ahead and he acual U.S. economy recession period, he shadow areas. Figure probabiliy of recessions and acual recession periods of U.S. Figure shows six periods of recessions in shadow areas. The probabiliy line maches he firs hree shadow areas well, ha is, model 9 can forecas he recessions happened from he firs quarer of 974 o he firs quarer of 975, from he second quarer of 980 o he hird quarer of 980 and from he fourh quarer of 98 o he fourh quarer of 982 well. Bu model 9 forecass he laer hree recessions abou one quarer laer han recession has happened. And we will noice ha model 9 forecass a recession happens a abou he firs quarer of 98 if he hreshold is 0.5, bu acually no recession happens a ha period. Overall, he in-sample forecasing of model 9 can give clear signals of recessions, which can be proved in he following mehod. The percenage of correc predicion can be checked by cross able 4, while he hreshold is fixed a 0.5. Table 4 he percenage of correc predicion of model 9 Observed Prediced Economy Correc Economy Expansion Recession Percenage Expansion % Recession % Overall Percenage 93.5% The overall percenage of correc predicion is high, so he fied AR logi model 9 o he U.S. daa is good, and model 9 can give clear signals in he predicing he 5

18 U.S. recessions Empirical analysis o Sweden We use he same process and crieria o implemen he selecion of he models fied o Swedish daa. Explanaory variables includes Sockholm PI index SPI, Swedish erm spread STS and German erm spread GTS according o he former secion of daa descripion. Thus he design marix of model 3 has he form X -h, SPI i, STS j, GTS k The proposed AR logi model should be logi p η αη µ y β β β β GTS η 0 SPI i 2STS- j 3 -k 0 We fi he daa using differen lagged values of explanaory variables and hen selec he mos saisfied one which has he smalles BIC value. The values of differen crieria are shown as following. Table 5 he summaries of model 0 wih differen lags Predicor Lag i, j, k,,,,2 2,, 2,2,2 inercep η y SPI i STS j GTS k Log-likelihood BIC Swedish daa including from he firs quarer of 97 o he fourh quarer of

19 From he above resuls, he sign of coefficien of Sockholm PI index is posiive which does no make economic sense. Meanwhile he coefficien is a lile larger han expeced. So, Sockholm PI index can no be used as explanaory variable and deleed from design marix. And he new proposed AR logi model is logi p η αη µ y β β β GTS η 0 STS-i 2 - j The nex sep is o decide he lagged values of explanaory variables, STS and GTS. Since when employed long lags, he degree of freedom will be sacrificed correspondingly, we only compare he models ha explanaory variables wih lags up o four for simpliciy. Afer simple calculaion, we find ha model wih lagged values STS, GTS -3-4 has he smaller BIC value and he signs of explanaory variables coefficiens make sense. And a good resul is ha he BIC is no declining when lags increase, ha is o say, he influence power of STS and GTS on he curren Swedish economy saus is increasing wihin one year and hen reducing as ime passes. The summary of he final model is shown as below. Table 6 he summary of model, explanaory variables wih lags 3, 4 Predicor Lag 3, 4 Crieria inercep Log-likelihood η -0.8 BIC 2.67 y 5.54 STS -.29 i GTS j Then he final model is logi p η η η y.29STS GTS-4 2 To illusrae he in-sample performance of model 2, figure 2 shows he 7

20 probabiliy ha he Sweden economy would be in a recession in one quarer ahead. The shadow area shows he acual Sweden economy saus. Figure 2 probabiliy of recessions and acual recession periods of Sweden Figure 2 shows wo periods of recessions in shadow areas, one from he hird quarer of 990 o he hird quarer of 993 and anoher from he hird quarer of 2008 o he fourh quarer of The probabiliy line maches he firs shadow area well, ha is o say, model 2 forecass he firs recession exacly. Bu model 2 forecass he second recession is a lile laer han recession has happened abou one quarer. The percenage of correc predicion of model 2 can be checked by cross able 7. The hreshold is fixed a 0.5. Table 7 he percenage of correc predicion of model 2 Observed Prediced Economy Correc Economy Expansion Recession Percenage Expansion % Recession % Overall Percenage 96.6% When he observed economy is in expansion, he model gives a 98.6% correc finess of he expansion. Meanwhile, when he observed economy is in recession, only wo prediced values of economy are no he same as he observed ones. Overall, he percenage of correc predicion of his model reaches a noably high level of 96.6%. The daa of ineres rae can only be used from he firs quarer of 987 8

21 3.2 Ou-of-sample resuls Ou-of-sample performance of he specified models will be mainly concerned, and his secion will check he prediced power of he models specified in he previous secion Ou-of-sample resuls of U.S. Based on previous secion, he model 9 can predic well in one quarer ahead in-sample U.S. recession forecass. So, consider using he same model, explanaory variables wih same lags, o check he ou-of-sample performance of AR logi model in U.S. recession forecasing. logi p η η αη µ β β β y 0 TS 2 2GTS-3 3 Because forecas horizon is wo, model 3 can make one quarer ahead and wo quarers ahead forecass. Using daa daed before forecasing quarer o esimae he parameers of model 3, and he ou-of-sample performance of model 3 one quarer ahead forecas is shown as figure 3. Figure 3 ou-of-sample forecass of U.S. one quarer ahead One quarer ahead recession forecass are made from he firs quarer of 990 o 9

22 he firs quarer of 200. Because he acual saus he firs quarer of 200, in recession or no, is no known, ha means i is no included in he using daa se, so using pink background o make i differen from he ohers. From figure 3, here are hree recession periods. The firs recession period from he fourh quarer of 990 o firs quarer of 99 can be prediced well by he model. Bu he laer wo recession period saring from he second quarer of 200 and he firs quarer of 2008 can no be prediced exacly abou one quarer lae han he acual saring quarers, respecively. And he economic recession prediced by model 3 will end a he firs quarer of 200. And ou of sample performance of model 3 wo quarers ahead forecas is shown as figure 4. Figure 4 ou-of-sample forecass of U.S. wo quarers ahead Two quarers ahead recession forecass are made from he firs quarer of 990 o he second quarer of 200. The unknown acual economy saus, in recession or no, is marked wih pink background. During he las wo quarers period saring from he fourh quarer of 990, U.S. economy is in recession, bu he esimaed recession probabiliy is lower han he hreshold value, 0.5, and a lile lag on he ime. The same siuaion of lower lagged esimaed probabiliy happens a he oher wo recession periods. Acually, here is only one period in recession, if he probabiliy hreshold is chosen a 0.5. Tha will be a kind of inconsisency wih he fac. Bu he volaile probabiliy can be checked direcly, ha is, when he recession probabiliy goes up 20

23 enough, economy recession happens in near periods. So, alernaive choice is decrease he hreshold o 0.3, and he predicion power of he model has been changed higher. Figure 4 also shows ha he recession saring from he firs quarer of 2008 will end a he firs quarer of 200 finally Ou-of-sample resuls of Sweden Based on model 2, consider using following model o check he ou-of-sample performance of AR logi model in Sweden recession forecasing. logi p η η αη µ β β β y 0 STS -3 2GTS-4 4 Model 4 can make one quarer ahead up o hree quarers ahead forecass due o is forecas horizon is hree. We sill use daa daed before forecasing quarer o esimae he parameers of model 4, and hen make he ou-of-sample forecas. One quarer ahead forecas of model 4 is shown as figure 5. Figure 5 ou-of-sample forecass of Sweden one quarer ahead Figure 5 shows he ou-of-sample predicive performance one quarer ahead. The forecas is daed from he firs quarer of 990 and end up o he firs quarer of 200. This model fis well when he firs recession occurs. However, as for he second recession happens a he hird quarer of 2008, he prediced resul given by his model is abou one quarer laer han he real recession happens. In addiion o he second 2

24 recession forecasing, he model maches well wih he real daa. Since he saus of economy a he firs quarer of 200 is unknown in he daa se, we use pink highligh o disinguish his period from ohers. Model 4 predics ha he recession saring from he hird quarer of 2008 will end a he firs quarer of 200. And he wo quarers ahead forecas of model 4 is shown as figure 6. Figure 6 ou-of-sample forecass of Sweden wo quarers ahead Figure 6 illusraes he ou-of-sample predicive resul when model 4 gives wo quarer ahead forecas. The forecas is daed from he firs quarer of 990 and end up o he second quarer of 200. When he firs recessions occur a he hird quarer of 990, he corresponding predicive period is abou one quarer lagged behind. When he firs recession ends, he predicive value is one quarer lagged behind as well. The same siuaion appears when he second recession occurs. Since we do no know when he second recession end, we could hardly say wheher he predicive value is lag or ahead o he real value. The predicive resul which is highlighed by pink color background suggess ha he recession are going o end a he firs quarer of 200 same as one quarer ahead forecas. As referred before, forecas horizon of model 4 is hree, so he hree quarers ahead forecas for Sweden business cycle is shown as figure 7. From figure 7, we can see ha he forecas is no very good which can be proved furher by cross able 8. 22

25 Table 8 he percenage of correc predicion hree quarers ahead Observed Prediced Economy Correc Economy Expansion Recession Percenage Expansion % Recession % Overall Percenage 87.5% When he observed economy is in recession, he model gives a 63.2% correc esimaion. Bu when observed economy is in expansion, hree prediced quarers are no consisen wih he observed economy. Overall, he percenage of correc predicion of his model is declining o 87.5%. Figure 7 ou-of-sample forecass of Sweden hree quarers ahead So, based on he above analysis, alhough model 4 can make hree quarers ahead forecas for Sweden economy, i is good choice o omi he hree quarers ahead forecas of model 4. 4 Conclusion In his paper, we examine ha he AR logi model can be used o forecas no only U.S. economic recessions bu also Sweden economic recessions. The processes of selecing design marix for U.S. and Sweden are same, bu he resuls are differen. The lagged value of AR erm is fixed a one, bu he lagged values of explanaory 23

26 variables are flexible. Generally, he deerminaion of he lagged values of explanaory variables is hrough he BIC value of each model. Afer selecing he bes fied model, we are more concern abou he in-sample and ou-of-sample forecasing performance. The in-sample forecasing of U.S. wih one quarer ahead shows a good finess of he model. Specifically, his model gives an overall 93.5% correcion of predicion. Thus, we can believe ha he model will also sugges a good resul for ou-of-sample forecasing. The in-sample forecasing performance of AR logi model when applying i o Sweden shows an overall 96.6% correcion of predicion. This resul indicaes ha he AR logi model is applicable in Sweden, wha s more, he percenage correcion of predicion even beer han U.S. Thus, we can conclude ha he AR logi model is quie suiable when modeling he economy recession in Sweden. Afer comparing he in-sample forecasing performance of AR logi model, he ou-of-sample forecasing is more concerned when applying he model in pracice. Since he forecas horizon in U.S. model is wo, we can implemen he ou-of-sample forecasing up o wo quarers ahead. We can ge he forecasing resul wih one quarer ahead direcly, bu when he sep of forecasing is larger han one, we use ieraive mehod o ge he forecasing resuls. Generally, he resuls of ou-of-sample forecasing are no as good as he ones of in-sample forecasing. To ge a beer visual figure, we mus limi he hreshold o smaller value han 0.5 someimes. Boh he forecasing resuls wih one quarer ahead and wo quarers ahead show a small probabiliy of recession which indicaes he economy in U.S. will recover from a deep recession. In oher words, he economy in U.S. will experience an expansion a leas unil second quarer in 200. Since he forecas horizon in Sweden model is hree, we can implemen he ou-of-sample forecasing up o hree quarers ahead. The forecasing wih one quarer ahead shows a beer performance han he ohers. Neverheless, all hese forecasing resuls show a small probabiliy of recession a he saring quarer of 200 which means ha he economy in Sweden will experience an expansion a leas up o he hird quarer in 200. Alhough he AR logi model can be considered as a powerful ool o model and 24

27 predic economic recessions, we can observe from he figures ha he prediced recessions mosly occur laer han he real ones. This phenomenon parly can be explained as he lag effec of models. Since all he models we proposed include an AR erm, i makes sense when we observe he prediced resuls reac laer han he real ones. Besides, i maybe ineres o add more explanaory variables o improve he performance of AR logi model or consider applying he AR logi model o describe he economy recession in oher homogeneous counries e.g. EEC is anoher possible way for furher sudy. 25

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