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W e d n e s d a y , M a r c h 6 , 2 0 1 3

ARDL Models - Part II've been promising, for far too long, to provide a post on ARDL models and bounds

testing. Well, I've finally got around to it

!ARDL! stands for !Autoregressive-Distributed Lag!. Regression models of t"is type"ave been in use for de#ades, but in more re#ent times t"ey "ave been s"o$n toprovide a very valuable ve"i#le for testing for t"e presen#e of long-runrelations"ips bet$een e#onomi# time-series.

I'm going to brea% my dis#ussion of ARDL models into t$o parts. &ere, I'm going todes#ribe, very briefly, $"at $e mean by an ARDL model. "is $ill t"en provide t"eba#%ground for a se#ond post t"at $ill dis#uss and illustrate "o$ su#" models #anbe used to test for #ointegration, and estimate long-run and s"ort-run dynami#s,even $"en t"e variables in (uestion may in#lude a mi)ture of stationary and non-stationary time-series.

In its basi# form, an ARDL regression model loo%s li%e t"is*

  yt +   /yt-/  ....... pyt-p  0)t  0/)t-/  01)t-1  ......... 0()t-(  2t

$"ere 2t is a random !disturban#e! term.

"e model is !autoregressive!, in t"e sense t"at y t is !e)plained 3in part4 by lagged

values of itself. It also "as a !distributed lag! #omponent, in t"e form of su##essivelags of t"e !)! e)planatory variable. 5ometimes, t"e #urrent value of ) t itself ise)#luded from t"e distributed lag part of t"e model's stru#ture.

Let's des#ribe t"e model above as being one t"at is ARDL3p,(4, for obvious reasons.

6iven t"e presen#e of lagged values of t"e dependent variable as regressors, 7L5estimation of an ARDL model $ill yieldbiased  #oeffi#ient estimates. If t"edisturban#e term, 2t, is auto#orrelated, t"e 7L5 $ill also bean inconsistentestimator, and in t"is #ase Instrumental 8ariables estimation $asgenerally used in appli#ations of t"is model.

In t"e /9:'s and /9;'s $e used distributed lag 3DL3(4, or ARDL3,(44 models a lot.o avoid t"e adverse effe#ts of t"e multi#ollinearity asso#iated $it" in#ludingmany lags of !)! as regressors, it $as #ommon to redu#e t"e number of parametersby imposing restri#tions on t"e pattern 3or !distribution!4 of values t"at t"e 0#oeffi#ients #ould ta%e.

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Per"aps t"e best %no$n set of restri#tions $as t"at asso#iated $it" t"e <oy#%3/9=>4 for t"e estimation of DL 3?4 model. "ese restri#tions imposed a polynomialrate of de#ay on t"e 0 #oeffi#ients. "is enabled t"e model to be manipulated intoa ne$ one t"at $as autoregressive, but $it" an error term t"at follo$ed a movingaverage process. oday, $e'd #all t"is an ARMA@ model. Again, Instrumental

8ariables estimation $as often used to obtain #onsistent estimates of t"e model'sparameters.

ran#es and van 7est 31>4 provide an interesting perspe#tive of t"e <oy#%model, and t"e asso#iated !<oy#% transformation!, = years after its introdu#tioninto t"e literature.

5"irley Almon populariBed anot"er set of restri#tions 3Almon, /9:=4 for t"e#oeffi#ients in a DL3(4 model. &er approa#" $as based on Weierstrass'sApproximation Theorem, $"i#" tells us t"at any #ontinuous fun#tion #an beappro)imated, arbitrarily #losely, by a polynomial of some order. "e only (uestionis !$"at is t"e order!, and t"is "ad to be #"osen by t"e pra#titioner.

"e Almon estimator #ould a#tually be re-$ritten as a restri#ted least s(uaresestimator. or e)ample, see 5#"midt and Waud 3/9;C4, and 6iles 3/9;=4.5urprisingly, t"oug", t"is isn't "o$ t"is estimator $as usually presented to studentsand pra#titioners.

Almon's approa#" allo$ed restri#tions to be pla#ed on t"e s"ape of t"e !de#aypat"! of t"e gamma #oeffi#ients, as $ell as on t"e values and slopes of t"is de#aypat" at t"e end-points, t+ and t+(. Almon's estimator is still in#luded in a number

of e#onometri#s pa#%ages, in#luding EViews. A ayesian analysis of t"e Almonestimator, $it" an appli#ation to Ee$ Fealand imports data, #an be found in 6iles3/9;;4, and 5"iller 3/9;C4 provides a ayesian analysis of a different type of distributed lag model.

D"rymes 3/9;/4 provides a t"oroug" and very general dis#ussion of DL models.

5o, no$ $e %no$ $"at an ARDL model is, and $"ere t"e term !Autoregressive-Distributed Lag! #omes from. In the next post on t"is topi# I'll dis#uss t"e modernappli#ation of su#" models in t"e #onte)t of non-stationary time-series data, $it"t"e emp"asis on an illustrative appli#ation $it" real data.

[Note: or an important update to t"is post, relating to EViews 9, see my 1/=post, here.G

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7HED5 55

S a t u r d a y , D e c e m b e r 2 7 , 2 0 1 4

"e Demise of a !6reat Ratio!7n#e upon a time t"ere $as a rule of t"umb t"at t"ere $ere 1 s"eep in Ee$Fealand for every person living t"ere. Jep, I %id you not. "e old adage used to be!C million peopleK : million s"eep!.

I li%ed to t"in% of t"is as anot"er important !6reat Ratio!. Jou %no$ - in t"e spiritof t"e famous !6reat Ratios! suggested by <lein and <osubod 3/9:/4 in t"e #onte)tof e#onomi# gro$t", and subse(uently analysed and augmented by a variety of aut"ors. "e latter in#lude 5imon 3/994, &arvey et al. 31C4, Attfield and emple31/4, and ot"ers.

After all, it's said t"at 3at least in t"e post-WWII era4 t"e e#onomies of bot"Australia and Ee$ Fealand !rode on the sheep's back!. If t"at's t"e #ase, t"en t"eEe$ Fealand 5"eep Ratio 3EF5R4 may "old important #lues for e#onomi# gro$t" int"at #ountry.

My interest in t"is matter rig"t no$ #omes from reading an alarming press releasefrom Statistics New ea!and, a fe$ days ago. "e latest release of t"e Agric"!t"ra! #rod"ction Statistics for E.F. revealed t"at t"e 3provisional4figure for t"e number of s"eep $as 3only 4 1:.9 million at t"e end of une 1/> -do$n > from 1/C.

I $as s"o#%ed, to say t"e least Worse $as to #ome. "e 1/> figure puts t"enumber of s"eep in E.F. at t"e lo$est level sin#e /9>C

I'm sure you #an understand my #on#ern. We'd better ta%e a #loser loo% at t"is,and $"at it all means for t"e EF5R*

Let's begin $it" t"e ra$ numbers, #ourtesy of 5tatisti#s E.F.'s free and user-friendly interfa#e, !$n%oshare!.

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I used t"e long-run "istori#al data for t"e 3"uman4 population, $"i#" is $"y t"atseries ends in 1//. More re#ent data are available, of #ourse, but I #ouldn't easily

mat#" t"em to t"e "istori#al series. "is is really of no #onse(uen#e for $"at I'mgoing to do "ere.

"e brea%s in t"e s"eep series are due to t"e fa#t t"at no agri#ultural survey $as#ondu#ted in #ertain years. I don't %no$ $"y not.

I'm going to do some unit root testing, so t"ese gaps "ave to be dealt $it". In line$it" my $or% $it" <evin Ryan 3as dis#ussed in an earlier post, here4, I've filled int"e four gaps in t"e series $it" t"e previous a#tual value.

Wit" t"e "uman population rising, and t"e s"eep population de#lining, you don't"ave to be a ro#%et s#ientist to %no$ $"at's been "appening to t"e EF5R*

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Arguably, t"e EF5R $as relatively stable up until about t"e end of t"e /9N's, butafter t"at..........t"e end of an era.

Let's analyBe t"e data more #arefully. All of t"e data I've used are available ont"e data page t"at a##ompanies t"is blogK and t"e 8ie$s $or%file is on t"e code

page.

"e first t"ing t"at I've done is to !tri#%! 8ie$s into giving me a##ess to t"estandard tests t"at $e use for stru#tural #"ange in a regression model. I regressedEF5R on Oust an inter#ept 3and no ot"er regressors4, using 7L5. "e estimatedinter#ept is Oust t"e sample mean for EF5R, and t"e residuals for t"e regresssionare Oust t"e EF5R data, e)pressed as deviations about t"is sample mean. Eo$ I #anapply tests for stru#tural #"ange, effe#tively to t"e mean-adOusted EF5R series. 3Imentioned t"is tri#% in an ear!ier post.4

5pe#ifi#ally, "ere are t"e results of applying t"e test suggested by ai and Perron3/99N4, using t"e !min. 5I! rule proposed by Liu et al. 3/99;4

Jou #an see t"at $e "ave eviden#e of stru#tural brea%s in /9=N, /99, and 1/."e se#ond of t"ese dates lines up ni#ely $it" my !eye-ball! #on#lusion from t"eprevious grap".

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Eo$ let's fo#us on t"e data for t"e period /9C= to /99. I'm going to test to see if t"ere is eviden#e of unit roots in t"e logarit"ms of P7P and 5&P. As t"ere is, I'mt"en going test if t"e logarit"m of EF5R is stationary. In ot"er $ords, I'll be#"e#%ing $e "ave #ointegration in t"e logarit"ms of t"e data.

At ea#" step, I'll allo$ for t"e fa#t t"at t"ere is eviden#e of a stru#tural brea% in/9=N. "e in#orporation of brea%point unit root tests is one of t"e many ni#e ne$features in t"e &eta release of 8ie$s 9, as noted in my previous post, here.

&ere is "o$ I've done t"is for t"e log3P7P4 series*

"is implements Perron's 3/9N94 modified AD tests, and "ere's t"e result*

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learly, $e #an't reOe#t t"e null of a unit root. In t"e #ase of t"e log35&P4 series,t"e #orresponding AD statisti# is -.9=9; 3p Q .=4, so $e #ome to t"e same#on#lusion.

"e log3EF5R4 series is trendless over t"is period, so t"is is ta%en into a##ount

$"en applying t"e modified AD test*

"e modified AD statisti# -C.=;/: 3p ./4. "e p-values for t"e modified ADtests are based on t"e asymptoti# distributions for t"e test statisti#s, and $e "ave + =:. Eone t"e less, $e "ave reasonable eviden#e t"at log3P7P4 and log35&P4

$ere #ointegrated over t"e period /9C= to /99. "eir log-ratio $as stationary.

"ere's our !6reat Ratio!

3or t"e re#ord, e)a#tly t"e same #on#lusions are rea#"ed if $e use t"e levels,rat"er t"an t"e logarithms, of t"e data.4

"e se#ond grap" given earlier ma%es it pretty #lear t"at t"e 36reat4 Ee$ Fealand5"eep Ratio is no longer $it" us. y $ay of #onfirmation, I've tested for a unit rootin log3EF5R4 over t"e full sample, /9C= to 1//. I used ea#" of t"e of t"e

brea%point unit root tests available in EViews 9 t"at allo$ for innovation outlierbrea%s. "ere are /: tests in total $"en you allo$ for t"e varous drfitStrendoptions.

y $ay of an e)ample, "ere are t"e results for Di#%ey-uller ma).-t test, allo$ingfor an innovation trend-brea%*

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or all of t"e tests applied to t"e full series for log3EF5R4, or for EF5R itself, verylarge p-values are a#"ieved and so $e #annot reOe#t a unit root in t"e data.

"e Ee$ Fealand 5"eep Ratio appears to be a t"ing of t"e past

T 1/>, David . 6iles

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F r i d a y , J a n u a r y 9 , 2 0 1 5

ARDL Modelling in 8ie$s 9My previous posts relating to ARDL models 3here and here4 "ave dra$n a lot of 

"its. 5o, it's great to see t"at EViews 9 3no$ in eta release - see t"edetails here4 in#orporates an ARDL modelling option, toget"er $it" t"e asso#iated!bounds testing!.

"is is a great feature, and I Oust %no$ t"at it's going to be a !$inner! for 8ie$s.

It #ertainly deserves a post, so "ere goes

irst, it's important to note t"at alt"oug" t"ere $as previously an 8ie$s !add-in!for ARDL models 3see here and here4, t"is $as (uite limited in its #apabilities.W"at's no$ available is a full-blo$n ARDL estimation option, toget"er $it" boundstesting and an analysis of t"e long-run relations"ip bet$een t"e variables beingmodelled.

&ere, I'll ta%e you t"roug" anot"er e)ample of ARDL modelling - t"is one involvest"e relations"ip bet$een t"e retail pri#e of gasoline, and t"e pri#e of #rude oil.More spe#ifi#ally, t"e #rude oil pri#e is for anadian Par at dmontonK and t"egasoline pri#e is t"at for t"e anadian #ity of 8an#ouver. Alt"oug" #rude oil pri#esare re#orded daily, t"e gasoline pri#es are available only $ee%ly. 5o, t"e pri#edata t"at $e'll use are $ee%ly 3end-of-$ee%4, for t"e > anuary 1 to /: uly1/C, in#lusive.

"e oil pri#es are measured in andian dollars per #ubi# meter. "e gasoline pri#esare in anadian #ents per litre, and t"ey e)#lude ta)es. &ere's a plot of t"e ra$data*

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"e data are available on t"e data page for t"is blog. "e 8ie$s $or%file is ont"e code page.

I'm going to $or% $it" t"e logarit"ms of t"e data* L76URHD and L76U6A5."ere's still a #lear stru#tural brea% in t"e data for bot" of t"ese series.5pe#ifi#ally t"ere's a stru#tural brea% t"at o##urs over t"e $ee%s ended N uly1N to C De#ember 1N in#lusive. I've #onstru#ted a dummy variable, RA<,t"at ta%es t"e value one for t"ese observations, and Bero every$"ere else.

"e brea% doesn't o##ur at Oust a single point in time. Instead, t"ere's a #"ange int"e level and trend of t"e data t"at evolves over several periods. We #all t"is an!innovational outlier!, and in testing t"e t$o time series for unit roots, I've ta%ent"is into a##ount.

In a recent post I dis#ussed t"e ne$ !rea%point Hnit Root est! options t"at areavailable in 8ie$s 9. "ey're perfe#tly suited for our #urrent situation. &ere's "o$

I've implemented t"e appropriate test of a unit root in t"e #ase of t"e L76URHDseries*

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"e result is*

We $ouldn't reOe#t t"e "ypot"esis of a unit root at t"e = signifi#an#e level, andt"e result is marginal at t"e / level. "e #orresponding result for t"e L76U6A5series is*

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In t"is #ase $e'd reOe#t t"e null "ypot"esis of a unit root at t"e = signifi#an#elevel, but not at t"e / level. 7verall, t"e results are some$"at in#on#lusive, andt"is is pre#isely t"e situation t"at ARDL modelling and bounds testing is designedfor. Applying t"e unit root tests to t"e first-differen#es of ea#" series leads to avery #lear reOe#tion of t"e "ypot"esis t"at t"e data are I314, $"i#" is important fort"e legitimate appli#ation of t"e bounds test belo$.

Eo$, let's go a"ead $it" t"e spe#ifi#ation and estimation of a basi# ARDL model

t"at e)plains t"e retail pri#e of gasoline in terms of past values of t"at pri#e, as$ell as t"e #urrent and past values of t"e pri#e of #rude oil. We #an do t"is in t"esame $ay t"at $e'd estimate any e(uation in 8ie$s, but $e sele#t t"e!stimation Met"od! to be !ARDL! 3see belo$4*

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Eoti#e t"at I've set t"e ma)imum number of lags for bot" t"e dependent variableand t"e prin#ipal regressor to be N. "is means t"at ;1 different modelspe#ifi#ations $ill be #onsidered, allo$ing for t"e fa#t t"at t"e current value of L76URHD #an be #onsidered as a regressor. Also, noti#e t"at I've in#luded t"eRA< dummy variable, as $ell as an inter#ept and linear trend as 3fi)ed4regressors. 3"at is, t"ey $on't be lagged.4

Hsing t"e 7PI7E5 tab, let's sele#t t"e 5#"$arB #riterion 354 as t"e basis fordetermining t"e lag orders for t"e regressors*

"e model $"i#" minimiBes 5 $ill be #"osen. "is results in a rat"erparsimonious model spe#ifi#ation, as you #an see*

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I mentioned in an ear!ier post on $n%ormation riteria t"at 5 tends to sele#t asimpler model spe#ifi#ation t"an some ot"er information #riteria. 5o, instead of 

5, I'm going to use A%ai%e's Information riterion 3AI4 for sele#ting t"e lagstru#ture in t"e ARDL model. "ere's a ris% of !over-fitting! t"e model, but Idefinitely don't $ant to under-fit it. &ere's $"at $e get*

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It's important t"at t"e errors of t"is model are serially independent - if not, t"eparameter estimates $on't be #onsistent 3be#ause of t"e lagged values of t"edependent variable t"at appear as regressors in t"e model. o t"at end, $e #anuse t"e 8IW tab to #"oose, R5IDHAL DIA6E75I5K 7RRL76RAM - V-5AI5I5, and t"is gives us t"e follo$ing results*

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"e p-values are only appro)imate, but t"ey strongly suggest t"at t"ere is noeviden#e of auto#orrelation in t"e model's residuals. "is is good ne$s

Eo$, re#all t"at, in total, ;1 ARDL model spe#ifi#ations $ere #onsidered. Alt"oug"an ARDL3>,14 $as finally sele#ted, $e #an also see "o$ $ell some ot"erspe#ifi#ations performed in terms of minimiBing AI. 5ele#ting t"e 8IW tab in t"eregression output, and t"en #"oosing M7DL 5LI7E 5HMMARJK RIRIA6RAP& from t"e drop-do$n, $e see t"e !op $enty! results*

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3Jou #an get t"e full summary of t"e AI, 5, &annan-Vuinn, and adOustedR1 statisti#s for all 72 model spe#ifi#ations if you sele#t RIRIA AL, rat"ert"an RIRIA 6RAP&.4

7ne of t"e main purposes of estimating an ARDL model is to use it as t"e basis forapplying t"e !ounds est!. "is test is dis#ussed in detail in one of my ear!ierposts. "e null "ypot"esis is t"at t"ere is no long-run relations"ip bet$een t"e

variables - in t"is #ase, L76URHD and L76U6A5.

In t"e estimation results, if $e sele#t t"e 8IW tab, and t"en from t"e drop-do$nmenu #"oose 7IIE DIA6E75I5K 7HED5 5, t"is is $"at $e'll get*

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We see t"at t"e -statisti# for t"e ounds est is C1.CN, and t"is #learly e)#eedseven t"e / #riti#al value for t"e upper bound. A##ordingly, $e strongly reject t"e"ypot"esis of !Eo Long-Run Relations"ip!.

"e output at t"is point also s"o$s t"e modified ARDL model t"at $as used toobtain t"is result. "e form t"at t"is model ta%es $ill be familiar if you've readmy ear!ier post on bounds testing.

In t"e estimation results for our #"osen ARDL model, if $e sele#t t"e 8IW tab,and t"en from t"e drop-do$n menu #"oose 7IIE DIA6E75I5K7IE6RAI7E AED L7E6 RHE 7RM, t"is is $"at $e'll see*

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"e error-#orre#tion #oeffi#ient is negative 3-.11N4, as re(uired, and is verysignifi#ant. Importantly, t"e long-run #oeffi#ients from t"e #ointegrating e(uationare reported, $it" t"eir standard errors, t-statisti#s, and p-values*

5o, $"at do $e #on#lude from all of t"is

irst, not surprisingly, t"ere's a long-run e(uilibrium relations"ip bet$een t"epri#e of #rude oil, and t"e retail pri#e of gasoline.

5e#ond, t"ere is a relatively (ui#% adOustment in t"e pri#e of gasoline $"en t"e

pri#e of #rude oil #"anges. 3Re#all t"at t"e data are observed weekly .4

"ird, a / #"ange in t"e pri#e of #rude oil $ill result in a long-run #"ange of ;in t"e pri#e of retail gasoline.

W"et"er or not t"ese responses are symmetri# $it" respe#t to pri#e in#reases andpri#e de#reases is t"e subOe#t of someon(going work o% mine.

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