Notes on cointegration of real interest rates and real exchange rates. ρ (2)

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1 Noe on coinegraion of real inere rae and real exchange rae Charle ngel, Univeriy of Wiconin Le me ar wih he obervaion ha while he lieraure (mo prominenly Meee and Rogoff (988) and dion and Paul (993)) e for coinegraion of real exchange rae and real inere rae, he heory acually implie explicily ha real exchange rae and real inere rae are no coinegraed. So fir I will demonrae ha claim. Aume ha uncovered inere pariy (UIP) hold, up o a ri premium, ρ, ha i aionary. (Meee and Rogoff aume UIP hold exacly, dion and Paul aume ha here i a aionary ri premium.) + ) ( = i i + ρ. () Here, i he log of he po exchange rae, i and i are he one-period domeic and foreign inere rae, repecively, and repreen expecaion condiional on ime informaion. Le = ε, o + ( + ) = i i + + ε + ρ (2) Since under raional expecaion ε + i an iid random error, and ρ i aionary, hen ρ + ε + mu be I(0). You can quicly ge a hin of where I am going wih hi. Since ρ + ε + i I(0), i mu be he cae ha + i coinegraed wih i i. If he exchange rae i I() a mo people eem o agree hen necearily he exchange rae i no coinegraed wih i i. Indeed, if he exchange rae i I(), he heory acually hen implie i i i aionary (ince if he exchange rae i I(), hen + i I(0), and + equal i i plu an I(0) error.) I will ranlae hi all ino erm of real exchange rae and real inere rae in a momen, bu le me fir poin ou ha here i rong empirical uppor for he long-run implicaion of he above heory. If I rearrange my equaion (2) above, I can wrie + = i i ε + Uing he covered inere pariy relaionhip, we ge + = f ρ (3) ρ ε (4) where f i he log of he one-period ahead forward rae. quaion (4) ell u ha he po and forward exchange rae mu be coinegraed if UIP hold up o a aionary ri premium. Tha i a relaionhip ha ha found rong empirical uppor for advanced counrie.

2 (Acually, here are a few paper ha claim ha po and forward exchange rae are no coinegraed. Zivo (2000) doe a good job of clearing ha iue up. In eence, he paper ha canno rejec he null of no coinegraion beween po and forward rae have eed for no coinegraion beween + and f. Thoe are lierally he variable ha equaion (4) ay hould be coinegraed. Bu if he exchange rae i I(), wha hould be an equivalen e i a e of no coinegraion beween and f. Thi e alway rejec he null of no coinegraion among advanced counrie. Zivo how ha he reaon he e ha ue + and f fail o rejec he null of no coinegraion i ha hey are much le powerful han he e ha ue and f. The ource of ha lac of power can be raced direcly o he high variance of ε +.) Reurning o equaion (2), ubrac he quaniy ( p + p ) ( p + p ) from boh ide, where p and p are he log of he home and foreign price level, repecively, o ge + ( p + p ) ( p + p ) = r r + + ε + ρ, (5) where r and r are he real inere rae a home and abroad, repecively, defined by r = i ( p + p )), and r = i ( p + p ). Define he error erm u by ( + + p ( p + p ) = ( p + p ) ( p + p ) + u+ p. (6) Then under raional expecaion, u + i an iid error erm. Plugging relaionhip (6) ino (5), we ge q + q = r r + + ε + + u + ρ, (7) where q i he real exchange rae defined by and + q = + p p. Since r 2 ρ i aumed aionary, ε and u + are iid, i follow ha q + q and r are inegraed of he ame order. Tha would be poible if q wa aionary and nonaionary, hen q and r r wa aionary. Bu if q i r r are necearily no coinegraed. So, o um up, merely uing he inere pariy condiion wih a aionary ri premium and raional expecaion, we conclude ha if q i no aionary, hen q and r r are necearily no coinegraed. I i riing hen ha he lieraure omehow hin ha he implicaion of he heory are ha if q i no aionary, hen q and r r are necearily coinegraed. Tha i, i eem lie he lieraure ha i exacly wrong. Where doe he lieraure go wrong? Le me reproduce exacly he equaion of Meee and Rogoff (988), and I hin i will be eay o ee wha he problem i: Fir i a real exchange rae adjumen equaion:

3 ( Then hey aume q i, in eence, a random wal: q + q + ) = θ ( q q ), 0 < θ < (8) Some manipulaion give hem q + = q (9) q = α ( q + q ) + q (0) where α /( θ ). Nex hey ue uncovered inere pariy a horizon : ( ) = i i + () where i and i are he nominal inere rae a ime on -horizon bond. Subracing expeced -horizon relaive inflaion rae, hey ge where he real inere rae i defined a: ( q q ) = r r + (2) r = i ( p + p )). Then uing equaion (0) and (2) hey ge ( r q = α ( r ) + q (3) Thi i he ame equaion ha dion and Paul (993) ue. (Alo, Obfeld and Rogoff (996) derive i in heir exboo, hough in ha derivaion hey aume q i conan.) There are wo poin I wan o mae. The main poin i hi: equaion (8) implie ha q q i aionary. Tha i rue no maer wha he order of inegraion of q and q hemelve. So, if we mainain equaion (8), hen equaion (3) necearily implie r r i aionary. Tha i why hee e of coinegraion baed on he above equaion are off bae. The heory implie r r i aionary, bu he empirical e alway rea i a nonaionary. If r r i nonaionary, he heory i rejeced. 3

4 There i anoher problem wih Meee and Rogoff, which i ha hey e o ee wheher r r ielf i coinegraed wih q. Tha i no only wrong becaue of my fir poin (ha r r i aionary), bu i i alo wrong becaue under heir aumpion, q and r r clearly are no coinegraed. Becaue hey aume q i I() in equaion (9), hen q mu be I(). Tha i, he difference q α ( r r ) i equal o q, which i aumed o be I(). Tha i why dion and Paul (993) include q in he coinegraing equaion. Bu even including q in he coinegraing relaion doe no olve he fir problem: ha he heory implie r r i aionary. So here are really wo poibiliie. One i ha cae q i aionary and r The heory above ill implie ha r r i aionary, o han r r 4 q i aionary, in which r i aionary. The oher poibiliy i ha q i no aionary. q i inegraed of a differen order. quaion (3) i a correc equaion given he aumpion ha lead o i, bu he empirical applicaion of he equaion are alway wrong ince hey alway aume r r i nonaionary. If I were rying o mae ene of wha Meee and Rogoff, or dion and Paul do, I would mae an argumen lie hi: q and q are nonaionary, bu coinegraed. Bu, q q i very perien (hough aionary), and r r i very perien hough aionary. So, i migh mae ene o rea q q and r r a if hey were nonaionary variable in examining dynamic, if he error erm in equaion (3) were much le perien han eiher q q or r r. The immediae problem wih hi argumen i ha here i no error erm in equaion (3). How could we modify he model o ha here wa an error erm? I eem o me here are hree poible way. Fir, here migh be an error erm in he price-adjumen equaion (0). In oher word, maybe price are no perfecly pree, or maybe hey are bu hi equaion doe no capure he relaionhip exacly. The econd poibiliy i ha uncovered inere pariy (equaion ()) doe no hold, o perhap here i a ri premium. Or perhap expecaion are no raional, o expecaional error do no have condiional mean of zero. In erm of he model, ince hoe are he only wo equaion, hoe are he only poible place where an error erm could ener. In pracice, in he economeric pecificaion, here could be an error erm if r r i meaured wih error (which would be a real problem, ince correlaion of r r and he error give you problem wih inerpreing andard error of coefficien wheher hi i a coinegraing relaionhip or a relaionhip beween aionary variable.) Then, if hi error erm wa no a perien a q q and r r, hen i migh mae ene o eimae an error-correcion model for q q and r r o capure he dynamic. Tha i, even hough we believe q q and r r are aionary, o ha in principle we could capure he dynamic in a VAR wih only lag of q q and r r, i migh mae ene in pracice o include an error correcion erm.

5 Reference dion, Hali J., and B. Dianne Paul, 993, A re-aemen of he relaionhip beween real exchange rae and real inere rae: , Journal of Moneary conomic vol. 3, no. 2, Meee, Richard, and Kenneh Rogoff, 988, Wa i real? The exchange rae-inere differenial relaion over he modern floaing-rae period, Journal of Finance, vol. 43, no. 4, Obfeld, Maurice, and Kenneh Rogoff, 996, Foundaion of Inernaional Macroeconomic (MIT Pre.) Zivo, ric, 2000, Coinegraion and forward and po exchange rae regreion, Journal of Inernaional Money and Finance vol. 9, no. 6,