IV. DOLLARIZATION AND PRICE DYNAMICS
3. Latin America and Long-Run Purchasing Power Parity
3.1. Long-Run Purchasing Power Parity Econometrics
which then becomes a declining trend during the sub-period 1972-2000 at an average rate of -1.244% per year. In both periods, no Latin American country has a level of real per- capita GDP above that of the U.S. If fact, the cross-country average relative real per- capita GDP declines, in log terms, from -1.313 in the first sub-period to -1.462 in the second sub-period.
context of CPI-based bilateral real exchange rates between the Latin American countries and the U.S. and after considering the existence of transaction costs, this definition would only require that the qi,t series calculated with equation (4.1) be stationarity. Temporary deviations from PPP are allowed because, in practice, disturbances in price levels are not immediately offset by changes in the nominal exchange rate (and vice-versa.) As Edison (1987) and Rogoff (1996) indicate, the consensus among economists is that PPP holds only in the long run.79
Abuaf and Jorion (1990) suggest that the time behavior of the real exchange rate of country i can be accurately modeled by a first-order autoregressive process such as:
t i t i i i t
i q q
q, = ,0+ρ ,−1+ε, (4.4)
where qi,0 is qi,t at t = 0, ρi is called the first-order autocorrelation coefficient and is assumed to be a constant parameter, and εi,t is a white-noise series of disturbances with zero mean. Equation (4.4) can be solved recursively going back an infinite number of time periods in the past to get:
(
...1 , 3
3 2 , 2 1 , ,
0 ,
, + + + + +
= − it i it− i it− i it−
i i t i
q q ε ρε ρ ε ρ ε
ρ
)
(4.5)The fraction on the right-hand side of equation (4.5) is the long-run equilibrium level of qi,t and is a constant. The terms in parentheses show the effect of the current and past disturbances, and are responsible for deviations of qi,t from its long-run equilibrium level.
If |ρi | < 1, the long-run level is well defined becoming the expected value of qi,t, and any deviations from this value tend to die out over time; that is, qi,t is mean-reverting or stationary, and long-run PPP holds. On the other hand, if |ρi | ≥ 1, qi,t is non-stationary and long-run PPP does not hold.
The typical test for PPP is an Augmented Dickey-Fuller (ADF) test for the presence of a unit root in the real exchange rate series, and consists of estimating the following regression:
t i j t i p
j j i t
i i i t
i q q
q i , ,
1 , 1
,
, =µ +α + β ∆ +ε
∆ −
=
−
∑
(4.6)For this test, Ho: α i = 0 indicates the presence of a unit root, and Ha: α i < 0 that the series is stationary. The lagged first difference terms in equation (4.6) are typically included to control for extra serial correlation in case the order of the autoregressive process is higher than 1. The optimal number of lags, pi, is chosen using the Schwarz Information Criterion. For a process like that of equation (4.4), α i = ρi – 1, so testing for α i = 0 is equivalent to testing for ρi = 1, and a result of α < 0 corresponds to ρ < 1. ADF tests will be done independently on each of the Latin American countries’ real exchange rate series using the longest period available, i.e. the full period 1950-2000.
The main drawback of the individual ADF test is its low power, which can result in the non-rejection of the null of a unit root when, in fact, the series is stationary. Abuaf and Jorion (1990), Frankel and Rose (1996), Oh (1996), Lothian (1997), and Papell (2002) suggest a multi-series approach to increase test power. For this reason, panel unit- root tests are done on the set of Latin American bilateral real exchange rates. These are the Levin, Lin, and Chu (2002), Breitung (2000), Im, Pesaran, and Shin (2003), Maddala and Wu (1999)80, and Hadri (2000) tests. For all tests, except the Hadri test, the null hypothesis is that all series have a unit root. For the Hadri test, the null hypothesis is that no series has a unit root, and the alternative hypothesis is that all series have a unit root.
80 Choi (2001) proposes a similar test as the Maddala and Wu (1999) test. These tests are based on an approach by Fisher (1932).
For the Levin-Lin-Chu and Breitung tests, the alternative hypothesis is that no series has a unit root, while for the Im-Pesaran-Shin and Maddala-Wu tests the alternative hypothesis is that some series do not have a unit root. The main disadvantage of the Levin-Lin-Chu, Breitung and Hadri tests is their assumption that all the series in the panel follow exactly the same autoregressive process (i.e. that ρi is the same for all i.) The Im-Pesaran-Shin and Maddala-Wu tests relax this assumption allowing the series to be heterogeneous.81
As a robustness check, a cross-country test for long-run relative PPP can also be performed. For this, equation (4.1) is first differenced, then averaged over time, and finally re-arranged to become the following country-specific equation:
i i
i p p e
q =∆ − −∆
∆ ( *) (4.7)
where
∑ ∑ ∑
=
=
=
∆
=
∆
−
∆
=
−
∆
∆
=
∆ N
t t i i
t t i N t i
N t
t i
i e
e N p
N p p p N q
q
1 ,
* , 1
* 1
,
), 1 1 (
) (
1 ,
, and N = 50 because the period is 1950-2000. Averaging eliminates the index t and all time variability.
According to equation (4.7), for each country i, the mean growth of its real exchange rate equals the difference between the mean inflation differential of that country with respect to the U.S. and the mean nominal exchange rate depreciation for the period of analysis.
PPP requires that ∆qi equal zero for each country. The cross-country test for long-run relative PPP, thus, consists of estimating the following two regressions:
i i
i p p
e =−µ+β∆ − +ε
∆ ( *) (4.8)
i i
i p e
p ) ' ' '
( − * =µ+β ∆ +ε
∆ (4.9)
where the intercepts, µ and µ', are to capture the mean value of ∆qi across countries, and εi and ε'i are assumed to be white noise. If PPP holds, inflation differentials must exactly offset changes in the nominal exchange rate, and vice-versa, particularly in the long-run.
This implies that the intercepts µ and µ' must be zero, and the coefficients β and β' must each equal unity. Equations (4.8) and (4.9) differ only in that the variables are transposed;
they are both estimated because there is no a-priori reason for having either variable as the dependent variable.