III. DOLLARIZATION AND PRICE DYNAMICS
3. Trends and Fluctuations in Real Exchange Rates
3.1. The Econometric Models
The first type of unit-root test used in this analysis is the typical univariate ADF test, and does not consider any breaks. The test is called “univariate” because it is applied to each relative price series, qt, independently.51 The test consists of estimating the following regression:
∑
= −− + ∆ +
+
=
∆ p
k
t k t k t
t q q
q
1
1 β ε
α
µ (3.3)
where the constant term, µ, is allowed because qt is required to be constant in steady state but not necessarily to equal zero. That is, relative rather than absolute LOP is required.
For this same reason, no trend is allowed in the regression. The Schwarz Criterion (SC) is
51 The subscript j indexing the commodity will be ignored for simplicity of notation.
used to select the optimum number of lags, p.52 The null and alternative hypotheses for this test are Ho: α = 0, and Ha: α < 0, with the null hypothesis indicating the presence of a unit root. Under Ho, the t-statistic for α does not follow the typical t-student distribution, so the critical values have to be calculated independently. Most statistical programs already provide MacKinnon (1996) critical values.
The second type of unit-root test used is still univariate but incorporates a mechanism to control for the regime changes. As indicated above, the demarcation of regimes is likely to be exogenous to the data generating process of the relative price series. Following Papell (2002), each qt series is allowed to have three breaks intended to control for the four hypothesized regimes. Once the break dates, i.e. months TB1, TB2 and TB3, are defined, each one of the 24 qt series is de-trended using the following regression:
t t t
t
t DT DT DT z
q =µ+γ1 1 +γ2 2 +γ3 3 + (3.4)
with the γ coefficients subject to the following two restrictions:
3 0
2
1+γ +γ =
γ (3.5)
0 ) 2 3 ( ) 1 3
( 2
1 TB −TB +γ TB −TB =
γ (3.6)
where DTit = (t – TBi) if t > TBi, and zero otherwise, for i = 1, 2, 3. Restrictions (3.5) and (3.6) make sure the resulting trend has the desired shape, i.e. with the same constant mean before TB1 and after TB3, and a triangular shape in between with the apex at TB2 (The break dates and the implied regimes are defined in the next section.) Each de- trended series consists of the residuals zt of equation (3.4). After obtaining the 24 residual series, the following regression is estimated:
∑
= −− + ∆ +
=
∆ p
k k t k t
t
t z z
z
1
1 β ε
α (3.7)
to test Ho: α = 0 versus Ha: α < 0, where Ho implies that the qt series has a unit root without breaks, and Ha implies that the qt series is stationary with breaks that are consistent with mean reversion. Under Ho, the t-statistic for α does not follow the typical t-student distribution, and because of the particular kind of trend assumed in equation (3.4), the critical values are not readily available from statistical packages.
The critical values for the univariate unit-root test with breaks are obtained through Monte Carlo simulation. The simulation consists of 50,000 iterations, each time generating a unit-root series of length 100, which is the number of months in the sample.53 The unit-root series is then de-trended using equation (3.4) subject to restrictions (3.5) and (3.6), and its residuals are regressed according to equation (3.7) to obtain the t-statistic of coefficient α. Using this procedure, a total of 50,000 t-statistic values are obtained. These values are sorted in ascending order, and the 500th, 2,500th, and 5,000th values are selected as the critical t values at the 1%, 5% and 10% significance level, respectively. These values are -3.516, -2.876, and -2.552, respectively.
The third type of unit-root test for this analysis is the Maddala and Wu (1999) test, which consists of combining the results of individual univariate unit root tests to increase statistical power. Univariate tests could fail to reject the null hypothesis of a unit root because of low statistical power. The solution, according to recent literature, is to use tests that exploit the structure provided in panel data, when available. Two usual unit-root tests for panel data are the Levin, Lin, and Chu (2002) and the Im, Pesaran, and Shin
53 Actually, 200 random-walk observations were generated with zero as initial value, and only the last 100 were used.
(2003) tests. However, although these tests could be applied to the series without controlling for structural breaks, they would not be appropriate for the series once de- trended to control for the breaks. The reason is that the trend assumed by equations (3.4), (3.5), and (3.6) is not typical, and so these tests applied to the de-trended series would produce the wrong critical values. The Maddala-Wu test is appropriate in both situations.54
The Maddala-Wu test consists of first obtaining the p-values for the t-statistics of the coefficients α estimated with the univariate unit-root tests for the individual commodity relative price series. For the de-trended series, the p-values are obtained using the list of sorted t-statistic values generated with the Monte-Carlo simulation explained above. For the series without breaks, a new Monte-Carlo simulation is needed, this time excluding the step where the unit-root series are de-trended. According to Maddala and Wu (1999), these p-values follow a Uniform distribution between 0 and 1, and -2Ln(p-value) follows a χ2 distribution with 2 degrees of freedom. Because of the additive property of the χ2 distribution, the p-values for the individual univariate tests can be combined so that the statistic:
∑
=−
−
= N
j
valuej
p Ln
1
) (
λ 2 (3.8)
has a χ2 distribution with 2N degrees of freedom. The null and alternative hypotheses of this test are Ho: αj = 0 for all j, and Ha: αj < 0 for some j, respectively, where j = 1, 2,…, N. Because in this test, the p-values are obtained from individual ADF tests on the commodity relative price series, the series are not assumed to follow the same
autoregressive process. That is, the αj coefficients are allowed to vary across the series.
Ho, which implies that all series have a unit root, is rejected if the test statistic λ is greater than the appropriate χ2 critical value.55
The Maddala-Wu test is not applied to the entire set of 23 commodity relative price series.56 If the series are heterogeneous, some of them could drive the result of the test.
According to the standard deviation values shown for the first-differenced series in Table 3.2 of section 2.2, a few commodities are much more volatile than the rest. In addition, some series could be subject to particular influences in a way that the trend described by equations (3.4), (3.5) and (3.6) does not apply to well. The test is, thus, applied to a sub- set of selected commodities. The selection consists of calculating the standard deviation of the de-trended series, determining the 75th percentile of the standard-deviation distribution, and eliminating those series with standard deviation above this percentile.57 The commodities excluded using this criterion are electricity, oranges, lemons, potatoes, tomatoes and spaghetti, leaving 17 commodities for the test.