Purchasing power parity, revisited

Recall that the principle of purchasing power parity said that, assuming the model

Average annual change inthe exchange rate = β0 + β1 ×

Difference in averageannual inflation rates

+ random error,

PPP is consistent with β0 = 0 and β1 = 1. We examined that before by looking at two

t–statistics, but that wasn’t quite correct; this really is a partial F–test problem, testing

the hypotheses

H0 : PPP is correct

vs.

Ha : a general linear relationship different from PPP is correct.

The appropriate F–statistic has the form

F =(Residual SSPPP − Residual SSgen)/2

Residual SSgen/(n − 2),

on (2, n − 2) degrees of freedom, where Residual SSPPP is the residual sum of squares

assuming the PPP model holds, while Residual SSgen is the residual sum of squares from

the general regression model (the one based on predicting Exchange rate change from

Inflation difference). The residual sum of squares based on the PPP model is not

obtained from a regression run in this case, since there are no estimated parameters;

rather, PPP implies that the fitted value for Exchange rate change for any case is exactly

Inflation difference, so the residual sum of squares is simply

∑

i

(Exchange rate changei − Inflation differencei)2,

which equals 5.677 for the industrialized country data. Recall the regression results for the

general model:

Regression Analysis: Exchange rate change versus Inflation difference

Analysis of Variance

Source DF Adj SS Adj MS F-Value P-Value

Regression 1 61.850 61.8496 183.49 0.000

Inflation difference 1 61.850 61.8496 183.49 0.000

c© 2015, Jeffrey S. Simonoff 1

Error 14 4.719 0.3371

Total 15 66.569

Model Summary

S R-sq R-sq(adj) R-sq(pred)

0.580583 92.91% 92.40% 90.91%

Coefficients

Term Coef SE Coef T-Value P-Value VIF

Constant 0.086 0.147 0.59 0.567

Inflation difference 0.9024 0.0666 13.55 0.000 1.00

Regression Equation

Exchange rate change = 0.086 + 0.9024 Inflation difference

The residual sum of squares for this model is 4.719, so the partial F–test has the form

F =(5.677 − 4.719)/2

4.719/14= 1.42;

on (2, 14) degrees of freedom, this has a tail probability of .274. Thus, this simultaneous

analysis says that we don’t have evidence that PPP doesn’t hold for the developed countries

(recall that Greece was an unusual country in this analysis, but it turns out that omitting

it doesn’t change this implication; in fact, the evidence against PPP becomes weaker).

We can do a similar calculation for the developing country data (I’ve already omitted

Brazil):

Regression Analysis: Exchange rate change versus Inflation difference

Analysis of Variance

Source DF Adj SS Adj MS F-Value P-Value

Regression 1 1309.89 1309.89 1359.33 0.000

Inflation difference 1 1309.89 1309.89 1359.33 0.000

Error 22 21.20 0.96

c© 2015, Jeffrey S. Simonoff 2

Total 23 1331.09

Model Summary

S R-sq R-sq(adj) R-sq(pred)

0.981645 98.41% 98.33% 98.06%

Coefficients

Term Coef SE Coef T-Value P-Value VIF

Constant -0.641 0.287 -2.23 0.036

Inflation difference 0.9637 0.0261 36.87 0.000 1.00

Regression Equation

Exchange rate change = -0.641 + 0.9637 Inflation difference

The residual sum of squares based on the general model is 21.2; using the PPP model,

it is 26.1 (this is determined by direct calculation). The partial F–test is then

F =(26.1 − 21.2)/2

21.2/22= 2.54;

this has a tail probability of .10, so we see marginal evidence to reject the PPP hypothesis,

which we see comes from the intercept term. Omitting Mexico, however, drops the p-value

of the F -test to less than .01, reinforcing that notions of whether PPP holds or not can be

strongly dependent on accidents of how the “long run” is defined.

c© 2015, Jeffrey S. Simonoff 3