s of the log of the real exchange rate:
*lnlnlnlnttttPPSR+−=
If PPP holds then the previous analysis indicates that the log real exchange rate should be a constant (identically zero if the price indices are based in the same year). Movements in the (log) real exchange rate are therefore equivalent to deviations from PPP. If PPP holds in the long-run then any deviations should be transitory – in other words, under PPP, the (log) real exchange rate should be a mean reverting process. In this context, a common (univariate) test of PPP involves simply testing the log real exchange rate for a unit root. If the null of a unit root is rejected in favour of the alternative then this
supports PPP: the implication is that the deviations from PPP are stationary and hence transitory.
However stationarity is a sufficient (but not a necessary) condition for mean reversion. For example if follows a fractional white noise process: tRln
()tdtLRε−−=1ln
then the process is mean reverting for 1<d (see lecture 7). In this context, however, it is possible for a non-stationary process to exhibit mean reversion (if15.0<≤d). This framework therefore provides a richer (and potentially more powerful) description of the long-run behaviour of the series. This may allow the data a better chance of distinguishing unit root behaviour (permanent shocks) from near unit root behaviour (highly persistent shocks) To test for mean reversion in the log real exchange rate we will therefore also apply the Geweke and Porter-Hudak (GPH) spectral regression to estimate the long memory parameterd(see lecture 7).
The analysis in this seminar will centre on univariate unit root testing of the nominal exchange rate and price indices. All the variables in the PPP relationship are price series so, à priori, we may expect them to be non-stationary processes. This univariate analysis forms an important pre-cursor to multivariate tests of PPP which we will carry out in the next seminar (‘testing for cointegration’). In this seminar, we will also carry out univariate tests of PPP based on the log real exchange rate. This involves applying unit root and long memory tests as discussed above.
2. Preliminary analysis
The analysis will involve the logarithms of the series so firstly generate the log transforms of the series
On the workfile toolbar click Genr and enter…
lns=log(s)
Repeat for lnp=log(p), lnpstar=log(pstar) and lnr=log(s)-log(p)+log(pstar) (the real exchange rate)
2.1 Visual inspection of the data
Figure 1: Graphs of PPP variables in levels 1.62.02.42.83.23.64.01975198019851990199520002005LNS2.02.53.03.54.04.55.01975198019851990199520002005LNP
We can obtain a lot of information about the non-stationarity of the series from a simple visual inspection of line graphs. 3.43.63.84.04.24.44.64.85.01975198019851990199520002005LNPSTAR2.83.03.23.43.63.84.01975198019851990199520002005LNR
Figure 2: Graphs of PPP variables in differences -.10-.05.00.05.10.15.20.251975198019851990199520002005DLNS-.03-.02-.01.00.01.02.03.04.051975198019851990199520002005DLNP
All of the series, including the real exchange rate, appear to be non-stationary in levels. However differencing the series makes them stationary. -.04-.02.00.02.04.061975198019851990199520002005DLNPSTAR-.10-.05.00.05.10.15.201975198019851990199520002005DLNR
2.2 Analysis of correlograms
Correlograms of PPP variables in levels
lns
Sample: 1973M01 2005M10In
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