The model presented in this section is similar to the cobweb models used by Engle, Hendry and Richard (1994) to analyse exogeneity issues. The phase average procedure is based on the mechanism detailed in Campos, Ericsson and Hendry (1990). The structure is a customary control rule for an output variable y„ being manipulated through the input variable z,. In our case y, would be the logarithm o f the output a given econom y at time /, while z, will be the size or activity in its financial system’^. From the analysis in chapter 1, i f w e consider a short enough time interval, the financial sector will not change until after the realisation y,.y is known.
y r = p i y , . j + p 2 Z , + £ , Z , = A y y ,./+ A 2 Z ,./ + V,
e , = p e , . i + (0 , with /p /< J and [ œ , , v j ~ i i d ( 0 , Q )
I f w e b elieved that the political system is also w eakly endogenou s to output, w e could think o f z, b ein g any m easurem ent reflecting p olitical, social or legal conditions. For ex a m p le, d em ocracy as m easured in a p ositive scale related to participation in election s or political instab ility in a n egative sca le m easured by
Our objective is to make correct inferences about the parameters in the first equation based on T time series observations. In particular, to make inferences about p2 that measures the
feedback from financial activity and possibly other weakly exogenous variables.
In order to deal with data problems as missing observations, and in order to smooth measurement errors in the variables, the observations o f y, and z, are often averaged and/or differentiated over time in empirical work. After transforming the data the relation to be estimated is:
yt = bi y,.k + b j z , + u,
The phase-averaging transformation is often applied to the data, see for example Barro et al. (1995) or King and Levine (1993). In their procedure, the original five-year frequency
data is averaged over a ten-year period^\ The parameter estimation method is then
applied using all the non-overlapping ten-year separated observations. The span o f the phase-average also implies that after averaging and smoothing the time observations, some o f the transformed data are dropped from the sample. A s an example, in Barro et al. (1995), the original panel data o f countries is available almost always in any five years period from 1960 to 1965, 1970, 1975, 1980, 1985 and 1990^"^. However, the parameter estimation stage is performed only over two elements comprising information from ten
num ber o f political assassinations.
K in g and L evin e (1 9 9 3 ) apply also a tw enty and thirty years p h ase-averaging, w h ile A tje and Jovanovic (1 9 9 3 ), Harris (1 9 9 7 ), Barro (1 9 9 1 ) and Barro (1 9 9 7 ) use m ore often a ten-year as w ell as other tim e intervals.
S o m e o f the variables are at the end o f the period as G D P , investm ent to product ratio, govern m en t expenditures and so on. Other observations are averages over the five years period; for exam p le p olitical stab ility or liq u id -liab ilities-to-G D P ratio.
years: 1965 to 1975 and 1975 to 1985. This transformation o f the data involved both procedures o f averaging the values over ten-year phases, and dropping the overlapping observations 1960 to 1970, 1970 to 1980 and 1980 to 1990.
The convenience o f averaging data before performing an estimation procedure is often related to alleged mitigation o f measurement errors. However, as has been summarised by Ericsson and Hendry (1994) phase average raises three questions: (i) what are the theoretical effects o f phase averaging? (ii) what are the observed effects o f phase averaging? and, (iii) what are the effect o f selecting the interval over which phase average?
A first negative effect we can identify, about phase averaging, is that it loses information. This can be seen in the frequency domain, where averaging the time series is equivalent to dropping some o f the frequencies o f the series. Therefore, unless there are good reasons to weight by zero these frequencies, phase averaging w ill be an inefficient method’^. It will not be advisable averaging the observations without knowing if it helps to obliterate these undesired measurement errors. A second problem with phase-averaged is that the tests over the parameters based in averaged data may be o f low power relative to those based in annual data. This is due to the decreased degrees o f freedom. Even if the corresponding annual parameters can be recovered from the phase average estimations so they can be
Barro (1 9 9 6 ) claim s that “ [T]he precise tim ing betw een growth and its determ inants is not w ell sp ecified at the high frequencies characteristics o f bu sin ess c y c le s .” (p. 13) He explain s that the underlying theory relates to long-term growth and that relationships at the annual freq uencies w ou ld be lik ely to be dom inated by m easurem ent error due to this m istim ing. H ow ever, he d o es not present an ex p lic it m odel for the kind o f data problem s present at the annual level and w h y these w ou ld disappear w hen taking the averages. In our m odel sin ce there is no relationship in the long-term w ithou t the relationship in the short term, averagin g and dropping observations increases m easurem ent error.
tested indirectly, the test procedures w ill be less sensitive to alternative hypothesis.
I f all the parameters o f interest, including mean values and standard deviations, can be recovered after the estimating from the phase-averaged data, the effects o f transformation would be unimportant. However, it is not likely that parameters related to granger non causality, short run variability or dynamic mechanisms whereby the econom y adjust to shocks can be recovered from estimation on phase-averaged data.
More importantly, weakly exogenous variables at annual observations level, might not longer be weakly exogenous at phase averaged level. This can be seen intuitively in the follow ing way. We pictured a system were growth y, depends on a predetermined variable Zt that measures for example: the activity o f the financial system or political instability. In an annual base, there can be some feedback from growth to variables like the size o f markets or to some social measurements. An economy that is growing faster this year may be able to delay or stop the detonation o f particularly explosive social issues. At the same time, a faster-growing econom y at a given year will push markets to operate or to get more involved in the economy. This possibly small annual feedback after phase averaging becom es a large simultaneity between the two variables. At the ten-year average level financial markets are not predetermined anymore.
The basic properties o f phase averaging is illustrated in a simple time series model, taken from Campos, Ericsson and Hendry (1990), let:
y , = p o + P 2 Z , + £ ,
£ , = p £ , . i + C Q
Z r = Xiy,.i+X2Z,.i+v, where, /p /< l and [cp , v j ~ N (0,D iag(o^ ,G v))