<!-- Start of picture text --> Box 1.1: Defining and Measuring Consumption Volatility<br><!-- End of picture text --> <!-- Start of picture text --> In this report, we assume that per capita consumption follows a geometric Brownian motion with drift represented by<br>- =, d +a. dZwhere dZis a standard Brownian motion with expectation and variance equal to 0 and dt,<br>c<br>respectively." 11 The expected instantaneous per capita consumption growth is u while a' (resp. a) represents the<br>variance (resp. standard deviation) of consumption growth. The variable a is usually referred to as the volatility of<br>per capita consumption. Using Ito's lemma, it can be shown that d(Logc,) =(p 1 .a2J dt+a dZ. Hence,<br>c, =c0 .exp[ (P_.aJ)+a.[Z(t+l)-Z(t)]] and Log( '"±)=( -.a2.)+a.[Z(t+l)-Z(t)]. Consequently, the<br>variance of per capita consumption growth and the expected instantaneous consumption growth are respectively given<br>b'y ai=Var[Log(c"')] and u=E[Log(C'+i)]+! .awhere Varand E refer to the unconditional variance and<br>I C, )J L lc,) 2<br>expectation. This consumption process -- which has the advantage of not violating the assumption of non-negative<br>consumption -- can be derived as the optimal outcome in a general equilibrium model with constant return-to-scale<br>technologies defined by = a dt + s -dZ where K represents the stock of capital, dY is the instantaneous output<br>K<br>and the technological coefficients {a,s} are exogenously specified constants with s > 0 (Auffret, 2002).<br><!-- End of picture text --> The lognormal model is consistent with the properties of historical consumption data. Indeed, historical data show that both the skewness and kurtosis of per capita consumption growth are not statistically different from those of the nornal distribution (skewness and kurtosis of the normal distribution are 0 and 3 respectively). The Table below reports sample statistics for yearly per-capita consumption growth. The skewness estimates are negative for all regions but are not statistically different from 0. Indeed, since there are 30 observations, the standard error for the skewness estimate under the null hypothesis of normality is 6/30 = 045 .3 Also, yearly per-capita consumption growth have a kurtosis which, in most regions, is not significantly different from the one of the normal distribution. Indeed, the standard error for the kurtosis estimate is 4/3-0 = 0 89 .

Table: Per Capita Consumpti on Statistics: Regional Comparison <!-- Start of picture text --> Region Per Capita Consumption<br>(unweighted average) Growth (%) Variance Skewness Kurtosis<br>:1 __ E[Lg(5c`)-E[Lg(5C )]f [g( -E[g( C<br>Caribbean Region 2.45 0.0083 -0.19 0.58<br>Sub-Saharan Africa 0.90 0.0074 -0.21 1.78<br>Middle East & North Africa 1.60 0.0061 -0.07 1.13<br>Latin America 1.44 0.0035 -0.47 2.04<br>Europe and Central Asia 1.49 0.0030 -0.52 1.11<br>East Asia and Pacific 2.90 0.0018 -0.41 3.52<br>OECD 2.50 0.0006 -0.08 0.52<br><!-- End of picture text --> _'/_ The geometric Brownian motion is the prototypical process used m finance to model stock price movenents.

2/ This is equivalent to say that instantaneous per capita consumption growth follows a generalized Wiener process.

**3/** in large samples of normally distributed data, the skewness and kurtosis estimators are normally distnbuted with means 0 and 3 and vanances _6/T_ and 24/T , respectively (Stuart and Ord, 1987, Vol. 1).

_5_