This preview shows pages 1–3. Sign up to view the full content.
Chapter 6. The LongTerm Growth Rate as a
Random Variable, with an Application to the
U.S. Economy
A growth process is far from linear. The economy is submitted to random
shocks and undergoes cycles, their length and amplitude hardly predictable.
make an estimate about the future yearly growth rates of an economy; each
of these growth rates is considered as a random variable, with given mean and
variance. What can we infer from those estimates about the
n
year horizon
expected growth rate and its variance? The answer is far from intuitive.
Indeed, we might be tempted to say that the expected longterm growth rate
is the expected yearly growth rate over that horizon. That this is not so will
be illustrated in the following example.
Consider two sectors of an economy, A and B. In sector A, the expected
yearly growth rate is 10% per year, with a standard deviation of 20%. In
sector B the expected yearly growth rate is 9% with a standard deviation
of 10%. We suppose that in each sector the yearly growth rates are inde
pendent and identically distributed, although we do not make any particular
hypothesis about the probability distribution of each growth rate. Table one
summarizes these estimates for each sector.
Table 1. Expected yearly growth rates
and their standard deviations
Sector A Sector B
Expected yearly growth rate
10%
9%
Standard deviation
20%
10%
What then is the 10year expected yearly growth rate in each sector?
The surprise is that sector B fares better: its expected 10year growth rate is
8.59% per year, while sector A±s expected 10year growth rate is 8.40% only.
1
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document The aim of this chapter is to explain how to determine such results. We
will also show how we can estimate the probability distribution of the long
term growth rate even if we do not know the probability law governing the
yearly growth rates.
growth rates if we want to infer anything about yearly growth rates. We will
longterm evolution of the US economy.
1
From daily to yearly growth rates
We will use the following notation;
j
always refers to a day;
t
refers to a year.
.
S
0
value at the beginning of a year
.
S
j
value at the end of day
j
(
j
= 1
;:::;
365)
.
.
S
j
=S
j
1
daily growth factor
.
S
j
S
j
1
S
j
1
=
R
j
1
;j
daily growth rate (compounded once a day)
.
log(
S
j
=S
j
1
) =
r
j
1
;j
continuously compounded daily growth rate
(see chapter 9).
.
S
365
=S
0
=
X
t
1
;t
=
yearly growth factor
.
S
This is the end of the preview. Sign up
to
access the rest of the document.
This note was uploaded on 06/16/2010 for the course MS&E 249 taught by Professor Olivierdelagrandville during the Fall '08 term at Stanford.
 Fall '08
 OLIVIERDELAGRANDVILLE

Click to edit the document details