L_CHAPTER6

L_CHAPTER6 - Chapter 6. The Long-Term Growth Rate as a...

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Chapter 6. The Long-Term 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 long-term 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 10-year expected yearly growth rate in each sector? The surprise is that sector B fares better: its expected 10-year growth rate is 8.59% per year, while sector A±s expected 10-year growth rate is 8.40% only. 1
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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 long-term 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
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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.

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L_CHAPTER6 - Chapter 6. The Long-Term Growth Rate as a...

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