ECON 213 Midterm, Spring 2008

Economic Statistics I Dr. Nisreen Salti American University of Beirut Fall 2006 Midterm Solutions 1 Problem 7 Suppose you are interested in the growth rates of di¤erent countries and assume that for any given country in any given year t, the growth rate gt is normally distributed with mean � and standard deviation 0.05. Assume also the growth rates are independent over time. Suppose that half the countries are high growth countries, with mean �H = 0:1 and the other half are low growth countries, with mean �L = 0:01 but you do not observe whether a country is low growth or high growth. (a) You randomly select a country. What is the probability that its growth rate is 0:1 or higher? ANSWER: Half the countries are high growth countries. For a high growth country, the probability of observing growth of 0:1 or higher is 0:5 (since growths are normally distributed with mean 0:5). The other half of countries are low growth countries. For low growth countries, the probability of observing growth of 0:1 or higher is: P(g � 0:1jlow growth) = P � Z � 0:1 � 0:01 0:05 � = 0:036 So the overall probability of randomly selecting a country with growth rate 0:1 or higher is: P(g � 0:1) = 0:5 � P(g � 0:1jhigh growth) + 0:5 � P (g � 0:1jlow growth) P(g � 0:1) = 0:268 (b) Suppose you are told a given country’s growth rate is positive. What is the probability that it is a low growth country? ANSWER: P (low growthjg � 0) = P(g � 0jlow growth)P(low growth) P(g � 0jlow growth)P(low growth) + P(g � 0jhigh growth)P (high growth) = 0:5 � P � Z � 0�0:01 0:05 � 0:5 � P � Z � 0�0:01 0:05 � + 0:5 � P � Z � 0�0:1 0:05 � = 0:37 (c) Question (b) tells you what the probability is that a country is a low growth country if you observe a positive growth rate. Use your answer to question (b) to calculate the probability 1 that a country that has a positive growth rate this year will have a negative growth rate next year. ANSWER: The growth rates over time are independent. The probability that it experiences negative growth next year is: P(gt+1 � 0) = P(gt+1 � 0jlow growth)�P(low growth)+P(gt+1 � 0jhigh growth)P (high growth) But the probabilities that the country is high or low growth are not 0:5 in this case, but instead, are updated using the information that gt � 0 this year. We calculate in part b the probability that a country is low growth, given that its growth rate is positive this year. So the probability that it is high growth is 1 � 0:37 = 0:63. So: P(gt+1 � 0) = 0:37 � P(gt+1 � 0jlow growth) + 0:63 � P(gt+1 � 0jhigh growth) P(gt+1 � 0) = 0:37P � Z � 0 � 0:01 0:05 � + 0:63P � Z � 0 � 0:1 0:05 � P(gt+1 � 0) = 0:17 2