) The cumulative probability distribution shows the probability

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1 Problem Set ECONOMETRICS Prof Òscar Jordà Due Date: Thursday, April 0 STUDENT S NAME: Multiple Choice Questions [0 pts] Please provide your answers to this section below: ) The cumulative probability distribution shows the probability a that a random variable is less than or equal to a particular value b of two or more events occurring at once c of all possible events occurring d that a random variable takes on a particular value given that another event has happened Answer: a ) The expected value of a discrete random variable a is the outcome that is most likely to occur b can be found by determining the 50% value in the cdf c equals the population median d is computed as a weighted average of the possible outcome of that random variable, where the weights are the probabilities of that outcome Answer: d 3) The conditional distribution of given X = x, Pr( = y X = x), is a b c d Pr( = y) Pr( X = x) l Pr( X = xi, = y) i= Pr( X = x, = y) Pr( = y) Pr( X = x, = y) Pr( X = x) Answer: d

2 4) Two random variables X and are independently distributed if all of the following conditions hold, with the exception of a Pr( = y X = x) = Pr( = y) b knowing the value of one of the variables provides no information about the other c if the conditional distribution of given X equals the marginal distribution of d E( ) = E[ E( X)] Answer: d 5) Assume that is normally distributed N ( µ, σ ) To find Pr( c c), where c < c ci µ and di =, you need to calculate Pr( d Z d) = σ 5) Φ( d) Φ( d) 6) Φ(96) Φ( 96) 7) Φ( d) ( Φ( d)) ( Φ( d ) Φ( d )) 8) Answer: a 6) The central limit theorem a states conditions under which a variable involving the sum of,, n iid variables becomes the standard normal distribution b postulates that the sample mean is a consistent estimator of the population mean µ c only holds in the presence of the law of large numbers d states conditions under which a variable involving the sum of,, n iid variables becomes the Student t distribution Answer: a 7) In econometrics, we typically do not rely on exact or finite sample distributions because Answer: d a we have approximately an infinite number of observations (think of re-sampling) b variables typically are normally distributed c the covariances of i, jare typically not zero d asymptotic distributions can be counted on to provide good approximations to the exact sampling distribution 8) The central limit theorem states that

3 3 µ a the distribution for becomes arbitrarily well approximated by the σ standard normal distribution p b µ c the probability that is in the range µ ± c becomes arbitrarily close to one as n increases for any constant c > 0 d the t distribution converges to the F distribution for approximately n > 30 Answer: a 9) Two variables are uncorrelated in all of the cases below, with the exception of a being independent b having a zero covariance c σ X σxσ d E ( X ) = 0 Answer: c 0) The correlation between X and a cannot be negative since variances are always positive b is the covariance squared c can be calculated by dividing the covariance between X and by the product of the two standard deviations cov( X, ) d is given by corr( X, ) = var( X ) var( ) Answer: c Problems [40 pts] Instructions: The goal of the problem set is to understand what you are doing rather than just getting the correct result Please show your work clearly and neatly Please write your answers in the space provided ) The following problem is frequently encountered in the case of a rare disease, say AIDS, when determining the probability of actually having the disease after testing positively for HIV (This is often known as the accuracy of the test given that you have the disease) Let us set up the problem as follows: = 0 if you tested negative using the ELISA test for HIV, = if you tested positive; X = if you have HIV, X = 0 if you do not have HIV Assume that 0 percent of the population have HIV and that the accuracy of the test is 095 in both cases of (i) testing positive when you have HIV, and (ii) testing negative when you do not have HIV (The actual ELISA test is actually 997 percent accurate when you have HIV, and 985 percent accurate when you do not have HIV)

4 4 (a) Assuming arbitrarily a population of 0,000,000 people, use the accompanying table to first enter the column totals Test Positive (=) Test Negative (=0) Total HIV (X=) No HIV (X=0) Total 0,000,000 Answer: Test Positive (=) Test Negative (=0) Total HIV (X=) 0,000 No HIV (X=0) 9,990,000 Total 0,000,000 (b) Use the conditional probabilities to fill in the joint absolute frequencies Answer: Test Positive (=) Test Negative (=0) Total HIV (X=) 9, ,000 No HIV (X=0) 499,500 9,490,500 9,990,000 Total 0,000,000 (c) Fill in the marginal absolute frequencies for testing positive and negative Determine the conditional probability of having HIV when you have tested positive Explain this surprising result Answer: Test Positive (=) Test Negative (=0) Total HIV (X=) 9, ,000 No HIV (X=0) 499,500 9,490,500 9,990,000 Total 509,000 9,49,000 0,000,000 Pr(X= =) = 0087 Although the test is quite accurate, there are very few people who have HIV (0,000), and many who do not have HIV (9,999,000) A small percentage of that large number (499,500/9,990,000) is large when compared to the higher percentage of the smaller number (9,500/0,000) (d) The previous problem is an application of Bayes theorem, which converts Pr( = y X = x) into Pr( X = x = y) Can you think of other examples where Pr( = y X = x) Pr( X = x = y)? Answer: Answers will vary by student Perhaps a nice illustration is the probability to be a male given that you play on the college/university men s varsity team, versus the probability to play on the college/university men s varsity team given that you are a male student

5 5 ) The expectations augmented Phillips curve postulates p = π f( u u), where p is the actual inflation rate, π is the expected inflation rate, and u is the unemployment rate, with indicating equilibrium (the NAIRU Non-Accelerating Inflation Rate of Unemployment) Under the assumption of static expectations (π = p ), ie that you expect this period s inflation rate to hold for the next period ( the sun shines today, it will shine tomorrow ), then the prediction is that inflation will accelerate if the unemployment rate is below its equilibrium level The accompanying table below displays information on accelerating annual inflation and unemployment rate differences from the equilibrium rate (cyclical unemployment), where the latter is approximated by a five = year moving average ou think of this data as a population which you want to describe, rather than a sample from which you want to infer behavior of a larger population The data is collected from United States quarterly data for the period 964: to 995:4 Joint Distribution of Accelerating Inflation and Cyclical Unemployment, 964:-995:4 ( u u) > 0 ( u u) 0 Total ( = 0 ) ( = ) p p > ( X = 0 ) p p ( X = ) Total (a) Compute E( ) and E( X ), and interpret both numbers Answer: E ( ) = percent of the quarters saw cyclical unemployment EX ( ) = percent of the quarters saw decreasing inflation rates (b) Calculate E ( X= ) and E ( X= 0) If there was independence between cyclical unemployment and acceleration in the inflation rate, what would you expect the relationship between the two expected values to be? Given that the two means are different, is this sufficient to assume that the two variables are independent? Answer: E ( X= ) = 0356 ; E ( X= 0) = 07 ou would expect the two conditional expectations to be the same In general, independence in means does not imply statistical independence, although the reverse is true (c) What is the probability of inflation to increase if there is positive cyclical unemployment? Negative cyclical unemployment?

6 6 Answer: There is a 344 percent probability of inflation to increase if there is positive cyclical unemployment There is a 70 percent probability of inflation to increase if there is negative cyclical unemployment (d) ou randomly select one of the 59 quarters when there was positive cyclical unemployment ( ( u u) > 0) What is the probability there was decelerating inflation during that quarter? Answer: There is a 656 percent probability of inflation to decelerate when there is positive cyclical unemployment 3) Show that the correlation coefficient between and X is unaffected if you use a linear * * transformation in both variables That is, show that corr( X, ) = corr( X, ), where * * X = a+ bx and = c+ d, and where a, b, c, and d are arbitrary non-zero constants Answer: * * * * cov( X, ) bdcov( X, ) corr( X, ) = = corr( X, ) * * var( X ) var( ) b var( X) d var( ) 4) The textbook formula for the variance of the discrete random variable is given as Another commonly used formulation is k = ( yi ) pi i= σ µ Prove that the two formulas are the same k = yi pi y i= σ µ Answer: k k k = yi pi = yi + yi pi = yi pi + pi yipi i= i= i= σ ( µ ) ( µ µ ) ( µ µ ) Moving the summation sign through results in k k k = yi pi + pi yipi i= i= i= But i= σ µ µ giving you the second expression after simplification k pi = and µ = yp, i i k i=

7 7 EViews Exercise [40 pts] For this exercise you will need to download the file: pswf from my web-site It contains macroeconomic yearly data on the US The range is ou can click each series to get a more detailed description Please note that the instructions on what you need to turn in are very precise Turning in more pages than requested will result in an automatic deduction of 5 points in your score! ou will need to manipulate some of the series as follows: Calculate the annual rate of inflation by taking the log-difference of the GDP deflator as follows: Inflation = log(deflator t ) log(deflator t- ) This can be accomplished by typing the following line in the command window and hitting enter: genr inflation = d(log(deflator))*00 Remark: This is an approximation to calculating percentage changes that is common in econometric analysis To convince yourself, generate the inflation series by calculating the annual percentage change of the GDP deflator Type the following in the command window and then hit enter: genr inflation = ((deflator - (deflator(-))/deflator(-))*00 Plot both series on a graph Are you convinced? Give this graph the name inf_graph ou need to do the same to calculate output growth using the series GDP Call the new series "output" and "output" Plot both and name the graph gdp_graph Calculate the ratio of the budget_deficit to GDP ratio Note that the budget_deficit data are measured in real terms for the fiscal year Make sure the base year is the same as that used to calculate the GDP data, which is in real terms also In addition, notice that both variables are measured in billions of dollars so that their ratio multiplied by 00 will give the percentage of a year s GDP that the budget deficit represents The budget_deficit to GDP ratio can therefore be calculated by typing the following in the command window: genr deficit_ratio = budget_deficit*00/real_gdp Plot and give the graph the name deficit_graph Things you need to turn in: [05 pts] Select the graphs you named above, namely, inf_graph, gdp_graph,and deficit_graph By doing this, they will appear together in the same window Make sure each graph is properly titled and try to display each graph differently (play around with EViews!) Please print only one page with these three graphs

8 8 [0 pts] Calculate basic statistics for Unemployment, Output, Inflation, and Government Deficit (as a percentage of GDP) and the Trade Deficit (as a percentage of GDP) Now subset your sample according to whether the president was a democrat or a republican and calculate the same sample statistics for each variable What can you say about the differences? Please report using the appropriate units Fit all of these statistics in three tables (general, democrat president, republican president) and print only one page 3 [0 pts] Graph each series (Unemployment, Output Growth, Inflation, and Government Deficit) for the whole sample in individual graphs (although all graphs need to be in the same page) Shade the periods corresponding to democrat presidents and label each variable correctly Remember to turn one page only 4 [5 pts] ou are asked by the member of congress (for which you are interning) to report on the relative economic success of democrat presidents versus republican presidents Based on the analysis you have conducted, write a one-paragraph memo entitled: In support of administrations (you may choose republican or democrat according to your findings or political affiliation, see below) Make sure you mention other factors that may affect your conclusions (wars, oil crises, etc) Make sure you state your conclusion clearly In class, you may be asked to defend your perspective using the data in this analysis only!

9 Answers to Problem Set # (Empirical) Plots of inflation, output, and budget deficit variables Inflation Rate Real GDP Growth INFLATION INFLATION OUTPUT OUTPUT Budget Deficit to GDP Ratio DEFICIT_RATIO

10 Basic statistics for unemployment, output, inflation, and government budget deficit General: UNEMPLOMENT OUTPUT INFLATION DEFICIT_RATIO Mean Median Maximum Minimum Std Dev Skewness Kurtosis Observations Democrat Sub-Sample: UNEMPLOMENT INFLATION OUTPUT DEFICIT_RATIO Mean Median Maximum Minimum Std Dev Skewness Kurtosis Observations Republican Sub-Sample: UNEMPLOMENT INFLATION OUTPUT DEFICIT_RATIO Mean Median Maximum Minimum Std Dev Skewness Kurtosis Observations Based on the tables above, unemployment and the budget deficit ratio is lower, while output and inflation are higher when a Democrat is President While Democrats are usually associated with expansionary policies, the data suggests Republican presidents actually tend to have higher budget deficit ratios than Democrat presidents This analysis ignores, however, the composition of Congress Given Congress power in determining the budget, this could explain why spending has been higher under Republican presidents compared with Democratic ones

11 3 Graph each series and shade Democrat president Time Series of Macroeconomic Variables (ears corresponding to Democrat presidents shaded) WHITE_UNEMPLOMENT INFLATION OUTPUT DEFICIT_RATIO

Introduction to Econometrics. Review of Probability & Statistics

Introduction to Econometrics. Review of Probability & Statistics 1 Introduction to Econometrics Review of Probability & Statistics Peerapat Wongchaiwat, Ph.D. wongchaiwat@hotmail.com Introduction 2 What is Econometrics? Econometrics consists of the application of mathematical