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Module Name: Introductory Econometrics

Code: P12205

Credits: 10

Semester: Spring 2011/12

Delivery: 16 one-hour lectures + 4 one-hour workshops

Aims:

The main aims of this module are: to introduce students to the principles, uses and interpretation of regression analysis most commonly employed in applied economics; to provide participants with sufficient knowledge of regression methods to critically evaluate and interpret empirical research.

On completion of this module students should be able to: demonstrate understanding of the assumptions and properties underlying regression analysis and the principle of ‘least squares’; interpret and manipulate the coefficients of multiple regression and performance criteria; conduct diagnostic checking of the validity of regression equations coefficients; appreciate the problems of misspecification, multicollinearity, heteroscedasticity and autocorrelation.

Content:

1. Simple Regression Analysis

2. Multiple Regression Analysis

3. Dummy Variables

4. Heteroscedasticity

5. Autocorrelation

Main Textbook:

Dougherty, C. (2011). Introduction to Econometrics, 4th edition, Oxford.

2.

Module Name: Computational Finance

Code: P12614

Credits: 10

Semester: Spring 2011/12

Programme classes: 12 1-2 hour lectures/workshops

Aims:

The module aims to describe and analyse the general finance topics and introduces students to implement basic computational approaches to financial problems using Microsoft Excel. It stresses the fundamentals of finance; provides students with a knowledge and understanding on the key finance subjects such as money market, return metric, portfolio modelling, asset pricing, etc.; and equips students with the essential techniques applied in financial calculations.

Contents:

1. Lecture Topic 1: Money Market Instrument : Introduction to the course; Interest rate types;…...

...Outline 4 Probability – the chance that an uncertain event will occur (always between 0 and 1) Impossible Event – an event that has no chance of occurring (probability = 0) Certain Event – an event that is sure to occur (probability = 1) Assessing Probability probability of occurrence= probability of occurrence based on a combination of an individual’s past experience, personal opinion, and analysis of a particular situation Events Simple event An event described by a single characteristic Joint event An event described by two or more characteristics Complement of an event A , All events that are not part of event A The Sample Space is the collection of all possible events Simple Probability refers to the probability of a simple event. Joint Probability refers to the probability of an occurrence of two or more events. ex. P(Jan. and Wed.) Mutually exclusive events is the Events that cannot occur simultaneously Example: Randomly choosing a day from 2010 A = day in January; B = day in February Events A and B are mutually exclusive Collectively exhaustive events One of the events must occur the set of events covers the entire sample space Computing Joint and Marginal Probabilities The probability of a joint event, A and B: Computing a marginal (or simple) probability: Probability is the numerical measure of the likelihood that an event will occur The probability of any event must be...

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... stresses the fundamentals of finance; provides students with a knowledge and understanding on the key finance subjects such as money market, return metric, portfolio modelling, asset pricing, etc.; and equips students with the essential techniques applied in financial calculations. Contents: 1. Lecture Topic 1: Money Market Instrument : Introduction to the course; Interest rate types; Forward-forward rate; Forward rate agreement; Hedging strategies using FRAs; Swaps; Zero-coupon rate and Yield curve. 2. Lecture Topic 2: Econometrics for Finance 1: Regression Analysis 3. Lecture Topic 3: Econometrics for Finance 2: Time Series Analysis and Forecasting. 4. Lecture Topic 4: Return and Performance Evaluation: Single period returns: discrete, continuous, real, nominal, domestic, foreign; Multiperiod buy and hold returns of an asset; Multiperiod buy-and-hold returns of a portfolio; Portfolio Performance Measurement. 5. Lecture Topic 5: Portfolio Modelling: Expected return (mean return) and variance of a portfolio. Mean-variance efficient frontier. Capital market line (CML), Capital asset pricing model (CAPM), Arbitrage pricing theory (APT), Fama–French three-factor model. Main textbook: B. Steiner (2008), Mastering Financial Calculation, Second Edition, FT Press. Final Marks: 66% 3. Module Name: Calculus and its applications Code: HG1M01 Credits: 10 Semester: Autumn 2011/12 Delivery: 22 one-hour lectures + 8 one-hour workshops Aims: To......

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... the consumer is $25.00. (a) What is the probability that a package chosen randomly from the production line is too heavy? (b) For each 10,000 packages sold, what proﬁt is received by the manufacturer if all packages meet weight speciﬁcation? (c) Assuming that all defective packages are rejected 62 and rendered worthless, how much is the proﬁt reduced on 10,000 packages due to failure to meet weight speciﬁcation? 2.72 Prove that Chapter 2 Probability P (A ∩ B ) = 1 + P (A ∩ B) − P (A) − P (B). 2.6 Conditional Probability, Independence, and the Product Rule One very important concept in probability theory is conditional probability. In some applications, the practitioner is interested in the probability structure under certain restrictions. For instance, in epidemiology, rather than studying the chance that a person from the general population has diabetes, it might be of more interest to know this probability for a distinct group such as Asian women in the age range of 35 to 50 or Hispanic men in the age range of 40 to 60. This type of probability is called a conditional probability. Conditional Probability The probability of an event B occurring when it is known that some event A has occurred is called a conditional probability and is denoted by P (B|A). The symbol P (B|A) is usually read “the probability that B occurs given that A occurs” or simply “the probability of B, given A.” Consider the event B of getting a perfect square when a die is tossed...

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...Probability, Mean and Median In the last section, we considered (probability) density functions. We went on to discuss their relationship with cumulative distribution functions. The goal of this section is to take a closer look at densities, introduce some common distributions and discuss the mean and median. Recall, we define probabilities as follows: Proportion of population for Area under the graph of p ( x ) between a and b which x is between a and b p( x)dx a b The cumulative distribution function gives the proportion of the population that has values below t. That is, t P (t ) p( x)dx Proportion of population having values of x below t When answering some questions involving probabilities, both the density function and the cumulative distribution can be used, as the next example illustrates. Example 1: Consider the graph of the function p(x). p x 0.2 0.1 2 4 6 8 10 x Figure 1: The graph of the function p(x) a. Explain why the function is a probability density function. b. Use the graph to find P(X < 3) c. Use the graph to find P(3 § X § 8) 1 Solution: a. Recall, a function is a probability density function if the area under the curve is equal to 1 and all of the values of p(x) are non-negative. It is immediately clear that the values of p(x) are non-negative. To verify that the area under the curve is equal to 1, we recognize that the graph above can be viewed as a triangle...

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... more than three will withdraw. 1-BINOM.DIST(3,20,0.2,TRUE) =0.5885511 Q4. Compute the expected number of withdrawals. E(X)=NP=4 Problem 4 Trading volume on the New York Stock Exchange is heaviest during the first half hour (early morning) and the last half hour (late afternoon) of the trading day. The early morning trading volumes (millions of shares) for 13 days in January and February were recorded in the data file Volume.xls (Barron’s, January 23, 2006; February 13, 2006; and February 27, 2006). The probability distribution of trading volume is approximately normal. (1) Compute the mean and standard deviation to use as estimates of the population mean and standard deviation (please round the mean and the standard deviation to the nearest integer, e.g., 199.98≈200, 26.04≈26) . Mean | 199.6923077 | Standard Error | 7.222103372 | Median | 201 | Mode | 211 | Standard Deviation | 26.03966403 | Sample Variance | 678.0641026 | Kurtosis | 2.480119959 | Skewness | 1.051066108 | Range | 102 | Minimum | 163 | Maximum | 265 | Sum | 2596 | Count | 13 | (2) What is the probability that, on a randomly selected day, the early morning trading volume will be less than 180 million shares? NORM.DIST(180,200,26,TRUE)= 0.22087816 (3) What is the probability that, on a randomly selected day, the early morning trading volume will exceed 230 million shares? NORM.DIST(230,200,26,TRUE)=0.87571838 1-0.87571838= 0.12428162 (4) What is the...

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...PROBABILITY 1. ACCORDING TO STATISTICAL DEFINITION OF PROBABILITY P(A) = lim FA/n WHERE FA IS THE NUMBER OF TIMES EVENT A OCCUR AND n IS THE NUMBER OF TIMES THE EXPERIMANT IS REPEATED. 2. IF P(A) = 0, A IS KNOWN TO BE AN IMPOSSIBLE EVENT AND IS P(A) = 1, A IS KNOWN TO BE A SURE EVENT. 3. BINOMIAL DISTRIBUTIONS IS BIPARAMETRIC DISTRIBUTION, WHERE AS POISSION DISTRIBUTION IS UNIPARAMETRIC ONE. 4. THE CONDITIONS FOR THE POISSION MODEL ARE : • THE PROBABILIY OF SUCCESS IN A VERY SMALL INTERAVAL IS CONSTANT. • THE PROBABILITY OF HAVING MORE THAN ONE SUCCESS IN THE ABOVE REFERRED SMALL TIME INTERVAL IS VERY LOW. • THE PROBABILITY OF SUCCESS IS INDEPENDENT OF t FOR THE TIME INTERVAL(t ,t+dt) . 5. Expected Value or Mathematical Expectation of a random variable may be defined as the sum of the products of the different values taken by the random variable and the corresponding probabilities. Hence if a random variable X takes n values X1, X2,………… Xn with corresponding probabilities p1, p2, p3, ………. pn, then expected value of X is given by µ = E (x) = Σ pi xi . Expected value of X2 is given by E ( X2 ) = Σ pi xi2 Variance of x, is given by σ2 = E(x- µ)2 = E(x2)- µ2 Expectation of a constant k is k i.e. E(k) = k fo any constant k. Expectation of sum of two random variables is the sum of their expectations i.e. E(x +y) = E(x) + E(y) for any...

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... | 40 | 65 | | | Value of X | 1 | 0.02 | 0.05 | 0.10 | 0.03 | 0.01 | 0.21 | | | 5 | 0.17 | 0.15 | 0.05 | 0.02 | 0.01 | 0.40 | | | 8 | 0.02 | 0.03 | 0.15 | 0.10 | 0.09 | 0.39 | | Probability distribution of Y | 0.21 | 0.23 | 0.30 | 0.15 | 0.11 | 1.00 | (a) The probability distribution is given in the table above. (b) The conditional probability of Y|X 8 is given in the table below Value of Y | 14 | 22 | 30 | 40 | 65 | 0.02/0.39 | 0.03/0.39 | 0.15/0.39 | 0.10/0.39 | 0.09/0.39 | (c) 2.11. (a) 0.90 (b) 0.05 (c) 0.05 (d) When then (e) where thus 2.13. (a) (b) Y and W are symmetric around 0, thus skewness is equal to 0; because their mean is zero, this means that the third moment is zero. (c) The kurtosis of the normal is 3, so ; solving yields a similar calculation yields the results for W. (d) First, condition on so that Similarly, From the law of iterated expectations (e) thus from part (d). Thus skewness 0. Similarly, and Thus, 2.15. (a) where Z ~ N(0, 1). Thus, (i) n 20; (ii) n 100; (iii) n 1000; (b) As n get large gets large, and the probability converges to 1. (c) This follows from (b) and the definition of convergence in probability given in Key Concept 2.6. 2.17. Y = 0.4 and (a) (i) P( 0.43) (ii) P( 0.37) (b) We know Pr(1.96 Z 1.96) 0.95, thus we want n to satisfy and Solving these inequalities yields n 9220...

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...PROBABILITY ASSIGNMENT 1. The National Highway Traffic Safety Administration (NHTSA) conducted a survey to learn about how drivers throughout the US are using their seat belts. Sample data consistent with the NHTSA survey are as follows. (Data as on May, 2015) Driver using Seat Belt? | Region | Yes | No | Northeast | 148 | 52 | Midwest | 162 | 54 | South | 296 | 74 | West | 252 | 48 | Total | 858 | 228 | a. For the U.S., what is the probability that the driver is using a seat belt? b. The seat belt usage probability for a U.S. driver a year earlier was .75. NHTSA Chief had hoped for a 0.78 probability in 2015. Would he have been pleased with the 2003 survey results? c. What is the probability of seat belt usage by region of the Country? What region has the highest seat belt usage? d. What proportion of the drivers in the sample came from each region of the country? What region had the most drivers selected? 2. A company that manufactures toothpaste is studying five different package designs. Assuming that one design is just as likely to be selected by a consumer as any other design, what selection probability would you assign to each of the package designs? In an actual experiment, 100 consumers were asked to pick the design they preferred. The following data were obtained. Do the data confirm the belief that one design is just as likely to be selected as other? Explain. Design | Number of Times Preferred | 1 | 5 | 2 | 15...

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...Prof. Dr. Somesh Kumar Department of Mathematics Indian Institute of Technology, Kharagpur Module No. #01 Lecture No. #07 Random Variables So, far we were discussing the laws of probability so, in the laws of the probability we have a random experiment, as a consequence of that we have a sample space, we consider a subset of the, we consider a class of subsets of the sample space which we call our event space or the events and then we define a probability function on that. Now, we consider various types of problems for example, calculating the probability of occurrence of a certain number in throwing of a die, probability of occurrence of certain card in a drain probability of various kinds of events. However, in most of the practical situations we may not be interested in the full physical description of the sample space or the events; rather we may be interested in certain numerical characteristic of the event, consider suppose I have ten instruments and they are operating for a certain amount of time, now after amount after working for a certain amount of time, we may like to know that, how many of them are actually working in a proper way and how many of them are not working properly. Now, if there are ten instruments, it may happen that seven of them are working properly and three of them are not working properly, at this stage we may not be interested in knowing the positions, suppose we are saying one instrument, two instruments and so, on...

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... | 11 | 7 | 5 | 100 | Total | 7 | 17 | 30 | 26 | 48 | 35 | 43 | 32 | 32 | 21 | 9 | 300 | Table showing result from the three experiments Analysis of Data Probability tree showing the theoretical way of obtaining the probability of two dice We can use the table to see that there are six ways to get a sum of seven with two dice: (1,6),(2,5),(3,4),(4,3),(5,2),and (6,1).There are a total of 36 outcomes. Probability=Possible outcome Total outcome =6 36 =1 6 From this experiment the table shows that there are 35 way in which the sum of seven was obtained with two dice. There are a total of 300 outcomes. Probability=Possible outcome Total outcome =35 300 =7 60 This experiment compare the theoretical probability from the observed probability for 100 throws under each of three different conditions. These condition have significant influences in the outcomes of these throws.1 the size of the cup chosen may have a particular influence on these outcomes.2 the inside surface of the two types of cups chosen are also factors that may influence these outcomes. These simple experiment were intended to give some idea of the theory of theoretical probability. The range between the theoretical......

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...CS 70 Discrete Mathematics and Probability Theory Fall 2009 Satish Rao,David Tse Note 11 Conditional Probability A pharmaceutical company is marketing a new test for a certain medical condition. According to clinical trials, the test has the following properties: 1. When applied to an affected person, the test comes up positive in 90% of cases, and negative in 10% (these are called “false negatives”). 2. When applied to a healthy person, the test comes up negative in 80% of cases, and positive in 20% (these are called “false positives”). Suppose that the incidence of the condition in the US population is 5%. When a random person is tested and the test comes up positive, what is the probability that the person actually has the condition? (Note that this is presumably not the same as the simple probability that a random person has the condition, which is 1 just 20 .) This is an example of a conditional probability: we are interested in the probability that a person has the condition (event A) given that he/she tests positive (event B). Let’s write this as Pr[A|B]. How should we deﬁne Pr[A|B]? Well, since event B is guaranteed to happen, we should look not at the whole sample space Ω , but at the smaller sample space consisting only of the sample points in B. What should the conditional probabilities of these sample points be? If they all simply inherit their probabilities from Ω , then the sum of these probabilities will be ∑ω ∈B Pr[ω ] = Pr[B...

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... theories, the theory of estimation is an illustration of the probabilistic ideas in official terms, which is in form, may be taken differently from their implication. These official terms are adjusted through the mathematical principles and logic, and any outcomes are translated or inferred back to the issue domain. Probability theory is utilized daily in risk evaluation and in trade on good markets. However, with different applications in the society, probabilities are not examined separately or appropriately very sensibly (Grinstead & Snell, 2010). References: Grinstead, C., and Snell J. (2010). Introduction to Probability. Retrieved on October 19, 2011, from Olofsson, Peter (2005). Probability, Statistics, and Stochastic Processes. New York, NY: Wiley- Interscience.[pic]...

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...PROBABILITY SEDA YILDIRIM 2009421051 DOKUZ EYLUL UNIVERSITY MARITIME BUSINESS ADMINISTRATION CONTENTS Rules of Probability 1 Rule of Multiplication 3 Rule of Addition 3 Classical theory of probability 5 Continuous Probability Distributions 9 Discrete vs. Continuous Variables 11 Binomial Distribution 11 Binomial Probability 12 Poisson Distribution 13 PROBABILITY Probability is the branch of mathematics that studies the possible outcomes of given events together with the outcomes' relative likelihoods and distributions. In common usage, the word "probability" is used to mean the chance that a particular event (or set of events) will occur expressed on a linear scale from 0 (impossibility) to 1 (certainty), also expressed as a percentage between 0 and 100%. The analysis of events governed by probability is called statistics. There are several competing interpretations of the actual "meaning" of probabilities. Frequentists view probability simply as a measure of the frequency of outcomes (the more conventional interpretation), while Bayesians treat probability more subjectively as a statistical procedure that endeavors to estimate parameters of an underlying distribution based on the observed distribution. The conditional probability of an event A assuming that B has occurred, denoted ,equals The two faces of probability introduces a central ambiguity which has been around for 350 years and still leads to disagreements...

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...Probability Question 1 The comparison between the bar chart and histogram are bar graphs are normally used to represent the frequency of discrete items. They can be things, like colours, or things with no particular order. But the main thing about it is the items are not grouped, and they are not continuous. Where else for the histogram is mainly used to represent the frequency of a continuous variable like height or weight and anything that has a decimal placing and would not be exact in other words a whole number. An example of both the graphs:- Bar Graph Histogram These 2 graphs both look similar but however, in a histogram the bars must be touching. This is because the data used are number that are grouped and in a continuous range from left to right. But as for the bar graph the x axis would have its individual data like colours shown in the above. Question 2 a) i) The probability of females who enjoys shopping for clothing are 224/ 500 = 0.448. ii) The probability of males who enjoys shopping for clothing are 136/500 = 0.272. iii) The probability of females who wouldn’t enjoy shopping for clothing are 36/500 = 0.072. iv) The probability of males who wouldn’t enjoy shopping for clothing are 104/500 = 0.208. b) P (AᴗB) = P(A)+P(B)-P(AᴖB) P (A|B) = P(AᴖB)P(B) > 0 P (B|A) = PAᴖBPA PAᴖBPB = PAᴖBPA PAPB = 1 P(A) = P(B) P(AᴗB) = 1 P(A)+P(B) = 1 P(B) > 0.25 Question 3 1. Frequency Distribution of Burberry...

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...Massachusetts Institute of Technology 6.042J/18.062J, Fall ’02: Mathematics for Computer Science Professor Albert Meyer and Dr. Radhika Nagpal Course Notes 10 November 4 revised November 6, 2002, 572 minutes Introduction to Probability 1 Probability Probability will be the topic for the rest of the term. Probability is one of the most important subjects in Mathematics and Computer Science. Most upper level Computer Science courses require probability in some form, especially in analysis of algorithms and data structures, but also in information theory, cryptography, control and systems theory, network design, artiﬁcial intelligence, and game theory. Probability also plays a key role in ﬁelds such as Physics, Biology, Economics and Medicine. There is a close relationship between Counting/Combinatorics and Probability. In many cases, the probability of an event is simply the fraction of possible outcomes that make up the event. So many of the rules we developed for ﬁnding the cardinality of ﬁnite sets carry over to Probability Theory. For example, we’ll apply an Inclusion-Exclusion principle for probabilities in some examples below. In principle, probability boils down to a few simple rules, but it remains a tricky subject because these rules often lead unintuitive conclusions. Using “common sense” reasoning about probabilistic questions is notoriously unreliable, as we’ll illustrate with many real-life examples. This reading is longer than usual . To keep things in...

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