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Lecture 4 Data Analysis Today’s Agenda Announcements Recap of Markets Qualitative Analysis Quantitative Analysis Announcements Recap of Markets Qualitative Analysis Quantitative Analysis Other Updates Tutorials are on this week! Assignment #1 to be posted by the end of the week o Remember, you can work in groups of up to 3 o You do not need to work with students in the same tutorial Announcements Recap of Markets Qualitative Analysis Quantitative Analysis What did we learn about markets? Markets are composed of ________________________ Individual buyers and sellers have their own __________ ____________________ A buyers demand is driven by her _________________ ___________ (as well as the other factors discussed last week) A sellers supply is driven by her ___________________ Announcements Recap of Markets Qualitative Analysis Quantitative Analysis What did we learn about markets? To get the demand and supply for the entire market, we ___________________________________ Aggregating horizontally implies that we add up ___________________________________ Aggregate demand and supply schedules may have a ________ in them Announcements Recap of Markets Qualitative Analysis Quantitative Analysis What did we learn about markets? Our measure of _________________ is concerned with _____________ Total surplus is the ____________________________ __________________________ When total surplus is maximized, we say the market is _______________________ Announcements Recap of Markets Qualitative Analysis Quantitative Analysis What did we miss out in on markets? Some of the institutions: 1. Posted Prices 2. Uniform Prices 3. One-sided Sequential Auctions 4. Double Auctions Announcements Recap of Markets Qualitative Analysis Quantitative Analysis Posted Prices This market institution is most often found in _________________________ Sellers quote prices on a take-it-or-leave-it basis in many ___________________________ Posted prices became common in the last century in large stores in which the owner/managers had to rely on numerous sales clerks Government regulation in industries such as alcohol sometimes require that prices be posted with the regulatory agency and that discounts not be granted Announcements Recap of Markets Qualitative Analysis Quantitative Analysis Posted Prices: Experimental Rules At the beginning of each trading period, sellers decide on a price offer which they write down and submit on cards Buyers will then be free to make bids to purchase from sellers Buyers will be ordered randomly, and allowed to make bids one at a time Once all buyers have had a chance to make a bid, the trading period is over Announcements Recap of Markets Qualitative Analysis Quantitative Analysis Posted Prices: Effect on Market Consider the simplest possible market with one buyer and one seller of a single unit Suppose the seller has a cost of $1 and the buyer a valuation of $2 In a market with decentralized negotiations, one would expect traders to reach an agreement somewhere in the middle (i.e. $1.50) In a posted price market, theory tells us that the seller could sell it for __________ Announcements Recap of Markets Qualitative Analysis Quantitative Analysis Uniform Prices The price discrimination that occurs in posted price institutions can cause a sense of regret among buyers The sale of all units at a uniform price, in contrast, can create an appearance of fairness In practice, a ____________ is a form of uniform price auction where both buyers and sellers submit bids The auctioneer then constructs the corresponding demand and supply schedules in order to determine the uniform price (i.e. the equilibrium price) Recap of Markets Announcements Qualitative Analysis Quantitative Analysis One-sided Sequential Auction This institution can be used when we have a single seller of a single good, along with multiple buyers There are 2 types of such auctions: 1. ___________________ o 2. Price starts high and is lowered until a buyer makes a bid ___________________ o Price starts low and increases until there is only 1 buyer willing to pay at that price Announcements Recap of Markets Qualitative Analysis Quantitative Analysis Double Auction Both buyers and sellers can actively post and accept prices _______________ of all the market institutions Thousands of experiments have been done investigating this institution and have found that neither complete information (i.e. buyers and sellers knowing each others costs and valuations) nor large numbers of traders is a necessary condition for convergence to competitive equilibrium outcomes Announcements Recap of Markets Qualitative Analysis Quantitative Analysis Double Auction: Experimental Rules During a trading period, any buyer is free at any time to raise his/her hand and make a bid to buy a unit at a specified price Any seller is free at any time to raise his/her hand to state an asking price for a unit at a specified price Any bid or asking price will remain on the blackboard until it is either accepted or improved (i.e. replaced by a higher bid or a lower asking price) When a bid or ask is accepted, all outstanding bids and asks are cancelled and the market is re-set Announcements Recap of Markets Qualitative Analysis Quantitative Analysis Qualitative vs Quantitative Analysis ___________________ gives us an overview of what the data have to say Basis Tools 1. 2. _______________ _______________ Goals of Summary Data 1. 2. Allows you to see _________________________ that require further investigation Guide subsequent __________________________ experimental data usually are ________________, so a descriptive summary is essential Announcements Recap of Markets Qualitative Analysis Quantitative Analysis Qualitative vs Quantitative Analysis ________________ gives specific answers to specific questions e.g. “Does treatment X affect outcome Y?” o Basis Tools 1. ______________________ Goals of Inferential Statistics 1. Determine whether there is a statistically significant difference between treatments 2. Determine whether there is a statistically significant difference between experimental outcomes and theoretical outcomes Announcements Recap of Markets Qualitative Analysis Quantitative Analysis Learning Statistics by Example For the rest of this lecture, we are going to work with some of the experimental data from our tutorials to learn the necessary statistical concepts and tools In particular, we are going to be working with the following market: Buyer’s Valuation Seller’s Cost Announcements Recap of Markets Qualitative Analysis Quantitative Analysis What does this market look like? Experimental Market 12 Price 10 8 Supply 6 Demand 4 2 0 0 1 2 3 4 5 6 7 8 Quantity Notice that the equilibrium quantity for this market is 5, while the equilibrium price is somewhere between ____________ Announcements Qualitative Analysis Recap of Markets Quantitative Analysis How do we draw this in Excel? 1. Need to organize the data properly Demand Supply Quantity 10 2 0 10 2 1 9 3 1 9 3 2 8 4 2 8 4 3 7 5 3 7 5 4 7 5 4 7 5 5 5 7 5 5 7 6 5 7 6 5 7 7 Announcements Recap of Markets Qualitative Analysis Quantitative Analysis How do we draw this in Excel? 1. Need to organize the data properly 2. Highlight all the data and select the Chart Wizard i. Choose “XY scatter” as the Chart Type ii. Choose “scatter with data points connected by lines without markers” as the Chart Sub-type 3. If at this point the graph does not look right, click on the “series” tab and adjust what data represents the x and y coordinates 4. Add a title and label the axes, then click “finish” Announcements Recap of Markets Qualitative Analysis Quantitative Analysis What is the maximum surplus of this market? Remember, surplus is maximized at the ___________ ______________________ In this market, we know the __________________, but the ___________________ could exist anywhere from $5 to $7 In such a case, we will ALWAYS assume the theoretical equilibrium price is at the ____________ of this range which is ____ CS = _______________________________________ PS = _______________________________________ TS = _______________________ Announcements Recap of Markets Qualitative Analysis Quantitative Analysis Graphic Presentation of Data In our example, there are 6 trading rounds Suppose we want to see how often the total surplus reaches certain levels To do this, we first have to calculate each __________ ______________________ (i.e. surplus) and then add them all up for each trading period We can represent this information graphically with something called a ______________, also known as a ____________________ Announcements Recap of Markets Qualitative Analysis Quantitative Analysis Graphic Presentation of Data Histogram of Total Surplus 5 Frequency 4 3 2 1 0 18 20 Total Surplus 22 Announcements Recap of Markets Qualitative Analysis Quantitative Analysis Graphic Presentation of Data A histogram is a ______________ that shows the amount of data that belongs to each “category” o In this example, the “categories” are the different levels of total surplus, namely ________________ In other words, a histogram shows us the ______________ of our outcome of interest What we notice from this histogram is that most often, a total surplus of _______ is achieved This tells us that most often, our market is ______________ total surplus, however we ___________________ with any statistical significance Announcements Recap of Markets Qualitative Analysis Quantitative Analysis Numerical Descriptive Statistics A histogram gives us a nice visual of the distribution of our outcome of interest Another way to get an understanding of the distribution of outcomes is to calculate some numerical statistics: 1. __________________ 2. __________________ 3. __________________ 4. __________________ 5. __________________ 6. __________________ Some of these statistics will become very useful when we want to start inferring the results of our experiment Announcements Recap of Markets Qualitative Analysis Quantitative Analysis Numerical Descriptive Statistics Mean The mean is also known as the _______________ The mean is calculated by ____________________________ _________________ and dividing this sum by the _________ ______________ Mathematically, we represent the mean as follows: o ______________________________ In our example, the mean is: o ______________________________ Announcements Recap of Markets Qualitative Analysis Quantitative Analysis Numerical Descriptive Statistics Median The median is the value of the data that occupies the middle position when the data are ____________________ Notice that the _______________________ is important when calculating the median o If there are an ______ number of observations, there is a unique ______________, and the median is just the value of the number occupying that position o However, if there are an ________ number of observations, the median is the _______________________________ In our example, we have 6 observations: o 18,20,20,20,20,22 o Thus the median is _________________________ Announcements Recap of Markets Qualitative Analysis Quantitative Analysis Numerical Descriptive Statistics Mode The mode is the value of y that occurs the ________________ The histogram provides an easy way to pick out the mode, as the mode will be the “category” that occurs _______________ In our example, ____ is obviously the mode Announcements Recap of Markets Qualitative Analysis Quantitative Analysis Numerical Descriptive Statistics Range The range is simply the difference between the ____________ __________________ In our example, the range is _____________ Announcements Recap of Markets Qualitative Analysis Quantitative Analysis Numerical Descriptive Statistics Variance The variance is a measure of __________________________ Mathematically, we calculate variance as follows: o ___________________________ In our example, the variance is: o s2 = _________________________________________________ o s2 = ______________________________ Announcements Recap of Markets Qualitative Analysis Quantitative Analysis Numerical Descriptive Statistics Variance Notice how the variance would ____________ if each of our outcome values (i.e. 18,20,22) each occurred twice o ӯ = ______________________________ o s2 = _________________________________________________ o s2 = ______________________________ Announcements Recap of Markets Qualitative Analysis Quantitative Analysis Numerical Descriptive Statistics Standard Deviation The standard deviation is also a measure of the ___________ ______________ The standard deviation is directly related to the ___________: o _____________ The standard deviation shows how much ________________ ______________________ In our example, the standard deviation is: o s =______________ Announcements Recap of Markets Qualitative Analysis Quantitative Analysis Why do we need quantitative analysis? Suppose our graphs and descriptive statistics do not give crystal clear answers to some of our questions even though our experimental design and descriptive statistics are well chosen At this point we need to turn to the second phase of the data analysis: ______________________________________ Formal tests are meant to provide specific answers to questions: o ____________________________________________________ o ____________________________________________________ Announcements Recap of Markets Qualitative Analysis Quantitative Analysis Why do we need quantitative analysis? In the context of our example, we could ask if the amount of experience subjects have affects the efficiency of the market The most obvious way to answer this question would be to compare the mean market efficiency in say period 1 with the mean market efficiency in some later period If we see a difference in the mean market efficiency across periods, we might be tempted to conclude that there is some sort of learning taking place But our conclusions might be incorrect because of ____________ ____________________ In the next few slides we will build some of the conceptual tools for understanding the sources and consequences of experimental error Announcements Recap of Markets Qualitative Analysis Quantitative Analysis Basic Concepts Observation Statistical procedures begin with a collection of observations The _______________ for an observation is not always clear For example, in market experiments, is it: 1. 2. 3. 4. A single _____________________ A single _______________________ A ______________________________ A single _________________________ o o If we want to see if people learn over time, then the market efficiency from one period in a single tutorial group would comprise a single observation However, if we wanted to see whether one tutorial group performed better than another, then the average market efficiency across all periods for a given tutorial group would be our unit of observation Announcements Recap of Markets Qualitative Analysis Quantitative Analysis Basic Concepts Suppose we have picked an appropriate definition of trial and now have a set of observations to analyze The fundamental problem we now have to deal with is imperfections in our set of observations To the extent that we get different results on replication, our analysis must deal with ____________ ___________________ Announcements Recap of Markets Qualitative Analysis Quantitative Analysis Basic Concepts Experimental Error Sources: 1. _________________ 2. _________________ ___________ error is conceptually straightforward to deal with ___________ error will require a much more extensive discussion Announcements Recap of Markets Qualitative Analysis Quantitative Analysis Basic Concepts Measurement Error Types: 1. 2. Recorded values for observations may not be exactly the actual values o Misheard an oral action o Made a mistake writing it down Experimenter loses experimental control o Inadvertently give player 1 private information about player 2 Announcements Recap of Markets Qualitative Analysis Quantitative Analysis Basic Concepts Measurement Error Solutions: 1. Careful choice of lab procedures, automating data capture and transmission, and building in redundancy should minimize the amount of erroneous data 2. Using our data summaries, check the raw data for large outliers and other anomalies, and check whether the anomalies are actually measurement error…when you detect erroneous data, you should throw them out before you run statistical tests Announcements Recap of Markets Qualitative Analysis Quantitative Analysis Basic Concepts Sampling Error The main idea with sampling error are concerns with when our “sample” is not representative of the underlying “population” The collection of all possible outcomes given our treatments can be thought of as ___________________________ There is always some variability in the population because of uncontrolled nuisances This variability induces a ______________________ Announcements Recap of Markets Qualitative Analysis Quantitative Analysis Basic Concepts Sampling Error If we knew the _____________________, our inferential task would be trivial o For example, if the population mean for double auction market efficiency is higher than the mean for posted offer markets, then we would correctly conclude that the double auction institution is, on average, more efficient The problem is, the population distribution can ____________ _________ since we will only be able to run a _____________ _________ and not necessarily be able to observe all possible outcomes Announcements Recap of Markets Qualitative Analysis Quantitative Analysis Basic Concepts Sampling Error Nevertheless, we do have some useful information about the population distribution because we have actually run a subset of all possible trials and have recorded the outcomes Thus our actual data constitute a ________________ from the population distribution ________________ arises to the extent that our sample is not representative of the underlying population Announcements Recap of Markets Qualitative Analysis Quantitative Analysis Basic Concepts Sampling Error You cannot expect to get a perfect sample, whose distribution exactly reproduces the population distribution, but with some care you can minimize sampling error within the bounds of your finite resources That is, we can take steps to ____________ samples and to ____________ samples Announcements Recap of Markets Qualitative Analysis Quantitative Analysis Good Samples and Bad Samples Ways to get good samples 1. _____________________ o 2. Each point in the population has the same probability of being picked _____________________ o Subdivide the population into several segments and draw observations from each segment with a frequency proportional to the weight of the segment in the population distribution o For example, we could segment the population by age, sex, education, location of residence, income, etc. balanced sampling tends to produce ___________ sample errors than a random sample Announcements Recap of Markets Qualitative Analysis Quantitative Analysis Good Samples and Bad Samples Folk wisdom among experimental economists: o An empirical regularity becomes ______________ when it is replicated with ____ different groups of subjects, preferably from different pools and in different labs Announcements Recap of Markets Qualitative Analysis Quantitative Analysis Inferential Testing Hypothesis test Hypothesis tests are how we will formally answer our questions of interest In hypothesis testing, we cannot prove something is ____, only that something is _______ This is inline with the deductive view of the world commonly attributed to the great philosopher Popper Popper said that we never no what is true, only what is not true Announcements Recap of Markets Qualitative Analysis Quantitative Analysis Inferential Testing Hypothesis test Formally, when we want to write down our hypothesis, there will be two components: o ______________________ o ______________________ The __________________ will always be what we want to prove or disprove The __________________ will often express what we think might be going on Announcements Recap of Markets Qualitative Analysis Quantitative Analysis Inferential Testing Hypothesis test Suppose what we are interested in investigating is the efficiency of the decentralized bargaining institution Furthermore, suppose we think this market institution will not result in the efficient competitive equilibrium outcome Instead of setting up our hypothesis test to try to prove the decentralized bargaining institution is not efficient, what we will do is try to show that it is efficient Remember, with hypothesis testing, we cannot show what is true, but we can show what is false Announcements Recap of Markets Qualitative Analysis Quantitative Analysis Inferential Testing Hypothesis test In this example, we will write our hypothesis test as follows: o ______________ o ______________ Here μ represents the _______________ in our experimental periods, or in other words, the _______________________ in each trading period In our example, the mean efficiency is ____, but this does not necessarily mean that we will reject the null hypothesis in favour of the alternative In order to answer our hypothesis test, we will need to construct a special test statistic Announcements Recap of Markets Qualitative Analysis Quantitative Analysis Inferential Testing Reference Distribution Assess the probability that differences in observed outcome across treatments are due to sampling error rather than due to differences in the underlying population distributions Such an assessment requires a _________________ to proxy for the population distribution In this class, our reference distribution of choice is known as the _________________ Note, to properly describe a distribution, we need to know both the mean and variance Announcements Recap of Markets Qualitative Analysis Quantitative Analysis Inferential Testing t-distribution The t-distribution is very similar to a ____________________, meaning that most of its density is found near the mean and decreases quickly as we move away from the mean Most courses at university have a normal distribution of grades, meaning that most people are around the average, and only a few get really high/low marks o In many economics courses, the class average is a B- o Usually 80% of the grades are between C- and B+ o Thus, those that fail or ace the course make up a small number of students