Introduction to Data and Probability for Economics
Statistics and probability are the most applicable methods for analyzing data in economics. And the likelihood is perhaps the most important analytic tool that describes any system where uncertainties are involved, whereas statistics offer a mathematical foundation in modeling situations that involve uncertainty. Probability helps economists choose the most probable course or solution for a problem. The most critical features for modern economics and finance is quantitative analysis, which has become widely applicable. Quantitative research is comprised of mathematical modeling in economic theory and empirical study of data in economics. And this comes from the cumulative effort of several generations of economic enthusiasts who have always tried to establish the best ways to make economic decisions.
For generations, it has been stated that mathematics cannot exist alone in the real world. This is why whey economic theory is formulated through mathematical tools, though it can achieve its logical consistency within assumptions, ideas, and its impacts. As Karl Max states, mathematics is a representation of the natural state of the science. Any economic theory will not be mathematical if it fails to explain important empirical stylizes facts that predict what happens in the future of economic development. This means the idea must validate economic models through the use of economic phenomena, mostly as data. Hence, mathematical tools cannot achieve this objective, which is why statistical tools have been fundamental.
The historical development of economics is a continuous process that will continue happening over and over again. Considering the economic cycles, we can then say that existing economic theory cannot explain new empirical stylized facts, rather economists develop new thoughts that explain newly acquired data. But these processes will not be complete or give proper results if there are not laid out empirical analytic tools.
For modern economics, the primary empirical analysis tool is econometrics. This term is defined as the statistical analysis of economic data using economic theories. Probability comes in as a natural quantitative tool when we want to explain uncertain outcomes in economics. Traditionally, probability gets its motivation from a game of chance. But later, scholars started applying its theory to actuarial issues, as well as in some areas of society. This was followed by the introduction of probability in physics by L. Boltzmann, J. Gibbs, and J. Maxwell, who found its application in different phases of human lives, determining that an element of uncertainty or risk drive human life. Therefore, the probability theory stands as the best tool to describe any system that involves uncertain situations.
Why is probability important?
Let us begin by considering the basic economic theory. Here, consumers will want to maximize their utility based on what is more important. This follows that law of supply and demand, where scarcity is a constant. Hence, for people to make the maximum utility decision, they have to involve that likelihood and possible outcomes. In other words, they have to consider the probable outcome of their decisions, which may be good or bad. Even so, research has shown that consumers don’t always account for their choices based on a rational or cold evaluation of a product. In most cases, people will always consider two risky alternatives when making consumption decisions; they will do this in relation to the perceived importance of outcomes and probabilities before committing to a choice. They will look at the option with the best outcome and choose it where the effect is more important to them. And if they feel like probability is more critical, they will still go for the option with the highest probability of the best outcome.
We are all face with decision making situations in life. This is true, especially when it comes to how we consume utilities. And because of scarcity, consumers don’t always get everything they want. The consumer theory states that one will go for a more useful and affordable product according to their budget and sacrifice the other. There are always two options, and people maximize the utility based on their perceived importance on the probable outcome. In other words, probability serves as a very important approach to deciding between two risky options.
It is not the only consumer who is affected by the need to make the most critical choices. The producer theory also states that firms will offer services and products based on the best outcome. They use data from consumers to determine whether or not bringing a new product on the market will work. It is still a game of chance that uses statistical data and probability in combination, considering the current economic statutes to make the line of action with most probable desired results.
As we can see, statistical inference is a method that helps us learn about the characteristics of a population through the analysis of samples of elements from an area. Now, consider a situation where a friend comes to you with a business idea and asks you to invest $50000. It may seem like a good idea, but you wouldn’t want to just jump in without considering the risks. So, you investigate and discover he has started about four other ventures that have failed. What are the chances that the current proposal a more than 50/50 chance of success? While answering this question, you may want to look at the four failures as sample elements and use them to compare with the methods of execution that the friend uses in their business ventures, where over half of the population is likely to succeed.
Such thinking is the basis of statistical inference since you will constantly be asking yourself about the likelihood of a certain outcome if the population is, in fact, what is thought to be. Statistical inferencing involves to a large extend, making a hypothesis about the nature of the population (which becomes null hypothesis in the end), as well as checking whether the sample has a big or less chance of happening, in case the hypothesis is try.
Probability in sample spaces and events
The best way to understand probability is by using the characteristic of a population that is familiar to use. There are many situations in the economy that indicate uncertain causes and outcomes. A simple example is tossing a coin and observing whether it islands on the head or the tails. As we know, a coin has only two sides, and it can never lie on both. You can toss an infinite sequence of tosses using a single coin to establish a relevant population. Each toss comes with the uncertainty of the results being either head or tails. This is an excellent reference for a random trial or experiment, which is the activity of getting two or more results with uncertainty in advance of what will come next or prevail. In a random trial, a set of basic outcomes is known as the sample space. For the case of the coin, we can say the sample space (S) contains two major results, head(H) and tail (T). This is a representation of a sample of one from an uncountable population of many single coin tosses – hence the basic outcomes can be denoted as S= (H, T).
The basic outcomes, as shown above, are also known as sample points, also simple events. They are mutually exclusive, meaning only one can occur at any given time, and at least one must happen. But what if we had two coins and tossed them at the same time to see whether they fall on heads or tails. It is very easy to think there will be three possible outcomes – say two heads, head, and tail, or two tails. In fact, there are four simple events since the head-tail combination can occur in two ways; head then tail or tail then head.
Based on this, the sample space for trial becomes S= (HH, HT, TH, TT). An event is a subset of simple events. It could, for instance, contain outcomes like E1 = (HH, HT, TH), which contains three sample points out of the four. Also, another result could be ‘both faces,’ which can be represented as E2 = (HH, TT).
There could also be a complimentary event, which is a set of outcomes not found within an even event Ej, denoted as Ecj. Thus, we can say the complementary chances for E1 and E3 are (TT) and (HT, TH), respectively.
These events only show how probability occurs. But there is more to it since you still need to follow the possibilities until you discover the most favorable out. In economics, the use of probability to determine more useful outcomes helps policymakers come up with ideas that help build a sustainable economy. As stated above, making decisions is not something we can avoid; whether with producers of producers, it always has to happen.
What does probability really mean in real life?
For long, probability has been used as a background of studying statistics. In other words, you need to understand the possible outcomes in uncertain events and combine them with statistical data before making a decision. We have already seen above that random trial is an experiment in which the outcome must be a set of sample points. We have also defined events as a collection of the same sample points, and that an event happens when the sample points or the basic outcome in it happens. Therefore, the probability is a value linked to a sample point or event that shows the likelihood of the event occurring. And there are rules that must be followed when looking at these events.
The probability of any outcome or event having a set of primary outcomes must be between zero and one. This means, probability can never be negative, and it cannot be more than one.
In any set of events if the sample space S, and event space, the probability of an event occurring is the sum of the probabilities that basic outcomes in that even will happen. This is to say, even will happen when the basic results happen.
It is always certain that at least one of the sample points/ elements will occur in the sample space, where P(S) = 1.
The null event cannot occur, which means P infinite = 0.
Is probability useful in economics?
We have seen the complications in modern economics, which also involves complex mathematics, with probability distribution fueling many events. As seen above, the probability is the proportion of times the event happens within a larger number of trials. A good example is tossing a coin, as seen in the previous section. We can also consider the probability of wooded homes catching fire in a certain region to be 0.01. this means 1% of the house will burn, based on experience, and past events. But it does not mean the 1% will remain constant all the year; it could be true or false this year. Over a long period, however, the average of the percentages will be 1%.
Economists can turn this information into the cost of fire damages. This will establish a case for insurance to the house owner against the risk of fire, information that can use by wooden homeowners in the region to spread the risk when setting up a fund. This means every owner may contribute a certain proportion of the money required to cover the costs. In the end, these homeowners will be members of a support group that shields them from the impact of fire damages.
This is only a simple instance where probability can be used to make decisions for the future. In economics, however, we don’t handle the homogenous case. There are not many repeated events. Hence, there is no member of any class. Also, using probability distribution may be hard to establish. Consider entrepreneurial activities; for instance, some scholars argue that we would not need entrepreneurs if they were homogenous. It is true that entrepreneurs arrange their activities to determine the consumer’s future behaviors. However, people’s needs are never constant; they keep changing, which means the entrepreneur must follow a specific line of thinking. Nevertheless, the probability is an important aspect of economics that lays the right background for statistics and data analyses. It is, therefore, necessary.
Author: James Hamilton