# Chi-Square (χ^2 ) Statistic

Markets provide various information relating to stocks, bonds, material prices, Mergers & Acquisitions and Initial Public Offerings. Financial analysts conclude trends and try to build financial models from existing data. This helps firms to identify statistical influences in areas such as financial performance, market research and monitoring changes in the business environment. Mathematical methods can be used for evaluations and predictions. The Chi-Square statistic is a frequently used method, used by financial analysts to test models.

Chi-square is defined as a goodness-of-fit parameter, it measures how good the model is comparing with the actual observed data. It is calculated as shown below: Where:

Oi is the observed value

Ei is the expected value

Chi-square tests the goodness-of-fit of the data to decide whether there is a difference between the observed (experimental) data and the expected (predicted) data. A suitable theory could be found by trying out different models such as linear or quadratic relations.

Chi-square also tests the independence and relationships of two attributes. Assume there is a linear correlation between the number of swimming suits sold at a shop and the temperature (that is, the higher the temperature, the more swimming suits would be sold), and collect data on the two chosen variables (number of swimming suits sold and the temperature). This model can be tested using chi-square minimisation. If the model is excepted, then there is a linear dependence between the two variables.

Minimised Chi-square gives a direct measurement of how good the model fits the actual observed data. Typically, the predicted data from the model would be close to the observed data when the minimised Chi-square statistic is close to one, for a given the sample size. Additionally, the best-fit parameter remains for which chi-squared is minimised. For example, the best proportionality factor for a linear model can be found by minimising Chi-square. It is calculated as shown below: Where v is the degree of freedom

There are various factors that can affect the chi-square statistic. Firstly, the degree of freedom can affect the chi-square statistic. This is directly associated with minimised chi-square; it can be determined using the sample size minus the number of parameters (constraints). Secondly, sample size can affect the chi-square statistic. Chi-square statistic requires a large amount of data in order to reduce bias and minimise the impact of outliers. Lastly, the model used is a factor that affects the chi-square statistic. If the model does not fit the characteristics of the data points, the minimised chi-square will be bigger than one.

By Jing Xu

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March 5, 2022

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