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FI4003
UK
University of Aberdeen
Regression is a statistical technique used in investing, finance, and other disciplines to determine the character and strength of the relationship between a dependent variable (customarily represented by Y) and a series of other variables (called independent variables). Regression assists financial managers and investors value assets and comprehend the rapport between capricious stocks of businesses dealing with commodities and commodity prices. There are two types of regression, multiple linear regression and linear regression, but there are non-linear regression methods for complex data analysis. In simple linear regression technique, one independent variable is essential to predict or explain the result of the dependent variable Y. multiple linear regression utilizes two or more independent variables to forecast the outcome. Regression is also helpful in predicting a company's sales based on GDP growth, previous sales, or other sorts of conditions. The (CAPM) capital asset pricing model is a frequently used regression model in business finances to discover capital and pricing assets costs. A panel data (longitudinal data) is a genus of data where one can get observations from the same set of entities at several periods.
In its most elementary form, regression analysis is the approximation of the ratio between two variables. For instance, one may want to estimate the progress in meat sales based on economic growth. If previous data indicates the improvement in meat sales is approximately one and a half times the growth in the economy, the regression would be represented as MS growth= 1.5(GDP growth) (Hall, 2021). If the relationship is between multiple variables, it will also be constant if meat sales are trending up, even if it is a one percent growth in a stagnant economy, MS= -1.5+1(GDP Growth).
The variable to be estimated is a dependent variable, whereas the variable used in the model to predict the dependent variable is called the independent variable. A regression has only one dependent variable. On the other hand, the number of possible independent variables is limitless, and such a model is called multiple regression if it entails some independent variables (Welc & Esquerdo, 2017). Regression models can also identify more complex variable relationships. At some point, a particular model can use square root, square, or any other power of either one or more independent variables to predict the dependent one. This makes it a non-linear regression; for instance, MS growth = (square root of GDP growth) 1/2(NISHIZAWA & ONODERA, 2018).
The importance of panel data are hereby listed below:
From the above graph why it represents the The panel data which has taken the time of division in individual cumulatively.
The implementation of panel data is immense in nature; since it is applicable in the field of social science, medicine, economics, finance, epidemiology, and physical sciences; which are briefly discussed as follows:
The role played by fixed effect models in panel data regression is immensely important. In this approach of the fixed effect model, the variable's effects are not estimated since the variable’s value does not change radically as time lapses.
One of the significant characteristics attributes of a fixed effect model is that it completely eradicates the bias that is relevant with variables by evaluating changes that have taken place in a group across time. This is done by a fixed effect model by integrating dummy variables for substituting the missing variable.
Random effect models always estimate the effect of individual variables that are time invariant in nature. However the estimates produced by the random effect model have the probability of being biased since all the omitted variables are not taken under regulations. In the field of statistics, random effect models are commonly referred to as the variance component model is a sort of statistical model, where the parameters of the model are characterized with random variables. The main purpose of random effect models is panel analysing the hierarchical data where the notion of a fixed effect is not considered.
Supposing an observation is made on the dataset with a very low or a very high value compared to other comments in the data to mean it is out of range, such remark is called an outlier. In other words, it is an extreme value. An outlier is a challenge since, in most cases, it interferes with the results that one expects.
When the variability of a dependent variable does not match the values of an independent variable, it is said to be heteroscedasticity. For instance, with an increase in one's income, the erraticism of food ingesting will upsurge. A more deficient person is likely to spend a relatively constant sum of money by continuously eating cheap food. On the other hand, a wealthier person may sporadically buy more affordable food and frequently consume expensive food (Pre-symposium proceedings: Containing summaries of the papers to be presented at the international symposium on rainfall-runoff modeling, May 18-21, 1981 Mississippi State University, Mississippi State, Mississippi). Generally, those people with higher earnings exhibit more considerable variability in food consumption.
If an independent mutable is highly correlated to each other, then that variable is a multi-collinear. Almost all regression methods assume an absence of multi-collinearity in the dataset. This is because it leads to challenges in variables ranking based on its importance. In other words, it makes it hard to choose the most significant independent factor.
If superfluous explanatory variables are used, overfitting may occur. Overfitting insinuates that the algorithm will work better on training, but it is impotent to work better on the test sets. It is also referred to as a problem of high variance (Shang et al., 2017). If the algorithm works unwell and cannot fit even the training set well, it can be under-fit data. This problem is also called the problem of higher bias.
Regression applies a parametric approach. Parametric insinuates that it makes assumptions on data for analysis purposes. As a result, reversal becomes restrictive. It fails to deliver quality results with data sets that do not have particular premises. This means that a successful regression analysis requires validation of certain assumptions (Hall, 2021); examples of such beliefs include, there should not exist a correlation error/residual terms. An additive and a linear relationship between response (dependent) and predictor (independent) variables should exist. There must be an even distribution of error terms—no correlation of independent variables (Hoffmann, 2016). Error terms must have a constant variance.
In conclusion, regression is a statistical technique used in investing, finance, and other disciplines to determine the character and strength of the relationship between a dependent variable (usually represented by Y) and a series of other variables (called independent variables). There are fundamentals of regression analysis, as mentioned earlier. Various applications of regression analysis include optimization of business processes, future prediction, decision making, among others. There are multiple regression analysis techniques, including linear regression, polynomial regression, quintile regression, logistic regression, elastic net regression, principal component regression, ordinal regression, and obit regression.
Another critical role of regression models is business processes optimization. For example, a factory manager might build a model to comprehend the relationship between shelf life and the oven temperature of the cookies baked using those ovens (Suarez et al., 2017). A company that operates a call centre may desire to understand the relationship between the number of complaints and wait times of callers (Weld & Esquerdo, 2017). An important driver of improved productivity in business and hasty economic advancement globally during the 20th century was the recurrent use of statistical methods in manufacturing and service industries.
One of the most common uses of regression in a business is predicting events that are yet to occur. Demand analysis, for instance, indicates the number of units that consumers will purchase. However, many other main parameters apart from demand are dependent variables in models of regression. Another prediction that can be used is the number of shoppers likely to pass in front of a given billboard or even the number of views likely to watch the Super Rowel. This will be useful to management in assessing what to pay for advertisement. Insurances companies sincerely rely on regression analysis to predict how many policyholders are likely to be victims of burglaries involved in accidents (Pal & Bharati, 2019). Advantages of predictive analytics include cost reductions, faster results, improved operational efficiency, fraud detection, risk management, optimization of marketing campaigns, and reducing the number of tools needed.
Even the most careful and careful manager can make mistakes in decisions and judgment. Regression analysis is essential to businesses and managers to correct and recognize errors. For instance, a manager of a retail store may feel that prolonging shopping hours might upsurge sales. Regression analysis may reveal that the expected rise of sales might not be sufficient to counterbalance the augmented labour cost and operating expenses, such as using more electricity (George et al., 2013). Therefore, with regression analysis, a manager can determine that increasing working hours will not necessarily increase profits. This will, in turn, assist the managers in avoiding making costly mistakes.
Observing data may provide and fresh and new insights. Many businesses are in a position to collect a lot of information about their clients. However, the data may be useless if proper regression analysis is not done to find the relationship among different variables for discovering a pattern. For example, scrutinizing data via regression analysis may designate a rise in sales during particular days of the week and a bead in sales for others. The management to compensate, such as ensuring there is enough stock during the peak days and bringing extra assistance to maximize sales, can then make adjustments. The command can also ensure that the best service or salespeople work during those days with increased sales.
Many companies, together with the top managers, are using regression analysis to make profitable business decisions. Regression analysis also assists managers in eradicating guesswork and spontaneous intuition. Regression assists businesses in adopting a scientific angle in their strategic management. There usually is more data bombarding the large and the small businesses. Therefore, regression analysis becomes helpful in managers to examine and pick the correct variables to make the most accurate decisions.
The significance of regression analysis is all about data. Data means figures and numbers defining a given business. The advantages include prediction of sales both in long and short terms, understand demand and supply, understanding why sales have dropped, and determining whether to do an advertisement or not. It is also used in forecasting, predictive analysis, operation efficiency, correcting errors, and making new insights.
Every regression method must have some assumptions accompanying it that one needs to meet before any analysis. These techniques vary in terms of the independent and dependent variables as well as the distribution. The following are the types of regression:
This is the simplest type of regression. It is a type of regression where the dependent variable is naturally continuous. In nature, it is believed that the relationship between the independent and dependent variables is linear. There are various assumptions associated with linear regression, including the absence of heteroscedasticity—no auto-correlation and multi-collinearity. There must exist a linear relationship between independent and dependent variables—distribution of error terms with a mean of zero always with a constant variance. No outliers, and finally, sample observations must be disconnected.
In this type of regression, the dependent variable is naturally binary. That is, it has two categories. Independent variables may be either binary or continuous. One can have more than two sorts in their independent variable when it comes to multinomial logistic regression. The reason why linear regression is not applicable in this case is the uneven distribution of errors, violation of homoscedasticity assumption, and the fact that y follows a binomial distribution, therefore not expected. An example of logistic regression is HR Analytics; Information Technology (IT) companies recruit significant people but unfortunately face a challenge. After accepting the offered job, most of the candidates do not join (Illukkumbura, 2020). The result of this is over-run costs since the firm has to repeat the whole process. Using logistic regression, a company can easily predict whether the applicant is likely to seam them (through binary outcome- that is, join/not join).
This method helps fit a non-linear equation by using polynomial functions of the independent variables. In other words, it occurs in a situation where the relationship between the independent and dependent variables appears to be non-linear.
It is essential to comprehend the regularization concept before directly going to ridge regression.
It is crucial in working outfitting problems implying a model that performs well during training data but performs poorly on validation data. Regularization helps solve such a problem by adding a penalty term to the function's objective and regulating the complex model using the same penalty term.
Here we try to reduce the objective function by adding the penalty term to the total sum of the squares of the coefficients. This is also referred to as the least absolute deviation method. A regression technique called Lasso utilizes L1 regularization.
Here the main goal is to minimize the objective function by adding penalty terms to the summation of squares of coefficients. Shrinkage regression or ridge regression uses L2 regularization.
Therefore, ridge/shrinkage regression, a constraint usually is added to the summation of squares of regression coefficients ("Ridge regression in theory and applications," 2019). On the other hand, in linear regression, the main objective is to minimize the sum of square errors.
It is an extension of linear regression and is applicable when heteroscedasticity, outliers, and high skewness exist in the data. In linear regression, the mean prediction of the dependent variable is made for particular independent variables. Modelling the mean is not a complete description of the relationship between independent and dependent variables since the standard does not describe the entire distribution. Therefore, quantile regression is preferred since it predicts a percentile or a quantile for any given independent variables (Table 3: Multivariate logistic regression models Fitting results for dimensions of resilience with SH by different types of childhood maltreatment," n.d). The dependent variable must be continuous in quintile regression and attempt to estimate the quantile of the dependent variable when values of X are provided. There are several advantages of quantile over linear regression. They include robust to outlies. When the data is skewed, it tends to be more beneficial than linear regression, quite advantageous in the presence of heteroscedasticity data. Additionally, the distribution of dependent variables can be described through various quantiles.
While using quintile regression, a point to note is that the coefficients that one gets in quintile regression should vary from those obtained from linear regression. This is done by observing the confident interims of regression coefficients of estimations obtained from both regressions.
When dealing with highly correlated data, elastic net regression is preferred over lasso regression and ridge regression. It is a combination of L2 and L1 regularization. It does not assume normality, just as in ridge.
This is a valuable regression technique for multi-collinearity or where there are many independent variables in any given data. It involves two steps, getting the components of the principal and running regression analysis on principal components. The most common features of main components regression are removal of multi-collinearity and reduction of dimensionality. This is a valuable statistical technique in extracting new features if the original parts are highly correlated.
It is a substitute regression method of the principal component when the independent variable is highly correlated. It is also crucial when there is a significant number of independent variables.
It utilizes the L1 regularization technique in the objective function. Lasso is an abbreviation for the minor absolute shrinkage and selection operator. It has an advantage over the ridge regression since one can perform in-built variable selection and parameter shrinkage. In ridge regression, one can end up getting all the variables, though, with parameters that have shrunk.
They are used in the prediction of ranked values. In other words, ordinal regression is appropriate when the dependent variable is naturally ordinal.
Poisson regression is applicable when the dependent variable has count data. Applications of Poisson distribution include estimating the number of emergency service calls during an event and predicting the number of markets in customer care concerning a particular product. Conditions that dependent variables in Poisson regression must meet are counts cannot be harmful, numbers must be whole numbers, and the dependent variable must have a Poisson distribution.
It can solve both linear and non-linear models of regression. Support vector regression uses appropriate non-linear functions, such as polynomial, to calculate optimal results for any given non-linear model (Smith, 2014). The sole idea of support vector regression is to reduce errors and individualize the hyper plane, optimizing the margin.
It is appropriate for a particular time to event data. Examples of time to event data include the time from first heart bout to the next, time after the treatment of cancer till demise, and time between a customer opening the account until abrasion("prediction of stock returns with regression approaches and feature extraction," 2016). Logistic regression will use a binary dependent variable but will ignore the events timings. This technique has some limitations, for instance, in an OLS make, which has heteroskedastic errors; the standard of estimated errors is too minute.
They were used to estimate linear relationships among variables if censoring is present in the dependent variable. Censoring means that if a person observes an independent variable for the entire observation, he/she only knows the actual value of the dependent variable for a restricted range of comments.
It is another method for negative binomial regression. It is also valuable for over dispersed count data. The two algorithms give the same results. Some differences exist in estimating the effects of covariates. The variance of a negative binomial model is an example of a quadratic function, whereas the conflict of a quasi-Poisson model is an example of the mean process.
N/B simple linear regression enables researchers to estimate components of two or more connecting variables. Regression is helpful in finance, economics, accounting, and many other vital calculations of businesses (Ludbr ook, 2010)
Appendix S2: Statin exposure models- Variables included in logistic regression and Cox proportional hazards regression analysis. (n.d.). https://doi.org/10.7717/peerj.1902/supp-2
Fox, W. P. (2017). Regression and advanced regression models. Mathematical Modeling for Business Analytics, 217-245. https://doi.org/10.1201/9781315150208-6
George, E., Bowman, D., & An, Q. (2013). Regression models for analyzing clustered binary and continuous outcomes under the assumption of exchangeability. Analysis of Mixed Data, 93-107. https://doi.org/10.1201/b14571-8
Hall, F. (2021). Valuing businesses using regression analysis: A quantitative approach to the guideline company transaction method. John Wiley & Sons.
Hall, F. (2021). Valuing businesses using regression analysis: A quantitative approach to the guideline company transaction method. John Wiley & Sons.
Hoffmann, J. P. (2016). Regression models for the categorical, count, and related variables: An applied approach. University of California Press.
Illukkumbura, A. (2020). Introduction to regression analysis. Independently Published.
Ludbrook, J. (2010). Linear regression analysis for comparing two measurers or methods of measurement: But which regression? Clinical and Experimental Pharmacology and Physiology, 37(7), 692-699. https://doi.org/10.1111/j.1440-1681.2010.05376.x
NISHIZAWA, S., & ONODERA, H. (2018). Design methodology for variation tolerant D-flip-Flop using regression analysis. IEICE Transactions on Fundamentals of Electronics, Communications, and Computer Sciences, E101.A(12), 2222-2230. https://doi.org/10.1587/transfun.e101.a.2222
Pal, M., & Bharati, P. (2019). Estimating calorie poverty rates through regression. Applications of Regression Techniques, 59-84. https://doi.org/10.1007/978-981-13-9314-3_4
Pre-symposium proceedings: Containing summaries of the papers presented at the international symposium on rainfall-runoff modeling, May 18-21, 1981 Mississippi State University, Mississippi State, Mississippi. (1981).
Ridge regression in theory and applications. (2019). Wiley Series in Probability and Statistics, 143-169. https://doi.org/10.1002/9781118644478.ch6
Shang, S., Nesson, E., & Fan, M. (2017). Interaction terms in Poisson and log-linear regression models. Bulletin of Economic Research, 70(1), E89-E96. https://doi.org/10.1111/boer.12120
Smith, H. (2014). Regression models, types of. Wiley StatsRef: Statistics Reference Online. https://doi.org/10.1002/9781118445112.stat03233
Suárez, E., Pérez, C. M., MartÃnez, M. N., & Rivera, R. (2017). Applications of regression models in epidemiology. John Wiley & Sons.
Table 3: Multivariate logistic regression models Fitting results for dimensions of resilience with SH by different types of childhood maltreatment. (n.d.). https://doi.org/10.7717/peerj.9800/table-3
The prediction of stock returns with regression approaches and feature extraction. (2016). Journal of Administrative and Business Studies, 2(3). https://doi.org/10.20474/jabs-2.3.1
Welc, J., & Esquerdo, P. J. (2017). Applied regression analysis for business: Tools, traps, and applications. Springer.
Welc, J., & Esquerdo, P. J. (2017). Applied regression analysis for business: Tools, traps, and applications. Springer.
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