Exploring Practical Applications of Linear and NonLinear Models In Business Research Dr.Jeelan Basha

[00:00:03][Music] Very good morning to all of you. I am CMA Dr. Jilan Bashavi, Professor Department of Studies in Commerce, Vijayagra Sri Krishna Devara University, Balari.

[00:00:24]Now I have been instructed my students and my teachers to deliver a lecture on exploring practical applications of linear and nonlinear models in business research. Now this is a more about mathematical applications which we need to apply it in our business research.

[00:00:51]This is a very interesting topic provided the commerce student should have known basic mathematics. Let us try our best to understand how the mathematical application can be used in business research methods.

[00:01:07]Now let us go to my objective of presentation. What is my purpose behind my pres presentation?

[00:01:17]The purpose is how to develop, how to evaluate, how to compare linear and nonlinear models in business research. This is my objective. Now many of common student may not know developing they may know it how to develop a linear model but not nonlinear models that I'll try my best to make you to understand here a comm students presentation for nonlinear model is quite different from a mathematical student so that I'll try my best to understand my students how far they can understand let us come to First what is model? Model is nothing but a simplified simplified representation of a reality which is used which is used for explaining which is used for predicting which is used for understanding a phenomena. This is called model. We explain in general sense. Let me repeat once again. Model is a simplified representation of a reality which is used to explain which is used to understand which is used to predict the phenomena. Phenomenum is in a it is an observable event or occurrence.

[00:02:45]Now this model can be explained in different context in different angles as I am presenting on linear and nonlinear model. Let me say what is a model from the point of view of mathematics and statistics. When it comes to statistics point of view, model is representing the relationship between the variables. It represents the relationship between two or more than two variables. Here one variable is a dependent variable.

[00:03:19]Another variable is a independent variable. We want to know the relationship between the dependent and independent variable that we are going to take it forward for my further presentation. Now the same model we explain in our business.

[00:03:37]Model is nothing but a company creates or delivers a models to capture the value.

[00:03:48]It develops something to create a value to deliver a value to capture a model to capture a value. This is called a model in a business sense or in usual business people will speak it.

[00:04:04]Now let us come to once the model is over let us come to the types of model be before that we'll go to the what is the purpose of model is used for testing the hypothesis model is used for predicting the observations model is used for supporting decision making is also used for guiding the collection of data Model is also used to illustrate the theories. These are the purposes behind usage of model. But here from our point of view here model we are using for the purpose of testing hypothesis. Model is also used for the for the purpose of supporting our decision making. For model is also used for prediction. Let us come to the there are various types of models we have.

[00:05:03]like models from the point of view of theory, theoretical model, statistical model, conceptual model, again sim simultaneous model, again empirical models. Here we categorize the model from the point of view of the relationship between the variables. Though there are many types of models from different context but we are taking the model which we can classify the observation into two categories based on the relationship that is called uh dependent and independent variable.

[00:05:41]Based on relationship between variables using dependent and independent variables there are two types of models. One is a linear model another one is a nonlinear model.

[00:05:53]What is linear model? The linear model is one in which the parameters in the linear are constant and additive. It has always a straight line relationship. It has always a straight line. When when the entire data is presented on a graph, it will be visualized as a straight line. the effect of independent variable. It is going to study the effect of independent variable on the dependent variable that is called linear model. It has always a straight line relationship or in other way we can say the proportionate change in a dependent variable due to change in independent variable that is called linear model. We take a practical examples of linear model. Profit depends on expenses. There is a proportionate change in profit due to change in the expenses. If the expenses are increased, there may be a positive relation, there may be negative relation. Not necessarily that there should always be a positive. But the thing is here a constant change will be there. That is called proportionate change is very important. With the increase in expenses, there is a proportionate decrease in profit. With the decrease in expenses, there is a proportionate increase in profit. So the profit depends on expenses. Another example also we can take. Salary depends on experience. With the increase in experience, there is a increase in salary. There is a proportionate. With the one year experience, you will be paid. For example, with the one year experience, you will be paid 10,000 rupees. With the 2 years experience, you will be paid 20,000 rupees. With the 3 years experience, you will be paid 30,000. This is what we can call as a proportionate change. Even the it can also be used in many areas not only in finance, in accounting, even in uh logistic also we can use it. This one with the increase in distance there is a increase in delivery cost. The delivery cost as well as distance has linear relationship. This is called linear model. This is a real time examples we come across in our day-to-day life. But in order to adopt this linear model, there are some assumption we need to assume it. First assumption is linearity. Our model is a linearity.

[00:08:37]It means that our model when it is presented on a graph either through scatter plot or through residual plots partial residual plots or through your calculation mathematical calculation. This linearity can be tested either by visualization or by mathematical equation. If it is visualized there we see a scatter plot or partial residual plot. We see it there should be a we come across there is a straight line. When it is a straight line we come to a conclusion that is a linearity. If you are not sure with the visualization you can go for mathematical calculation. There are any number of mathematical equations are there.

[00:09:28]equations are there for testing the linearity. Very popular is we can call it a ROM test. We can call say a reset.

[00:09:36]ROM say reset test. We use it in order to determine whether your data is a linear or nonlinear. This is the first assumption of your linear model. Second one is your data must be independent. Your data must be independent. The whatever the variables you are considering for your calculation or estimation should be independent to each other. They should not be dependent on one another.

[00:10:07]Whatever the data variables you are collecting, you are collecting at on at random. They are not collected at at a particular point of time where there is a cluster or group. You are taking randomly from one group one variable one observation another group another observation like that you are taking independently each observation when you take it the observation or variables independently they are not dependent on each other this is a must if they are dependent there is a problem of colinearity that I'll say you later what is that so that your data must be independent second assumption third assumption your data must be homosasticity. What is homo means same?

[00:10:54]Schedasticity means variance. There must be equal variance of your data. What is equal variance? Whatever the data you take when you calculate variance there should be there must be same variance. No, if not same at least at 5% of level of significance there must be same.

[00:11:15]Whether if you take a 10 observation if you want to calculate variance if out of 10 observation if your variance comes five example even if you take half of the observation if you calculate variance there must also be five even if you take 1/4 of the data for the calculation of variance there there is also be there will also be the answer is five so at every level at every broken fragmented data you will get the same variance that is called homoscadacity. This can be visualized this can also be calculated with a mathematical calculation.

[00:11:56]Now if you want to look at this whether the how kind what type of data if you look at visualization we look at QQ plot we'll see normal Q plot will decide whether your data is homoscapacity or not otherwise you can go for mathematical calculation like bruish pagen test BP test we can use it in order to determine whether your data is a homo scadastacity or not. This is the third assumption. One more fourth assumption we have that is called normality. Whether your data is a normally distributed or not. What is normality? We say very simple. Normality you get when the mean, median and mode of the data is same. When you calculate the mean of the data, the median of the data and mode of the data almost same. If the mean, median, mode are same for that data, we say that data is normally distributed. This can also be tested by visualization as well as by mathematical calculations.

[00:13:14]If you want to use it visualization, we can use an histogram. Histogram is one of the methods of visualization to determine the normality. This it will have a bell-shaped curve. When you plot it on a graph, it will find a bell-shaped curve.

[00:13:33]Your frequency curve will be bell-shaped curve should be there. When bell-shaped curve is divided into two parts, half of the left side should always be equal to the right side. Half of the right side that of right side that is called normality. Now this can also be tested by mathematical calculation. There are any number of tests we have. The popular is we can call it say a shapiro test or we can also go it for many tests. We have like a jarco test is also there. We use it either sharpiro test or jcoa test to determine whether your data is normally distributed or not. Now next one is autocorrelation.

[00:14:27]Now here autocorrelation means your current observation has any kind of correlation with its own previous observations. Your current observation has any relation with its own previous observation for which we calculate autocorrelation. We should see that the data must be free from autocorrelation.

[00:14:56]There must not be autocorrelation. This can be calculated with the help of mathematical calculations, mathematical test that we can call it is a Derbin Watson test. Derbin Watson test. This can also be tested through Bruish Godfrey test also we use it in order to determine whether your data is autocorrelated or not.

[00:15:21]Now with respect to setting of hypothesis for those test I'll tell you in my pre next presentation is the next two s slides otherwise here only we can say now in order to test it whether your data is a linear or nonlinear your null hypothesis linear your alternative hypothesis nonlinear in order to test your homoscadacity Your null hypothesis is a homoscaracity. Your alternative hypothesity. In case of normality test, your null hypothesis, your data is a normally distributed. Your alternative hypothesis is not normally distributed.

[00:16:06]In case of autocorrelation, your ho is not that is called null hypothesis. There is no autocorrelation.

[00:16:17]Alternative is there is autocorrelation. So that these are all hypothesis we need to set when we adopt mathematical equation. Mathematical equation example let me repeat once again one small linearity. What we said the hypothesis is there is no there is a linearity. Ho is there is a linearity. H1 is there is no linearity.

[00:16:49]If a calculator ROM say reset test your P value, your P value, this is a P value is going to determine whether null hypothesis accepted or alternative hypothesis is accepted. To accept null hypothesis, your p value should should be more than 5% level of significance. We are testing the hypothesis at 5% level of significance. It is nothing but 95% confidence level. We are testing the hypothesis. It is a universally accepted. Accepting or rejecting at 5% level of significance. Even you can test it for much more accuracy. For more significance, you can go even for 1% level of significance also you can test.

[00:17:41]But by default almost all of them will accept at 5% level of significance. Hy is there is a linearity. H1 is there is no linearity.

[00:17:52]In order to accept the null hypothesis, your p value must be more than 5% level of significance. That is called your p value must be more than more than 0.05.

[00:18:07]In case alternative hypothesis to be accepted is to be accepted your p value must be less than 0.05 in case your p value is less than 0.05 05. We can say that there is no linearity like that. We have to test we have to set the hypothesis for each of the mathematical tests or for each test we need to set what is HO what is H1 like that for homoscarity like that for normality even for autocorrelation. Now let us come to the what may what is the mathematical equation for this linear model. We say very simple y = a + b x plus error e. y is a dependent variable. Here dependent variable from our example what has been shown on the slide like profit is dependent variable y we can take it as a dependent variable. A is we can call it as a alpha or we can also call it as a intercept.

[00:19:21]What is intercept? What is alpha? This is a contribution to the dependent variable when the contribution of all the when the contribution of all these slopes are [Music] zero. This can be explained very easily. The dependent variable is influenced by three things. One is internal, second one is external, third one is unexpected.

[00:19:50]One is internal events, external and unexpected events. Internal we can call it is a alpha that is called intercept.

[00:20:00]What is a contribution is emerging out of your internal source that is called a alpha. Next one is beta that is also we can call it a slope. Now what is the contribution of x to y? What is a contribution of X variable to Y variable that is nothing but that is explained with the help of coefficient alpha beta error term these are all we can call as a coefficient. Now this is what external how much X is contributing as a outside variable to Y variable which is explained with the help of B. The third one is unexpected error term. This is unexpected contribution to Y variable.

[00:20:45]This we cannot predict it. Hence we assume that it is a zero. So this external variables may be more than one. Here we have taken only B X1. If there are more than one external variables are there. B2 X2 B3 X3 plus plus we we go on adding dot dot dot B X1 B1 X1 plus B1 B2 X2 plus B3 X3 plus B 4 X4 B5 X5 like that this will be continued here this will be continued this this go on increasing this further like that plus plus like that this is called linear equation in the mathematics. I hope you got it. What is linear model? It is very simple. I can say further repeat it. It is a straight line relationship. There is a additive model. Why it is additive model? In each in the equation there is a plus signs are there. A plus B1 X1 plus B2 X2 plus B3 X3 plus. There are no product. There are no any other mathematical signs like square, like log function, like exponential function, like power function. There is no such functions in the equation. If such functions are appeared before before coefficients, then we can say that that is called nonlinear model.

[00:22:24]So nonlinear model is one where the relationship between dependent and independent variable is straight and constant. Again the residuals of these variables may not have any pattern. They are all random. One more characteristics we find from the linear model that the residuals what is residuals? There difference between expected and actual. The difference between actual minus expect uh predicted or expected that we can call as a residuals. The residuals of this data must be random. When this characteristics are found we can say that linear model or this can also be directly we can calculate with the alpha that is what Ram say reset test that will tell you even there are many like harvest is also there for nonlinear calculation. How many are there? I'm giving one or two so that you should not get much confusion by saying many tests of same category. Now this is what I say what is linear model. Now question comes when to use linear model. This is very important when the relationship among the variables that is called between dependent and independent variables are very clear. They are very simple and they are straight line. When the relationship among the variables are clear, simple and straight line, we use linear model. That is what proportionate change in dependent variable due to change in independent variable. If you look at characteristic of linear model, it is easier to interpret and estimate. The reason is that there is a additive model. There are no any mathematical signs which are very complex to understand. Sometimes power cubed, power of four, raised to the power of five or exponential sign or some other log function, many other function square like that many functions are there. We as a non-mathematic students we cannot much understand it becomes more complex. So linear model is very easy to interpret and estimate. It is a faster computation. computation calculation is also very faster in comparison with your nonlinear model. It is a very rigid model. You cannot change anything. It is it will not have much flexibility in its calculation. Again it is a OS estimation ordinary le square estimation. Now this all when to use this linear model. Now application of linear models in business. I have taken few real time examples that we know it many of us in case of marketing predicting sales from advertisement spending. This can be used for linear model. You can predict sales from your advertisement you are spending over a period of time. Sales you can take it for a period of 10 years. Advertisement you spent take it for 10 years. You can predict for the 11th year. coming 11th year and 12th year you can predict it.

[00:25:43]These sales and advertisements have linear relationship. In finance we can take a capa model that we can say capital asset pricing model. This is what single factor model market risk is considered in order to measure the return of a security.

[00:26:02]So here returns of a security is influenced by market risk. Higher the market risk higher the returns. So market risk is a independent variable which is influencing the returns. So CFM model is also a linear model capital asset pricing model. There are now many factors model has come like three factor model, four factor, five factor, six factor form of French models are there but we don't want to explain.

[00:26:33]CFP is a single factor model is very easy to understand even in operations also we can use it cost minimization from usage of inputs. This also we can use it as a linear model. HR also we can use it predicting salary from the experience of employees. The salary of 10 years you can take it. How the salary has been changed over a period of 10 years and his experience you can take it. Salary is a y variable experience is a x variable you can take it. You can calculate linear model regression.

[00:27:12]Simple regression you can calculate. You can predict with the help of that you can also predict what may be the salary with the further increase in experience. These are all practical real time example we come across in our linear mode. Let us come to the nonlinear pattern. This may be a pattern when it is plotted on a graph your data there will be a straight line. If there is a curve then it is called nonlinear. If you find curves in the straight line then we immediately say that this will have nonlinear pattern. So this is a visualization of linear pattern. Let us come to the nonlinear model. What is nonlinear model? I told you the change in the dependent variable may not be proportionate to the change in independent variable. They may not be proportionately changing. There may be disproportionate change in dependent variable due to change in independent variable. Here the relationship is not constant. The relationship is not of additive. The relationship maybe of log function, maybe of polomial function, maybe of exponential function, maybe of logistic function. The relationship may be of different category. It is what we can call it is a curved type of relationship. When you plot the data on a graph, you may not have straight line what it was shown previously. You will have curves. Then nonlinear models are of two various categories that I'm explaining very simple function form of linearity.

[00:28:59]Nonlinearity time series models of nonlinearity machine learning models of nonlinearity. There are three categories of nonlinear models we have. First one is very simple. Second one is much more complex. Then third one is the most complex. The simple way we say that functional form of linearity we use exponential science we use it.

[00:29:25]E value 2.7183 exponential we use it before your independent variable.

[00:29:36]Rather than using like plus or multiplication which will be there that we use it exponential form of function.

[00:29:45]Logarithmic form of function natural log we use it here. Polomial function is a hyperbolic function. It is where we use it powers power of one power of two with a increase in independent variable the power will also be increased or changed.

[00:30:04]Power log function, logistic function, cyclical function, more fluctuating function. These are all one category of nonlinear models. Second category of nonlinear models are time series model that we can call arch and garch model. Auto reggressive conditional hoscadacity.

[00:30:26]Garch model there is a family of arch is there arch g there are many are there. I'm not going to explain much here. Here that's not our topic to discuss. All arch and g mod models are time series models of nonlinearity. And one more th model threshold auto reggressive model. Here the resume shift will be there. The threshold means a particular limit is fixed before that how the resume before that how it was after that resume change. Threshold changes comes when threshold is changed when limit is changed. What may be the change that is what tar model we use it again one more star model is also there smooth auto smooth threshold auto reggressive model.

[00:31:13]Again MSM model also we use it that is called marco switching models. These are all categories of time series models.

[00:31:22]The third is a much more complex models that is called machine learning models.

[00:31:29]Why it is called machine learning? It is very difficult to calculate manually. We use softwares like we use Python, we use R, we use MATLAB, we use EVs, we use SAS. There are different softwares which are available for calculating machine learning models.

[00:31:53]We use it because it's all iterative model. Your answer will not be arrived at a single shot. You need to calculate again often. There may be one where there is optimization. You need to calculate trial and error method till you get your optimum model till you get the best model. In case of machine learning model, there are few categories we have taken for my presentation like one is neural network. Next one is random forest, decision tree models, gradient G- boosting models, support vector regression models. They're all very complex models. Sometimes it is beyond our knowledge. Sometimes you feel but if you develop it it is not so tough we can also do it we can also write an article using this machine learning models now I told you what is nonlinear what are the categories there when to use nonlinear model I told you when it comes to linear when it is easy simple and straight relationship is there we use linear When the opposite comes, when the relationship among the variables are very complex or threshold limits exist, then we use when your answer may not be arrived at at single calculations. We need to do it many calculation till we get optimization model. When that is there, we always go for nonlinear model. You cannot get your answer at a single shot.

[00:33:44]This is what we can call say nonlinear model when the relationship among the between that is what dependent and independent variable is not straight line it is a curved relationship I I'll show the graphs of those then you will understand what it is and there is not constant change there is a varied change in the input that is called dependent variable due to change in independent variable. Now look at the characteristics of this nonlinear model.

[00:34:20]We say that it is it is the most difficult to estimate. It requires iteration again and again trial and error till you get a optimal model. We use it in uh especially in finance like IR internal rate of return.

[00:34:39]We use it. We have to find out that rate at which present value of cash inflows is equal to present values of cash outflows. We never get exactly we cannot imagine also this may be the exact answer for that irra rate. We are we go on doing it till we get there are some hints are there. There are some clues are there which we adapt it in order to arrive at that right rate of return.

[00:35:06]These are all machine learning models are there. Many are there that I'll explain you that later practically how it can be done. This requires computation requires I told you optimization logarithm algorithm. It is very flexible. You can change the variables. If you don't get you can change power rise to the power of two to raise to the power of six. If it is not obtained you can go for exponential form. You can go for logarithmic form. Like that there is a transformation of functions mathematical function can be used transmission the transformation can be made but linear we cannot made that's why that linear model is not flexible but this is a flexible but most powerful model it is a best for classification as well as volatarity. So these are the characteristic of nonlinear model. Now nonlinear pattern the pattern may not be straight line it may be some it is a curve the curves are of different types it is a U-shaped curve it is S-shaped curve or you get there are V curve there are many curves are there that based on that curve pattern we can say what type of model it is what type of model it is also we can decide so when in your when you present your data on a graph when you don't ified straight line you say that it is a nonlinear it is a curved one so that here we say that many math functional these are all types of nonlinear p exponential this exponential pattern is also there when you are there is a rapid change increase or decrease is there in your data then we can call it as a say exponential exponential growth or exponential decay if the If there is increase is there we can say that exponential growth. If there is a decrease is there we can say exponential decay. When there is a rapid change or whether rapid increase or decrease we come from any data we say that it is exponential pattern. This will have always a J-shaped curve. This always have Jshaped curve. The examples of this are population growth is a exponential growth pattern. Again viral marketing spread is also an example of exponential pattern.

[00:37:40]Now equation when you look at it mathematical equation Y is always dependent variable. X is always independent variable. y = alpha into exponential power of bx. Here exponential mathematical function is used instead of using the plus simple without using but is a power is there.

[00:38:08]So is is something complex. Now earlier very easy because in linear model there is no any such complex things only plus+ plus will be there additive. So it was easy for us to do it but this is a something complex it has come some new mathematical sign has been appeared before your coefficient of independent variable then it becomes some complexity has come. So this is what we can call say exponential pattern and exponential mathematical equation. Next one is loarithmatic pattern. Logarithmic function of nonlinear model.

[00:38:46]Here how the data of this category means there is a rapid increase thereafter it levels off. It will firstly it will increase then it will be stabilized. Rapid changes at first then levels off. This shape we can call it as a concave curve. Now this you can find it on the slide also. This is what this is what all concave curve is.

[00:39:21]This is what concave curve. This is what concave curve. We have a shape of concave curve. This is a shape of concave curve. This is what concave curve. This is what how this is what concave curve. This is what the concave curve we use it when it is a logarithmic function. The equation is y = a + b into log x. The log is used as a new mathematical sign for nonlinear model. This is applied in case of practical application is in case of learning curves we use it. In case of calculation of diminishing return, constant return and increasing return we use it. So this is what third one again we'll go to. Now this is practically applied in case of learning curves as well as diminishing m diminishing returns which we use in economics. So this is what we have one. Then next we have third one power law distribution. This is another pattern of nonlinear model. Here small values are more and larger values are few. When that is there we use this model. This shape is we can call it as a straight on log lock plot. We can call the shape of this is it is a straight just like a straight it will be found the way you are finding here. It will be just like a linear we find it. If it comes exactly straight is a linear but slightly it is a curvy. When slide curve is there, we can call it as a log lock plot curve. Straight on log lock plot curve. It is used practically in case of wealth distribution and in case of city sizes where this can be used. The mathematical equation of this is y = a a into x ^ of b.

[00:41:33]Here there is no additive model. It is a product model. The multiplication is there. Again B E has become a power of X is it now power of X has come. So that this is what another pattern we come across. The next is fourth pattern of nonlinear one. This is called quadratic or polomial pattern. It is nothing but parabolic shape or curvy linear shape. This is what shape as you come across in our graph either it may be downward or it may be upward also not necessarily that is always downward U-shaped curve we say pattern or inverted U- shape inverted U-shaped curve this is practically used in case of estimating cost curves or in analyzing performance versus stress we use it this quadratic pattern of mathematic quadratic pattern of nonlinear model. The mathematical equation is y = a + b x + c x ^ of 2 x to the power of two. This is what equation mathematical equation we use it for quadratic nonlinear pattern. This is another pattern we have. The fifth pattern is sigmoidal pattern or sshaped pattern of nonlinear model. It is not it is also called it as logistic curve. Here this is a shaped curve. Slow will be there again fast will be there again slow will be there. It will be upward or it may also be downward but slowly it will rise thereafter fast it will rise then slow it will rise slow fast slow increase. This scurve this is more used in case of product adoption in case of population saturation this model is extensively used. Now this this is a another pattern we have. The last pattern is mathematical function which we use it for nonlinear model is cyclical or periodic pattern. Cyclical or periodic pattern.

[00:43:50]This is used when there is a fluctuations upside downs. When you come across in any data that is called wavelike fluctuations when you come across we use this cyclical or periodic pattern. The shape is wavelike pattern or it is also we can call it a senu sodial pattern. So part the example we use it in case of estimation of business cycles and seasonal trends we use it. Y is equal to A + C sin into B X + C. This is a mathematical equation we use it for this cyclical pattern of non logistic models we use more for the for the customer retention. In order to study the customer retention whether the customer retention we go for this logistic model even for pricing strategy we use it polomial model in case of finance for garch calculation of for the estimation of garch models that is used for volatility forecasting how fluctuation how there is a variation in a data that can be forecasted with the help of garch model. This is used in finance. Even risk models for which we also use this nonlinear models in finance. In consumer behavior for in the subject of consumer behavior, we use neural network. In supply chain management, we use power law models for demand prediction. For neural network, for logistic optimization, we use it in econometrics. We use it threshold models for economic cycle. That is called up to a threshold limit there is a one chain after that beyond that there's another chain that is called we can call it as a resime chain resign shift we can call these are all we use it in case of nonlinear models. Now let us come to now practical example we use it for nonlinear models. This is a how we can use those models in practical in research. I have taken my own example that is called d I want to find out or know what are the determinants of dividend policy. This is what my topic I have taken for study it. Now I took the data for this estimation.

[00:46:31]The data is for a period of 2002 2024 around 15 years annual data I have taken the company I have taken I have taken 23 companies these are all BSC Sensex company I have taken out of that seven companies they are related to finance and banking that I have eliminated because of some nonavailability of some variables for which those have been forced to be removed from the my 30 companies. I have taken only 23 companies data. The variables I have chosen 13 variables for the estimation of determinance of dividend policy. in these 13 variables. Now, DPS, dividend per share, dividend per share is a dependent variable which is also called it as response variable. Others are independent variable. Out of 13 variables, one is a dependent variable which is known as DPS. The other 12 are independent variables. They are also called it as predictors.

[00:47:46]They are earning per share, return on net worth, return on total assets, quick ratio, debt to equity, debt turnover ratio, credit hours turnover ratio, price to earning ratio P ratio, price to book ratio, PB ratio, yield ratio, assets turnover ratio and size. size is decided based on the total assets amount of total asset. So the data is collected from the CMI data center for monitoring Indian economy that is called proves we have taken this is a data we have taken in order to estimate the determinance of dividend policy.

[00:48:38]Now when we have taken all the 12 variables as a independent variable all 12 variables are not suitable for considering them for estimation before that I have to check whether there is any problem of colinearity multiolinearity among this 12 independent variables if there is a strong correlation whether it is a positive or negative among this independent variable there is a problem of multiolinearity If there is a strong relationship is found among the variables, it is very difficult to say what is the contribution of that those variables to the dependent variable. So in order to know clarity in order to know genian results we need to avoid the problem of colinearity. The problem of colinearity is calculated with the help of VIF. Variance inflation factor. Variance inflation factor is going to decide whether your independent variable has multiolinearity or not. One more test is also there that is called tolerance test is also is also used for this multiolinearity.

[00:49:58]Now once we get the VAF we divide the VAF into three categories. Less than five is a low colinearity. 5 to 10 is a moderate colinearity. About 10 is a high colonarity. We see that our independent variable may not have high colinarity and moderate colonity. they may have low colinearity. In order to have low colinearity, our independent variables VAF should be less than five. So when I have taken those 12 variables for calculation of VAF, we find we found only five variables whose VAF is less than five. EPS has a 2.74. It is a low colinearity. Quick ratio has a 2.91.

[00:50:54]It is also low colinarity debt equity ratio 1.84 credit credit turnover ratio 3.82 and size is 4.99 with this I came to conclusion that these independent five are suitable for the estimation of determinance of dividend policy. So I have finally taken those five variables as a independent variable for calculation. This is a visualization of those five variables using VAF as a base for data present uh graph presentation.

[00:51:37]This is what almost all of them are less than five is there. So that we have gone for this.

[00:51:45]Now I have taken one linear model, four nonlinear models for the estimation of determinance of dividend policy. I have taken totally five models for the estimation of dividend determinance of dividend policy. Out of five, one is a linear model others other four are nonlinear models. First one is a linear model. That linear model I have taken mathematical equation to know it because when we are using some mathematical equation we need we need to know how that can be presented in our research articles that is called DPS is a Y variable that is called dividend per share. I have taken DPS is equal to BY is nothing but here is alpha that is called intercept plus B1 is a coefficient of first independent variable EPS coefficient of first EPS is B1 that is called beta 1 that is also called it as slope one beta_2 of QR fick ratio that is a second coefficient second beta of QR now B3 of D B4 of CTR B5 of size plus erat what this beta say now a one unitit change in EPS independent variable what will be the change in dependent variable that is explained by beta let me repeat once again the coefficient beta 1 says that what may be the change in dependent variable due to change in one unit of EPS. Similarly for other independent variable like B2 will say what may be the change in dependent variable that is called DPS due to one unit of change in earthquake ratio.

[00:53:50]There may be a positive relation, there may be negative relation need worry what sign it is. There may be negative relation may also be there like example example I'm seeing a one unit decrease in QR quick ratio there is a increase in dependent variable that is called DPS here decrease in quick ratio there is a increase in DPS they may also be there not necessary that all independent variable have positive relation with the dependent variable there may one or two negative may be there. You should not worry about negative signs that relation may come. So second is a nonlinear model we are taking that is called log transformation model. Here B 0 that is called intercept. B1 is a coefficient of log. Here we have used be clear. We have not used a log for all the variables. We can use it log for some of the independent variables. It is you to decide because if you want you can use it all independent variable in log function also you can use it. Log of EPS we have used for first independent variable. For second P ratio we we have not used any log function but for date equity also we we have not used LO function. When it comes to log of size, we have used only for two independent variable. We have you we have transformed the the plane function into log function. Now similarly log wherever log is used that we can call as a log transform model which comes under one of the categories of nonlinear model. Third model is polomial model. It is a hyperbolic model. Here your power is go your power goes on increasing in first independent variable power of one second independent variable power of two like that there may be change in power with the in with the increase in your subsequent independent variables. Again here also we it doesn't require that powers can be used for all the independent variable you can decide unless you get till you get your model is good fit till your model is good fit because I have used polomial function only for EPS that is two EPS comma 2 mean power of two the same I have not used for other independent variable they are all plain B2 B2 is a coefficient of QR. B3 is a coefficient of D. B4 is a coefficient of CTR. B5 is a coefficient of size. Here I have no I have used a polomial function only for the EPS independent variable.

[00:56:53]So whenever we use any mathematical function either for one or for all this becomes a nonlinear function not necessary that you should use it mathematical functions for all the variables which you have considered. Second one is gam model.

[00:57:10]This is a general additive model. Here smoothing parameter is used as a lambda that is going to be adjusted by its own till it gets maximum till it gets optimum. But what is that parameter? How much it is that smoothing parameter that I have not known because I have used our program for my estimation that that itself will adjust it by using iteration method. Then what may be the smoothing parameter each it is taking for each independent variable. Now DPS is equal to a plus. Now this is what DPS is equal to A. A is a intercept. Then here smoothing parameter has come. Now what what is that value? I don't know because it may be automatically adjusted to as a coefficient to the EPS in such a way that it has to give optimum number like that smoothing parameter for QR, smoothing parameter for D, smoothing parameter for CTR, smoothing parameter for again size.

[00:58:18]This is what we can call to say generalized additive model. This is another category of nonlinear model. The last nonlinear model is nonlinear model.

[00:58:30]Nonlinear model least actually that is full form is nonlinear least square model and ls here also we use it sometimes exponential here exponential we have used in this not necessary that exponential to be used for all the independent variables it is you to decide sometimes what we do we go on applying e to one by one for each independent variable Till we get optimum first we apply to one we'll see what may be our measure of value as a good fit we look at it whether this model is good fit for with that one exponent with the second exponent we we see again our good fit measure is satisfied or not that we'll see it whether our model is becomes good fit or not so like that we are going to do it this is what how it is now this mathematical equation when it is replaced with the real values with our data to our data this has come. Now for linear model DPS is equal to 17.91 it is a intercept that is called alpha plus 0 1 08 it is a coefficient of EPS beta 1 then there is a plus sign when it comes to QR there is a negative relation be clear there is a negative coefficient minus 0.4976 4976 coefficient of QR Q ratio. Again for D also there is a negative negative relation with this dependent variable of DPS D. Even CT are also negative relationship. See here CT are also negative relationship. Size is a positive. No doubt. Let me say what it says. Now first one is 17.91. It is contributing to the it is a baseline. It is a contributing to the DPS. When the contribution of all the independent variables become zero when the contribution of independent variables is zero, what may be the contribution of your internal source that is called alpha. That internal source efficiency is going to say that is called 17.91. It is nothing but we say in our cap SAPM security analysis portfolio model while doing performance evaluation models where we use it one model is a Jensen's alpha Jensen's alpha is going to measure the management's ability whichever firms are alpha is high we say that that firm's management ability is high nothing but internal efficiency is high so that 17.91 it is contributing positively to the DPS when the contribution of all independent variables are zero.

[01:01:30]Now let us come to the second first independent variable EPS. How it contributes for one unit change in EPS. There is a change in positive change in DPS by 0.108. This is what it is. This is a change in DPS. Let me this is what I say.

[01:02:00]What I'm saying here now the change in DPS by 0.108 108 due to one unit change in EPS if the EPS is increased by 1 rupee in the DPS it will be increased by 10% 1 rupee in EPS and DPS is going to be increased by dividend per share is going to be increased by 10% approximate I'm saying there value similarly for a one unitit change in Q ratio if the kick ratio is changed from five to six one unit change there is a negative change in DPS the DPS is going to be decreased by almost half 0.49 49 means 0.5 if the one unit increase in Q ratio there is a decrease in half in the DPS this is what how this can be interpreted by by your coefficients like that all will be same what may be the change in DPS due to change in one unit of EPS one unit of QR one unit of D one unit of CTR, one unit of size. Again here we have given some here stars three stars is more significant variable. First star is we say that listen to me first wherever green color is there that is a very very significant variable in contribution to dependent. Again yellow color says that it is a very significant very very significant most excellent significant is green very significant is yellow pink is significant. Now here the two bps and size are the most significant variables in contributing to DPS.

[01:04:08]QR and CTR are only very significant contributors to DPS. D is only significant. However, all five variables are significant in contribution irrespective of their degree of contribution irrespective of their degree magnitude. But similarly lo transformation also we have used with that equation. We got it. Now you see butuh here intercept has become negative contribution intercept has become negative it is not contributing internal source to the DPS if it is a 99 minus it comes there is a decline in one so there is a negative contribution internal efficiency is not showing anything but when it comes to EPS again there's a one unit change in EPS there is a change in DPS by 2.65044. So is very good like that same calculation there may be some positive some may be negative. So in that most significant independent variable for contributing to a dependent variable is green is only one that is called size.

[01:05:29]Now next is pink. Only size and pink color only. First is size. Next is EPS.

[01:05:36]Only two are significant variables in contributing to the dependent variable.

[01:05:43]Whereas whereas quick ratio DE and CTR are not significant variables in contributing to the DPS like that polomial mathematical equation we have used. Then with the result we got it from our software this is what 22.79 again it is a intercept it is a positively contributing when the contribution of your independent variables are zero. Now polomial EPS is also contributing to the extent of 228.932 dependent variable. So here size is also contributing to the extent of 0.0000 00 0 for one unit change in size like that. Here EPS and size are the most significant variable in contribution to dependent variable. CTR is very significant. DE is significant.

[01:06:41]Again, GAM model also we have used like EPS is more contributing. This we have found it from your what we can call nominal deviance model figure result we look at looked at it with which we have come to know that this is also more contributing now again nonlinear le square model that also says that contribution here only two variables are more contributing EPS and size we have taken out of this now the question comes which is the best among these five which is the best model in predicting your DPS that we have taken. This is what developing model that is our objective is to develop to evaluate to compare the models of linear and nonlinear in business research. This is our objective. This the first one is what we are saying with respect to development of model. Second one we are coming evaluation. When it comes to evaluation, which is the best model among the five, we look at different parameters, different metrics in order to determine which is the best. Here I have taken one is adjusted R square nothing but what is the amount of total contribution of your independent variable to the dependent variable? What is the contribution?

[01:08:11]The total percentage of variance in dependent variable is explained by adjusted aspect. Percentage of total percentage of total variance in a dependent variable is explained by independent variables. nothing but we have taken again one more parameter is there like we can also take like information criteria like AIC and BIC that is also another parameter to determine whether your model is good fit or not that despite having that data we have not taken here for presentation like adjusted R square we have taken again errors we have taken standard error like this is called accuracy errors This is going to say precision of your prediction. Precision of your prediction. Lesser the RMSSE, higher the prediction. Lesser the RMSSE, the higher the contribution. The higher the R square, higher the contribution. P value. If the P value is less than 0.05, you say that the model is overall good fit. In these three measures, whichever R square is the highest, that is the best. In case of RMS, whichever is the lowest, that is the best. In case of P value, whichever has the least value less than 0.05, that is the best in case of all these model. Now, which is the best? GAP model is contributing to the extent of 97.94%. Almost all 98% all independent variable together are contributing to the dependent variable of DPS. The 97.94% of the total variance of DPS is explained by independent those independent five variables that is called EPS. Again pick ratio again debt equity ratio credit or turnover ratio and size these are all five put together are contributing to the DPS to the extent of 98%. Hardly 2% is left out. It is a very good model. It is a best model. If any model is more than in finance we say more than 60 to 70 this is a good model good fit model RAP in case of marketing 3 to 5 30 to 50% in marketing is all rule of thumb there is no any hard and fast rules in social sciences like history and other sciences less than 30% is good 30% is also working but in finance we say up to 80 70 is good but it is a 90 it is just like in pure sciences we'll get in physics or chemistry in those sciences we get adjusted R square while calculating their nonlinear model they get so this is very good model even RMS is the lowest in gamma model only 1 309 here 1 is the lowest so among this all the types of models the least RMS that is [Music] called 1.309 30. Are you getting your mind? Root mean square error that we can call R means root mean square error. This is used for precision of prediction. Accuracy error. Lesser the accuracy error, more the prediction. More the precession of for prediction. Lesser the accuracy error, more the precession for prediction. Next one is P value. In all the cases P value is less than 0.05. Hence these models are good fit.

[01:12:24]No doubt all the models are good fit in all the cases adjusted R square are more than 90%. No doubt all the models are very good. It is lucky enough that I have got this good result. I should say proudly because we don't get this much good result in many of the data because this is a data I took it I got it it is one of my research student is doing research on this topic so we took the data and we have used so here best performance is coming from the GAM model so this model can be used for your estimation for prediction of your DPS. No, this is what now next we have here certain diagnostic test we use it diagnostic test we use it in order to validate our model in order to confirm the best fit model we use some more resial analysis resial test we use it that also we can call it a diagnostic test after getting the results of regression We use some diagnostic tests which is also known as residual test. These are used to validate our result.

[01:13:54]To know the geninity of our result to confirm our results are best.

[01:14:02]We should decide from our diagnostic test for which we use three tests as you know which I said in the beginning also.

[01:14:13]One is linear normality test we use it.

[01:14:17]These tests are used these diagnostic test we are using from the residuals residials not from the original data. We should calculate residials for all the observations you have taken. How do you calculate residuals? I told you actual observation minus expected that that we can call as a ratio. The ratios you have to get from those ratios you have to calculate normality test, homoadastic set test and autocorrelation test.

[01:14:52]Now other test I told you linear multiolarity for determining whether which whether or which independent variable is considered for regression colinearity that is called VF again one more test I told you in the beginning that is called linearity now some people what they do whether I want to go for linear linear model or nonlinear model for that you can initially test it to your data whether your data is suitable for linear model or nonlinear model using I told you Ram say reset test or holier harier test there are many are there these are the test we use it to decide whether this data is suitable for linearity or nonlinearity if the test is giving us giving a hint for linearity then we go we go for directly linearity if the test is giving symptom for nonlinearity directly we go for nonlinear instead of wasting our time with the linear model.

[01:16:00]So here we test diagnostic tests, residial analysis test we do it in order to decide whether my results are validated or not. For which we do three tests, normality test, homoscadasticity test and autocorrelation test. In all the three cases we need null hypothesis to be accepted. In all the three cases we need null hypothesis to be accepted in order to say my results are valid, my results are gen. First we go with normality test.

[01:16:42]Here we have done reset test because I told you here after doing all the different types of linear and nonlinear model now we are calculating reset test.

[01:16:52]Usually people will do at the beginning itself whether I want to go for linear and nonlinear because I am comparing the model. Hence I have taken at the end why I should calculate in the initial itself to know it only linear linear are suitable nonlinear are suitable let me calculate at the end so that after calculating all the types of models linear and nonlinear results we'll see now reset this is what first this reset test is used to determine whether my model is whether my data is suitable for linear or nonlinear model Here we set the hypothesis here.

[01:17:32]Now in the hypothesis, HY is there is a linearity. HY is there is a linearity. To get linearity, null hypothesis should be accepted. Whose p value should be more than 0.05. To get null hypothesis to be accepted, your p value must be greater than 0.05.

[01:17:58]H1 is there is no linearity nothing but nonlinearity that is called alternative to get alternative hypothesis to be accepted your PO p value must be less than 0.05 0.05 05. Now by testing this one we from the reset test our p value is less than 0.05 it supports it supports for nonlinear model. Even in our calculation results also our best good fit measures are nonlinear model. So our test supports our results. Our test supports our result. P value is less than 0.0. 05 hence it supports nonlinear model so this is a one we have done it it should have been done in the beginning itself if you want but I don't want to tell you there itself the what is the best what is not the best so this one we do it this we usually do it at the beginning in the research that is what we do it now other three test we do it that is what I told you one is normality second One is homoscaracity that is homoscaracity is a null hypothesis. Alternative hypothesis is a hrocarasticity hyrocar. Third one is autocol. First I am doing here normality test. This can be tested with the help of chapiro bilk test.

[01:19:40]There are many are there. Many tests are there like jco is there. Anderson something is there. Many are there.

[01:19:47]Depending upon the type of the data, how your data is there, how your data is based on that also we can decide which normality test we want to do it. If the robustness if you want we can go for Anderson something is there that test we use it in case of normality but otherwise we always go for like if you adopt like eSh adopts Jarcoa test if you go for like Python and R they adopt Shappiro test like that each software adopts a particular category of test but all no doubt almost same answer. Now here when I do the Shapiro test the P value has come to 0.594 5949. This is what the P value is very important in all the tests when you do research P value is a must with the P value it has become easier for the researcher to do his research.

[01:20:51]Earlier we didn't get a P value. We used to get one is a table value. Another one is a calculated value. Table value is called it as critical value. Table value is called as critical value. Another one is a calculated value. Your calculated value is compared with the critical value. If your calculated value is more than is more than critical value, we say that null alternative hypothesis accepted. In case calculated value is more than critical value, we say that alternative hypothesis is accepted.

[01:21:30]If calculated value is less than your critical value, null hypothesis is accepted. For which we need to take the tables, the respective tables. The tables we should refer it under three situations. One is at single tile or two tile. One you have to see. Next one is significance level.

[01:21:55]You have to see at what level of significance you are testing. You have to see again degree of freedom you have to see. All these three things should be simultaneously seen in order to know your critical value. That critical value must be compared with your calculated value. Then you can decide it whether your test is null is accepted or rejected. Such problem is not there. Now the reason is that the p value is readily available in almost all the softwares. Even the p value is calculated using some Excel commands. Here p value is 0.5949. So it is a more than 0.05. 05.

[01:22:45]So it accepts null hypothesis. Null hypothesis says that there is a normality. Your data is normally distributed. So this is the one of the basic requirements for satisfying your data to be valid. Your results should be valid. Excuse me. Sorry, not data. your results what you got what you told almost all your adjusted R square are more than 90 what you said in order to say my results those results are correct very correct this is very much required now this can also be visualized with the help of what we can call uh even histogram this can be visualized normality test even normal Q plot your all circles original data is lied on is lying on your straight line they are very closer to straight line so that your data is a normally distributed. So this one test we tested it as a diagnostic test this is satisfied. Next we go to another test that is what this test this graph is used for your component residual plot for whether your data is a linear or nonlinear but certain variables are linear. We find it like here it is there. It is a very much like straight line is there. It is a linear. Uh this is not linear but this is also almost linear. Uh these two are almost linear. This also third is a very closer to linear. When it comes this one is not linear but those are not coming because straight line is different. Your original is somewhere outside. Here is much more different. They are not together. So this is also diagram we use it to decide it whether your diagram says whether linear or nonlinear one that is what is a diagram name of the diagram is component plus ratial plots are used to determine whether your model is linear or nonlinear then this is a this is a visualization this can be tested with the help of ROM say reset test that has already been told that has already that I told you already. Now next one is another diagnostic test we use it that is called homoscaracity or hetroscaracity. We need our data to be homoscaracity. We need our data to be homoscaracity for which null hypothesis has to be accepted.

[01:25:28]So to accept null hypothesis our p value must be more than 0.05. This is a common. To accept null hypothesis your p value must be greater than 0.05. To accept your alternative hypothesis your p value must be less than 0.05. Here your p value has become more than 0.05.

[01:25:56]Hence null hypothesis is accepted. So our data our residuals are homoscaracity. So this is another diagnostic test has been satisfied. So we have taken two tests for our for our this data. Third one, one more is there autocorrelation for which either we can go for what we can call Derbin Watson test or we can also go for brewish godfrey test we can do it that I have not done because only two tests have done it with these two tests we have satisfied our data to be a valid one so this is what we have done the all calculation we have done one more test people will do it in softwares is available whether we have whether our data data has some outliers whether our data has any extreme values that can be tested that can be known with the help of cooks distance cooks distance whether those extreme value has any influence on your other non-extreme values that can be checked with this residual diagnostic only one or two are there as extremes it will not have much impact on your other non extreme values. If many are there outliers or extreme values then there is a possibility that having influence on the non- influential non extreme values.

[01:27:29]So that extreme two two are there. This is what the this is what bigger these what these two are extreme values. These extreme values may not have much influence on other variables. So that is what it may not have that also we can see is there any influence or not of extreme values that can also be checked. Now these are all we have done using our linear and nonlinear model. Now these are we have developed the model with a mathematical equation. We also evaluated using good fit measures are metrics using adjusted R square using RMSSE using P value. Another metric is also there like that is what we can call say information criteria. AIC Aka information criteria again BIC basian information criteria we also use it where we need the lowest value will be the best lowest AIC or BIC is the best model good fit so that I have not that is there in our data it has come results but I have not taken due to the non aility of space in displaying our results so these are all we got it from our this one. Now finally we can say that from our result I told you already but let me repeat once again. GAM model provided the highest adjusted R square and lowest RMSC indicating the B best predictive performance. It is followed by others are also there like log transfer model linear log transform model is also there nonlinear le square model is also there.

[01:29:22]Other models are also there but the best model is GAM model.

[01:29:26]Generalized additive model where adjusted R square is the highest, RMS is the lowest and P value is the least. So even we used residial test in which also it has satisfied to say our results are the best recommendation. We say that prefer gap model for predictive purpose otherwise use lock transfer model for simple interpretation if necessary. These are all my presentation which I have taken for this practical purpose.

[01:30:10]Hopefully you have understood. In case any doubts, I'll give my mobile number as well as my email number just to contact me. I'll try my best to provide the answers for the queries.

[01:30:28]Thank you. Thank you very much.

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