Regression analysis is the analysis of the relationship between a response variable and another set of variables. The relationship is expressed through a statistical model equation that predicts a response variable from a function of regressor variables and parameters. In a linear regression model the predictor function is linear in the parameters. The parameters are estimated so that a measure of fit is optimized. For example, the equation for the observation might be: where Y_i is the response variable, Χ_i is a regressor variable, β_0 and β_1 are unknown parameters to be estimated, and ε_i is an error term. This model is termed the simple linear regression model, because it is linear in β_0 and β_1 and contains only a single regressor …show more content…
The equation for the i^th observation might be:
There are many cases where the dependent variable is restricted to take on a limited range of values, for example only values 0 or 1 (binary logistic regression). In this case the regression model is slightly different.
We introduce the idea of Random Utility Model. Assume that an individual has to make a choice between two alternatives. Let U_ij be the utility that individual i (for i=1... N) gets if alternative j (for j=0, 1) is chosen. The individual makes choice 1 if U_1i≥U_0i and makes choice 0 otherwise. Since U_1i≥U_0i is equivalent to U_1i-U_0i≥0, the choice can be seen to depend on the difference in utilities across the two alternatives and we define this difference as
Y_i^* should depend on an individual’s characteristics (in our case LoanValue, EmpStatus, Age, HouseholdStatus etc.). Idea that a variable depends on some characteristics (explanatory variables) is a multiple regression model:
To make the derivations easier, we write some of the formulae below in terms of the simple regression model
The problem with this regression is that we do not observe individual’s utility and, thus, Y_i^* is unobservable. The logistic regression model can be interpreted as this regression, where the errors are assumed to satisfy all the classical assumptions except one. The exception is that the errors are assumed to have a logistic distribution.
Y_i^* is unobservable,
17 In regression analysis, the coefficient of determination R2 measures the amount of variation in y
* Independent variable coefficient – This is the measured effect the independent variables have on the dependent variable. This is the main output of the regression analysis.
4) Use cubic regression to determine an equation for the data (or lwh where (12 – x) represents the sides and (x) represents the height of the box).
~ From the example above, the dependant variable is banana quality and the independent variable is time
Linear regression is an approach for modeling the relationship between a scalar dependent variable Y and explanatory variables (or independent variables) denoted X. Function $f(X,W)=Y$ (shown below) can be learned to predict future values.
Project two in ECON-E 281 - APPLIED STAT FOR BUS & ECON II consisted of the students evaluating three independent variables such as Pickup Time, Delivery Time, and Mileage. The dependent variable was Cost. Only one independent variable could be selected when applying “ONLY” the p-value approach. The first step, I selected the Data then the Data Analysis tool. Next select Regression. The Input Y Range I selected the Cost data. The Input X range I selected the first independent variable Pickup time. Then check marked the boxes Labels, Confidence Level 95%, New Worksheet Ply, Residuals, and Residuals Plots. After checking the boxes I pressed OK. This gave me my first regression model. I used this process for the next two independent variables
"There are several different kinds of relationships between variables. Before drawing a conclusion, you should first understand how one variable changes with the other. This means you need to establish how the variables are related - is the relationship linear or quadratic or inverse or logarithmic or something else" ("Relationship Between Variables ", n.d)
6. Furniture. The presence or absence of furniture is recorded for each unit, and represented with a single dummy variable (1 if the unit was furnished and 0 if
Binary logistic regression makes no assumption about the distribution of the independent variables. The relationship between the predictor and response variables is not a linear function in logistic regression; instead, the logistic regression function is used, which is the logit transformation of p:
Also known as “causal” models involve the identification of variables that can be used to predict another variable of interest. They are based on the assumption that the historical relationship between "dependent" and"independent" variables will remain valid in future and each independent variable is easy to predict. This form of analysis can take several months and is used for medium-term forecasts for products in their growth or maturity phase.
If this is present it means there is a violation of the constant variance assumption.
We can then look at the T-Statistics to see how individual variable can explain the dependent variable, by doing a T-test comparing it with the Critical value T. By doing a 10%,5% or 1% test.
Explain how regression can be used to analyse the relationship between political freedom and real GDP per capita? -Regression is be used to show whether there is a positive or negative relationship between political freedom and GDP per capita. The coefficient of b determines the slope of the line and how an increase in X affects Y. The regression also determines the R squared which shows how closely the X and Y variable are correlated within the regression. The confidence levels indicate how confident political freedom has an impact on GDP at a particular confidence level. The P value explains whether to accept the null hypothesis or to reject it.
• Error values (ε) are statistically independent • Error values are normally distributed for any given value of x