The main idea of a multiple regression analysis is to understand the relationship between several independent variables and a single dependent variable. (Lind, 2004) A model of the relationship is hypothesized, and estimates of the parameter values are used to develop an estimated regression equation.(abyss.uoregon.edu) The multiple regression equation used to describe the relationship is: Y' = a + b1X1 + b2X2 + b3X3 + . + bkXk. It is used to estimate Y given selected X values and k independent variables. (Lind, 2004) There are two components of a multiple regression analysis that are important; the coefficient of correlation (multiple R) and coefficient of determination (R²). The coefficient of correlation is the degree to which two or more independent variables are related to the dependent variable. The coefficient of determination is "the proportion of the total variation in the dependent variable that is explained by the independent variable. It can assume any value between 0 and +1.00 inclusive." (Lind, 2004)
In this assignment, we are concern with the multiple R and R² of two sets data from the "Research Methods for Managerial Decisions" simulation exercise, in which we performed a multiple regression analysis on a company called Coffee Time. We needed to determine, which, produced a better multiple regression model, and how to further optimize this model. Then the team needed to answer the following scenario: Tourism is one consideration for Coffee Time's future. A survey of 1233 visitors to Mumbai last year revealed that 110 visited a small café during their visit. Laura claims that 10% of tourist will include a visit to a café. Use a 0.05 significance level to test her claim. Would it be wise for her to use that claim in trying to convince management to increase their advertising spending to travel agents?
Reference:
University of Oregon. Regression and correlation analysis. October 3, 2007....
...1. If the correlation coefficient between the variables is 0, it means that the two variables aren’t related. – TRUE
2. In a simple regressionanalysis the error terms are assumed to be independent and normally distributed with zero mean and constant variance. – TRUE
3. The difference between the actual Yvalue and the predicted Yvalue found using a regression equation is called the residual (ε) – TRUE
4. In a multipleregressionanalysis with N observations and k independent variables, the degrees of freedom for the residual error is given by (Nk). – FALSE (correct answer Nk1)
5. From the following scatter plot, we can say that between y & x there is _______. – Negative correlation
6. According to the graph, X & Y have ________. – Virtually no correlation
7. A cost accountant is developing a regression model to predict the total cost of producing a batch of printed circuit boards as a function of batch size (the number of boards produced in one lot or batch.) The explanatory variable is called the _______. – Coefficient of determination
8. In the regression equation, y = 75.65 + 0.50x, the intercept is ______. – 75.65
9. The assumptions underlying simple regressionanalysis include ______. – The error terms are independent
10. The proportion of variability of the dependent...
...MULTIPLEREGRESSION
After completing this chapter, you should be able to:
understand model building using multipleregressionanalysis
apply multipleregressionanalysis to business decisionmaking situations
analyze and interpret the computer output for a multipleregression model
test the significance of the independent variables in a multipleregression model
use variable transformations to model nonlinear relationships
recognize potential problems in multipleregressionanalysis and take the steps to correct the problems.
incorporate qualitative variables into the regression model by using dummy variables.
MultipleRegression Assumptions
The errors are normally distributed
The mean of the errors is zero
Errors have a constant variance
The model errors are independent
Model Specification
Decide what you want to do and select the dependent variable
Determine the potential independent variables for your model
Gather sample data (observations) for all variables
The Correlation Matrix
Correlation between the dependent variable and selected independent variables can be found using Excel:
Tools / Data Analysis… /...
...Chapter3
MultipleRegressionAnalysis: Estimation
Key drawback of SLR: all other factors affecting y are unrelated
with x, as is unrealistic.
Multipleregression allows us to control for many other
factors to explain dependent variable, which is useful both for
testing economic theories and for drawing the ceteris paribus
conclusion.
In addition, MR can incorporate fairly general functional form and
build better models for predicting the regressand.
Econometrics_buaa_Phd, Ma
1
3.1
Motivation for multipleregression
3.2* Mechanics and Interpretation of
MultipleRegression
3.3
The expected value of OLS estimators
3.4** Variance of the OLS Estimators
3.5 The GaussMarkov Theorem
Econometrics_buaa_Phd, Ma
2
3.1 Motivation for multipleregression
1. Taking exper out of u and put it explicitly in the equation:
wage = β0+ β1educ+ β2exper+ u
Now, we can hold exper fixed to evaluate the ceteris paribus effect
of educ on wage, other than assume exper is unrelated with educ.
2. MR is useful for genralizing functional relationship between
variables, for example (a quadratic function):
cons = β0+ β1inc+ β2inc2+ u
Note: we can not hold inc2 fixed while inc changes.
The marginal propensity of income to consumption depends on β1
as well as β2 and the level of income.
3. The definition of the independent variable is crucial...
...referring to the recent boom in house prices in many developed countries following a sharp bust in 2008. Researches and policy makers alike have realized that housing has significant influences on the business cycle. This paper tries to figure out the determinants of the selling price of houses in Oregon. The data set used in this paper has been retrieved from the case study titled “Housing Price” (Case #27  Practical Data Analysis: Case Studies in Business Statistics Marlene A. Smith & Peter G. Bryant)
The most important factor in determining the selling prices ofhouses is to know the features that drive the selling prices of the house. People tend to have more interest in houses with multiple bed rooms/bathrooms, fireplace, garage for multiple cars and location while choosing a house. So, a house that meets this requirement tends to be priced more and the house with these features being absent is priced low. According to the survey conducted by Marlene A. Smith & Peter G. Bryant while forming their case study titled “Housing Price” (Case #27  Practical Data Analysis: Case Studies in Business Statistics), 10 variables were selected to find out their impact in determining the housing price. A sample of 108 houses wasselected from East Ville, Oregon along with their characteristics on 10 selected selected variables. The variable set for the study is:
Selling Price of House
Area
No of Bed rooms
No of Bath rooms...
...MultipleRegressionAnalysis of exchange rate with the determinant factors
RegressionAnalysis: USD versus GDP Growth, FER, FDI Growth, Interest Rate, Money Supply, Terms Of Trade
The regression equation is
USD = 41.5  1.95 GDP Growth + 0.000943 FER  0.139 FDI Growth + 0.048 Differential Interest Rate + 0.000067 Money Supply + 0.166 Terms of Trade  0.000097 External Debt 
Predictor T PConstant 2.32 0.039GDPGrowth 3.43 0.005 FER 1.01 0.332FDIGrowth 1.55 0.146Differential Int Rate 0.11 0.913Money Supply 0.89 0.393Terms of Trade 0.35 0.731External Debt 0.73 0.479 
Where,

T is t stat. Tstat is a measure of the relative strength of prediction (is more reliable than the regression coefficient because it takes into account error). 
The pvalue is a percentage. It tells you how likely it is that the coefficient for that independent variable emerged by chance and does not describe a real relationship.
A pvalue of .05 means that there is a 5% chance that the relationship emerged randomly and a 95% chance that...
...Multipleregression: OLS method
(Mostly from Maddala)
The Ordinary Least Squares method of estimation can easily be extended to models involving two or more explanatory variables, though the algebra becomes progressively more complex. In fact, when dealing with the general regression problem with a large number of variables, we use matrix algebra, but that is beyond the scope of this course.
We illustrate the case of two explanatory variables, X1 and X2, with Y the dependant variable. We therefore have a model
Yi = α + 1X1i + 2X2i + ui
Where ui~N(0,σ2).
We look for estimators so as to minimise the sum of squared errors,
S =
Differentiating, and setting the partial differentials to zero we get
=0 (1)
=0 (2)
=0 (3)
These three equations are called the “normal equations”. They can be simplified as follows: Equation (1) can be written as
or
(4)
Where the bar over Y, X1 and X2 indicates sample mean. Equation (3) can be written as
Substituting in the value of from (4), we get
(5)
A similar equation results from (3) and (4). We can simplify this equation using the following notation. Let us define:
Equation (5) can then be written
S1Y = (6)
Similarly, equation (3) becomes
S2Y = (7)
We can solve these two equations to get:
and
Where =S11S22 – S122. We may therefore obtain from equation (4).
We can...
...Introduction
Team D will examine positive relationship of wages with multiple variables. The question is, are wages dependent on the gender, occupation, industry, years of education, race, years of work experience, marital status, and union membership. We will use the technique of linear regression and correlation. Regressionanalysis in this case should predict the value of the dependent variable (annual wages), using independent variables (gender, occupation, industry, years of education, race, and years of work experience, marital status, and union membership).
RegressionAnalysis
Based on our initial findings from MegaStat, we built the following model for regression (coefficient factors are rounded to the nearest hundredth):
Wages (Y) = 12,212.98 + 167.51(Industry) + 71.13(Occupation) + 3,085.27(Years of Education) – 6,172.13(NonWhite) + 1,857.06(Hispanic) – 11,822.96(Gender) + 356.27(Years of Experience) + 4,589.46(Marital Status) – 4,018.87(Union Member)
Global Test:
Ho: All regression coefficients for the variables in the population are zero
H1: Not all regression coefficients are zero
Significance level = 0.05
Decision rule: Reject Ho if pvalue < 0.05
The pvalue generated by the regressionanalysis is nonzero (4.42x107), therefore we reject Ho and conclude that regression...
...Multipleregression, a timehonored technique going back to
Pearson's 1908 use of it, is employed to account for (predict)
the variance in an interval dependent, based on linear
combinations of interval, dichotomous, or dummy independent
variables. Multipleregression can establish that a set of
independent variables explains a proportion of the variance in a
dependent variable at a significant level (through a significance
test of R2), and can establish the relative predictive importance
of the independent variables (by comparing beta weights).
Power terms can be added as independent variables to explore
curvilinear effects. Crossproduct terms can be added as
independent variables to explore interaction effects. One can
test the significance of difference of two R2's to determine if
adding an independent variable to the model helps significantly.
Using hierarchical regression, one can see how most variance in
the dependent can be explained by one or a set of new
independent variables, over and above that explained by an
earlier set. Of course, the estimates (b coefficients and constant)
can be used to construct a prediction equation and generate
predicted scores on a variable for further analysis.
The multipleregression equation takes the form y = b1x1 + b2x2
+ ... + bnxn + c. The b's are the regression coefficients,...
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