Submitted By marco100

Words 1464

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Words 1464

Pages 6

Allin Cottrell∗

1

The simple linear model

Suppose we reckon that some variable of interest, y, is ‘driven by’ some other variable x. We then call y the dependent variable and x the independent variable. In addition, suppose that the relationship between y and x is basically linear, but is inexact: besides its determination by x, y has a random component, u, which we call the ‘disturbance’ or ‘error’. Let i index the observations on the data pairs (x, y). The simple linear model formalizes the ideas just stated: yi = β0 + β1 xi + ui The parameters β0 and β1 represent the y-intercept and the slope of the relationship, respectively. In order to work with this model we need to make some assumptions about the behavior of the error term. For now we’ll assume three things: E(ui ) = 0 2 2 E(ui ) = σu E(ui u j ) = 0, i = j u has a mean of zero for all i it has the same variance for all i no correlation across observations

We’ll see later how to check whether these assumptions are met, and also what resources we have for dealing with a situation where they’re not met. We have just made a bunch of assumptions about what is ‘really going on’ between y and x, but we’d like to put numbers on the parameters βo and β1 . Well, suppose we’re able to gather a sample of data on x and y. The task ˆ of estimation is then to come up with coefﬁcients—numbers that we can calculate from the data, call them β0 and ˆ1 —which serve as estimates of the unknown parameters. β If we can do this somehow, the estimated equation will have the form ˆ ˆ yi = β0 + β1 x. ˆ We deﬁne the estimated error or residual associated with each pair of data values as the actual yi value minus the prediction based on xi along with the estimated coefﬁcients ˆ ˆ ui = yi − yi = yi − β0 + β1 xi ˆ ˆ In a scatter diagram of y against x, this is the vertical distance between observed…...

...Introduction Regression analysis was developed by Francis Galton in 1886 to determine the weight of mother/daughter sweet peas. Regression analysis is a parametric test used for the inference from a sample to a population. The goal of regression analysis is to investigate how effective one or more variables are in predicting the value of a dependent variable. In the following we conduct three simple regression analyses. Benefits and Intrinsic Job Satisfaction Regression output from Excel SUMMARY OUTPUT Regression Statistics Multiple R 0.616038 R Square 0.379503 Adjusted R Square 0.371338 Standard Error 0.773609 Observations 78 ANOVA df SS MS F Significance F Regression 1 27.81836 27.81836 46.48237 1.93E-09 Residual 76 45.48382 0.598471 Total 77 73.30218 Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Lower 95.0% Upper 95.0% Intercept 2.897327 0.310671 9.326021 3.18E-14 2.278571 3.516082 2.278571 3.516082 X Variable 1 0.42507 0.062347 6.817798 1.93E-09 0.300895 0.549245 0.300895 0.549245 Graph Benefits and Extrinsic Job Satisfaction Regression output from Excel SUMMARY OUTPUT Regression Statistics Multiple R 0.516369 R Square 0.266637 Adjusted R Square 0.256987 Standard Error 0.35314 Observations 78 ANOVA ...

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...Introduction to Multiple Regression Dale E. Berger Claremont Graduate University http://wise.cgu.edu Overview Multiple regression is a flexible method of data analysis that may be appropriate whenever a quantitative variable (the dependent or criterion variable) is to be examined in relationship to any other factors (expressed as independent or predictor variables). Relationships may be nonlinear, independent variables may be quantitative or qualitative, and one can examine the effects of a single variable or multiple variables with or without the effects of other variables taken into account (Cohen, Cohen, West, & Aiken, 2003). Multiple Regression Models and Significance Tests Many practical questions involve the relationship between a dependent or criterion variable of interest (call it Y) and a set of k independent variables or potential predictor variables (call them X1, X2, X3,..., Xk), where the scores on all variables are measured for N cases. For example, you might be interested in predicting performance on a job (Y) using information on years of experience (X1), performance in a training program (X2), and performance on an aptitude test (X3). A multiple regression equation for predicting Y can be expressed a follows: (1) [pic] To apply the equation, each Xj score for an individual case is multiplied by the corresponding Bj value, the products are added together, and the constant A is added to......

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...Regression Models Student Name Grantham University BA/520 – Quantitative Analysis Instructor Name April 6, 2013 Abstract This paper will refer to regression models and the benefits that variables provide when developing and examining such models. Also, it will discuss the reason why scatter diagrams are used and will describe the simple linear regression model and will refer to multiple regression analysis as well as the potential uses for this type of model. Regression Models Regression models are a statistical measure that attempts to determine the strength of the relationship between one dependent variable (usually denoted by Y) and a series of other changing variables (known as independent variables). Regression models provide the scientist with a powerful tool, allowing predictions about past, present, or future events to be made with information about past or present events. Inference based on such models is known as regression analysis. The main purpose of regression analysis is to predict the value of a dependent or response variable based on values of the independent or explanatory variables. According to Render, Stair, and Hanna (2011) they are two reasons for which regression analyses are used: one is to understand the relation between various variables and the second is to predict the variable's value based on the value of the other. Variables provide many advantages when creating models. One of the......

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...Intercontinental University Unit 5 Individual Project BUSN311-1301B-10: Quantitative Methods and Analysis Instructor Leonidas Murembya April 23, 2013, Abstract This paper will be discussing regression analysis using AIU’s survey responses from the AIU data set in order to complete a regression analysis for benefits & intrinsic, benefits & extrinsic and benefit and overall job satisfaction. Plus giving an overview of these regressions along with what it would mean to a manager (AIU Online). Introduction Regression analysis can help us predict how the needs of a company are changing and where the greatest need will be. That allows companies to hire employees they need before they are needed so they are not caught in a lurch. Our regression analysis looks at comparing two factors only, an independent variable and dependent variable (Murembya, 2013). Benefits and Intrinsic Job Satisfaction Regression output from Excel SUMMARY OUTPUT Regression Statistics Multiple R 0.018314784 R Square 0.000335431 The portion of the relations explained Adjusted R Square -0.009865228 by the line 0.00033% of relation is Standard Error 1.197079687 Linear. Observations 100 ANOVA df SS MS F Significance F Regression 1 0.04712176 0.047122 0.032883 0.856477174 Residual 98 140.4339782 1.433 Total 99 140.4811 Coefficients Standard Error t Stat P-value Lower......

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...Q-2b: Find the equation of the least squares regression line that models the relationship between square footage and rental amount and interpret the meaning of the coefficients. Q-2c: Predict the monthly rent for an apartment with 1000 square feet. Q-2d: Explain why it would not be appropriate to use the model to predict the monthly rent for apartments that have 500 square feet. Q-2e: You are considering signing a lease for an apartment in this residential neighborhood. You are deciding between two apartments, one with 1,000 square feet that rents for $1,275 and the other with 1,200 square fee that rents for $1,425. Which is a better deal? Explain. Page 1 2 Page 2 2 Q-2a Rent vs. Size 2500 2000 Rent in $ 1500 1000 500 0 0 500 1000 1500 2000 2500 Size in sq. feet y = 1.0651x + 177.12 R² = 0.7226 Q-2b Regression equation = (slope) (x) + (y-intercept) = (1.0651) (x) + (177.12) The slope is 1.0651. This means the rent increases by $1.07 for every sq. foot increase in size Q-2c Regression equation = (slope) (x) + (y-intercept) = (1.0651) (1000) + (177.12) = $1,242.22 According to the data provided, the monthly rent for a 1,000 square feet is $1,242.22 Q-2d It would not be appropriate to use the model to predict the monthly rent for apartments that have 500 square feet because the smallest apartment from the data provided is 700 square feet; 200 square feet is a significant difference . Q-2d According to data Regression equation Apartment # 1 when x =......

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...Q1: All the regressions were performed. Output can be made available if needed. See outputs for Q2 in appendix. Q2: Select the model you are going to keep for each brand and explain WHY. Report the corresponding output in an appendix attached to your report (hence, 1 output per brand) We use Adjusted R Squared to compare the Linear or Semilog Regression. R^2 is a statistic that will give some information about the goodness of fit of a model. In regression, the Adjusted R^2 coefficient of determination is a statistical measure of how well the regression line approximates the real data points. An R2 of 1 indicates that the regression line perfectly fits the data. Brand1: Linear Regression R^2 | 0.594 | SemiLog Regression R^2 | 0.563 | We use the Linear Regression Model since R-squared is higher. Brand 2: Linear Regression R^2 | 0.758 | SemiLog Regression R^2 | 0.588 | We use the Linear Regression Model since R-squared is higher Brand 3: Linear Regression R^2 | 0.352 | SemiLog Regression R^2 | 0.571 | We use the Semilog Regression Model since R-squared is higher Brand 4: Linear Regression R^2 | 0.864 | SemiLog Regression R^2 | 0.603 | We use the Linear Regression Model since R-squared is higher Q3: Here we compute the cross-price elasticity. Depending on whether we use linear or semi-log model, Linear Model Linear Model Semi-Log Model Semi-Log Model ` ...

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...relationships between the variables. The relationships can either be negative or positive. This is told by whether the graph increases or decreases. Benefits and Intrinsic Job Satisfaction Regression output from Excel SUMMARY OUTPUT Regression Statistics Multiple R 0.069642247 R Square 0.004850043 Adjusted R Square -0.00471871 Standard Error 0.893876875 Observations 106 ANOVA df SS MS F Significance F Regression 1 0.404991362 0.404991 0.50686 0.478094147 Residual 104 83.09765015 0.799016 Total 105 83.50264151 Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Lower 95.0% Upper 95.0% Intercept 5.506191723 0.363736853 15.13784 4.8E-28 4.784887893 6.2274956 4.7848879 6.22749555 Benefits -0.05716561 0.080295211 -0.711943 0.47809 -0.21639402 0.1020628 -0.216394 0.10206281 Y=5.5062+-0.0572x Graph Benefits and Extrinsic Job Satisfaction Regression output from Excel SUMMARY OUTPUT Regression Statistics Multiple R 0.161906 R Square 0.026214 Adjusted R Square 0.01685 Standard Error 1.001305 Observations 106 ANOVA df SS MS F Significance F Regression 1 2.806919 2.806919 2.799606 0.097293 Residual 104 104.2717 1.002612 Total 105 107.0786 Coefficients Standard Error t Stat P-value Lower 95% Upper......

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...Linear Regression deals with the numerical measures to express the relationship between two variables. Relationships between variables can either be strong or weak or even direct or inverse. A few examples may be the amount McDonald’s spends on advertising per month and the amount of total sales in a month. Additionally the amount of study time one puts toward this statistics in comparison to the grades they receive may be analyzed using the regression method. The formal definition of Regression Analysis is the equation that allows one to estimate the value of one variable based on the value of another. Key objectives in performing a regression analysis include estimating the dependent variable Y based on a selected value of the independent variable X. To explain, Nike could possibly measurer how much they spend on celebrity endorsements and the affect it has on sales in a month. When measuring, the amount spent celebrity endorsements would be the independent X variable. Without the X variable, Y would be impossible to estimate. The general from of the regression equation is Y hat "=a + bX" where Y hat is the estimated value of the estimated value of the Y variable for a selected X value. a represents the Y-Intercept, therefore, it is the estimated value of Y when X=0. Furthermore, b is the slope of the line or the average change in Y hat for each change of one unit in the independent variable X. Finally, X is any value of the independent variable that is......

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...STATISTICS FOR ENGINEERS (EQT 373) TUTORIAL CHAPTER 3 – INTRODUCTORY LINEAR REGRESSION 1) Given 5 observations for two variables, x and y. | 3 | 12 | 6 | 20 | 14 | | 55 | 40 | 55 | 10 | 15 | a. Develop a scatter diagram for these data. b. What does the scatter diagram developed in part (a) indicate about the relationship between the two variables? c. Develop the estimated regression equation by computing the values and. d. Use the estimated regression equation to predict the value of y when x=10. e. Compute the coefficient of determination. Comment on the goodness of fit. f. Compute the sample correlation coefficient (r) and explain the result. 2) The Tenaga Elektik MN Company is studying the relationship between kilowatt-hours (thousands) used and the number of room in a private single-family residence. A random sample of 10 homes yielded the following. Number of rooms | Kilowatt-Hours (thousands) | 12 9 14 6 10 8 10 10 5 7 | 9 7 10 5 8 6 8 10 4 7 | a. Identify the independent and dependent variable. b. Compute the coefficient of correlation and explain. c. Compute the coefficient of determination and explain. d. Test whether there is a positive correlation between both variables. Use α=0.05. e. Determine the regression equation (used Least Square method) f. Determine the value of kilowatt-hours used if number of rooms is 11. g. Can you use the model in (f.) to predict the kilowatt-hours if number of......

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...MULTIPLE REGRESSION After completing this chapter, you should be able to: understand model building using multiple regression analysis apply multiple regression analysis to business decision-making situations analyze and interpret the computer output for a multiple regression model test the significance of the independent variables in a multiple regression model use variable transformations to model nonlinear relationships recognize potential problems in multiple regression analysis and take the steps to correct the problems. incorporate qualitative variables into the regression model by using dummy variables. Multiple Regression 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… / Correlation Can check for statistical significance of correlation with a t test Example A distributor of frozen desert pies wants to evaluate factors thought to influence demand Dependent variable: Pie sales (units per......

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...Acts 430 Regression Analysis In this project, we are required to forecast number of houses sold in the United States by creating a regression analysis using the SAS program. We initially find out the dependent variable which known as HSN1F. 30-yr conventional Mortgage rate, real import of good and money stock, these three different kinds of data we considered as independent variables, which can be seen as the factors will impact the market of house sold in USA. Intuitively, we thought 30-yr conventional mortgage rate is a significant factor that will influences our behavior in house sold market, which has a negative relation with number of house sold. When mortgage rate increases, which means people are paying relatively more to buy a house, which will leads to a decrease tendency in house sold market. By contrast, a lower interest rate would impulse the market. We believe that real import good and service is another factor that will causes up and down in house sold market. When a large amount of goods and services imported by a country, that means we give out a lot of money to other country. In other words, people have less money, the sales of houses decreased. Otherwise, less import of goods and services indicates an increase tendency in house sold market. We can see it also has a negative relationship with the number of house sold. Lastly, we have money stock as our third impact factor of house sold. We considered it has a positive relationship with the number of...

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...Regression analysis is a statistical process for estimating the relationships among variables. It includes many techniques for modeling and analyzing several variables, when the focus is on the relationship between a dependent variable and one or more independent variables. More specifically, regression analysis helps one understand how the typical value of the dependent variable (or 'criterion variable') changes when any one of the independent variables is varied, while the other independent variables are held fixed. Local Government Engineering Department (LGED) is a public sector organization under the ministry of Local Government, Rural Development & Cooperatives. The prime mandate of LGED is to plan, develop and maintain local level rural, urban and small scale water resources infrastructure throughout the country. Here, I considered LGED as the organization and considering a projects eight districts “available fund” as Independent variable and “development (length of development of road in km)” as dependent variable. The value of the variables are- Districts Fund, X (lakh tk) Development,Y (km) Panchagar 450 10 Thakurgaon 310 6.8 Dinajpur 1500 32 Nilphamari 1160 24.5 Rangpur 1450 31 Kurigram 450 9 Lalmonirhat 950 16 Gaibandha 1550 33 For the two variables “available fund” and “development”, the regression equation can be given as: Y= a + bX Where, Y = Development X = Fund b = rate of change of development a...

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...Introduction Simple linear regression is a model with a single regressor x that has a relationship with a response y that is a straight line. This simple linear regression model is y = β0 + β1x + ε where the intercept β0 and the slope β1 are unknown constants and ε is a random error component. Testing Significance of Regression: H0: β1 = 0, H1 : β1 ≠ 0 The hypotheses relate to the significance of regression. Failing to reject H0: β1 = 0 implies that there is no linear relationship between x and y. On the other hand, if H0: β1 = 0 is rejected, it implies that x is of value in explaining the variability in y. The following equation is the Fundamental analysis-of-variance identity for a regression model. SST = SSR + SSRes Analysis of variance (ANOVA) is a collection of statistical models used in order to analyze the differences between group means and their associated procedures (such as "variation" among and between groups), developed by R. A. Fisher. In the ANOVA setting, the observed variance in a particular variable is partitioned into components attributable to different sources of variation. P value or calculated probability is the estimated probability of rejecting the null hypothesis (H0) of a study question when that hypothesis is true. VIF (the variance inflation factor) for each term in the model measures the combined effect of the dependences among the regressors on the variance of the term. Practical experience indicates that if any of...

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...A) Estimated regression equation – First Order: y = β0 + β1x1 + β2x2 + ε Output of 1st Model | | | | | | | | | | | | | | Regression Statistics | | | | | | Multiple R | 0.763064634 | | | | | | R Square | 0.582267636 | SSR/SST | | ̂̂̂ | | | Adjusted R Square | 0.512645575 | | | | | | Standard Error | 547.737482 | | | | | | Observations | 15 | | | | | | | | | | | | | ANOVA | | | | | | | | df | SS | MS | F | Significance F | | Regression | 2 | 5018231.543 | 2509115.772 | 8.363263464 | 0.005313599 | | Residual | 12 | 3600196.19 | 300016.3492 | | | | Total | 14 | 8618427.733 | | | | | | | | | | | | | Coefficients | Standard Error | t Stat | P-value | Lower 95% | Upper 95% | Intercept | -20.35201243 | 652.7453202 | -0.031179101 | 0.975639286 | -1442.561891 | 1401.857866 | Age (x1) | 13.35044655 | 7.671676501 | 1.740225432 | 0.107375657 | -3.364700634 | 30.06559374 | Hours (x2) | 243.7144645 | 63.51173661 | 3.837313819 | 0.002363965 | 105.334278 | 382.0946511 | B) equation | ŷ= -20.3520124320994 + 13.3504465516772 x̂1 + 243.714464532425 x̂2 | C) Interpretation of β β̂1 = 13.35044655, If number of hours worked (x2) held fixed, we can estimate that every one-year increase in age (x1) the mean of annual earnings will increase by 13.35044655. β̂2 = 243.7144645, If age (X1) held fixed, we can estimate that every one hour (x2) of work increase, the mean of......

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...Regression Paper Team RES/342 Research and Evaluation Teacher Date The Hypothesis Team C’s hypothesis is that the more years of education one receives the more a person can potentially earn in salary. The team will use the process of linear regression analysis to explain how the information is used and conduct a five-step test to see if the hypothesis proves true or false. Linear Regression Analysis Team C’s purpose of this research paper is to use a linear regression analysis test to determine if a significant linear relationship exists between an independent variable which is X, level or years of education, and a dependent variable Y, salaries earned or potentially earned. “It is used to determine the extent to which there is a linear relationship between a dependent variable and one or more independent variables,” (Statistically Significant Consulting, 2010, para. 1). Learning Team C will use the salary and education levels from the Wages and Wage Earners Data Set collected through access to the e-source link of University of Phoenix. For this test the dependent variable, Y, will represent the salary of the 100 participants and the independent variable, X, will represent the education of the 100 participants. How the Information is used This information will be used in a linear regression test to see if there is enough evidence to reject the null hypothesis that a higher education does not......

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