Samenvatting voor het vak Statistics for Business and Economics op de Rijksuniversiteit Groningen. Hoofdstuk 12, 13, 15, 16, & 17.
Simple regression (see chapter 11) can predict a dependent variable as a function of a single independent variable. But often there are multiple variables at play. In order to determine the simultaneous effect of multiple independent variables on a dependent variable, multiple regression is used. The least squares principle fit the model.
This chapter focusses on topics that add to the understanding of regression analysis. This includes alternative specifications for these models, and what happens in the situations where basic regression assumptions are violated.
There are situations and experiments that require processes to be compared at more than two levels. Data from such experiments can be analysed using analysis of variance or ANOVA.
Time series data involves measurements that are ordered over time, in which the sequence of observations is important. Most procedures for data analysis cannot be used for this data, as these procedures are based on the assumption that the errors are independent. Thus, different forms of analysis are needed.
There are various ways of sampling a population, according to research and analysis goals.
Regression objectives are either to predict the value of the dependent variable, or to estimate the marginal effect of each independent variable.A population multiple regression model is a model that includes multiple independent variables.Standard multiple regression assumptions include the four standard simple regression assumptions, plus a fifth one: It is not possible to find a set off nonzero numbers such that the sum of the coefficients equals zero.Multiple regression models include an error term, ε, that represents variability caused by variables not included in the model.In multiple regression coefficients are estimated using least squares, but these estimates become less reliable the higher the correlations between independent variables are.Any regression coefficient in a multiple regression model is dependent on all independent variables, and are thus referred to as conditional coefficients.Mean square regression (MSR) shows the proportion of the variability by the dependent variable that can be explained by the regression model.In a multiple regression model the sum-of-squares (SST; or sample variability) can be split into the sum of squares regression (SSR; or explained variability) and the sum of squares error (SSE; or unexplained variability). This is referred to as sum-of-squares decomposition.The coefficient of determination, R2, describes the strength of the linear relationship between the independent variables and the dependent variables, and is calculated by 1 – SSE/SST.Adding more independent variables leads to a misleading increase in R2, which can be avoided by calculating the adjusted coefficient of determination.The coefficient variance estimator, s2b, is calculated as:The square root of s2b is the coefficient standard error.Multiple regression models can be transformed into non-linear models, namely quadratic models and logarithmic models.Dummy variables can be used to represent categorical data...
yn = -59.31 + 4.983x1 + 2.198x2 + 3.816x3 - 0.310x4 11.1562 10.2102 12.0632 10.3302-0.886x5 + 3.215x6 + 0.85x7 13.0552 11.5682 10.3542R2 = 0.766where:yn = new business starts in the industryx1 = population in millionsx2 = industry sizex3 = measure of economic quality of lifex4 = measure of political quality of lifex5 = measure of environmental quality of lifex6 = measure of health and educational quality of lifex7 = measure of social quality of lifeThe numbers in parentheses under the coefficients are the estimated coefficient standard errors.a. Interpret the estimated regression coefficients.b. Interpret the coefficient of determination.c. Find a 90% confidence interval for the increase in new business starts resulting from a one-unit increase in the economic quality of life, with all other variables unchanged.d. Test, against a two-sided alternative at the 5% level, the null hypothesis that, all else remaining equal, the environmental quality of life does not influence new business starts.e. Test, against a two-sided alternative at the 5% level, the null hypothesis that, all else remaining equal, the health and educational quality of life does not influence new business starts.f. Test the null hypothesis that, taken together, these seven independent variables do not influence new business starts. ...
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