Multiple regression, 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. Multiple regression 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 multiple regression equation takes the form y = b1x1 + b2x2 + ... + bnxn + c. The b's are the regression coefficients,
representing the amount the dependent variable y changes when the corresponding independent changes 1 unit. The c is the
constant, where the regression line intercepts the y axis,
representing the amount the dependent y will be when all the independent variables are 0. The standardized version of the b coefficients are the beta weights, and the ratio of the beta coefficients is the ratio of the relative predictive power of the independent variables. Associated with multiple regression is R2, multiple correlation, which is the percent of variance in the dependent variable explained collectively
...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...
...0905section2.QX5
7/12/04
4:10 PM
Page 140
13 MultipleregressionMultipleregression
In this chapter I will briefly outline how to use SPSS for Windows to run multipleregression analyses. This is a very simplified outline. It is important that you do
more reading on multipleregression before using it in your own research. A good
reference is Chapter 5 in Tabachanick and Fiddell (2001), which covers the
underlying theory, the different types of multipleregression analyses and the
assumptions that you need to check.
Multipleregression is not just one technique but a family of techniques that
can be used to explore the relationship between one continuous dependent variable
and a number of independent variables or predictors (usually continuous). Multipleregression is based on correlation (covered in Chapter 11), but allows a more
sophisticated exploration of the interrelationship among a set of variables. This
makes it ideal for the investigation of more complex reallife, rather than
laboratorybased, research questions. However, you cannot just throw variables
into a multipleregression and hope that, magically, answers will appear. You
should have a sound theoretical or conceptual reason for the analysis and, in
particular, the order of...
...MultipleRegression Analysis of exchange rate with the determinant factors
Regression Analysis: 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 the relationship is real. ...
...
Logistic regression
In statistics, logistic regression, or logit regression, is a type of probabilistic statistical classification model.[1] It is also used to predict a binary response from a binary predictor, used for predicting the outcome of acategorical dependent variable (i.e., a class label) based on one or more predictor variables (features). That is, it is used in estimating the parameters of a qualitative response model. The probabilities describing the possible outcomes of a single trial are modeled, as a function of the explanatory (predictor) variables, using a logistic function. Frequently (and subsequently in this article) "logistic regression" is used to refer specifically to the problem in which the dependent variable is binary—that is, the number of available categories is two—while problems with more than two categories are referred to as multinomial logistic regression or, if the multiple categories are ordered, as ordered logistic regression.
Logistic regression measures the relationship between a categorical dependent variable and one or more independent variables, which are usually (but not necessarily) continuous, by using probability scores as the predicted values of the dependent variable.[2] As such it treats the same set of problems as doesprobit regression using similar techniques.
Fields and examples of...
...Regression Analysis: A Complete Example
This section works out an example that includes all the topics we have discussed so far in this chapter.
A complete example of regression analysis.
PhotoDisc, Inc./Getty Images
A random sample of eight drivers insured with a company and having similar auto insurance policies was selected. The following table lists their driving experiences (in years) and monthly auto insurance premiums.
Driving Experience (years) Monthly Auto Insurance Premium
5 2 12 9 15 6 25 16
$64 87 50 71 44 56 42 60
a. Does the insurance premium depend on the driving experience or does the driving experience depend on the insurance premium? Do you expect a positive or a negative relationship between these two variables? b. Compute SSxx, SSyy, and SSxy. c. Find the least squares regression line by choosing appropriate dependent and independent variables based on your answer in part a. d. Interpret the meaning of the values of a and b calculated in part c. e. Plot the scatter diagram and the regression line. f. Calculate r and r2 and explain what they mean. g. Predict the monthly auto insurance premium for a driver with 10 years of driving experience. h. Compute the standard deviation of errors. i. Construct a 90% confidence interval for B. j. Test at the 5% significance level whether B is negative. k. Using α = .05, test whether ρ is different from zero.
Solution a. Based on theory and intuition, we...
...Topic 4. Multipleregression
Aims
• Explain the meaning of partial regression coefficient and calculate and interpret multipleregression models • Derive and interpret the multiple coefficient of determination R2and explain its relationship with the the adjusted R2 • Apply interval estimation and tests of significance to individual partial regression coefficients d d l ff • Test the significance of the whole model (Ftest)
Introduction
• The basic multipleregression model is a simple extension of the bivariate equation. • By adding extra independent variables, we are creating a multipledimensioned space, where the model fit is a some appropriate space. , p , • For instance, if there are two independent variables, we are fitting the points to a ‘plane in space’. trick. • Visualizing this in more dimensions is a good trick
Model specification – scalar version
• The basic linear model: • Yi = ß0 + ß1 X1i+ ß2X2i+ ß3X3i +….+ ßkXki +ui …. u • If bivariate regression can be described as a line on a plane, multipleregression represents a kdimensional object in a k+1 d dimensional space. l
Matrix version
• We can use a different type of mathematical g structure to describe the regression model Frequently called Matrix or Linear Algebra • The multiple...
...Chapter3
MultipleRegression Analysis: 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 for any...
...are
i. Simple correlation
ii. Multiple correlation
iii. Partial correlation
iv. Linear and nonlinear correlation
i. Simple correlation:
Correlation only between two variables is called simple correlation. For example, correlation between income and expenditure. Simple correlation is of three types, they are
a. Positive Correlation;
b. Negative correlation;
c. Zero Correlation.
a. Positive Correlation:
If the two correlated variables are varying in the same direction, i.e. if one variable is increasing the other on an average is also increasing or, if one variable is decreasing the other on an overage is also decreasing, correlation is said to be positive. For example, correlation between income and expenditure.
b. Negative Correlation:
If the two correlated variables are varying in the opposite direction, i.e. if one variable is increasing the other on an overage is also decreasing or, if one variable is decreasing the other on an overage is also increasing, correlation is said to be negative. For example, correlation between price and demand.
c. Zero correlation:
Out of two correlated variables if one variable increase or decreases then other variable remain constant, correlation is said to be zero. For example, correlation between height and income.
ii. Multiple Correlations:
Correlation between more than two variables is called...