Introduction And Abstract Of Research Study 11.1

Front 1

Define a Regression Model

Back 1

A regression model provides the user with a functional relationship between the response variable and explanatory variables that allows the user to determine which of the ex- planatory variables have an effect on the response. The regression model allows the user to explore what happens to the response variable for specified changes in the explanatory variables.

Front 2

reference to future values

Back 2

prediction

Front 3

reference to current or past values

Back 3

explanation

Front 4

In order for prediction to work there must be a _______ between x and y.

Back 4

For prediction (or explanation) to make much sense, there must be some connection between the variable we’re predicting (the dependent variable) and the variable we’re using to make the prediction (the independent variable).

Front 5

In order for prediction to work there must be a _____.
i.e there should be an entity that relates the two variables.

Back 5

Unit of Association

Front 6

With time-series data, the unit of association may simply be ______.

Back 6

time

Hint 6

UOA

Front 7

For _______, an economic or physical entity should connect the variables.

Back 7

cross-sectional data

Hint 7

Type of Data

Front 8

The variables may be measured at the same time period or, for genuine prediction, the independent variable may be measured at a time period _____ the dependent variable.

Back 8

before

Front 9

simple linear regression analysis, in which there is a __________ and the equation for predicting a dependent variable y is a linear function of a given independent variable x.

Back 9

single independent variable

Front 10

yˆ=2.0+3.0x
The constant term, such as the 2.0, is the _____ and is interpreted as the predicted value of y when x =0.

Back 10

intercept term

Front 11

yˆ=2.0+3.0x
The coefficient of x, such as the 3.0, is the _____ of the line, the predicted change in y when there is a one-unit change in x.

Back 11

slope

Front 12

what is the prediction equation ?

Back 12

yˆ= b0ˆ+b1ˆx
where b0ˆ is the intercept and b1ˆ is the slope

Front 13

The basic idea of simple linear regression is to use data to fit a prediction line
that relates a dependent variable __ and a single independent variable ___.

Back 13

y , x

Front 14

According to the assumption of linearity, the _____ of the equation does not change as x changes.

Back 14

slope

Front 15

corresponding to each y we introduce a random error term ei and assume the model
y=b0 +b1x+e. Why?

Back 15

Assuming linearity, we would like to write y as a linear function of x: y=b0+ b1x.
However, according to such an equation, y is an exact linear function of x; no room is left for the inevitable errors (deviation of actual y values from their predicted values). So we introduce εi

Front 16

We assume the random variable y to be made up of a predictable part (a linear function of x) and an unpredictable part, the ________.

Back 16

random error term. εi
y=b0 +b1x+ε

Front 17

The coefficients __ and ___ are interpreted as the true, underlying intercept and slope.

Back 17

b0 and b1

Front 18

The error term ε includes the effects of all other factors, ___ or ____.

Back 18

known or unknown.
factors that might have been used in prediction but were not
The combined effects of unpredictable and ignored factors yield the random error terms ε.

Front 19

In regression studies, the values of the independent variable (the xi values) are usually taken as predetermined _______, so the only source of randomness is the ei terms

Back 19

constants

Front 20

If the data are based on a random sample of applicants, xi (as well as yi) is a ________ variable.

Back 20

random

Front 21

If the xis are random, we can simply regard all probability statements as conditional on the _______ xis.

Back 21

observed

Front 22

Formal assumptions of regression analysis are? there are 4.

Back 22

1. The relation is, in fact, linear, so that the errors all have expected value zero: E(εi) = 0 for all i.

2. The errors all have the same variance: Var(εi) = σ^2 for all i.

3. The errors are independent of each other.

4. The errors are all normally distributed; εi is normally distributed for all i.

Front 23

The actual values of the dependent variable are distributed ______, with _______ values falling on the regression line and the same ______ at all values of the independent variable.

Back 23

normally, mean, standard deviation

Front 24

This is simply a plot of each (x, y) point, with the independent variable value on the horizontal axis, and the dependent variable value measured on the vertical axis.

Back 24

scatterplot of the data

Front 25

scatterplot: Look to see whether the points basically fall around a straight ______ or whether there is a definite ______ in the pattern. Also look to see whether there are any evident _______ falling far from the general pattern of the data.

Back 25

line, curve, outliers

Front 26

_______ have been developed to sketch a curve through data without necessarily assuming any particular model.

Back 26

smoothers

Front 27

If such a smoother yields something close to a straight line, then ___________ is reasonable

Back 27

linear regression

Front 28

Roughly, a smoother takes a relatively narrow “slice” of data along the x axis, calculates a line that fits the data in that slice, moves the slice slightly along the x axis, recalculates the line, and so on. Then all the little lines are connected in a smooth curve.

Back 28

LOWESS (locally weighted scatterplot smoother).

Front 29

LOWESS: The width of the slice is called the _______.

Back 29

bandwidth

Front 30

It can be understood as taking a narrow slice of data, fitting a curve (often a cubic equation) to the slice, moving to the next slice, fitting another curve, and so on. The curves are calculated in such a way as to form a connected, continuous curve.

Back 30

spline fit. Another type of scatterplot smoother

Front 31

Many economic relations are not linear. For example, any diminishing returns pattern will tend to yield a relation that ________, but at a ________ rate.

Back 31

increases, decreasing

Front 32

If the scatterplot does not appear linear, by itself or when fitted with a LOWESS curve, it can often be “straightened out” by a ________ of either the inde- pendent variable or the dependent variable.

Back 32

transformation

Front 33

We can try several transformations of the independent variable to find a scatterplot in which the points more nearly fall along a straight line. Three common transformations are ?

Back 33

square root, natural logarithm, and inverse(one divided by the variable).

Front 34

Finding a good transformation often requires______ and ______.

Back 34

trial and error.

Front 35

There are two key features to look for in a scatterplot. What are they?

Back 35

First, is the relation nonlinear? Second, is there a pattern of increasing variability along the y (vertical) axis? If there is, the assumption of constant variance is questionable.

Front 36

choosing a transformation:
If the plot indicates a relation that is increasing but at a decreasing rate, and if variability around the curve is roughly constant then..

Back 36

transform x using square root, logarithm, or inverse transformations.
Symmetric where the relation is decreasing at a decreasing rate

Front 37

choosing a transformation:
If the plot indicates a relation that is increasing at an increasing rate, and if variability is roughly constant then..

Back 37

try using both x and x2 as predictors. This method uses two variables, the multiple regression methods are needed.
Symmetric where the relation is
decreasing at an increasing rate

Front 38

choosing a transformation:
If the plot indicates a relation that increases to a maximum and then decreases, and if variability around the curve is roughly constant,

Back 38

try using both x and x2 as predictors.

Front 39

choosing a transformation:
If the plot indicates a relation that is increasing at a decreasing rate, and if variability around the curve increases as the predicted y value increases,

Back 39

try using y2 as the dependent variable.
Symmetric where the relation is decreasing at a decreasing rate

Front 40

choosing a transformation:
If the plot indicates a relation that is increasing at an increasing rate, and if variability around the curve increases as the predicted y value increases,

Back 40

try using ln(y) as the dependent variable. It sometimes may also be helpful to use ln(x) as the independent variable. Note that a change in a natural logarithm corresponds quite closely to a percentage change in the original variable. Thus, the slope of a transformed variable can be interpreted quite well as a percentage change.
Symmetric where the relation is
decreasing at an increasing rate

Front 41

The assumptions we made in this section allow us to make _______ about the true parameter values from the sample data.

Back 41

inferences

Front 42

The population intercept, slope, and error variance all have to be _______ from limited sample data

Back 42

estimated

Front 43

For a simple regression with a single predictor, careful checking of a _____, ideally with a smooth curve fit through it, will help avoid serious blunders.

Back 43

scatterplot

Front 44

Once we have decided on any mathematical transformations, we must estimate the actual equation of ________.

Back 44

the regression line