Lecture 14 Summary of previous Lecture Regression through the origin Scale and measurement units.

14
Lecture 14 Summary of previous Lecture Regression through the origin Scale and measurement units

Transcript of Lecture 14 Summary of previous Lecture Regression through the origin Scale and measurement units.

Page 1: Lecture 14 Summary of previous Lecture Regression through the origin Scale and measurement units.

Lecture 14

Summary of previous Lecture

Regression through the origin

Scale and measurement units

Page 2: Lecture 14 Summary of previous Lecture Regression through the origin Scale and measurement units.

Food for today

• Special features of scaling and units

• Regression on standardized variables

• Functional form

Page 3: Lecture 14 Summary of previous Lecture Regression through the origin Scale and measurement units.

Special features of scaling and units

1- if w1 = w2, that is, the scaling factors are identical, the slope coefficient and

its standard error remain unaffected in going from the (Yi , Xi) to the (Y*i , X*i )

scale.

2- The intercept and its standard error are both multiplied by w1.

3- If the X scale is not changed (i.e., w2 = 1) and the Y scale is changed by the

factor w1, the slope as well as the intercept coefficients and their respective

standard errors are all multiplied by the same w1 factor.

4- Finally, if the Y scale remains unchanged (i.e., w1 = 1) but the X scale is

changed by the factor w2, the slope coefficient and its standard error are

multiplied by the factor (1/w2) but the intercept coefficient and its standard error

remain unaffected.

Note: transformation from the (Y, X) to the (Y*, X*) scale does not affect the

properties of the OLS estimators.

Page 4: Lecture 14 Summary of previous Lecture Regression through the origin Scale and measurement units.

Example

• value is invariant to changes in the unit of measurement, as

it is a pure, or dimensionless, number.

Page 5: Lecture 14 Summary of previous Lecture Regression through the origin Scale and measurement units.
Page 6: Lecture 14 Summary of previous Lecture Regression through the origin Scale and measurement units.

A Word about Interpretation

• Since the slope coefficient β2 is simply the rate of change, it is measured in

the units of the ratio: Units of the dependent variable/ Units of the

explanatory variable

• The interpretation of the slope coefficient 0.3016 is that if GDP changes by

a unit, which is 1 billion dollars, GPDI on the average changes by 0.3016

billion dollars.

• In regression a unit change in GDP, which is 1 million dollars, leads on

average to a 0.000302 billion dollar change in GPDI.

• The two results are of course identical in the effects of

GDP on GPDI; they are simply expressed in different units of measurement.

Page 7: Lecture 14 Summary of previous Lecture Regression through the origin Scale and measurement units.

Regression on standardized variables Units in which the regressand and regressor(s) are expressed affect the

interpretation of the regression coefficient.

Standardized variables help in avoiding this situation

A variable is said to be standardized if we subtract the mean value of the

variable from its individual values and divide the difference by the

standard deviation of that variable.

Page 8: Lecture 14 Summary of previous Lecture Regression through the origin Scale and measurement units.

Properties and interpretation of standardized variables

Properties:

1. zero mean

2. Standard deviation is one

The regression coefficient of standardized variables are called Beta

Coefficients in literature.

Interpretation of Beta Coefficients in standard form:

If the independent variable changes by one standard deviation, on

average, the dependent variable changes by β*2 standard deviation

units.

Unlike the traditional model the effect is measured not in the original

units but of X and Y but in standardization unit.

Page 9: Lecture 14 Summary of previous Lecture Regression through the origin Scale and measurement units.
Page 10: Lecture 14 Summary of previous Lecture Regression through the origin Scale and measurement units.

Advantages of standardized variables

More useful in multiple regression analysis.

By standardizing all regressors, we put them on equal basis

and therefore can compare them directly.

For example: If the coefficient of a standardized regressor is

larger than that of another standardized regressor appearing in

that model, then the latter contributes more relatively to the

explanation of the regressand than the latter.

In other words, we can use the beta coefficients as a measure

of relative strength of the various regressors

Page 11: Lecture 14 Summary of previous Lecture Regression through the origin Scale and measurement units.

More about regression on standardized variables

• There is an interesting relationship between the β coefficients

of the conventional model and the beta coefficients.

• Example:

Page 12: Lecture 14 Summary of previous Lecture Regression through the origin Scale and measurement units.

FUNCTIONAL FORMS OF REGRESSION MODELS

• We assumed that models should be linear in parameter.

• It may take different forms and with the change in form the interpretation

of the parameter changes.

• 1. The log-linear model

• 2. Semi-log models

• 3. Reciprocal models

• 4. The logarithmic reciprocal model

Page 13: Lecture 14 Summary of previous Lecture Regression through the origin Scale and measurement units.

Choice of functional form

The choice of a particular functional form is easy in two variable case. Just

plot the variable and guess.

The choice becomes much harder when we consider the multiple regression

model.

Great deal of skill and experience are required in choosing an appropriate

model for empirical estimation.

2- Find slope and elasticities of different models and compare. Here are

formulas

Page 14: Lecture 14 Summary of previous Lecture Regression through the origin Scale and measurement units.

Some guide lines can help in the choice of model.

1-The underlying theory may suggest a particular functional form.

2- The coefficients of the model chosen should satisfy priori expectations. For

example in demand analysis the slope coefficient should be negative.

4- Sometime more than one model may fit a given set of data reasonably well.

In both cases the coefficients may be significant.

However, go for the coefficient of determination.

But make sure that in comparing two r2 values the dependent variable, or

the regressand, of the two models is the same; the regressor(s) can take

any form.