Regression Through the Origin. The intercept is absent or zero. It can be shown that:. Regression Through the Origin. Characteristics of this model:In calculating the parameter we use raw and cross products instead of mean-adjusted sums of squares and cross productsThe degrees of freedom are n-1 rather than n-2The formula for r2 includes an intercept. Therefore you should not use this formula, or you will obtain nonsensical results, like negative r2The sum of the residuals is always zero,9462
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1. Class Outline Functional Forms of Regression Models
Regression Through the Origin
Log Linear Model
Comparing Linear and Log Linear Models
Multiple Log-Linear Models
The Semilog Model
The Lin-Log Model
Polynomial Regression Models
Summary of Functional Forms
Reading: Chapter 6
3. Regression Through the Origin Characteristics of this model:
In calculating the parameter we use raw and cross products instead of mean-adjusted sums of squares and cross products
The degrees of freedom are n-1 rather than n-2
The formula for r2 includes an intercept. Therefore you should not use this formula, or you will obtain nonsensical results, like negative r2
The sum of the residuals is always zero, but in the case of a model with intercept this could not be the case.
4. Log-Linear Model Remember our example with the Lotto regression.
Assume that our expenditure function is as follows:
Where Y is the expenditure on Lotto and X is personal disposable income. The model is nonlinear in the variable X
5. Log-Linear Model We can transform the equation in logarithms:
For estimation purposes we can write this model as
6. Log-Linear Model This is a linear regression model for the parameters ?1 and ?2.
Observe that the slope coefficient ?2 measures the elasticity of Y with respect to X, that is, the percentage change in Y for a given percentage change in X.
7. Log-Linear Model We can define the elasticity as:
8. Log-Linear Model
9. Comparing Linear and Log-Linear Models Assume we run a linear model and a log linear model for the same dataset, which one to choose?
Plot the data and see if you can determine the functional form
We cannot compare r2 of log linear and linear models. By definition in the linear model r2 measures the proportion of the variation in Y explained X, while in the log linear model r2 explains the proportion of the variation of lnY explained by lnX. These measures are different
10. Comparing Linear and Log-Linear Models The variation in Y is a absolute change, while the variation in log of Y measures the relative or proportional change.
Even if the dependent variables of two models is the same, we should not choose our models based on the highest r2 criterion. This is because this measure changes with the addition of variables.
11. Multiple Log-Linear Regression Models Example: The Cobb Douglas Production Function
12. How to Measure the Growth Rate: The Semilog Model When we are interested on the growth rate of some economic variables we can use this model.
Example: we want to measure the growth rate of population over the period 1970-1999
Y0: beginning value of Y
Yt: Y’s value at time t
r: the compound rate of growth over time
13. How to Measure the Growth Rate: The Semilog Model Let’s transform this equation as follows:
14. The Lin-Log Model: When the Explanatory Variable is Logarithmic In this case, the independent variable, X, is expressed in logarithm
Example: we want to find how expenditure on services (Y) behaves if total personal consumption expenditure (X) increases
15. The Lin-Log Model: When the Explanatory Variable is Logarithmic
This equation states that the absolute change in Y is equal to ?2 times the relative change in X.
16. Reciprocal Models
These models are used when the functional form have some asymptotic characteristics
17. Reciprocal Models
18. Polynomial Regression Models
We can estimate this model with OLS. The only problem is the presence of multicollinearity, but usually this problem should not be too important in this case because the explanatory variables are not linear functions of X.
19. Polynomial Regression Models
20. Summary of Functional Forms