1. Linear Regression

2. METHODOLOGY OF ECONOMETRICS
1. Statement of theory or hypothesis.
2. Specification of the mathematical model of the theory
3. Specification of the statistical, or econometric, model
4. Obtaining the data
5. Estimation of the parameters of the econometric model
6. Hypothesis testing
7. Forecasting or prediction
8. Using the model for control or policy purposes.

3. 1. Meaning of Regression

4. Meaning of Regression
Examine relationship between dependent and independent variables
Ex: how is quantity of a good related to price?
Predict the population mean of the dependent variable on the basis of known independent variables
Ex: what is the consumption level , given a certain level of income

5. Meaning of Regression Also test hypotheses:
Ex: About the precise relation between consumption and income
How much does consumption go up when income goes up.

6. 2. Regression Example

7. Example

8. Regression Example
Assume a country with a total population of 60 families.
Examine the relationship between consumption and income.
Some families will have the same income
Could split into groups of weekly income ($100, $120, $140, etc)

9. Regression Example
Within each group, have a range of family consumption patterns.
Among families with $100 income we may have six families, whose spending is 65, 70, 74, 80, 85, 88.
Define income X and spending Y.
Then within each of these categories, we have a distribution of Y, conditional upon a certain X.

10. Regression Example For each distributions, compute a conditional mean:
E(Y|(X=X i).
How do we get E(Y|(X=X i) ?
Multiply the conditional probability (1/6) by Y value and sum them
This is 77 for our example.
We can plot these conditional distributions for each income level

11. Regression Example
The population regression is the line connecting the conditional means of the dependent variable for fixed values of the explanatory variable(s).
Formally: E(Y|Xi)
This population regression function tells how the mean response of Y varies with X.

12. Population Regression Line

13. Regression Example What form does this function take?
Many possibilities, but assume its a linear function: E(Y|Xi) = Xi
1 and 2 are the regression coefficients (intercept and slope).
Slope tells us how much Y changes for a given change in X.
We estimate 1 and 2 on the basis of actual observations of Y and X.

14. 3. Linearity

15. Linearity Linearity can be in the variables or in the parameters.
Linearity in the variables
Conditional expectation of Y is a linear function of X -
The regression is a straight line
Slope is constant
Can't have a function with squares, square root, or interactive terms- these have a varying slope.

16. Linearity We are concerned with linearity in the parameters
The parameters are raised to the first power only.
It may or may not be linear in the variables.

17. Linearity Linearity in the parameters
The conditional expectation of Y is a linear function of the parameters
It may or may not be linear in Xs.
E(Y|Xi) = 1 + 2Xi is linear
E(Y|Xi) = 1 + 2Xi is not.
Linear if the betas appear with a power of one and are not multiplied or divided by other parameters.

18. 4. Stochastic Error

19. Stochastic Error Example
Model has a deterministic part and a stochastic part.
Systematic part determined by price, education, etc.
An econometric model indicates a relationship between consumption and income
Relationship is not exact, it is subject to individual variation and this variation is captured in u.

20. What Error Term Captures
Omitted variables
Other variables that affect consumption not included in model
If correctly specified our model should include these
May not know economic relationship and so omit variable.
May not have data
Chance events that occur irregularly--bad weather, strikes.

21. What Error Term Captures
Measurement error in the dependent variable
Friedman model of consumption
Permanent consumption a function of permanent income
Data on these not observable and have to use proxies such as current consumption and income.
Then the error term represents this measurement error and captures it.

22. What Error Term Captures
Randomness of human behavior
People don't act exactly the same way even in the same circumstances
So error term captures this randomness.

23. 11. Hypothesis Testing

24. Hypothesis Testing
Set up the null hypothesis that our parameter values are not significantly different from zero
H0:2 = 0
What does this mean?:
Income has no effect on spending.
So set up this null hypothesis and see if it can be rejected.

25. 12. Coefficient of Determination--R2

26. Coefficient of Determination
The coefficient of determination, R2, measures the goodness of fit of the regression line overall

27. Correlation Coefficient
The correlation coefficient is the square root of R2
Correlation coefficient measures the strength of the relationship between two variables.

28. 13. Forecasting

29. If the chosen model does not refute the hypothesis or theory under consideration, we may use it to predict the future value(s) of the dependent, or forecast, variable Y on the basis of known or expected future value(s) of the explanatory, or predictor, variable X.

31. the estimated consumption function is:
suppose we want to predict the mean consumption expenditure for The GDP value for 1997 was billion dollars.
Putting this GDP figure on the right-hand side of we obtain:

32. Forecast Error
given the value of the GDP, the mean, or average, forecast consumption expenditure is about 4951 billion dollars.
The actual value of the consumption expenditure reported in 1997 was billion dollars.
The estimated model thus overpredicted the actual consumption expenditure by about billion dollars. We could say the forecast error is about billion dollars.

33. what is important for now is to note that such forecast errors are inevitable given the statistical nature of our analysis.