Linear Regression

METHODOLOGY OF ECONOMETRICS

1. Statement of theory or hypothesis.2. Specification of the mathematical model of the

theory3. Specification of the statistical, or econometric,

model4. Obtaining the data5. Estimation of the parameters of the econometric

model6. Hypothesis testing7. Forecasting or prediction8. Using the model for control or policy purposes.

1. Meaning of Regression

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

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.

2. Regression Example

Example

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)

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.

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

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.

Population Regression Line

Regression Example

What form does this function take? Many possibilities, but assume its a linear function:

E(Y|Xi) = 1 + 2Xi

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.

3. Linearity

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.

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.

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.

4. Stochastic Error

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.

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.

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.

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.

11. Hypothesis Testing

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.

12. Coefficient of Determination--R2

Coefficient of Determination

The coefficient of determination, R2, measures the goodness of fit of the regression line overall

Correlation Coefficient

The correlation coefficient is the square root of R2 Correlation coefficient measures the strength of the

relationship between two variables.

13. Forecasting

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.

the estimated consumption function is:

suppose we want to predict the mean consumption expenditure for 1997. The GDP value for 1997 was 7269.8 billion dollars.

Putting this GDP figure on the right-hand side of we obtain:

ˆ 184,08 0,7064 iY X

1997ˆ 184,0779 0,7064(7269,8) 4951,3167Y

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 4913.5 billion dollars.

The estimated model thus overpredicted the actual consumption expenditure by about 37.82 billion dollars. We could say the forecast error is about 37.82 billion dollars.

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