The model is Linear in the Coefficients and the Error Term

Means:

• You must always write your model so it is Linear in the coefficients
• You Assume an error term is added on the end

Problem:

• Ols will give you no solution

Problem Found in:

• Equations which in theory cannot be written linearly

Solution:

• "Iterative" Computer techniques called Non-Parametric Methods

The Error Term has a Zero Population Mean

Means: The distribution of the error term must have an expected value of zero.

Problem: An error term which has a mean other than zero will influence the estimated coefficients. The error term is zero tso that we can assume all the changes in the dependent variable have to do with independent variable.

Problem Found in: All Linear Regressions

Solution: Use a constant term

No independent variable is correlated with the error term

Means:

• There is no relationship between the error term and the independent variables
• All independent variables have to be determined outside of the model and not with each other or the independent variable.

Problem: Simultaneous equation bias. Coefficients are biased. (example supply and demand together determine effect of price on quantity)

Problem found in: The dependent variable could be in a second regression model that explains an independent variable.

Solution: Create instrumental variables by "Two-Stage" least squares instead of OLS

Classical Assumption #4

Error term observations are not correlated with each other

Means: The error for one observation should in no way influence the error for the next observations.

Problem: Serial Correlation

1. Pure Serial Correlation: Comes from theory, not biased, increased variance
2. Impure Serial Correlation: When you leave out an important variable. Biased. Increased Variance.

Problem Found In: Time Series Models

Solutions: First, test for serial correlation, then:

1. Pure: Use Generalized Least Squares, not OLS
2. Impure: Find the missing variable

The error term has a constant variance.

Means: The variance of the error term will stay the same, regardless of independent variables used.

Problems: Two Types

1. Pure Heteroskedasticity: Comes from theory. Not biased. Increases Variance.
2. Impure Heteroskedasticity: When you leave out an important variable. Biased. Increase Variance.

Problem Found in: Cross-Sectional Data

Solutions: First, test for heteroskedasticity, then:

1. Pure: Redefine variables or use Weighted Least Squares
2. Impure: Find the Missing Variable

Independent Variables are not perfect linear functions of each other

1. Means: There is no relationship between any two or more independent variables

Problem: Multicollinearity

1. Perfect Multicollinearity: Exact mathematical relationship, cannot solve for coefficients
2. Imperfect Multicollinearity: Strong fuctional relationship, unbiased. Increase variance for affected variables.

Problem Found In: Both timer series and cross sectional models.

1. Perfect: Comes from specification
2. Imperfect: May come from chance of samples or two independent variables are really related

Solutions: First, test for multicollinearity, then:

1. Perfect: Drop one of the perfect multicollinearity variables.
2. Imperfect: DO NOTHING (avoid specification bias)

Classical Assumption #7

(Not necessary, but used in Hypothesis testing)

The error term is normally distributed

Means: The error term will only have a bell-shaped distribution (this allows for t and F tests)

Problems: When this doesn't hold, we cant use the simple t and F tests for significance.

Problem Found In: Models where theory tells you assuming normal is inappropriate.

Solutions: Assume normal or assume some other more theoretically appropriate distributions

1. Pure: Comes from theory. Not Biased.  Increased Variance

• Use generalized least squares, not OLS

2. Impure: When you leave out an important variable. Biased. Increased Variance

• Find the missing variables

1. Pure: Comes from Theory. Not Biased. Increased Variance.

• Redefine the variables or use weighted least squares

2. Impure: When you leave out an important variable. Biased. Increased variance

• Find the Missing Variable

Perfect: Exact mathematical relationship. Cannot solve for coefficients

• Drop one of the variables

Imperfect: Strong functional relationship. Unbiased. Increased variance for affected variables

• Do nothing to avoid specification bias

= √ ε^e²/n-3

ε(_X1i-x̄₁)2(1-r² ₁₂)

Substitute for theoretically desired variables when data on variables are incomplete or missing. Must move proportional to variable being measured.

(Ex: Zip code as a prozy quite successfully for income)

When using timer series data, a certain variable in one period may be affected by something that happened in a previous period.

β_t-1

Taking qualitative measurements and converting them into quantitative variables for use in OLS

Two Methods:

From a baseline - always pick a base and drop it

Incremental Change

Sometimes you are not interested in the total value of a variable, but how it changes from one period to the next.

Δβ