Answer to Question #317433 in Economics of Enterprise for given

Question

explain the small sample properties of ordinary least squares under the assumptions such that the estimate are unbaised ,the variance of the estimators are constant and the estimators can be used to make stastical inference

Answer 1

The Ordinary Least Squares (OLS) approach is commonly used in econometrics to estimate the parameters of a linear regression model. While running linear regression models, assumptions are applied to ensure the validity of OLS estimations.

1. The regression model's coefficients are linear, and the error term is linear.

This assumption is concerned with the model's functional form. In statistics, a regression model is linear if all of the variables in the model are either constants or parameters multiplied by independent variables. The model equation is constructed only by adding the components together.

2. The observations of the error term are unrelated to one another.

A positive correlation, for example, exists when the error for one statement is positive and consistently raises the likelihood that the following error will be positive. A negative correlation exists when the subsequent mistake is more likely to have the opposite sign. This issue is known as serial correlation as well as autocorrelation. In time series models, serial correlation is most likely to occur.

3. The population mean for the error term is zero.

The error term accounts for the variance in the dependent variable that is not explained by the independent variables. The values of the error term should be determined by random. The average value of the error term must equal 0 for your model to be unbiased.

4. The error term is unrelated to any of the independent variables.

If an independent variable is associated with the error term, we may use the independent variable to forecast the error term, which contradicts the idea that the error term indicates unpredictability. We need to figure out how to include that data into the regression model.

5. The variance of the error term is constant (no heteroscedasticity)

The error variance should be consistent across all observations. In other words, the conflict does not vary for any given statement or set of data. This favored state is referred to as homoscedasticity (same scatter). When the variance fluctuates, we call this heteroscedasticity (different scatter).

6: The error term has a regular distribution

To provide unbiased estimates with the lowest variance, OLS does not need that the error term has a normal distribution. However, meeting this assumption allows you to test statistical hypotheses and obtain valid confidence and prediction ranges.

These assumptions are critical because any deviation from them renders OLS estimations inaccurate and wrong. A violation might, for example, result in erroneous signs of OLS estimates or unexpected variance of OLS estimates, resulting in confidence intervals that are too large or too tight.