OLS

The coefficient

The residuals or error term

Linearity, Full rank, Regressors are exogenous, Homoskedasticity and no autocorrelation, Non-stochastic regressors, Normal disturbances

The model species a linear relationship between y and x

x has full column rank. There is no exact linear relationship between the di¤erent independent variables

E (e\x) = 0 The mean of the residuals (holding x xed) is zero.

The residuals have a fixed variance sigma^2 and the residual for observation i is uncorrelated with the residual for observation j.

x is non-stochastic as in experimental data.The conometricianchooses the values of the regressors before observing y. For example, where x is fertilizer and irrigation and y is agricultural yield. In this case, we do not need to condition on x in the assumptions discussed above

We assume that the residual is normally distributed e\x ~ N(0,sigma^2*I)

Total sum of squares = regression sum of squares + residual sum of squares

Values between 0 and 1

It measures the proportion of the total variation in y that is accounted for by variations in the regressors. Values closer to 1 indicate that variation in the regressor contributes highly to the variation in the dependent variable.

As the number of regressors increase the R2 in the longer regression cannot be smaller.

However R2 does not provide an absolute basis for comparison and a high value of R2 depends on the context. In other words the variation in the dependent variable can be very di¤erent in di¤erent regression models.

Whether the R-hat^2 rises or falls with an additional regressor depends on whether the improvement in t due to the additional regressor more than o¤sets the correction for the loss of an additional degree of freedom.

In a linear regression model, the least squares estimator ˆb is the minimum variance linear unbiased estimator of b. The OLS estimator is the most e¢ cient in the class of linear unbiased estimators. Other unbiased estimators may exist but they have a larger variance. Requires assumptions A1 to A4, but not normality.

If disturbances are normally distributed the OLS estimator is also the maximum likelihood estimator (MLE). Means that OLS is asymptotically e¢ cient among consistent and normally distributed estimators. Large sample counterpart to Gauss Markov (Cramer Rao Lower bound).

The object of interval estimation is to present an estimate of the parameter with a measure of uncertainty attached to it–i.e. b-hat +- sampling variability

Wald Test: The Wald test measures how close Rb - q is to zero

As the sample size grows, the t-distribution approaches the normal distribution and the F-distribution approaches the Chi-squared from above. This suggests that in moderate samples, the t and the f distributions provide a conservative approximation.

To answer questions, To test economic theories

Note that beta stands for the change in H when Y changes by 1 unit. Note that this is not the elasticity.

Distance between line and data points is minimised.

Regression with multiple variables

y = xb + e

Independent variable

The coefficient

The residuals or error term

Linearity, Full rank, Regressors are exogenous, Homoskedasticity and no autocorrelation, Non-stochastic regressors, Normal disturbances

The model species a linear relationship between y and x

x has full column rank. There is no exact linear relationship between the di¤erent independent variables

E (e\x) = 0 The mean of the residuals (holding x xed) is zero.

The residuals have a fixed variance sigma^2 and the residual for observation i is uncorrelated with the residual for observation j.

x is non-stochastic as in experimental data.The conometricianchooses the values of the regressors before observing y. For example, where x is fertilizer and irrigation and y is agricultural yield. In this case, we do not need to condition on x in the assumptions discussed above