Question

Regression analysis consists of two major tasks: (i) estimation of population parameters and (ii) hypothesis testing (e.g., t-test, F-test) or the application of inferential statistics to the estimated parameters. We learned OLS (ordinary least square) principle as the major estimation tool (thereby fulfilling the first task). There is a critically important assumption that we have to make in order to perform the second task (that is, to conduct hypothesis testing with respect to estimated parameters). Identify and discuss it. (10 points)

Answer 1

We derive tests about the coefficients of the normal linear regression model. In this model the vector of errors is assumed to have a multivariate normal distribution conditional on with mean equal to and covariance matrix equal to where is the identity matrix and is a positive constant.

It can be proved (see the lecture about the normal linear regression model) that the assumption of conditional normality implies that:

• the OLS estimator beta b^{^}  is conditionally multivariate normal with mean beta b and covariance matrix

• the adjusted sample variance of the residuals is an unbiased estimator of (sigma)²; furthermore, it has a Gamma distribution with parameters N-K and (sigma)²

• b^{^} is conditionally independent of (sigma)^{2^}

• The dependent variable Y has a linear relationship to the independent variable X.
• For each value of X, the probability distribution of Y has the same standard deviation σ.
• For any given value of X,
• The Y values are independent.
• The Y values are roughly normally distributed (i.e., symmetric and unimodal). A little skewness is ok if the sample size is large.
• The dependent variable Y has a linear relationship to the independent variable X. To check this, make sure that the XY scatterplot is linear and that the residual plot shows a random pattern. (Don't worry. We'll cover residual plots in a future lesson.)
• For each value of X, the probability distribution of Y has the same standard deviation σ. When this condition is satisfied, the variability of the residuals will be relatively constant across all values of X, which is easily checked in a residual plot.

These are the requirements for parameters testing