Question

why cross sectional regression rule out perfect collinearity among the regressors

Answer 1

In time series data, may be that the regressor includeed in the model share a common trend ,that is, they all increase or decrease over time. Thus , in the regression of consumption expenditure on income, wealth, and population, the regressors income, wealth, and population may all be growing over time at more or less the same rate, leading to collinearity among these variables.

A variant of the extraneous or a priori information technique is the combination of cross sectional and. Suppose we want to study the demand for automobiles in the United States and assume we have time series data on the number of cars sold, average price of the car, and consumer income. Suppose also that

log(Y_t)= b₁ + b₂*log(p_t)+b3*log(I_t)+u_t

where Y = number of cars sold, P = average price, I = income, and t = time.

Out objective is to estimate the price elasticity β2 and income elasticity β3. In time series

highly collinear. Therefore, if we run the preceding regression, we shall be faced with the usual multicollinearity problem. A way out of this has been suggested by Tobin. He says that if we have cross-sectional data (for example, data generated by consumer panels, or budget studies conducted by various private and governmental agencies), we can obtain a fairly reliable estimate of the income elasticity β3 because in such data, which are at a point in time, the prices do not vary much. Let the cross-sectionally estimated income elasticity be. Using this estimate, we may write the preceding time series regression as

Y_t^*= b₁ + b₂*log(p_t)++u_t

_where Y_t^*= log(y_t)-b3*log(I_t)