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Consider a random sample on consumption and disposable income, $$ ( CON_i, YD_i ) $$, $$ ( i = 1, 2, \ldots, n ) $$. Suppose the joint distribution of $$ ( CON_i, YD_i ) $$ (which is the same across $$ i $$ because of the random sample assumption) is normal. Clearly, Assumption 1.3 is satisfied; the rank of $$ \mathbf{X} $$ would be less than $$ K $$ only by pure accident. Show that the other assumptions, Assumptions 1.1, 1.2, and 1.4, are satisfied. Hint: if two random variables, $$ y $$ and $$ x $$, are jointly normally distributed, then the conditional expectation is linear in $$ x $$, i.e.,
$$
\mathrm{E} ( y \mid x ) = \beta_1 + \beta_2 x,
$$
and the conditional variance, $$ \mathrm{Var} ( y \mid x ) $$, does not depend on $$ x $$. Here, the fact that the distribution is the same across $$ i $$ is important; if the distribution differed across $$ i $$, $$ \beta_1 $$ and $$ \beta_2 $$ could vary across $$ i $$.