3 the error process has zero mean that is eu t 0 for

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(3) The error process has zero mean. That is, E(u t ) = 0 for all t. (4) The errors terms, u t , t=1,..,T, are serially uncorrelated. That is, Cov(u t ,u s ) = 0 for all s not equal to t. (5) The errors have a constant variance. That is, Var(u t ) = σ 2 for all t. (6) Each regressor is asymptotically correlated with the equation disturbance, u t . We sometimes wish to make the following assumption: (7) The equation disturbances are normally distributed, for all t. Sometimes it is more convenient to use matrix notation. The regression model can be written in matrix notation for all T observations as Y X u = + β (A) Here, Y is a (T × 1) vector of observations on the dependent variable, X is a (T × k) matrix of T observations on k possibly stochastic but stationary explanatory variables, one of which will usually be an intercept. β is a (k × 1) vector of parameters, and u is a (T × 1) vector of disturbance terms. Using the notation x / t to denote the t th row of the matrix X (and so is a (k × 1) containing one observation on each of the k explanatory variables), we can also write the model in matrix notation for a single (t th ) observation as
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3 t t t Y = x + u t = 1,2,...,T β For the simple k=2 variable case, this can be written as t 1 t 2 t t Y = X + X + u t = 1,2,...,T β β 1 2 , , If X 1 is an intercept term, then we can more compactly write this as t 1 2 t t Y = + X + u t = 1,2,...,T β β In matrix terminology, the assumptions in Table 4.1 would be re-expressed as in Table 4.2. TABLE 4.2 (1) The dependent variable is a linear function of the set of possibly stochastic but stationary regressor variables and a random disturbance term as specified in (A). No variables which influence Y are omitted from the regressor set X , nor are any variables which do not influence Y included in the regressor set. In other words, the model specification is correct. (2) Lack of perfect collinearity (the T*k matrix X has rank k) (3) The error process has zero mean ( E( u ) = 0) (4) The errors terms, u t , are serially uncorrelated ( E(u t ,u s ) = 0 for all s not equal to t). (5) The errors have a constant variance ( E(u t 2 ) = σ 2 ) for all t. In matrix terms, (4) and (5) are written as Var( u ) = σ 2 I T (6) plim(1/T{X / u}) = 0 and if we wish to use it: (7) The errors are normally distributed. THE LINEAR REGRESSION MODEL WITH STOCHASTIC REGRESSORS A variable is stochastic if it is a random variable and so has a probability distribution; it is non-stochastic if it not a random variable. Some variables are non-stochastic; for example intercept, quarterly dummy, dummies for special events and time trends are all non-random. In any period, they take one value known with certainty. However, many economic variables are stochastic. Consider the case of a lagged dependent variable. In the following regression model, the regressor Y t-1 is a lagged dependent variable (LDV): t 1 2 t 3 t-1 4 t-1 t Y = + X + X + Y + u , t = 1,...,T β β β β Clearly, Y t = f(u t ), and so is a random variable. But, by the same logic, Y t-1 = f(u t-1 ), and so Y t-1 is a random or stochastic variable. Any LDV must be a stochastic or random variable. If our regression model includes one, the assumptions of the
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