OLS regession Essay

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Cross-sectional OLS

Classical Assumptions
Asymptotic Assumptions
(Any Sample Size)
(Large Sample)
MLR.1 Linear in parameters
The model in population can be written as y = β0 + β1 x1 + . . . + βk xk + u, where β0 , β1 , . . ., βk are the unknown parameters of interest and u is an unobservable random error.
MLR.2 Random sampling
We have a random sample of n observations, {(xi1 , xi2 , . . . , xik , yi ) : i = 1, 2, . . . , n} from the population model.
MLR.3 Zero conditional mean
MLR.3 Zero mean and zero correlation
The error u has an expected value of zero, given any values of the E(u) = 0 and Cov(xj , u) = 0 for j = 1, 2, . . . , k independent variables. In other words, E(u|x1 , x2 , ...xk ) = 0
MLR.4 No perfect collinearity
In the sample (and therefore in the population), none of the independent variables is constant, and there are no exact linear relationships among the independent variables.
MLR.5 Homoskedasticity
The error term, u, conditional on the explanatory variables, is the same for all combinations of outcomes of the explanatory variables.
V ar(u|x1 , . . . , xk ) = σ 2
MLR.6 Normality
The population error u is independent of the explanatory variables x1 , x2 , . . . , xk and is normally distributed with zero mean and variance σ 2 : u ∼ N ormal(0, σ 2 )

1

2

Time-series OLS

Classical Assumptions
Asymptotic Assumptions
(Any Sample Size)
(Large Sample)
TS.1 Linear in parameters
TS.1 Linearity and weak dependence
The stochastic process {(xt1 , xt2 , . . . , xtk , yt ) : t = 1, 2, . . . , n} follows Same as TS.1 except we must also assume that {(xt1 , xt2 , . . . , xtk , yt ) : the linear model yt = β0 + β1 xt1 + . . . + βk xtk + ut where {ut : t = t = 1, 2, . . . , n} is weakly dependent.
1, 2, . . . , n} is the sequence of errors or disturbances. Here n is the
number

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