1. Consider a simple linear regression model Under what assumptions, will OLS es
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1. Consider a simple linear regression model Under what assumptions, will OLS estimators of unknown coefficients 0 and be unbiased? List these assumptions and briefly discuss (in plain English) their relative importance. 2. Consider the savings function where e, is a random error with E[eilinci)-Ek/ = o and varfellnGJ-2 . That is, our usual so-called error term ut is in fact a function of the "true" error €1. (a) Show that E[u, line.] = 0, so that the key zero conditional mean assumption (Assumption SLR.4) is satisfied. violated. In particular, the variance of sav, increases with inc with family income (b) Show that var[u, linc.] = 2 x1nq, so that the homoskedasticity Assumption SLR.5 is (c) Provide a discussion that supports the assumption that the variance of savings increasesExplanation / Answer
For a simple linear regression model, y=beta zero +beta one*x+u;
Basic assumptions for unbiased OLS estimators (of unknown coefficients beta-zero and beta-one) are:
1. Linearity in relationship : The dependent variable should be a linear function of independent variable.
[if not, the regression equation, which is linear will not be able to capture non-linearity and the coefficients will be wrong]
2. Randomness in error: The errors (here, u) must be random (showing no specific pattern), with expected value (i.e., mean zero)
[otherwise, the intercept beta zero will be biased]
3. Error non-correlation with dependent variable & independently distributed
[this means no spatial-correlationship (which means correlationship with sequence/position of observation), nor auto- or serial-correlationship (which means correlationship with regard to time-sequence), which add bias to the coefficients]
4. Homoskedasticity : The variance of the error term, i.e., V(u), should be constant, not change over time
[since variance is uncertainty in estimation, if it varies over time, also called heteroskedasticitythat means, the actual measurements and the estimations should be more variable with time, because then the estimation will be biased)
5. Deterministic independent variable: The independent variable should not be stochastic (or random) and hence uncorrelated with error terms.
[Without this, model will not be truly predictable, because errors will not be random)
6. Multicolinearity : This is applicable for more number of independent variables, who are correlated with each other.
[Because of mutual correlationship between independent variables, the coefficients will be more biased towards these correlated variables, than the non-correlated variable].
Besides these, as in case of any statistical modeling, the measurement errors should be as minimum as possible.
Hope, the explanations are clear !
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