Regression with a Single Regressor
7 important questions on Regression with a Single Regressor
What about two-sided hypotheses concerning B1?
- Compute standard error
- Compute t-statistic
- Compute P-value (smallest significance level) or compare with critical value
T-statistic = ( estimator - Hypothesis value ) / Standard error of the estimator
In large samples the sampling distribution of Y is approximately normal.
Difficult formula for standard error: 1/2 * ((1/(n-2)* sum(Xi-Mean(X))^2* error^2) / (1/n * SUM(Xi - MEan(x))^2)^2.
What about one-sided hypotheses concerning B1?
H0= certain value, H1 < or > then a certain value
In practice, one-sided alternative hypotheses should be used only when there is a clear reason for doing so (economic theory). But ambiguity/skepticism often leads econometricians to use two-sided tests.
What about confidence levels?
95% CL (+/- 1.96*SE) means the set of values that cannot be rejected using a two-sided hypothesis test with a significance of 5%. Or 95% probability that we have the "true" value.
Can also be used to make a predicted effect by multiplying low level and high level by delta X.
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What about a binary variable?
What about hetero- and homoskedasticity?
OLS estimators remain unbiased and consistent, even if homoskedastic or heteroskedastic. Moreover, in large samples asymptotically normal.
What about efficient estimator (Gauss-Markov)?
Note: some use this as a default!! Underestimate t-statistic, even in normal distribution and large samples. Therefore heteroskedastic-robust SE always more conservative.
Economic theory rarely gives any reason to believe errors are homoskedastic. Therefore be more prudent and conservative unless you have compelling reasons to believe otherwise.
What if the sample size is small?
STUDENT is more prudent but in larger samples the difference is negligible. Moreover rarely homoskedastic and normally distributed so get large datasets :p
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