Question

Please I need the mathematical proves in all questions and

Question 1 15 MARKS Suppose that we have two possible models that we are considering. Although we don't realize it, the first model (M1) is the correct model, in the sense that its error term satisfies all of the usual assumptions: Mi: y= XB+€ (0,-1.) (Correct Model). The other model (M2) is an incorrectly specified model: M2: y = 2 + (False Model). In each case we have the same dependent variable, observed for the same sample of data. The two models may contain different numbers of regressors. Now, suppose that I decide to use the following rule to try and select the correct model: fit each model by OLS, and then choose the model that has the larger R2. (a) Show that this is equivalent to choosing the model that has the smaller (usual unbiased) estimator for the error variance. (b) Using your answer to part (a), prove that this rule will lead to the selection of the correct model (M1), on average. (c) What are the practical limitations of such a rule for model selection?

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Answer 1

Given that Correct model MI: yo XB+E; En Co, on? In) → false model M2: y = Z8+Y @ x = xßte + correct y = 22 +4 → incorrect en N(2,0?!n) and variance correlation (9) - (22)" ( 2 492) (2 e)" matrix of empor-valiance from the sandwitch proposition I var. Correlation C4). varrcor (E) PD positive definite matria

Therefore the and small vadiance (&) is satisfies "BLUE" Choosen if the variance value is below for low ESS Ess is high, - Ess is also high, it means higher the R² higher the explained sum of square (Ess) And less than residual Sum of square TIS 3 In better model Rss is low , R² is high and the correct model the choose

© Limitations i * In this model add a redudant regneson Then R² is tend to increase but do not increase adjusted R² * If take the decision at the base basis of adjusent R² rather than R² value If the specification of the model is incorrect then we should not loose into R² if say we take incorrect mathematical form