Linear Regression Model - Assignment Example

Question One You want to estimate the following linear regression model: With a) Assume that Can you compute the least squares estimator in this model? Explain your answer and show your work. SOLUTION (a) (b) From (b), satisfies (c) And satisfies We have however been given so substituting yields; b) Assume again that : Can you obtain a univariate model: that is equivalent to the multivariate linear regression model (1)? If this is possible, can you compute the least squares estimator for this univariate model? Explain your answer and show your work. SOLUTION Least squares estimates c) Consider the alternative model: Assume again that : Can you compute the least squares estimator in this model? Explain your answer and show your work. SOLUTION We have the estimates as; We have however been given so substituting yields; d) Assume again that X2i=2+X1i: Can you obtain a univariate model: that is equivalent to the multivariate linear regression model (2)? If this is possible, can you compute the least squares estimator for this univariate model? Are your answers in c: and d: different from your answers in a and b:? If yes, why? SOLUTION Looking at the results it is evident that answers in a and b are different this is because in a we are estimating a multivariate model while in b we are estimating a univariate model. Similarly, the results shows that answers in c and d are different since we c we estimated for a multivariate model while in d we estimated for a univariate. Question Two Assume that the variable Yi is defined as with corr (Wi; ui) = 0 and corr (Xi; ui) 6= 0: All the other assumptions of the linear regression model and the assumption of the extended model hold. a) You have data on the variables Y , X and W, and you decide to estimate model (3) by ordinary least squares. Is the least squares estimator bMC consistent in this case? Explain your answer. SOLUTION The least squares estimator bMC is not consistent in this case since there exists autocorrelation between the independent variable and the error term ui i.e. i.e. the disturbances are pairwise correlated. This is referred to as autocorrelated disturbances. b) You have data on the variables Y , X, W and Z. The variable Z satisfies the "exogeneity" condition and the "relevance" condition . You decide to estimate model (3) using the instrumental variables method. Is the instrumental variables estimator bIV consistent in this case? Explain your answer. SOLUTION The instrumental variable estimator bIV is consistent in this case since the instrumental variable is uncorrelated with the error term though there exists a correlation between it with the exogenous variable. c) You have data on the variables Y, X and Z, but you dont have data on W. The variable Z satisfies the "exogeneity" condition and the "relevance" condition . You decide to omit the variable W from the regression and to estimate the model: using the instrumental variables method. i) If Zi and Wi are not correlated, is the instrumental variables estimator bIV consistent? Explain your answer. SOLUTION The instrumental variable estimator bIV is inconsistent in this case; for the consistency in the instrumental variables Zi and Wi should be correlated. In this given case, the conditions for consistency have not been met hence leading the instrumental variables estimator bIV to be inconsistent. ii) If Zi and Wi are correlated, is the instrumental variables estimator bIV consistent? Explain your answer. SOLUTION The instrumental variable estimator bIV is consistent in this case because the variables Zi and Wi are correlated to each other thus satisfying the conditions for consistency d) You have data on the variables Y , X, W, Z1 and Z2. Assume that for all variables you sample is very large. The variables Z1 and Z2 satisfy the "relevance" condition and . You decide to use the overidentification test to check the "exogeneity" condition, and you obtain a value for the J statistic of 15.7. i) What is the interpretation of this value of the J statistic? Do we reject the hypothesis of exogeneity of the variables Z1 and Z2? (you will need to look at the critical values in the table for the distribution of to answer this question). SOLUTION The interpretation for the J statistic is that that all instruments are uncorrelated with ui. With the value of J statistics being 15.7, we reject the null hypothesis and conclude either one or more of the instruments are invalid or that the structural model is specified incorrectly ii) Does the value of the J statistic suggest that ; or , or both? Explain your answer. SOLUTION The J statistic neither suggest ; or but it rather suggests since the statistic test that under the null hypothesis all instruments are uncorrelated with ui Question Three: Empirical Question. Descriptive statistics a) You want to study the effects of family income on the level of education. You decide to estimate the following linear regression model: where the variable bytest is included to account for the innate abilities of each individual. Is b1 (the least squares estimator of ) significant at the 5% level? Explain your answer. The p-value of the coefficient of incomehi is less than 5% significance level (p-value=0.000 Read More