1. A researcher at a Central Bank models aggregate production in the economy using the following linear regression model where y is a T1 vector with tth element equal to the natural logarithm of aggregate production in year t,X is the T3 matrix with tth row xt=(1,xt,2,xt,3) where xt,2 is the natural logarithm of the aggregate labour input in year t and xt,3 is the natural logarithm of aggregate capital stock in year t, and u is the T1 error vector. Assume the model in (1) satisfies Assumptions SR1-SR6 given in lectures. The researcher estimates the model via Ordinary Least Squares (OLS) based on a sample of size T=25 and obtains the following results y^t=(0.321)1.360+(0.123)0.632xt,2+(0.081)0.317xt,3,R2=0.946, where the numbers in parentheses are the associated standard errors. (a) Based on these estimation results, calculate a 95% confidence interval for the elasticity of output with respect to labour. Be sure to explain your calculation. [5 marks] (b) The researcher wishes to test the null hypothesis that the aggregate production function exhibits constant returns to scale and asks for your help. In addition to the estimated regression equation above, the researcher shares with you the following results all based on the same sample: - the OLS-estimated regression equations: y^t=(0.407)1.246+(0.069)1.061xt,2,R2=0.909 y^t=(0.398)2.224+(0.052)0.693x3,t,R2=0.882 v^t=(0.125)1.121+(0.071)0.287zt,R2=0.410 where vt=ytxt,2 and zt=xt,3xt,2 - the summary statistics: Sy.y=t=1T(yty)2=12.911,S2,3=t=1T(xt,2x2)(xt,3x3)=14.074,S2,2=t=1T(xt,2x2)2=10.424,Sy,2=t=1T(yty)(xt,2x2)=11.064,S3,3=t=1T(xt,3x3)2=23.718,Sy,3=t=1T(yty)(xt,3x3)=16.437. where y=T1t=1Tyt,x2=T1t=1Txt,2, and x3=T1t=1Txt,3. Use this information to perform the test of whether the aggregate production function exhibits constant returns to scale at the 5% significance level. Be sure to specify the null and alternative hypotheses, the decision rule, the formula for the test statistic and how you have calculated the test statistic from the information provided. [15 marks] 1. A researcher at a Central Bank models aggregate production in the economy using the following linear regression model where y is a T1 vector with tth element equal to the natural logarithm of aggregate production in year t,X is the T3 matrix with tth row xt=(1,xt,2,xt,3) where xt,2 is the natural logarithm of the aggregate labour input in year t and xt,3 is the natural logarithm of aggregate capital stock in year t, and u is the T1 error vector. Assume the model in (1) satisfies Assumptions SR1-SR6 given in lectures. The researcher estimates the model via Ordinary Least Squares (OLS) based on a sample of size T=25 and obtains the following results y^t=(0.321)1.360+(0.123)0.632xt,2+(0.081)0.317xt,3,R2=0.946, where the numbers in parentheses are the associated standard errors. (a) Based on these estimation results, calculate a 95% confidence interval for the elasticity of output with respect to labour. Be sure to explain your calculation. [5 marks] (b) The researcher wishes to test the null hypothesis that the aggregate production function exhibits constant returns to scale and asks for your help. In addition to the estimated regression equation above, the researcher shares with you the following results all based on the same sample: - the OLS-estimated regression equations: y^t=(0.407)1.246+(0.069)1.061xt,2,R2=0.909 y^t=(0.398)2.224+(0.052)0.693x3,t,R2=0.882 v^t=(0.125)1.121+(0.071)0.287zt,R2=0.410 where vt=ytxt,2 and zt=xt,3xt,2 - the summary statistics: Sy.y=t=1T(yty)2=12.911,S2,3=t=1T(xt,2x2)(xt,3x3)=14.074,S2,2=t=1T(xt,2x2)2=10.424,Sy,2=t=1T(yty)(xt,2x2)=11.064,S3,3=t=1T(xt,3x3)2=23.718,Sy,3=t=1T(yty)(xt,3x3)=16.437. where y=T1t=1Tyt,x2=T1t=1Txt,2, and x3=T1t=1Txt,3. Use this information to perform the test of whether the aggregate production function exhibits constant returns to scale at the 5% significance level. Be sure to specify the null and alternative hypotheses, the decision rule, the formula for the test statistic and how you have calculated the test statistic from the information provided. [15 marks]