Exercise 7 To examine the quantity theory of money, Brumm (2005) [‘‘Money Growth, Output Growth, and Inflation: A Reexamination of the Modern Quantity Theory’s Linchpin Prediction,’’ Southern Economic Journal, 71(3), 661–667] specifies the equation: where INFLAT is the growth rate of the general price level, MONEY is the growth rate of the money supply, and OUTPUT is the growth rate of national output. According to theory we should observe that and . The data used in this paper is contained in the file brumm.gdt. It consists of 1995 year data on 76 countries. a) Estimate the model by OLS and interpret all the parameters. ols Inflation const Money Output b) Test the joint hypothesis that and . What do you conclude? restrict b[1] = 0 b[2] = 1 b[3] = -1 end restrict We reject H0, therefore, restrictions do not hold c) Examine the least squares residuals for the presence of heteroskedasticity related to the variable Money. series resid=$uhat gnuplot sq_resid Money modtest –white (this tests all explanatory variables for heteroskedasticity Manually: genr mout=Money*Output ols sq_resid const Money Output sq_Money sq_Output mout Graphical user interface, text, application, Word Description automatically generated LM= R^2*n=0.3794*76=28.83 Critical value at 1% significance Therefore, we reject the hypothesis that there is no heteroskedasticity with respect to the variable Money d) Obtain robust standard errors for the model and compare them to the OLS standard errors. Does your conclusion change in part (b) after using robust standard errors? ols Inflation const Money Output –robust Conclusion does not change – they are jointly not equal to the theoretical parameters e) It is argued that Output may be endogenous. Four instrumental variables are proposed, INITIAL = initial level of real GDP, SCHOOL = a measure of the population’s educational attainment, INVEST = average investment as a share of GDP, and POPRATE = average population growth rate. Using these instruments, obtain instrumental variables (2SLS) estimates of the inflation equation (do the two stage procedure). First stage: ols Output const initial poprate school invest Money series Output_hat=$yhat Second stage: ols Inflation const Money Output_hat Graphical user interface, text, application, Word Description automatically generated Alternatively, we can use Gretl command tsls Inflation 0 Output Money ; 0 initial invest poprate school Money OR tsls Inflation const Output Money ; const initial invest poprate school Money f) Are the instruments strong? Only invest predicts the Output significantly, other variables are weak instruments. The theoretical parameters are again jointly rejected. The impact of output on the inflation is now lower than before. According to F test, instrument is weak because it falls below 10.27, where bias could have been roughly around 10% only but this is beyond the scope of our course Graphical user interface, text, application, email Description automatically generated