ln(RM2) = b1 + b2ln(RGDP) + b3ln(RTB) + e
Since the data for "money demand" is usually not available, the quantity of money supply M2 has been used to approximate the demand for money. This approximation will work only for the case of equilibrium in money market, that is demand equals to supply.
As in the previous homework, let RM2 = M2*100/PGDP be the real money supply and RGDP = GDP*100/PGDP is the real GDP. RTB denotes the 3-month TB interest rate. The equation for M2 money demand in log form is written as:
ln(RM2*) = b1 + b2ln(RGDP) + b3ln(RTB) + e ............ (A)
where RM2* is the unobservable demand for real balance that is not necessary the same as real money supply RM2 for which data is available. Equation (A) is usually referred as the "long-run" demand equation (at equilibrium) for money.
To allow for disequilibrium in the money market, we introduce the (logarithmic) partial adjustment hypothesis for the quantity of money demanded as follows:
ln(RM2) - ln(RM2-1) = d (ln(RM2*) - ln(RM2-1)) ........ (B)
where the variable with subscript (-1) indicates the lagged one- period variable. Coefficient d is interpreted as the coefficient of adjustment. Equilibrium is obtained only when d = 1. In general, the market may not clear (that is, demand will not equal to supply) and d lies between 0 and 1.
(1.1) Using quarterly data series, estimate and interpret regression results from the model combining equations (A) and (B) above. The resulting equation is called the "short-run" demand equation for money as follows:
ln(RM2) = a1 + a2ln(RGDP) + a3ln(RTB) + a4ln(RM2-1) + e ............ (C)
This is a regression model with lagged dependent variable. The use of lagged dependent variable introduces the potential problem of inconsistency in parameter estimates. Not only the explanatory variables are correlated with the error term, but also the errors are potentially serially correlated.
Test and correct for the problem of first order serial correlation when there is a lagged dependent variable presented in the model. Hint: using instrumental variable estimation is recommended, but the correction for serial correlation may be more involved.
(1.2) Compute and interpret mean lag and median lag of the money demand. What are the long-run and short-run GDP and interest rate elasticities respectively?
(1.3) If there is a classical error term added to the partial adjustment process (B), the model may suffer problems of autocorrelation in its theoretical specification and in the process of empirical estimation. Answer (1.1) and (1.2) above with the appropriate adjustment and correction of serial correlation.
ln(RM2) = a + b ln(RGDP) + c ln(RTB) + e
Let
b ln(RGDP) = b0 ln(RGDP) + b1 ln(RGDP-1) + b2 ln(RGDP-2) + b3 ln(RGDP-3) ...
and
c ln(RTB) = c0 ln(RTB) + c1 ln(RTB-1) + c2 ln(RTB-2) + c3 ln(RTB-3) ...
are the corresponding distributed lag structure of two explantory variables.
(2.1) Find the best fitted polynomial lag structure for the above model. That is, identify the most appropriate number of lags, orders, and end-point restrictions up to the 4th lags and 4th orders.
(2.2) Estimate and interpret the best model identified from (2.1). Compute and interpret median lag, mean lag, long-run and short-run elasticities.
(2.3) Test and correct for the problem of autocorrelation.
Hint: ln(RGDP) and ln(RTB) may not have the same lag structure. First, use OLS (with or without autocorrelation) to determine the maximum number of significant lags (up to 4) for each variable. Secondly, given the best lags identified, determine the maximum number of significant orders (up to 4) and various end-point restrictions. Warning: There is no way you can try out all possible combinations of about 3600 cases!