Goal for this lab:

Review some of the key concepts in Problem Set #7:

1. Run model interactively and view a number of plots of the relationships among key variables
2. Review simple model from basic Monetary Economics (equilibrium interest rate determination as a function Money Demand and Money Supply)
3. Examine consequences of failure to account for serial correlation in the errors (by using plain OLS)
4. Examine how goodness of fit is dramatically improved by the use of a more appropriate AR model
5. Examine AR(1) error specification:

   e_t = rho*e_t-1 + E_t

   where E is a pure "white noise" random error.
6. Examine AR(4) error specification (due to quarterly data):

   e_t = rho₁*e_t-1 + rho₂*e_t-2 + rho₃*e_t-3 + rho₄*e_t-4 + E_t

Details:

Keep this Web page active while you also access the usual SSC Lab Economics 143 window. You will need to "toggle" back and forth to receive instructions about which commands to enter for your SHAZAM program. This page will then offer some interpretation of what you see on the SHAZAM screen.

1. Sketch a plot of r versus M with a vertical Money Supply curve (determined by central bank policies) and a downward-sloping Money Demand curve. This simple model of Money Demand and Money Supply suggests that interest rates depend inversely on Money Supply and that outward shifts of the Money Demand curve (resulting from higher incomes) cause equilibrium interest rates to rise. View all of the empirical results in this exercise from the point of view of these expected effects. In a model to explain equilibrium real interest rates, the coefficient on real GNP should be positive, and the coefficient on real money supply should be negative.
2. Get into SHAZAM for Windows and open the file called intlab9.sha. You will also need the data in int2.dat.
3. Browse down through the program intlab9.sha and notice that it does several things. (You may want to read this as you first look at the program file, and then go over it again as you execute the program.)
   
   
   1. A date variable is created to convert the uninformative observation numbers into years, so we can readily identify the different time periods. (We should have done something like this with the credit.sha program, so the crucial May 1987 observation could have been readily identified.)
   
   
   
   2. The program uses the time series for the price index to calculate the quarterly rate of inflation on scale of 1992=100. This quarterly rate is multiplied by 4 to yield the corresponding annualized rate (since the nominal interest rate is quoted as an annualized rate).
   
   
   
   3. The real interest rate is calculated by taking the nominal interest rate and subtracting off the rate of inflation.
   
   
   
   4. The nominal data for money supply (M1 and M2), and for GNP, are also converted into real terms (constant 1992 dollar amounts). Note that since the price index is based on 100.0, rather than 1.000, we should multiply the (nominal amount/price index) by 100 so that we have not fundamentally changed the scale of measurement. Notice, however, that such a change of scale would not change the statistical significance of any variable (the t-ratio). It would affect the magnitude of the coefficient and the magnitude of the standard error on that coefficient, but not the variable's significance in the regression. The scale of the coefficient would simply adjust to offset the change in scale of the variable.
   
   
   
   5. We then do some preliminary simple regressions to see the pairwise relationships between interest rates and money supply, or interest rates and gnp. This will serve as background for the multiple regressions to come later. We will see that these simple regession show each explanatory variable, entered separately, to be statistically significant.
   
   
   
   6. For the OLS regression of realr on both realyp and realm2, note that we have exercised the "resid=" option and the "predict=" option on OLS, which allows us to save (a) the fitted sample regression function errors in "e" and (b) the fitted values of the dependent variable (the Y-hat values) for each observation in a variable which we choose to call "fitols" in this instance. We have also asked for the exact Durbin-Watson test statistic for first-order serial correlation in the errors, and for some diagnostics for multicollinearity.
   
   
   
   7. The we generate a number of lagged values of the error term, in order to "line up" the current and lagged values of e in rows of the augmented data set. We then look at a plot of e_t against e_t-1 and then e_t against e_t-4 as examples. We will be looking at the plots for visual evidence of serially correlated errors in these data.
   
   
   
   8. Finally, we look at the results from an AUTO command, used instead of OLS. The default option is "order=1" which is first-order serial correlation in the errors. We then try analogous models that allow for AR(2) errors and then AR(4) errors, just to see whether the fit of the model is substantially improved by these even greater degrees of generalization.
   
   
   
   9. Recall what is going on in the background if you specify an AUTO regression: SHAZAM does a preliminary OLS command and saves information about the errors to be used in constructing an initial estimate of rho, the correlation between current and last-period's errors. (Recall that rho can come from a STAT command on these two errors, from a regression of e_t on e_t-1, or from the DW test statistic, which is approximately equal to 2(1-rho).) SHAZAM then uses this initial estimate of rho to form the "generalized differences" of every variable in the regression (including the constant term). The transformed data allow a regression of the form:

      (Y_t - rho*Y_t-1) = b₁(1-rho) + b₂(X_t - rho*X_t-1) + (e_t - rho*e_{t- 1})

      If the original error term was e_t = rho*e_t-1 + E_t, then the error term in the transformed model is just E_t, which has perfectly acceptable properties (if, indeed the error process is AR(1)). SHAZAM then conveniently reports the results of the model (point estimates, standard errors, t-ratios, as though the original OLS specification had been used. With OLS, however, the standard errors were wrong. Now, they are correct because they come from a model that has, by construction, acceptable error terms. Notice the Durbin-Watson test statistic for the transformed model. It is very close to 2, which is the expected value of the DW test statistic if, in fact, the true rho value is zero.
   
   
   
   10. We also look at the second-order and fourth-order AUTO results, each model being a little more general in that it does not set to zero the effects of greater-lagged error terms on the value of the current error term. The fourth- order model captures the important relationship between e_t and e_{t- 3} that shows up in the regression of current error on the full set of four different lags of the error term. Watch what happens to the t-ratios on the slope and intercept parameters as we go to more-and-more general models of the structure of the error process. The more serial correlation we manage to purge from the model, the more "correct" are the inferences to be drawn. Note the iterations that the program undertakes to find the best value for rho. Iterations generally belie a nonlinear optimization process in the background when you ask for a particular procedure (like AUTO) to be done. Some serially correlated error models estimate rho at the same time as the regression slopes and intercept, treating all unknown parameters equivalently. The optimization produces asymptotic t-test statistics that can be used to test whether any particular estimates rho value is statistically significantly different from zero.
   
   
   
   11. The last block in the program has a bunch of plot statements.



4. Proceed as follows for the plots:



1. look at how nominal interest, inflation, and real interest move

   plot r infl realr date / gnu line




2. plot all of the variables used in the model

   plot realr realyp realm2 date / gnu line

   Why is the line for realr coincident with the horizontal axis? Think about the magnitude of the realr variable, as compared to the realyp and realm2 variables.



3. check multicollinearity

   plot realyp realm2 / gnu line

   This is what I have been calling the "footprint" of the data for the model. The line that joins the points traces the time path of the joint evolution of these two variables. The lower left-hand end is the beginning of the data (1959), whereas the upper right is the most recent stuff (second quarter of 1997). Notice that there is fairly high multicollinearity. However, the two variable do display some independent variation, since the relationship is not a straight line.



4. look at the fit from an ols model

   plot fitols realr date / gnu line

   The variable "fitols" contains the fitted values of the dependent variable (the Y- hats) from the plain (naive) OLS model. This plot shows the variability in the actual data on realr, as opposed to the very smooth increase in realr predicted by the model. Would you call this a model with good predictive power? We'll assess this by looking at the R-squared value for this model when we run the program again and save the output to a file. Would you get an enthusiastic reception from the consortium of bankers who have pooled their resources to hire you as a consultant on the matter of predicting interest rates??? Probably not.



5. look for patterns in ols residuals

   plot e date / gnu line

   We saved the OLS residuals, so we can plot these as a function of time to see if there appear to by "cycles" in the errors--patterns that would lead to a suspicion of positive serial correlation.



6. look at the fit from an AR(1) error model

   plot fitauto realr date / gnu line

   When we fitted an AR(1) error, we saved the fitted values as "fitauto." Examine how well "fitauto" matches the actual data on real interest rates contained in realr (as contrasted with "fitols"). How would you characterize the fit now. Remember to check the R-squared value of the first AUTO command to see a measure of the goodness of fit of this specification.



7. look at the fit from an AR(2) error model

   plot fitauto2 realr date / gnu line

   We see similar much-improved fits from any of the AR models, as compared with the OLS model.



8. look at the fit from an AR(4) error model

   plot fitauto4 realr date / gnu line

   And again, a much better fit than OLS.



5. The key part of the output to look at is the last part. Here is the output from an OLS command with the / list option attached. After such a regression, SHAZAM automatically saves and prints the actual and predicted values of the dependent variable, followed by the implied sample regression errors associated with each observation in the sample. The latter information is then shown in a simple plot, where positive errors lie to the right of a center line, and negative errors lie to the left. By scanning down this listing, it is usually easy to discern visually whether there is a cyclical pattern to the errors. We already saw this information in one of the nice gnu line plots, but the /list option is one way to get better resolution on the error properties without having to run graphics plots to see these errors. Since the data are printed out vertically, there is no problem with the horizontal compression that obscures everything interesting in a conventional horizontal dot-matrix plot.