It does not appear that income has much effect on quantity no matter how it is incorporated into the model. It does not appear to be statistically significant if it enters linearly; nor is either term significant if it enters both in linear and quadratic form. Note, however, that residuals analysis following Regression A2 suggests that we should be cautious in drawing any inferences about anything from Regression A2.

The diagnos / het output provides just the relevant results from a set of regression of the squared errors from the previous regression on a variety of things to which they might be shown to be related. The squared errors from a preliminary naive OLS regression represent the best information we have about the actual sizes of the true s_i². If there is homoscedasticity, these conditional error variances should be independent of any other variable we consider. These results show that they are not. We reject the hypothesis of no relationship between the error variance and each thing except in one case. The exception is the test for ARCH (AutoRegressive Conditional Heteroscedasticity). This is only relevant if we have time series data, where there are patterns in the error variance over time (as opposed to patterns in the sign of the error over time). Since there is no particular order to these data, it is not surprising that the variance associated with adjacent observations is unrelated.

In the one-by-one regressions, the squared errors appear to be positively related to p and to p². They are unrelated to y. When we regress the squared errors on all three candidates, we see a high degree of multicollinearity between p and p2, such that it is impossible to distinguish, statistically, the independent contributions of these two variables to explaining the squared errors if both are used. It looks like a toss-up which we choose from among p and p2 to capture the variations in the error variance across observations. We definitely do not want to use y.

Since the error variance appears to vary directly with the magnitude of either p or p², and the weights should vary inversely with the error variance, therefore the weights should vary inversely with either p or p². The inappropriate weighting variable would be wt1. Either wt2 or wt3 is probably sufficient, with perhaps the slightest edge to 1/p, since the statistical significance of p as an explanatory variable for the squared errors is ever so slightly larger and the R-squared value for that model is slightly higher.

If my preferred model is taken to be Regression A9, the parameter point estimates change only very slightly (since different formulas are used to calculate them under WLS than under OLS). As for the standard errors, if we had the corrected OLS standard errors, they should be expected to be larger than the WLS standard errors--since WLS is more efficient than OLS under heteroscedasticity. However, all we get from SHAZAM is the uncorrected standard errors, which are just plain wrong, since they have been calculated using a formula that assumes you can factor out a common s², when you cannot. All the same, there appear to be no meaningful or surprising changes in the implications of this model when we correct for heteroscedasticity. Unfortunately, this is not a general result. You never know that heteroscedasticity correction will make little difference to your estimates or inferences until you do the weighted least squares model and find out.

In Regression A10, we have been careful to include the loglog option in the regression command, so SHAZAM knows that the dependent variable in this model has been logged. Thus the regression algorithm undertakes to adjust the formula for the log-likelihood to account for the logged dependent variable, thereby making the log-likelihood comparable to models which use the (raw) "levels" of the dependent variable. For the log-log model, the maximized log-likelihood is -253.223. For the levels model, it is -246.915. The higher value is obtained for the levels model, so it would be preferred. Recall that a log-likelihood can be interpreted as the log of the joint probability of observing the data that we have observed in our sample.

Residuals analysis following the log-log model in Regression A10 is conducted in Regression A11. The squared errors from the log- log model (note that e is redefined in Regression A10) are statistically significantly related to the magnitude of the log of price. Thus the log transformation is not successful, for these data, in eliminating the heteroscedasticity that plagues the levels data. Logging can sometimes eliminate heteroscedasticity, but only if the nature of the heteroscedasticity is exactly such that logging will get rid of it. If the heteroscedasticity is not of this kind, logging can fail to remedy the problem, or even make it worse.

Yes, the coefficient on the time trend variable T is positive and appears to be hugely statistically significant (although we'll have more on this later). The point estimate suggests that public school expenditure has been growing, on average, by $4.4 million per month over the time period from January 1964 through December 1995.

New public school construction appears to depend positively on the sizes of the first three cohorts (P1, P2, and P3), but not on the size of the oldest cohort of school-aged children (P4). Changes in the size of the P2 group appear to have the biggest influence on new public school construction. Again, we will see in a minute that we cannot really trust the t-ratios, although in this model they seem pretty "healthy."

There is considerable multicollinearity between the sizes of the four different age cohorts of school-aged children. One would expect this to compromise the individual statistical significance of the slope coefficients on these variables. The strong linear relationship between P4 and the others might indeed explain the statistical insignificance of its coefficient in the regression. (We'll argue in a minute that time-series hypothesis tests, before we have assessed the error properties, are always suspect. Thus, while it looks as though this multicollinearity has not ruined our ability to detect statistically- different-from-zero slope coefficients on three of the cohort-size variables, we cannot be sure. In any event, multicollinearity will make the parameter standard errors larger, meaning relatively more hypotheses will be deemed acceptable. I.e., our parameters estimates will offer less resolution than might have been possible with uncorrelated regressors. Here, unfortunately, there is no option of going back for a different sample. History is history. Although there might be hope for future data, say over the next 20 years.)

Inconveniently, whomever ran these regressions for you did not ask for a DWPVALUE (exact Durbin-Watson test statistic and p-value). All you get with this output is a point estimate of the D-W test statistic (0.2884) and a point estimate of the AR(1) error correlation (rho=0.85648). These look suspicious, even though you do not have d-tables in which to look up the critical values. Smells like positively serially correlated errors of at least AR(1), possibly more, since these are monthly data. Fortunately, Regression B2 provides pretty strong evidence of systematic relationships between the errors associated with different time periods at particular intervals of spacing. Positive serial correlation in the errors generally means that the standard errors on the parameter estimates are understated, leaving the t-ratios overstated, and the P-values too small. You end up rejecting a lot of hypothesis you really cannot reject. The point estimates are still unbiased, but your inferences are pretty worthless until you have corrected the problem.

Since we have monthly data and many processes have a regular annual cycle, we can expect a priori that 12th order autoregressive errors may be relevant in monthly data. Thus, we want to explore at least the relationship between current error and each of the first twelve lags of the error term. This is an interesting side-issue, intended to show if you understand serial correlation patterns in errors and can tie this to multicollinearity. If you have high-order serial correlation in the errors and you run this regression of current errors on twelve different lags of the same variable, then each e_t will be correlated with each e_t-1. If rho is large, there could be very substantial multicollinearity and it could get difficult to discern the individual coefficients in this regression. Despite this problem here, the first, eighth, and eleventh lags appear to be statistically significant. Note that there would be no way to "fix" this multicollinearity, since all "variables" are just different past observations on the same variable.

Consider an AR(1) model, where the presumption is that u_t = rho u_{t- 1} + e_t. SHAZAM comes up with an initial estimate of the correlation between current and lagged error terms and uses this to transform the regression equation by taking each variable (including the constant term), lagging it, and calculating the transformed variables according to the current-period value minus rho times the lagged-1-period value. SHAZAM then regresses uses these generalized differences in a regression where the error term is "fine" because the current minus rho-times-lagged error term is just e_t, which is pure noise and therefore fits the criteria for OLS. The "iteration" part: Next, SHAZAM takes the point estimates from this generalized difference regression and applies them to the raw X data to compute a fitted Y value for each observation. The difference between this fitted Y and the actual Y is a new estimate for e_t. If you take the correlation between this new e_t and its lagged value, you get an revised estimate of rho. Use this again in creating the generalized differences. Continue these iterations until some convergence criterion is achieved (either a stable largest-achievable maximized log-likelihood, or a stable smallest-achievable sum of squared errors). These final "fine-tuned" parameter estimates (intercept, slope, rho value(s)...and sigma-squared, of course) can then be reported along with their asymptotic standard errors and associated asymptotic t-ratios that allow us to test statistically the null hypotheses that individual rho parameters are zero.

Conveniently, there is an F-test provided to test the hypothesis that all coefficients on the P1 through P4 variables are simultaneously zero. This hypothesis is soundly rejected, even though ONLY P2 is now individually statistically significant. Just the P2 coefficient could account for this, however. The interesting test would have been to see a joint test for the coefficients that appear individually insignificant: P1, P3 and P4. You'd certainly want to try one of these. If it weren't for the multicollinearity issue, we'd probably conclude that public school construction only starts when there has been a surge in kindergarten-through-fourth-grade aged children. If preschool populations (P1) had an individually statistically significant effect on new school construction, you could say that construction anticipated enrollments. The insignificance of this coefficient could still be due to multicollinearity, however, so we cannot be entirely certain about this conclusion.

The left-out month is January, so the basic intercept term is the intercept for January. The coefficients on the other monthly dummies tell how much expected new public school construction differs in that month compared to it's expected level in January. For example, expected new public school construction in August is higher than in January by about $287 million. In February, it is lower by $15.9 million compared to January, although this difference is not statistically significantly different from zero. What is the overall pattern? Public school construction is highest in the summer months, when there is good weather and most kids are out of school. It is lowest in the winter months when weather is bad and attempts to build new structures at existing campuses would disrupt classes.