Using the person-year as the unit of observation (n = 12,864), we estimate parameters in yearly multivariate regressions with log mental health care costs as the dependent variable, and include as regressors demographic variables, a substance use disorder indicator variable, and indicator variables for various solo, double and triple combination treatment types. We then pool across fiscal years as permitted by results of hypothesis tests for parameter stability over the six-year 1994-95 through 1999-2000 time period.
Results from tests for parameter stability indicated decisive rejection of the null hypothesis of parameter equality across years for all coefficients other than the constant term (F85,12756 = 1.43, p-value 0.0059). Since parameter equality across years is rejected, pooling is not supported, and thus in we present parameter estimates from the yearly regressions. Recall that the omitted reference case is a male, caucasian, never had SSI or a substance use disorder, from Orlando, with a miscellaneous “all other” treatment type. Four sources of intertemporal instability merit particular attention.
PARAMETER ESTIMATES FROM LOGARITHMIC ANNUAL TREATMENT COST EQUATION
First, based on the yearly regressions, the evidence suggests that relative to caucasians, blacks are receiving increasingly less costly treatment, ceteris paribus
; the cost differential is about −16% in 1994-95, and increases to about −26% (exp(−0.298) = 0.742) by 1999-2000. For hispanics, the trend is not as clear. Relative to caucasians, individuals from all other races are also receiving increasingly less costly treatment, increasing from −13% in 1994-95 to −24% in 1999-2000. Related findings on treatment differentials by race have been reported in the literature by others.23
Second, the spending premium for treating schizophrenia in individuals who have ever had a substance use disorder is positive, but appears to be falling over time. Although this premium was about 18% and 23% in 1994-95 and 1996-97, respectively, ceteris paribus, by 1999-2000 it had fallen to about 5%.
Third, while the yearly treatment spending premia for only AP and only AT increased slightly between 1994-95 and 1999-2000 (by 2% and 6%, respectively), for only THY the annualized treatment costs fell by 75%, even as those for psychosocial rehabilitation fell 34%.
Fourth, although there are no general significant trends for annualized treatment spending premia involving the two dual combination therapies (any AP plus any THY, and any AT plus any THY), the two triple combination therapies display quite different trends. In particular, while the annualized spending premia for any AP plus any THY plus REHAB changes from insignificantly negative in 1994-95 to significantly positive in 1999-2000, for the any AT plus any THY plus REHAB triple combination therapy the negative premium doubles from −0.382 in 1994-95 to −0.786 in 1999-2000.
It is also worth noting that although the R2 varies from about 0.34 to 0.40, there is a general downward trend in the root mean squared error across the fiscal years, falling about 15% from 1.131 in 1994-95 to 0.984 in 1999-2000. Duan's smearing factor, which reflects the mean value of the exponentiated residuals from the yearly log-cost regressions, also declines over time, by 16.6 % from 1.926 in 1994-95 to 1.607 in 1999-2000, an AAGR of −3.56% per year..
Since the data do not support pooling across fiscal years, it is not possible to use exponentiated yearly indicator variable coefficient estimates to obtain quality-constant price index measures of treatment cost over time.24
Instead, we hold quality and other characteristic variables fixed over time, and then compare predicted spending. But at what years' values does one hold quality fixed? Should one employ base year quality and other characteristics fixed over time, those from the final year, or those from intervening years? We follow price index methodology, and for any bilateral time comparison, and take the geometric mean of price indexes based on the first and final years' values of the quality and characteristics variables.25
More specifically, we first take the composition of treatment types, demographic characteristics and substance use disorder data from the 1994-95 base year cohort. Holding these variables fixed over time for each individual, we then predict log annual mental health costs for each individual for each subsequent year, using the yearly parameter estimates. Since we hold the treatment type and other variables fixed over time at their base year levels, this is analogous to a fixed weight Laspeyres price index procedure, where our procedure is now interpreted as holding quality fixed at the 1994-95 base year levels. We then retransform from predicted log spending to natural cost units, take the mean of this predicted log spending, and then multiply by Duan's 
smearing factor estimate, separately for each year, to obtain predicted total mental health spending. This yearly predicted total mental health spending is then normalized by the 1994-95 base year predicted spending to convert the predicted spending series into a price index, with 1994-95 = 100.
We then do the polar extreme of this, using instead the composition of treatment types, demographic characteristics and substance use disorder values from the final 1999-2000 fiscal year in our study sample. We follow similar procedures in using the yearly parameter estimates to predict log- spending and retransform to natural spending units, then employ the yearly Duan smearing factor estimate, and finally, normalize to 1994-95. Use of this procedure is analogous to using the current time period fixed weights in the Paasche price index formulae, and is interpreted here as holding treatment quality constant at the 1999-2000 levels.
As has been emphasized by others, since in the consumer demand context the Laspeyres and Paasche price indexes bracket true changes in the cost-of-living, researchers have long advocated using the Fisher Ideal price index (the geometric mean of the Paasche and Laspeyres) as an even-handed compromise in choice of weights in making bilateral time series comparisons.26
We call these three price indexes the fixed weight Laspeyres, the fixed weight Paasche, and the fixed weight Fisher price index.
When the composition of treatment bundles changes considerably over time, as has occurred in the context of schizophrenia during our 1994-1995 through 1999-2000 time period, growth rates of quality-constant price indexes can vary considerably, depending on which years' weights are employed, i.e., those of 1994-1995, or of 1999-2000. To reduce this sensitivity to choice of fixed weights, price index researchers typically employ chained price indexes, in which weights are sequentially updated with each pair of adjacent time periods. Unlike the case with the fixed weight Laspeyres, fixed weight Paasche and fixed weight Fisher price indexes, with a chained Laspeyres index between fiscal years, say, 1996-1997 and 1997-1998, the bilateral time comparison would employ updated 1996-1997 base year weights, while the chained Paasche index would employ updated 1997-1998 current year weights. The chained Fisher Ideal price index is then computed as the geometric mean of the chained Laspeyres and chained Paasche price indexes. We compute both fixed weight and chained price indexes. Results are presented in .
QUALITY-CONSTANT FIXED WEIGHT AND CHAINED PRICE INDEXES
With the fixed weight Laspeyres index (first column), where treatment quality is held constant over all years at the 1994-95 values, the quality-constant price index drops about 14% in 1995-96, it drops another almost ten percentage points in 1996-97, fall slightly, then increases in 1999-2000, ending up at 75.25 in 1999-2000, a cumulative price decline of 24.75%, and an average annual growth rate (AAGR) of −5.53%.
For the fixed weight Paasche index (third column), where treatment quality is held constant over all years at the 1999-2000 values, the quality-constant price index drops even more to about 84 in 1995-96, but unlike the fixed weight Laspeyres that falls, the fixed weight Paasche increases to almost 86 in 1996-97. The fixed weight Paasche then falls to about 78 in 1997-98 and 1998-99, and then increases slightly, ending up at 81.85 in 1999-2000, a cumulative price decline of 18.15%, and an AAGR of −3.93%. Notice that the fixed weight Laspeyres declines considerably more rapidly than the fixed weight Paasche, −5.53% vs. −3.93%.
With the fixed weight Fisher Ideal price index (column 5, calculated as the square root of the product of the column 3 fixed weight Paasche and the column 1 Laspeyres indexes), there is a 15% decline in 1995-96, followed by four percentage point declines in 1996-97 and 1997-98, no change in 1998-1999, and an increase in 1999-2000. The fixed weight Fisher is always in between the fixed weight Laspeyres and Paasche indexes, ending up at 78.48 in 1999-2000, a 21.52% cumulative decline reflecting an AAGR of −4.73%.
With the chained price indexes, differences between the Laspeyres and Paasche price indexes become considerably smaller than with the fixed weight variants. As is seen in columns 2, 4 and 6 of , the AAGR for the chained Laspeyres at −5.68% is only very slightly different from that of the chained Paasche at −5.40%, with the Fisher being in between at −5.54%.
Aficionados of price index measurement will appreciate that a somewhat surprising result here is that the fixed weight and chained Laspeyres price index fall more rapidly than do their corresponding Paasche price indexes, counter to the usual inequality between these two indexes. A related empirical finding, where the Paasche price index increased more rapidly than the Laspeyres, has been reported by Berndt, Busch and Frank 
in the context of treatment costs for individuals diagnosed with acute phase major depression.27
The conventional wisdom result that Laspeyres indexes will increase by more (decrease by less) than the Paasche price indexes is based on the assumption of a constant utility framework with stationary and homothetic preferences, for in that context price relatives are negatively correlated with quantity relatives. As has been shown by, among others, Allen , in certain contexts (such as when viewed from the point of view of a multiproduct competitive supplier), price and quantity relatives can be positively correlated, resulting in the reversal of the usual Paasche increasing less than the Laspeyres inequality. Since observed price and quantity movements reflect the net outcome of changes in demand and supply, differing inequalities between measured Paasche and Laspeyres price index changes can occur over time, reflecting a variety of underlying shifts in demand and supply. In the current context, one can interpret the reversal of the usual Paasche – Laspeyres inequality as reflecting physicians' learning about the efficacy and increased tolerability of the more costly atypicals antipsychotics, changing their prescribing behavior towards the more costly atypicals, shifting “demand” curves to the right, thereby generating positive correlations between price and quantity relatives for the atypicals.28
In summary, although changes in mean mental health-related treatment costs for individuals diagnosed with schizophrenia, unadjusted for changes in treatment quality over time, have increased about 2.4% between 1994-95 and 1999-2000 (an AAGR of 0.5%), holding treatment quality constant over time results in a cumulative price decline of about 22% with the fixed weight Fisher Ideal index, an AAGR of −4.73%. Holding treatment quality constant in sequentially updated chained Fisher Ideal indexes yields a slightly larger cumulative decline of almost 25%, and an AAGR of −5.54%.