Un-COLA: Why Have Cost-of-Living Clauses Disappeared from Union Contracts and Will They Return?

Bratsberg, Bernt

James F. Ragan, Jr. [*]

Bernt Bratsberg [+]

For more than 20 years, unions have been trading cost-of-living adjustment clauses (COLAs) for other forms of compensation. Various explanations have been offered for the erosion of COLA coverage--including reduced inflationary uncertainty, lower union power, and structural shifts in the economy--but the relative importance of these and competing hypotheses remains untested. We investigate the reasons for the decline in COLA coverage using a pooled cross-sectional, time-series model that accounts for industry fixed effects and recognizes the multiyear nature of most union contracts. After assessing the relative importance of alternative hypotheses, we conclude with a discussion of the potential for a rebound in COLA rates.

1. Introduction

One of the major changes in collective bargaining in recent decades has been the gradual elimination of cost-of-living adjustments (COLAs) from union contracts. In 1976, 61% of union workers covered by major collective bargaining contracts had COLA provisions, but by the end of 1995, when the U.S. Bureau of Labor Statistics stopped collecting data on collective bargaining settlements, COLA coverage had fallen to 22%. [1] Although the decline in COLA rates has been studied extensively, there is no agreement as to which factors are primarily responsible for this decline. One view attributes the elimination of COLAs, or escalator clauses, to declines in inflationary uncertainty; a second view emphasizes the erosion of union power; yet another view focuses on structural shifts in the U.S. economy. Previous research has not determined the relative importance of these and other hypotheses.

Another reason to reassess COLA determination is that the economy has changed since the early research was completed. The bulk of studies to examine COLA incidence statistically have relied on sample periods that end in 1982 or earlier. [2] Since then, there have been major changes in union strength, inflationary uncertainty, and other potential determinants of wage indexation, including deregulation of certain industries. In addition, more than 90% of the decline in COLA rates has occurred since 1982. It is worthwhile to determine whether the factors deemed to be important in the early studies are still important.

We provide a comparison with previous research, formally assess the contributions of various factors to the erosion of COLA rates, and provide insights as to likely changes in COLA coverage in the future. If inflation picks up, and with it inflationary uncertainty, [3] will COLA rates rise to levels not seen since the 1970s? What would be the consequences of a rebound in union power, of further economic deregulation, of likely industrial and demographic shifts in the economy? Once the major causes of past changes in COLA coverage are understood, it becomes possible to project the consequences of changes in key economic variables. Given the concern in macroeconomics with wage indexation and the linkage between wages and prices (Gray 1976; Fischer 1986; Ball and Cecchetti 1991; Schiller 1997), the extent to which COLA coverage responds to economic factors has important implications for the economy.

Underlying the empirical analysis is a pooled cross-sectional, time-series model of COLA incidence that accounts for industry fixed effects. To estimate the model, we assemble a panel data set of COLA coverage and an array of characteristics for 32 private-sector industries over a 22-year period. Because COLA clauses are multiyear in nature, the model also recognizes that the percentage of workers covered by COLA clauses in a particular year depends on conditions in both the current year and previous years.

Previewing results, we find that the major cause of the decline in COLA coverage has been the reduction in inflationary uncertainty. Second in importance has been the erosion of union power. Both the percentage of the industry's workforce that is unionized and the estimated union wage premium in the industry are highly significant determinants of COLA coverage. But because the union wage premium has remained largely intact in most industries, changes in the premium are much less important than losses in unionization in explaining the overall decline in COLA provisions. Economic deregulation, industrial shifts, and an increased presence of women in the workforce have had modest effects on COLA rates. On the basis of our findings, a rebound in inflationary uncertainty to the levels experienced in the 1970s would raise COLA coverage by 10 percentage points.

2. Background

For workers in long-term contracts, COLAs provide at least partial protection from the consequences of unexpected inflation. For the employer, however, COLAs increase rigidity of real wages and relative wages. Whether or not a given union contract contains an escalator clause depends on the valuation of such a provision by both workers and the employer and, in turn, on risk preferences of the two parties.

In the efficient-contract model of Ehrenberg, Danziger, and San (1984), the union and the employer jointly maximize the weighted average of expected utility of the representative union member and expected profits, with weights based on the relative bargaining power of the two parties. In addition to determining whether or not the contract contains a COLA clause, a companion issue is the degree to which any COLA clause is indexed, where [epsilon], the elasticity of indexation, measures the extent to which higher prices translate into higher nominal wages. Ehrenberg, Danziger, and San show that the optimal elasticity of indexation depends on the risk preferences of workers and of the employer.

Workers are assumed to be risk averse. If the employer is risk neutral and if there are no costs of COLA provisions (administrative or otherwise), then all contracts contain COLA clauses and these contracts are fully indexed ([epsilon] = 1). When the employer is also risk averse, the degree of indexing depends on relative risk aversion. The greater the risk aversion of workers relative to the employer, the closer [epsilon] is to 1. [4]

Including COLA clauses in contracts is likely to entail costs. Once these costs are recognized, the probability that a contract contains a COLA clause depends on the costs of COLA as well as the benefits. As Ehrenberg, Danziger, and San observe, an increase in costs reduces the likelihood that a contract will contain a COLA clause. Another conclusion of their model is that increased risk aversion of workers increases the probability of a COLA. Finally, the model implies that "given that workers are risk averse, the more uncertain inflation is the greater is the gain to them of indexation and thus the greater is the likelihood of indexation" (p. 16). Consistent with this prediction, most studies find a strong positive relationship between inflationary uncertainty and COLA coverage (Cousineau, Lacroix, and Bilodeau 1983; Hendricks and Kahn 1983, 1985; Holland 1984, 1986; Prescott and Wilton 1992; Weiner 1993).

Figure 1 depicts movements in inflationary uncertainty, as proxied by the standard deviation of inflation forecasts from the Livingston Survey, and aggregate COLA coverage. Although movements in the standard deviation of inflation forecasts are more erratic, the two series display broadly similar patterns (the simple correlation coefficient is 0.73). When inflationary uncertainty picked up in the 1970s, so did the incidence of indexation. More recently, both series have declined. In light of the previously cited studies, it is likely that at least part of the erosion in wage indexation can be traced to the reduction in inflationary uncertainty.

The model of Ehrenberg, Danziger, and San does not explicitly consider how union bargaining power affects the decision of whether or not to index a contract. When Hendricks and Kahn (1985) recast the model so that bargaining power enters the solution for the optimal degree of indexation, they show that "higher bargaining power raises the incidence of indexation when [epsilon] [less than] 1" (p. 145). Because data in both Hendricks and Kahn and in Card (1986) indicate that [epsilon] tends to be less than 1, the implication is that greater union bargaining power can be expected to increase the likelihood of COLA coverage. Empirical evidence supportive of this proposition has been uncovered by Hendricks and Kahn (1983, 1985), Prescott and Wilton (1992), and Weiner (1993).

Another issue concerns the extent to which trends in aggregate COLA coverage result from changes in employment patterns across industries, rather than from trends in industry COLA rates. Devine (1996, p. 25) notes that union employment has declined faster in "the more COLA-prevalent manufacturing sector" than in nonmanufacturing, though she does not attempt to quantify the effect of changing employment patterns on overall COLA coverage. Davis et al. (1990) go further, asserting that shifts in employment patterns are the principal cause of declines in the aggregate COLA rate:

The proportion of workers with COLA coverage ... declined slowly from the end of 1976 through 1984 largely because of employment losses in industries in which COLA clauses were common. (p. 7)

Unfortunately, the study provides no empirical evidence to support its claim.

Lending credence to this view, however, are two earlier studies. Examining the modest two percentage point decline in COLA coverage between 1979 and 1984, Mitchell (1985, p. 595) finds that "changes in industry mix account for almost all of the change." Studying a longer time period, 1977 to 1986, Weiner (1986) reaches an apparently similar conclusion:

On net, 2.5 million workers lost their COLAs. Sixty-nine percent of this decline was attributable to employment shifts while 31 percent was attributable to COLA eliminations. (p. 15)

But unlike the Davis and Mitchell studies, Weiner focuses on the change in the absolute number of union workers covered by COLA clauses rather than the change in the percentage of workers covered. Over the period Weiner studies, the number of workers with COLA clauses fell for two reasons--a decline in the number of workers who belong to unions and a decline in the percentage of union workers with COLA clauses. Because Weiner's technique does not differentiate between these two sources of "employment shifts," it cannot be used to infer whether changing patterns of union employment are more or less important than declines in industry COLA rates. Stated differently, even if the pattern of union employment across industries had remained constant, the number of union workers covered by escalator clauses would have declined. Whereas the preceding four studies raise the possibility that shifting employment patterns may have contributed importantly to the erosion of COLA coverage since 1976, none of these studies p rovides compelling evidence in support of this hypothesis.

3. Empirical Framework

To examine the importance of alternative explanations for the decline in indexation, we develop an empirical model of COLA determination in the union sector. Because contracts containing escalator clauses cover a period of more than one year, the percentage of workers covered by COLA clauses depends not only on contracts negotiated that year, but also on contracts negotiated in previous years. For instance, assuming that contracts are three years in duration (Douty 1975; Mitchell 1985; Hendricks and Kahn 1986), the percentage of workers covered by COLA clauses at the end of 1995 is influenced not only by contracts negotiated in 1995, but also by contracts negotiated in 1994 and 1993.

More formally, let [COLA.sub.it] depict the fraction of union workers in industry i covered by contracts with COLA clauses as of period t. Further, let [X.sub.it] denote a (row) vector of characteristics influencing contract negotiations in period t. Given the three-year horizon, [COLA.sub.it] is a function of conditions in the current and previous two years:

[COLA.sub.it] = [[[sigma].sup.2].sub.s=0] ([[alpha].sub.ts][X.sub.i,t-s])[beta] + [u.sub.it], (1)

where [[alpha].sub.ts] is the fraction of workers covered by contracts in year t that were negotiated s periods prior to year t, that is, in year t - s. In other words, [COLA.sub.it] depends on a weighted average of present and past values of explanatory variables, with weights equal to the fraction of workers whose contracts were negotiated in a particular year. [5] Using available information on the distribution of union contracts, we calculate. the values of [[alpha].sub.ts] over a three-year time horizon. [6] The procedure for calculating values of [[alpha].sub.ts] is detailed in the appendix, and values of these weights are reported in Table 1.

Model Specification

According to the efficient-contract model, union contracts include a COLA clause when the weighted average of expected profits and workers' expected utility is greater with wage indexation than without. As implied by the Hendricks and Kahn formulation of the model, COLA coverage also depends on relative bargaining power of unions. On the basis of this framework, the fraction of workers with COLA clauses can be estimated as a function of union bargaining power and those factors that influence the costs and benefits of indexation. The major variables likely to influence COLA rates are described in the remainder of this section.

Union Bargaining Power

Union bargaining power in the industry is proxied by a pair of variables--the fraction of the industry's workforce belonging to a labor union and the estimated union wage premium in the industry. The estimated union wage premium is obtained from auxiliary wage regressions closely following the approach of Linneman, Wachter, and Carter (1990). [7]

Although the union wage premium may influence COLA coverage, it is important to recognize that this premium may be determined simultaneously with COLA coverage. In particular, during contract negotiations a union of given bargaining power may forgo a COLA clause for higher wages or vice versa. Such trade-offs generate a contemporaneous negative correlation between unobservable determinants of the union wage premium and COLA provisions. In turn, this correlation may lead to a negative bias in the estimated coefficient of the union wage premium in the COLA equation. For this reason, we examine the sensitivity of coefficient estimates to simultaneity bias in the model and report results with and without instrumenting for the union wage premium.

Inflationary Uncertainty

Assuming that workers are risk averse, the benefits of COLA provisions are directly related to inflationary uncertainty. When inflation is highly variable and therefore difficult to predict, the protection afforded by COLA clauses is high. But when inflation can be predicted with great confidence, as when inflation is steady, the value of COLA protection diminishes. Inflationary uncertainty is measured by the standard deviation of 12-month, year-end forecasts of inflation, obtained from the Livingston Survey, lagged one year. [8]

Industry Conditions

Economic deregulation has spurred competition in trucking, airlines, and other industries. In response, employers in these industries have pressured unions to accept lower compensation, and the cost of resisting that pressure--employment loss in the union sector--has increased. To the extent COLAS are less painful for unions to part with than other forms of compensation (base pay, pensions, etc.), COLA coverage is predicted to be lower in an industry after deregulation. [9] To capture the effect of deregulation, the model includes a dummy variable set equal to unity once an industry becomes deregulated. Deregulation is assumed to take place in 1978 for the airline industry; in 1980 for trucking, railroads, and finance, insurance, and real estate; and in 1984 for communications.

The cyclical conditions prevailing in an industry may also influence the percentage of workers with escalator clauses. We hypothesize that workers place a relatively greater premium on wage indexation when the labor market is strong. And when conditions deteriorate sufficiently that workers agree to lower real wages, it is easier for unions to accept the loss of indexation than a rollback in the base wage rate. Our measure of labor market tightness is the industry-specific unemployment rate, calculated from the Annual Demographic (March) files of the Current Population Survey.

Apart from the reasons previously articulated, COLA rates may vary from industry to industry. For example, Ehrenberg, Danziger, and San (1984) uncover evidence that industry-specific measures such as the elasticity of product demand with respect to unanticipated inflation and the industry-level replacement rate of unemployment insurance affect levels of COLA1 coverage across manufacturing industries. To the extent that unobserved industry characteristics are time invariant, we expect such factors to be captured by industry dummy variables. (For completeness, we also estimate the model without industry fixed effects.)

Other Factors

Demand for COLAs may depend on the sex and racial composition of union employees and the type of work they do. Not only may there be inherent differences across employees in risk aversion, but differences in the value of COLA clauses may also arise because of cultural differences, differences in wealth, and other factors. For example, given a shorter average work horizon, women may be less concerned than men about the effect of unexpected inflation on future wages. With respect to race, Ehrenberg, Danziger, and San (1984) argue that blacks have less access to capital markets than whites and thus have greater demand for COLAs to stabilize consumption over time.

Part-time employees, by virtue of their shorter job horizon, are likely to place a lower value on escalator clauses than do full-time employees. Compared with white-collar workers, blue-collar workers are less likely to have diversified portfolios, including home equity, and therefore may be more vulnerable to the consequences of unexpected inflation. In addition, if wages constitute a larger share of their income, the insurance value of COLA provisions may be greater for blue-collar employees. On the basis of this reasoning, COLA coverage is predicted to be positively related to the fraction of minority and blue-collar workers in the union sector of the industry and negatively related to the fraction of women and part-time workers. [10]

4. Data

We collected information on COLA coverage by industry from the U.S. Bureau of Labor Statistics, which tracked major collective bargaining settlements in the private nonagricultural sector through 1995. Because data on the distribution of length of union contracts (described in the Appendix) are not available for earlier years, the first year of our study is 1974. From published tabulations of COLA statistics, it is possible to construct consistent coverage series spanning the period 1974-1995 for 32 private-sector industries. Table 2 lists these industries and provides descriptive measures of COLA coverage for the sample period separately by industry.

Table 2 indicates considerable variation in COLA rates across industries. For example, union workers in both the transportation equipment industry (which includes auto manufacturing) and the tobacco industry averaged greater than 90% COLA coverage over the sample period. In contrast, union workers in the petroleum, paper, and lumber industries each had average coverage rates of less than 5%. Figure 2 plots COLA coverage in 1995 (the overall low year in the sample) against coverage in 1976 (the overall peak year) by industry and provides insight into industry trends in COLA coverage. It is of note that only five industries experienced increases in COLA rates between 1976 and 1995. Three industries--trucking, mining, and railroads--saw coverage rates drop from nearly 100% to zero.

The panel data set underlying the empirical analysis covers the 32 industries listed in Table 2 and the years 1974 through 1995, [11] resulting in 704 observations in the data set. Table 3 formally defines the variables used in the analysis and lists data sources and descriptive statistics.

5. Empirical Analysis

Table 4 presents estimates of the empirical model of COLA determination. The table first lists estimates of the baseline model, followed by four alternative model specifications. To assess the sensitivity of empirical findings to econometric assumptions, the table next lists results based on four alternative estimators or sample specifications (in turn, a tobit model, an instrumental variable estimator, a restricted sample, and a generalized least squares (GLS) procedure that accounts for an MA(2) process in the residuals).

Baseline Model

Consider first the baseline specification in column 1. As coefficient estimates reveal, the degree of COLA coverage in an industry depends importantly on the strength of unions in the industry. Whether we measure union power by the fraction of workers in the industry who are union members or the union wage premium, union strength is associated with a significantly higher incidence of COLAs. According to the parameter estimates, a one percentage point increase in the unionization rate boosts the industry COLA rate by 0.6 percentage points and a one percentage point increase in the union wage premium raises COLA coverage by 0.4 percentage points. Although other studies do not consider the effect of the union premium, the unionization rate has appeared in previous research. Ebrenberg, Danziger, and San (1984) and Hendricks and Kahn (1985) estimate that the unionization rate has a smaller impact than we find, whereas Weiner (1993) finds a modestly larger effect.

As expected, COLA rates vary positively and significantly with inflationary uncertainty. The standard deviation of inflation forecasts from the Livingston survey, our proxy for inflationary uncertainty, ranged between one-half and two percentage points over the period 1956 through 1995 (see Figure 1). On the basis of the estimated coefficient from column 1, a one percentage point increase in the standard deviation of inflation forecasts is associated with a 20 percentage point increase in COLA coverage. Inflationary uncertainty therefore emerges as a dominant factor in the decision of whether or not to bargain for COLA protection. Despite measuring uncertainty differently, other studies find roughly similar effects. Compared to our 20-point increase, the models of Cousineau, Lacroix, and Bilodeau (1983) and Weiner (1993) place the figure at 15 points, whereas the estimates of Hendricks and Kahn (1985) imply increases of 41 points in manufacturing and 23 points in nonmanufacturing industries.

The percentage of union workers with escalator clauses also depends on regulatory and economic conditions in the industry. As predicted, COLA incidence declines once an industry is deregulated. According to estimates from column 1, COLA coverage is a statistically significant 19 percentage points lower after deregulation. COLA coverage also has a pronounced cyclical component--a one percentage point increase in the industry's unemployment rate reduces the COLA rate by an estimated 1.3 percentage points.

The incidence of COLA provisions is inversely and significantly related to the female share of union employment in the industry, as in Hendricks and Kahn (1985). This result is consistent with the argument that women, because they have a shorter expected job tenure than men, are less concerned with protecting future wages from unexpected inflation and therefore place a lower value on COLA clauses. An alternative interpretation is that COLA incidence is lower in female-dominated industries because union power is lower in these industries. But given our efforts to control for union power, we discount this possibility. Although not statistically significant, the other demographic variable, BLACK, has the positive sign predicted by Ehrenberg, Danziger, and San. In addition, and as hypothesized, COLA coverage is inversely related to the fraction of union workers who work part-time and positively related to the fraction in blue-collar jobs. This latter finding is consistent with evidence from Hendricks and Kahn.

Supplementary Regressions

A Role for Expected Inflation?

Economic theory suggests that the expected rate of inflation should not affect indexation in wage contracts. This follows directly from the model of Danziger (1984). As Card (1984, p. 3) explains: "The uncertainty associated with a long term nonindexed contract depends on the predictability and not the level of inflation." For this reason, Card, Danziger, and others (Ehrenberg, Danziger, and San 1984; Hendricks and Kahn 1985) argue persuasively that COLA coverage depends on the variability of inflation but not the rate of inflation. Nonetheless, the limited empirical evidence to date is mixed. When Cousineau, Lacroix, and Bilodeau (1983) include measures of both expected inflation and variability of inflation in a probit model of COLA incidence in union contracts, coefficients of both variables are positive and significant. But when Prescott and Wilton (1992) estimate a similar specification, they find no evidence that the expected rate of inflation affects COLA coverage. [12] Moreover, certain studies (e.g. , Weiner 1993; Holland 1995) have used the rate of inflation as a proxy for inflationary uncertainty, relying on the previously observed correlation between the two measures. Given theoretical considerations that only inflationary uncertainty should matter, it would be reassuring if analysis of the data indicates that it is inflationary uncertainty rather than the expected rate of inflation that drives results.

To provide insights into this matter, we re-estimated the model, first replacing inflationary uncertainty with expected inflation (from the Livingston Survey) and then including both expected inflation and inflationary uncertainty. As anticipated, when expected inflation is the only measure of inflation, this variable is positively and significantly correlated with COLA coverage (see column 2). More important, when both variables are included in the model (column 3), only inflationary uncertainty is statistically significant. Consistent with the findings of Prescott and Wilton (1992), the implication is that--from an empirical perspective as well as theoretically--the component of inflation that influences COLA coverage is inflationary uncertainty.

Time Trend

To examine the sensitivity of results to inclusion of trend variables not already captured by the empirical model, we next augmented the baseline model with a time trend. This experiment also addresses the extent to which variables in the model simply capture a trend. As the results in column 4 indicate, the coefficient of the time variable is not statistically significant. [13] Although the addition of this variable reduces coefficient estimates of UNION and BLUECOLLAR and increases standard errors of these coefficients, indicating that these measures are correlated with TIME, results in general are robust to inclusion of the trend variable. The indication is that the variables in the baseline model adequately explain trends of the dependent variable and that the time trend does not belong in the equation. [14]

Unobserved Industry Effects

The baseline model in column I allows for fixed industry effects on COLA coverage. [15] For completeness, in column 5, we report coefficient estimates from a model that excludes industry effects. In a relatively short panel such as ours, this alternative specification relies on variation in dependent and independent variables between industries when identifying parameters of the model. In contrast, the fixed industry effects estimator in column 1 relies solely on changes in variables within industries. As such, a comparison of coefficient estimates in columns 1 and 5 provides insights into differences in COLA coverage between versus within industries and sheds light on how failure to control for unobserved industry effects can generate misleading results. Although most results in the two columns are qualitatively similar, certain coefficient estimates differ appreciably. For example, the estimated cyclical sensitivity is more than twice as great when the model excludes industry effects, indicating that indus tries with high unemployment rates are less likely than other industries to have COLA clauses.

More dramatic is the difference in parameter estimates for the deregulation variable. The positive coefficient estimate in column 5 reflects the fact that COLA coverage in industries that were eventually deregulated was higher than the economywide average before deregulation. But as column 1 reveals, COLA coverage dropped significantly in these industries after deregulation. Thus, the important negative effect of deregulation becomes apparent only after industry effects are included.

Also important is the sign reversal of the coefficient of the variable measuring female union membership. Results in column 5 suggest that, other things equal, higher female union membership increases COLA coverage, whereas fixed-effects estimates demonstrate that increases in the female share of union employment within an industry actually are associated with lower COLA coverage. The contrasting results stem from the fact that in a cross section, industries with above-average female union representation (such as communications) tend to have higher rates of COLA coverage than industries with low female union membership (such as construction). However, over the sample period, industries that experienced increases in female union membership (e.g., printing and trucking) also saw above-average reductions in COLA coverage rates.

What these results indicate is that industry fixed effects are important. Unless one accounts for inherent differences across industries in the tendency to index for inflation, one is likely to draw incorrect inferences as to the impact of key economic variables.

Econometric Issues

Tobit

Although results presented to this point support the baseline model of column 1, several econometric issues have yet to be examined. The first issue concerns the extent to which results are sensitive to treatment of censoring. Because the fraction of union workers in an industry covered by COLA clauses cannot be less than zero or greater than one, the dependent variable in the COLA equation is censored at zero and one. [16] Column 6 reports results obtained from a two-limit tobit model including industry fixed effects. Results are generally comparable to those of the baseline model (column 1). For this reason, and because further econometric checks are cumbersome in the tobit model, the remaining analysis ignores censoring of the dependent variable.

Endogeneity of PREMIUM

The model of COLA determination outlined in preceding sections points to a possible trade-off between wages and COLA provisions in collective bargaining--a trade-off that could negatively bias the estimated coefficient of the union wage premium in the COLA equation. This view is consistent with that of Hendricks and Kahn (1986), who argue that COLA coverage is endogeneous in their model of individual wage settlements in manufacturing. Indeed, when they account for endogeneity of the COLA variable, Hendricks and Kahn uncover evidence that unions accept lower wages in return for COLA provisions. [17]

We examine the robustness of the model to simultaneity bias by use of an instrumental variable estimator; results appear in column 7. [18] The coefficient of the union wage premium does become larger when we instrument for this variable--which is consistent with the preceding argument that there may be a temporal, negative correlation between unobservable determinants of wages and COLA coverage. Because failure to account for this correlation understates the effect of bargaining power, we proceed using the instrumental variable estimator. However, instrumenting for PREMIUM or dropping the variable from the model (results not shown) have only modest effects on estimates of other coefficients, thus giving no indication that endogeneity bias invalidates estimation of the impact of other determinants of COLA coverage.

Serial Correlation

A final econometric issue concerns possible serial correlation of the error term. From a theoretical perspective, the moving-average process implicit in the regressors of Equation 1 might also be present in the regression error. Moreover, the Durbin-Watson statistic for the baseline model, 0.476, provides strong indication of serially correlated errors. Although serial correlation does not induce biases in coefficient estimates, it does lead to misstated standard errors and invalid statistical inference.

We address the issue of serially correlated errors in two ways. First, we follow the approach of Ehrenberg, Danziger, and San (1984) and re-estimate the model using only observations that are three years apart, beginning with 1974 (see Table 4, column 8). Not surprisingly, this experiment raises standard errors; however, standard errors do not increase appreciably more than one would expect as a result of the large reduction in sample size. More importantly, perhaps, statistical inference is largely unchanged. [19]

Because reducing the sample size is an inefficient approach to dealing with serially correlated errors, we next adopt the GLS methodology of Rowley and Wilton (1974) and estimate the model accounting for an MA(2) process in the residuals and using the MA weights given in Table 1. [20] Specifically, the GLS approach assumes the error term in Equation 1 follows a moving-average process similar to that of the explanatory variables and that [u.sub.it] is a weighted average of the present and past two realizations of a white-noise error term, v:

[u.sub.it] = [[[sigma].sup.2].sub.s=0] [[alpha].sub.ts][v.sub.i,t-s] = [[alpha].sub.t0][v.sub.i,t] + [[alpha].sub.t1][v.sub.i,t-1] + [[alpha].sub.t2][v.sub.i,t-2] (2)

Results from the GLS estimator are reported in Table 4, column 9. [21] The Durbin-Watson statistic of the transformed MA(2) equation is 2.24, which means that we cannot reject the hypothesis of no autocorrelation at conventional levels of significance--the GLS approach appears to adequately clean the model for serially correlated errors. [22] Results also indicate efficiency gains from the GLS estimator, with standard errors generally smaller than in column 7. Coefficient estimates are remarkably stable across estimation methodologies, suggesting that serially correlated errors are not generated by omitted variables that are correlated with the explanatory variables of the model. Therefore, although we uncover evidence of autocorrelation in residuals, accounting for serial correlation does not impact coefficient estimates and leaves statistical inference unchanged.

Summary

The econometric robustness analysis shows that estimation of the COLA equation is not sensitive to treatment of censoring of the dependent variable, with results based on a two-limit to bit model comparable to results from a model that ignores censoring. On the other hand, the union wage premium appears to be endogenous in the COLA equation, prompting us to instrument for this variable. Although the estimated coefficient of the union wage premium is larger when the instrument is used, other results are unaffected. We also uncover evidence that residuals are serially correlated, although there is no indication that any bias in standard errors caused by serially correlated errors invalidates statistical inference. Given these findings, empirical results discussed in the next section are based on the instrumental variables estimator incorporating a GLS procedure to account for the MA(2) process in residuals (column 9 of Table 4).

6. Decomposing the Change in Aggregate COLA Coverage, 1976-95

Table 4 provides insights concerning various factors responsible for the decline in COLA coverage, but results thus far do not reveal the relative importance of each factor, nor do they shed light on the role of changing patterns of union employment. To address these issues, we decompose the change in aggregate COLA coverage over the period 1976-95 into three components: capturing the impacts of changes in the values of the explanatory variables in the COLA equation, changes in unexplained residuals, and changes in relative employment.

We start by observing that the aggregate COLA rate for union workers in year t can be written as the average of industry COLA rates, each weighted by relative union employment of the industry:

[COLA.sub.t] [equivalent] [[[sigma].sup.32].sub.i=1] [[phi].sub.it][COLA.sub.it] (3)

where [[phi].sub.it] equals the share of union employment in period t originating in industry i.

The aggregate COLA rate for union workers fell from a peak of 61% in 1976 to 22% in 1995, and this decline can be partitioned into different components:

[COLA.sub.95] - [COLA.sub.76] = [[sigma].sub.i] [[phi].sub.i95][COLA.sub.i95] - [[sigma].sub.i] [[phi].sub.i76][COLA.sub.i76]

= [[sigma].sub.i] [[phi].sub.i95][COLA.sub.i95] - [[sigma].sub.i] [[phi].sub.i95] + [COLA.sub.i76] [[sigma].sub.i] [[phi].sub.i95][COLA.sub.i76] - [[sigma].sub.i] [[phi].sub.i76][COLA.sub.i76]

= [[sigma].sub.i] [[phi].sub.i95]([COLA.sub.i95] - [COLA.sub.i76]) + [[sigma].sub.i] ([[phi].sub.i95]) - [[phi].sub.i76][COLA.sub.i76]. (4)

To proceed, let [X.sup.*].sub.it] denote the weighted (row) vector of regressors, where the weights are given in Table 1. Substituting [[X.sup.*].sub.it][beta] + [u.sub.it] for [COLA.sub.it],

[COLA.sub.95] - [COLA.sub.76] = [[sigma].sub.i] [[phi].sub.i95]([[X.sup.*].sub.i95] - ([[X.sup.*].sub.i76])[beta] + [[sigma].sub.i] [[phi].sub.i95]([u.sub.i95] - [u.sub.i76]) + [[sigma].sub.i] ([[phi].sub.i95] - [[phi].sub.i76])([COLA.sub.i76] (5)

where [u.sub.it] is the residual for industry i in period t.

The first term of the right-hand side of Equation 5 measures the estimated impact of changes in the values of explanatory variables. The second term captures changes in unexplained residuals, and the third term indicates the effect of changes in the pattern of union employment across industries.

Incorporating the preceding formula, Table 5 decomposes the overall change in COLA coverage between 1976 and 1995 using parameter estimates from Table 4, column 9. On the basis of this exercise, 72% of the decline in COLA coverage over this time frame is attributable to changes in the values of independent variables. The decline in inflationary uncertainty accounts for more of the decline in aggregate COLA coverage, 27%, than any other factor. [23] At 1976 levels of inflationary uncertainty, our model predicts that the aggregate COLA rate in 1995 would have been 32% rather than 22%.

Second in quantitative importance is the decline in union power, with almost all of the decline resulting from reduced unionization. In our data, the average industry unionization rate fell by 40% between 1976 and 1995. In contrast, the average union wage premium barely changed. As a result, it is the erosion in industry unionization rates rather than a reduced ability to influence wages (extract economic rents) that has contributed to falling COLA coverage.

Industry deregulation also has played a role, as evidenced by the disproportionate losses in COLA coverage after deregulation. Overall, deregulation is estimated to have contributed 12% of the decline in aggregate COLA coverage (4.7 percentage points). Although economic conditions in the industry also have an effect, industry unemployment rates were generally lower in 1995 than 1976 and therefore did not contribute to the decline in COLA rates over this time frame.

The most important change in the composition of the union workforce appears to be the increased relative employment of women. A relative decline in blue-collar employment and to a lesser extent a shift to part-time employment have also been associated with reduced COLA coverage. Altogether, changes in the type of jobs and the composition of employees account for an estimated 9% of the lost COLA coverage.

Consistent with the claims of previous studies (Mitchell 1985; Weiner 1986; Davis et al. 1990; Devine 1996), shifting industrial patterns of employment also have contributed to the decline in the aggregate COLA rate, although these shifts appear quantitatively smaller than suggested in these studies. By our calculations, changing patterns of union employment across industries resulted in approximately 11% of the overall decline.

7. Summary and Discussion

For more than 20 years, unions have been trading COLA clauses for other forms of compensation. After peaking in 1976 at 61% of the workers covered by major collective bargaining contracts, the overall COLA rate had fallen to 22% at the end of 1995, when COLA statistics were last collected. Previous studies offered competing hypotheses as to the reasons for this decline, but the relative importance of various factors influencing COLA rates was not assessed.

We estimate a model of COLA coverage based on 22 years of data for each of 32 private-sector industries. The model formally recognizes that an industry's COLA rate in a given year depends on conditions prevailing in the current and previous years and on the distribution of contracts across these years. Consistent with economic theory, empirical estimates reveal that COLA rates depend on inflationary uncertainty but not on the expected rate of inflation. COLA rates also are related significantly to union power (both unionization rate and union wage premium in the industry), economic deregulation, the gender mix of union workers in the industry, the composition of jobs, and labor market tightness (as measured by the industry unemployment rate).

On the basis of decomposition analysis, the decline in the aggregate COLA rate between 1976 and 1995 is partitioned into various categories. The primary reasons for reduced COLA coverage are lower inflationary uncertainty and reduced union power. Other factors, in descending order of importance, are economic deregulation, industrial shifts in union employment, an increasingly female composition of the union workforce, and a shift away from blue-collar and full-time jobs.

Given these findings, it is possible to provide some assessment as to the prospects for a rebound in COLA provisions. The first conclusion is that the economy is unlikely anytime soon, if ever, to experience COLA coverage approaching the levels of the mid- to late 1970s. Union power has declined appreciably since 1976. In our sample, the unionization rate is down by 40%. Because COLA provisions are rare among nonunion employees, lower unionization would reduce the extent to which wages in the United States are indexed to prices even if COLA rates held firm in the union sector. But as our analysis shows, lower unionization rates also depress COLA rates among union employees.

Just as there is no reason to anticipate a resurgence in union power, reregulation of industries that have been deregulated appears unlikely. The growing presence of women in the union workforce appears likely to continue, as do the less important shifts from blue-collar to white-collar jobs and from full-time to part-time jobs. If union employment continues to shift from manufacturing, with its relatively higher incidence of COLA provisions, to nonmanufacturing, the overall COLA rate is likely to erode further.

It would be a mistake, however, to conclude that lower COLA rates are inevitable. Inflation can be highly variable, as history shows. As recently as 1979, the inflation rate was 13.3%, and just three years earlier it was 4.9%. Between 1971 and 1974, the annual rate of inflation jumped by 9.0 percentage points. A rise in uncertainty from its 1995 value to the mean value of the sample period could be expected to raise COLA coverage by 7.5 percentage points. If inflationary uncertainty returned to its peak level of the sample period, a 19.9 percentage point increase in COLA coverage is predicted. In conclusion, structural shifts and declines in union power appear likely to restrain any rise in COLA coverage, but given the major role played by inflationary uncertainty and the historical variability of this measure, it would be presumptuous to conclude that COLA rates will not rebound.

(*.) Department of Economics, Kansas State University, Waters Hall 327, Manhattan, KS 66506-4001, USA; E-mail jfrjr@ksu.edu; corresponding author.

(+.) Department of Economics, Kansas State University, Waters Hall 327, Manhattan, KS 66506-4001, USA.

We acknowledge the helpful Comments of Dek Terrell and two anonymous referees. Part of this research was completed while Bratsberg was visiting the University of Oslo, whose hospitality is gratefully acknowledged.

Received January 1999; accepted March 2000.

(1.) All COLA statistics cited in the paper refer to COLA coverage in the private sector at year's end. This is in contrast to the Department of Labor's convention of computing a given year's COLA rate on the basis of statistics at the end of the preceding year. Thus, according to the Department of Labor, COLA coverage fell from 61% in 1977 to 22% in 1996.

(2.) These studies include Cousineau, Lacroix, and Bilodeau (1983); Hendricks and Kahn (1983, 1985); Ehrenberg, Danziger, and San (1984); and Card (1986). Studies that incorporate post-1982 data are Prescott and Wilton (1993), who analyze Canadian contracts over the period 1979-86; Weiner (1993); and Holland (1995), whose analysis is limited to the relationship between inflation and aggregate wage indexation.

(3.) Higher inflation generally corresponds with greater inflationary uncertainty. See Ball (1992), Golob (1994), and Grier and Perry (1998).

(4.) Prior research indicates that workers are risk averse (Farber 1978; Hendricks and Kahn 1986). Typically employers are assumed to be risk neutral (Baily 1974; Azariadis 1975; Holland 1984; Danziger 1984, 1988) or at least less averse to risk than workers (Gordon 1974; Shavell 1976). Card (1986) estimates parameters of relative risk aversion of unionized workers and firm owners and concludes (p. S163) that "workers are substantially more risk averse than owners."

(5.) In addition to past values of variables included in X, COLA may also be influenced by past realizations of unobservable determinants captured by the error term. In particular, the error term may well be described by a moving-average process similar to that of X. We return to this point later.

(6.) According to these data, the mean duration of union contracts over the period of study was approximately 33.5 months.

(7.) Specifically, we obtain year-by-year estimates of the union wage premium in an industry as the coefficient of an interaction term between union status and industry from a series of Mincerian wage regressions using samples drawn from the Current Population Survey (CPS). The wage regressions also control for schooling, potential experience and its square, race, gender, marital status, part-time work, standard metropolitan statistical area, census division, occupation, and industry. The regression samples are based on all CPS data files that include data on union status: March 1971, May 1973-1981, and Outgoing Rotation Groups 1983-1995. Unfortunately, the CPS did not include questions about union membership in 1972 and 1982. For these two years, we interpolate by averaging values for the previous and following years.

(8.) In studies of aggregate COLA coverage, Holland (1984, 1986) shows that lagged values of inflationary uncertainty outperform current values. We too obtain a better fit when the variable is lagged, though parameter estimates are similar.

(9.) Although the proxies for union power may capture some of the influence of deregulation, these proxies cannot be expected to capture bargaining power precisely. In addition, an external change imposed on the industry, such as deregulation, may make it easier for a union to relinquish COLA coverage.

(10.) We compute industry-level measures of the fraction of union members who are female, black, work part-time, or hold blue-collar jobs on the basis of the regression samples described in footnote 7.

(11.) Because of the lag structure of the empirical model, data series for the explanatory variables begin in 1972.

(12.) Both Cousineau, Lacroix, and Bilodeau and Prescott and Wilton examine COLA coverage in samples of Canadian union contracts but for different time periods. The sample of the former study covers 1967 through 1978, and that of the latter 1979 through 1986.

(13.) We reached the same conclusion when we included a quadratic time trend. Coefficients of both TIME and [TIME.sup.2] were insignificant, and a Wald test of joint significance yielded an F(2, 661) sample test statistic of 1.34 and a p value of 0.263.

(14.) Another factor correlated with COLA coverage is the length of contract--longer contracts are more likely to contain COLA clauses than are shorter contracts. But, as argued by Hendricks and Kahn (1983, p. 452; 1985, p. 165), contract length is not a determinant of COLA coverage. Rather, contract length and COLA provisions are determined jointly. (See Christofides 1990 for discussion of the link between length of contract and degree of indexation.) Nonetheless, to see whether results would be affected, the equation was re-estimated with mean contract length as an additional explanatory variable. The coefficient of contract duration was insignificant, and other results were similar to those reported in Table 4.

(15.) Diagnostic tests strongly favor the fixed industry effects formulation of the model. A Wald test of the joint significance of industry effects strongly rejects exclusion of such effects, with an F(31, 663) test statistic of 60.58 (critical value at the 1% level is 1.71). Furthermore, a Hausman specification test favors the fixed industry effects model over a random effects formulation (results not reported but available on request), yielding a [[chi].sup.2](9) test statistic of 116.97 (critical value at the 1% level is 21.67). The data therefore provide evidence that important unobservable industry characteristics influence COLA coverage and that these characteristics are correlated with the explanatory variables of the model.

(16.) Twelve percent of the sample is censored.

(17.) In the Hendricks and Kahn study, endogeneity of the COLA variable arises because COLA coverage is positively correlated with unmeasured union bargaining power. This induces a positive bias in the coefficient estimate of COLA in the union wage equation. In contrast, the present study uses the union wage premium to proxy for bargaining power. Thus we seek to "instrument away" the compensating wage effect uncovered by Hendricks and Kahn.

(18.) To construct the instrumental variable, we draw on the analysis of union wage premiums by industry in Bratsberg and Ragan (1999) and base the instrument on import penetration in the industry, lagged aggregate unemployment, and an indicator variable for deregulation of telecommunications. We obtained comparable results when we instead constructed the instrument from the lagged union wage premium, but in view of the upcoming analysis of autocorrelation of residuals, the lagged wage premium is unlikely to be a valid instrument. As a final robustness check, we dropped the potentially endogeneous variable from the model. This had little effect on other coefficient estimates.

(19.) We obtained similar results in the two other subsamples in which observations are three years apart. We report results for the subsample that begins in 1974 because it contains one more year of observations.

(20.) Evidence that errors are generated by a moving average rather than an autoregressive process comes from a Burke, Godfrey, and Tremayne (1990) test. As extended to panel models by Baltagi and Li (1995), this test rejects the AR(1) process in favor of an MA(1) process at the 1% level, yielding a sample test statistic of -2.56 (critical value at the one percent level is -2.33).

(21.) The GLS procedure nets out the last two terms of Equation 2 from [COLA.sub.it] to form the dependent variable of a second regression model. Because the error of this new equation is heteroscedastic, we weigh each observation of the transformed model by (1/[[alpha].sub.10]). We initialize presample values of v (1972 and 1973) as zero and then iterate on the transformed model, combining the unbiased estimate of [u.sub.it] from the specification in Table 4, column 7, and the residual ([[alpha].sub.t0][v.sub.it]) of the transformed model to improve estimates of [v.sub.i,72] and [v.sub.i,73]. Parameter estimates stabilize after a few iterations; results in column 9 are based on 20 iterations. Programs and complete results are available on request.

(22.) In contrast, when we estimated the model allowing for an AR(1) process in the error, the Durbin-Watson statistic of the transformed model was 1.20, indicating that serial correlation remains in the model after accounting for the AR(1) process.

(23.) Results are largely comparable if, instead of column 9, we base the decomposition analysis on alternative specifications. For example, on the basis of the baseline (column 1) and tobit (column 7) specifications, the contribution of explanatory variables is 71% and 77% and that of inflationary uncertainty is 28% and 34%, respectively. Note also that the decomposition is not unique: one could alternatively weigh changes in explanatory variables by relative union employment in 1976 and changes in relative union employment by COLA coverage rates in 1995. When we use this alternative decomposition, the estimated contributions to the decline in COLA coverage are 70% for explanatory variables in general, 27% for inflationary uncertainty, and 8% for the industrial pattern of employment.

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Appendix: Calculating Weights for the COLA Equation

Weights for the COLA equation--[[alpha].10], [[alpha].sub.11], and [[alpha].sub.12] -- were calculated from data on the distribution of union contracts, obtained from the U.S. Bureau of Labor Statistics, Current Wage Developments. Workers were assigned to one of three categories of contracts--one-year; two-year, or three-year--thereby permitting a partitioning of workers on the basis of the year in which their contract was negotiated.

As an illustration, consider the calculation of weights for the year 1995 (the last year listed in Table 1). That year, 1,508,000 workers were covered by new contracts negotiated in 1995. The previous year, 1,468,000 workers received new two- or three-year contracts, and in 1993, an additional 1,674,000 workers received new three-year contracts. Assuming that these workers were still covered by collective bargaining settlements in 1995, a total of 4,650,000 workers were under collective bargaining agreements in 1995. Of these workers, 32.4% (1,508,000 of 4,650,000) had their contracts negotiated in 1995, 31.6% had their contracts negotiated in 1994, and 36.0% had their contracts negotiated in 1993.

Weights for other years were calculated in a similar manner.