Unit root tests of sigma income convergence across US metropolitan areas

Journal of Economic Geography 4 (2004) pp. 583 595 doi:10.1093/jnlecg/lbh035 Unit root tests of sigma income convergence across US metropolitan areas Matthew P. Drennan*, José Lobo**, and Deborah Strumsky*** Abstract The standard deviation of metropolitan per capita personal income (PCPI) and metropolitan average wage per job (AWPJ) provide straightforward indicators of unconditional sigma convergence for metropolitan economies within the United States. Using data for all metropolitan areas in the continental United States for the period 1969 2001, we tested for the unconditional sigma income convergence hypothesis by applying two unit root tests to the time series of the two standard deviations. Our results indicate that the time series can be described as random walks with drift, thereby supporting the claim that income divergence among metropolitan economies is not decreasing. Keywords: metropolitan income convergence, unit root test JEL classifications: R, C1, C52 Date submitted: 6 January 2003 Date accepted: 20 April 2004 1. Introduction Income convergence means that income growth will tend to be slower in areas with higher than average income and faster in areas with lower than average income. Thus, in the longrun, relative income differences among locations (nations, states, cities) will become smaller; that is, incomes will converge. The income convergence hypothesis derives from the neoclassical economic growth model with diminishing returns, under which the mobility of labor and capital, combined with the maximizing behavior of workers and owners of capital, argues that differences in returns to labor or capital among different places will diminish over time. The fact that the evidence for rich and poor nations does not support the convergence hypothesis (Barro and Sala-i-Martin, 1991) has generated a great deal of attention among economists. Applied to metropolitan areas, the neoclassical account predicts that labor will move to locations where real income is higher. 1 * Department of City and Regional Planning, Cornell University, 213 West Sibley Hall, Ithaca, NY, USA. email <mpd12@cornell.edu> ** Author to whom correspondence should be addressed: Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, NM 87501, USA. email <jose@santafe.edu> *** Harvard Business School, E1-2A Gallatin Hall, Soldiers Field Road, Boston, MA 02163, USA. email <dstrumsky@hbs.edu> 1 One would need to take into account that in some urban areas housing and other amenities will be more expensive, and that to compensate for this wages must be higher, which in turn raises the price of local labor services (non traded goods). Journal of Economic Geography, Vol. 4, No. 5, # Oxford University Press 2004; all rights reserved.

584 Drennan, Lobo, and Strumsky The most common measure of convergence is beta convergence, either unconditional or conditional. Beta convergence occurs when places with initially low levels of per capita income or output experience faster rates of growth than places with initially high levels of per capita income or output. Thus the poorer places and the richer places will eventually converge. The standard test of unconditional beta convergence is a cross-section regression in which the growth rate of per capita income over a relatively long period is regressed on the initial level of per capita income. A negative and significant coefficient supports convergence because it implies that higher initial levels correspond to lower growth, and vice versa. Conditional convergence recognizes that initial values, which may not be equilibrium or steady-state values, may differ among places, either because wages and incomes adjust across places to equate unobserved utility across households, or because of differences in skills and mix of industries. Therefore, tests of conditional beta convergence often include other variables to capture differing initial conditions. For example, Barro and Sala-i-Martin (1991) test for unconditional beta convergence as well as conditional beta convergence for states, introducing regional dummy variables and a sectoral mix variable. Similarly, Mankiw et al. (1992) test for unconditional as well as conditional beta convergence for three samples of nations, introducing saving, population growth, and human capital variables. In those two studies, the equations for conditional beta convergence provide better results than the equations for unconditional beta convergence, in accord with the heterogeneity of places. Another measure of convergence is sigma convergence, which is the tendency of the variation of income among places to diminish over time (Quah, 1993). Sigma convergence is usually measured by the standard deviation of per capita income or output for places over time. A persistent decline in the annual standard deviations indicates sigma convergence. Both Friedman (1992) and Quah (1993) have argued that sigma convergence is the only valid measure of convergence because beta convergence tests are subject to Galton s fallacy of regression to the mean. Barro and Sala-i-Martin (1991) tested for sigma convergence using state per capita income data from 1880 to 1988. Their measure is the cross-sectional standard deviation of the log of per capita income. Their results support sigma convergence for all decades except the 1920s and the 1980s, which they dismiss as aberrations. The income convergence hypothesis is more plausible for places within a nation than among nations because of the assumed mobility of capital and labor, and because the legal system, language, currency, financial markets, and culture are likely to be homogeneous. In the 1960s, the convergence hypothesis was tested for states (Borts, 1960; Borts and Stein, 1964) and for regions (Perloff, 1963) of the United States. The results mostly support income convergence, but with some exceptions. That might have been the end of the story had it not been for evidence of income dispersion (the opposite of convergence) among states and regions of the United States in the 1980s. Beginning in the late 1980s, a flurry of articles appeared in the economics literature testing for convergence within the United States using regions, states, and metropolitan areas as the units of analysis (e.g., Browne, 1989; Garnick, 1990; Barro and Sala-i-Martin, 1991; Blanchard and Katz, 1992; Carlino, 1992; Mallick, 1993; Crihfield and Panggabean, 1995; Glaeser et al., 1995; Drennan et al., 1996; Sala-i-Martin, 1996; Vohra, 1996; and Drennan and Lobo, 1999). These studies, which focused mainly on unconditional beta convergence, found some evidence of dispersion in the 1980s. The paper by Drennan et al. (1996) presents descriptive evidence that the unconditional beta convergence seen among US regions in earlier decades was replaced by divergence in the 1980s (no statistical

Income convergence across US metropolitan areas 585 tests of beta convergence were performed). Drennan and Lobo (1999) address the issue of unconditional beta convergence among metropolitan areas by developing a conditional probability test for beta convergence which avoids the regression to the mean problem noted by Friedman (1992) and Quah (1993). Their results support unconditional beta convergence among metropolitan areas. Visual evidence of unconditional sigma divergence for metropolitan areas is also presented, but no formal tests are performed. The most thorough study, by Barro and Sala-i-Martin (1991), covers a period of 100 years and uses state per capita income data to test for conditional beta convergence, as well as unconditional sigma convergence. In this paper we address the issue of unconditional sigma convergence for metropolitan areas of the United States. As a measure of dispersion we use the standard deviation of the natural logarithm of metropolitan per capita personal income (PCPI ) and average wage per job (AWPJ ) over the past 33 years. Both measures exhibit a rising trend, that is, divergence rather than convergence. In order to formally test for sigma convergence, we apply two unit root tests the Augmented Dickey-Fuller test and the DF-GLS test to the time series of the standard deviations. We ask, in effect, whether our chosen measures of metropolitan income dispersion can be characterized as random walks with drift. If the measures follow a random walk, the effects of local and temporary shocks, such as changes in public policy or technological innovations, will not dissipate after a while, reverting to a long-run trend, but will instead be permanent. It is difficult to imagine income dispersion diminishing under such circumstances. 2 The unit root tests are applied to all metropolitan areas of the United States for the period 1969 2001. Although most of the convergence literature has dealt with states, we contend thatmetropolitanareas definedassinglelabormarkets are less artificial spatial unitswith which to study the question of convergence. Summarily stated, we find that the time series of the standard deviation of the natural logarithm for both metropolitan PCPI and AWPJ can be described as random walks with positive drifts, thereby supporting the visual result that neither per capita personal income nor the average wage per job is converging. But that result must be viewed as tentative because of the relatively short period, 33 years, and the low power of unit root tests. Although our concern is unabashedly empirical, we offer possible reasons for the apparent shift from income convergence to income divergence. 2. Measures of dispersion: metropolitan income and wages Barro and Sala-i-Martin (1991) measure dispersion in state per capita personal income using the standard deviation. If income and wages are approximately lognormally distributed across metropolitan areas, the standard deviation of the natural logarithm of either variable would be a natural measure of dispersion. We applied the Shapiro-Wilk test for normality (Shapiro and Wilk, 1965) to the natural logarithm of metropolitan PCPI and AWPJ and concluded that both variables are log-normally distributed. 3 We therefore use the standard deviation of LN(PCPI) and LN(AWPJ) as our measures of dispersion. 2 We know by now that a variety of macroeconomic time series appear to be random walks. See, for example, Nelson and Plosser (1982), Stulz and Wasserfallen (1985), Wasserfallen (1986), Campbell and Mankiw (1987), Gardner and Kimbrough (1989), Brunner and Hess (1993), Cheung and Chinn (1996), and Ben- David et al. (2003). 3 More precisely, we cannot reject the null hypothesis that LN(PCPI) and LN(AWPJ) are lognormally distributed. The results of the Shapiro-Wilk test are available upon request.

586 Drennan, Lobo, and Strumsky Standard Deviation (of Natural Log) 0.2 0.19 0.18 0.17 0.16 0.15 0.14 0.13 0.12 0.11 0.1 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 00 01 AWPJ Figure 1. Variation in US metropolitan per capita personal income and average wage per job, 1969 2001. Year PCPI Figure 1 plots the standard deviation of the natural logarithm of metropolitan PCPI and metropolitan AWPJ from 1969 to 2001. Over 33 years, if the theory of income convergence is correct, one would expect to see a persistent downward trend in both variables. For the standard deviation of LN(PCPI), the first seven years do show a downward trend (from 0.176 in 1969 to 0.156 in 1976), but from the mid-1970s through the entire 1980s, the standard deviation rises. From 1989 through 1994, which includes a recession, the standard deviation of metropolitan LN(PCPI) declines somewhat, but then resumes its upward trend during the period 1994 2001 (the value for 2001 is 0.188; the highest value, 0.191, is observed in the year 2000). Although the pattern is somewhat different for the standard deviation of LN(AWPJ), the conclusion is the same. The standard deviation declined for the first 14 years, going from 0.145 in 1969 to 0.122 in 1983, and then commenced a steady increase, even during the recession of the early 1990s (to 0.164 in 2001). A casual examination of Figure 1 seems to indicate that dispersion in metropolitan incomes is not diminishing but is, on the contrary, increasing. If we were to regress either of the measures of income variation on a constant and on a time trend, making appropriate allowance for serial correlation, we would not expect to get a negative coefficient on the trend (and indeed, we do not). Thus, the hypothesis of unconditional sigma convergence appears implausible for the period covered. For the sake of greater formality, in this short note we propose a simple test for the unconditional sigma income convergence hypothesis by applying a unit root test to the time series of the standard deviation of metropolitan PCPI and AWPJ. More precisely, we test whether the coefficients have been diverging. The test is applied to all the metropolitan areas of the United States for the period 1969 2001. We want to address early on the concern about the low power of the unit root test. As discussed by Shiller and Perron (1985) and Ghysels and Perron (1993), the power of the unit root test depends more on the span of the data (the number of years covered by the data) than the number of observations. The 33-year period covered by our data is a reassuring span; however, we do not have a strong a priori reason to think that the process generating the data has been stable over the entire sample period. We therefore accept and interpret our results cautiously.

Income convergence across US metropolitan areas 587 3. Methodology: the unit root test Suppose the time series model for a variable y is described as an AR(1) process: y t ¼ a þ ry t 1 þ e t, where a and r are parameters and e t is Gaussian White Noise. If 1 < r < 1 then y is a stationary time series (a stochastic process is said to be stationary if its mean and variance are time-independent and if the covariance between any two periods depends only on the lag and not the actual time at which the covariance is calculated). If, however, r ¼ 1, then y is a nonstationary time series, and the stochastic process modeled by equation (1) is a random walk with drift (Brockwell and Davis, 1991; Gujarati, 1995; and Hamilton, 1994). When r ¼ 1, the process in equation (1) is referred to as a unit root process. (For a detailed discussion of the unit root issue, the ADF test, and the PP test, see Banerjee et al., 1993; Hamilton, 1994; Gujarati, 1995; Pindyck and Rubinfeld, 1999; and Smith, 1999.) The most commonly used test for the presence of a unit root is the Augmented Dickey- Fuller (ADF) test. The ADF test (Dickey and Fuller, 1979, 1981; Fuller, 1976) is carried out by estimating an equation with y t 1 subtracted from both sides of equation (1): ð1þ Dy t ¼ a þ gy t 1 þ e t, ð2þ where Dy t ¼ y t y t 1, g ¼ r 1, and the null and alternative hypotheses are, respectively, H 0 : g ¼ 0, H A ¼ g50: Due to the possible presence of non-stationarity, the t-statistic under the null hypothesis of a unit root does not have the conventional t-distribution. MacKinnon (1991, 1996), using extensive Monte Carlo simulations, estimated the Dickey-Fuller critical t-values for any sample size and for any number of right-hand variables. (Note that the ADF test is valid only if the time series is an AR(1) process.) If the series is correlated at higher order lags, the assumption of white noise disturbances is violated. The ADF test makes a parametric correction for higher-order correlation by assuming that the y series follows an AR( p) process (i.e., by adding lagged difference terms of the dependent variable y to the right-hand side of the regression) and by adjusting the test methodology (Said and Dickey, 1984). The ADF test was the first developed for testing the null hypothesis of a unit root, and it is the most commonly used. Other tests have been proposed, however, many of which have higher power than the ADF test. A test with higher power than the ADF test is more likely to reject the null hypothesis of a unit root against the stationary alternative when the alternative is true; thus, a more powerful test is better able to distinguish between a unit AR root and a root that is large but less than unity. One such test is the DF-GLS test, developed by Elliott et al. (1996). Essentially, the DF-GLS test is an Augmented Dickey- Fuller test, except that the time-series is transformed via a Generalized Least Squares (GLS) regression prior to performing the test. The DF-GLS test is performed in two steps. First, the intercept and trend are estimated using GLS, and these estimators are then used to compute a detrended version of the dependent variable. Second, the Dickey-Fuller test is used to test for a unit autoregressive root in the dependent variable (for a discussion of the DF-GLS test, see Stock and Watson, 2003). For the ADF test, the test statistics are the t-statistic and the z-statistic, both with their associated P-values, of the coefficient of the lagged dependent variable. For the DF-GLS test, the test statistic is the t-statistic on the lagged dependent variable.

588 Drennan, Lobo, and Strumsky Given the visible presence of a time trend in Figure 1, we choose the following as our basic model for the time series of the standard deviations of metropolitan PCPI and AWPJ: y t ¼ a þ ry t 1 þ dt þ e t, where y t is the coefficient of variation, at time t, a is a constant (intercept), and dt is a time trend (see Ayat and Burridge (2000)). We applied both the Augmented Dickey-Fuller test and the DF-GLS test to equation (3). The results, obtained using the statistical analysis software package STATA version 8.0, are presented in Section 5. ð3þ 4. Data We use two measures of metropolitan income, namely, per capita personal income (PCPI) and average wage per job (AWPJ); data for the two variables are provided by the Bureau of Economic Analysis Regional Economic Information Systems (REIS). 4 PCPI includes wages, salaries, and other labor income (accounting for 57% of total PCPI in 1999), dividends, interest, and rent (19%), proprietors net income (8%), and transfer payments (16%). AWPJ is defined as total annual earnings (wages, salaries, other labor income, and the net income of proprietors or the self-employed) divided by the number of full and part time jobs. The Bureau of Labor Statistics (BLS) estimates the number of local area jobs using State and Federal data on unemployment insurance and compensation. In order to provide complete coverage for all jobs in the United States, the unemployment insurance and compensation data are adjusted for underreporting and misreporting under these programs. We use average wage per job, instead of average wage per employee, in order to have a more comprehensive measure of wages. Both PCPI and AWPJ were deflated using the multiple versions of the urban Consumer Price Index (CPI) provided by the BLS. The BLS publishes monthly a consumer price index for urban areas classified by population size: all metropolitan areas of 1.5 million or more, metropolitan areas smaller than 1.5 million, and all non-metropolitan urban areas. Indices are also available for urban areas within each of four regions Northeast, Midwest, South, and West cross-classified by urban population size. The BLS also publishes a specific annual index for each of 24 metropolitan areas (see Appendix 1 for a list of these metropolitan areas). These CPI indices have a 1982 84 reference base. 5 In adjusting the yearly metropolitan income and wage data, we used either an available metropolitan-specific index, a regional index for the given population size, or the average urban CPI. Following the recommendation of the BLS, we used indices unadjusted for seasonal variation. We are well aware that the CPI for individual areas cannot be used to compare living costs among the areas. An individual area index measures how much prices have changed over a specific time period in that particular area and does not show whether prices or living costs are higher or lower relative to another area. Insofar as the regional and urban consumer price indices reflect the varying composition of market baskets as well as changes over time in the prices paid by urban consumers, then using these indices 4 http://www.bea.doc.gov/bea/regional/reis/ 5 The consumer price indices are available at http://www.bls.gov/cpi/home.htm.

Income convergence across US metropolitan areas 589 results in adjusted income measures that reflect local price conditions. In the absence of comparative data on urban cost of living with sufficient geographic and temporal coverage, this is probably the best that we can do in terms of constructing real income variables. The universe for our analysis consists of 318 metropolitan economies within the Continental United States, which includes all 245 Metropolitan Statistical Areas (MSAs), 11 New England Metropolitan Areas (NECMAs), and all 62 Primary Metropolitan Statistical Areas (PMSAs) as defined by the Office of Management and Budget (OMB) in 2002. The geographic definition of MSAs, NECMAs and PMSAs that is, their county composition is constant back to 1969. We are well aware that metropolitan area definitions are dynamic. Thus, by applying the 2002 metropolitan taxonomy to data from previous years, we are in effect treating counties that were not metropolitan in the past as if they were metropolitan (for a discussion of the dynamic nature of metropolitan definitions see Nucci and Long, 1995). Treating non-metropolitan counties as metropolitan, however, is likely to bias the data in favor of convergence, as non-metropolitan counties tend to be poorer than metropolitan counties. Descriptive statistics for metropolitan PCPI and AWPJ are presented in Table 1. There is a great deal of variation among places in levels of PCPI, and the variation has increased over time. In 1970 the location with the highest PCPI ($16,462) was 327% above the location with the lowest PCPI ($5,031); the coefficient of variation another possible measure of dispersion for this period is 0.170. In 2000 the wide range became even wider; the metropolitan area with the highest PCPI ($30,392) was 390% above the area with the lowest PCPI ($7,785). The coefficient of variation increased to 0.197. Metropolitan average wage per job (AWPJ) shows less variation than PCPI although the temporal pattern is similar. In 1970 the location with the highest AWPJ ($11,366) was 218% above Table 1. Descriptive statistics for metropolitan variables* 1970 1980 1990 2000 Per capita personal income (PCPI) Mean 9,808.80 11,776.33 13,913.58 15,855.49 Standard deviation 1,662.02 1,927.47 2,527.72 3,122.81 Coefficient of variation 0.169 0.164 0.182 0.197 Median 9,731.96 11,606.19 13,430.76 15,276.96 Maximum 16,461.54 19,692.79 23,841.03 30,391.79 Minimum 5,030.93 6,510.92 7,134.66 7,785.43 Observations 318 318 318 318 Average wage per job (AWPJ) Mean 16,606.51 16,111.38 16,255.72 17,643.27 Standard deviation 2,304.13 2,083.13 2,269.01 3,218.80 Coefficient of variation 0.139 0.129 0.140 0.182 Median 16,519.33 15,881.67 15,868.40 17,014.40 Maximum 24,814.43 26,033.98 24,708.49 36,843.03 Minimum 11,365.98 11,149.27 11,462.13 12,260.87 Observations 318 318 318 318 Notes: * Monetary figures expressed in 1982 1984 dollars (deflated using the BLS s urban CPIs).

590 Drennan, Lobo, and Strumsky the location with the lowest AWPJ ($11,128); the coefficient of variation for this period is 0.145. In 2000 the range became even wider; the metropolitan area with the highest AWPJ ($36,843) was 300% above the area with the lowest AWPJ ($12,261), while the coefficient of variation increased to 0.182. 5. Results The ADF test is performed on the following specification: Dy t ¼ a þ gy t 1 þ bdy t 1 þ dt þ e t, where y t denotes either SD(LNPCPI) t, the standard deviation of the natural logarithm of metropolitan PCPI at time t, orsd(lnawpj) t, the standard deviation of the natural logarithm of metropolitan AWPJ at time t, Dy t ¼ y t y t 1, and Dy t 1 ¼ y t 1 y t 2. Table 2 presents the results of applying the Augmented Dickey-Fuller Test to equation (4) using SD(LNPCPI) t. We fail to reject the null hypothesis that there is a unit root in this time series by examining either the MacKinnon (1996) approximate asymptotic P-value or the interpolated Dickey-Fuller critical values. The Augmented Dickey-Fuller z-statistic can be obtained by dividing the number of observations times the coefficient y t on by one minus the coefficient of Dy t 1 : 33ð 0:043Þ 1 þ 0:102 ð4þ ¼ 1:288: ð5þ The P value for the z-statistic, also based on the program in MacKinnon (1996), is 0.0646, which also implies a failure to reject the null hypothesis. Table 3 presents the results of applying the Augmented Dickey-Fuller Test to equation (4) using SD(LNAWPJ) t as the dependent variable. The value of the t-statistic of the coefficient of SD(LNPCPI) t 1, 0.64, and its finite sample P-value of 0.9694, Table 2. Results of Augmented Dickey-Fuller unit root test. Dependent Variable: DSD(LNPCPI) t ¼ SD(LNPCPI) t SD(LNPCPI) t 1 Test statistic 1% critical value 5% critical value 10% critical value 2.777 4.325 3.576 3.226 MacKinnon approximate (finite sample) P-value for test statistic: 0.2151 Variable Coefficient Std. error t Pt> SD(PCPI) t 1 0.258 0.093 2.777 0.010 D SD(LNPCPI) t 1 0.512 0.171 3.00 0.006 Constant 0.040 0.015 2.730 0.011 Trend 0.001 0.000 2.340 0.011 Number of observations: 33

Income convergence across US metropolitan areas 591 Table 3. Results of Augmented Dickey-Fuller unit root test. Dependent variable: DSD(LNAWPJ) t ¼ SD(LNAWPJ) t SD(LNAWPJ) t 1 Test statistic 1% critical value 5% critical value 10% critical value 0.641 4.325 3.576 3.226 MacKinnon approximate (finite sample) P-value for test statistic: 0.9694 Variable Coefficient Std. error t Pt> SD(LNAWPJ) t 1 0.043 0.066 0.64 0.527 D SD(LNAWPJ) t 1 0.102 0.294 0.35 0.732 Constant 0.002 0.009 0.250 0.802 Trend 0.0002 0.000 2.640 0.014 Number of observations: 33 means that we fail to reject the null hypothesis. The Augmented Dickey-Fuller z-statistic is given by: 33ð 0:043Þ 1 þ 0:102 ¼ 1:288: ð6þ The value of the test statistic, and its P value of 0.8735 (based on MacKinnon, 1996), provide a less forceful failure to reject the null hypothesis. The results of the ADF unit root test would seem to imply that the time series of the standard deviations of the natural logarithm of PCPI and AWPJ follow a random walk, which in turn would seem to provide evidence against the convergence hypothesis. One must be cautious, however, so as not to overstate support of the random walk hypothesis. Recall that our data is limited to 33 observations. And as Pindyck and Rubinfeld (1999, p.510) note, Although the Dickey-Fuller test is widely used, one should keep in mind that its power is limited. It only allows us to reject (or fail to reject) the hypothesis that a variable is not a random walk. A failure to reject (especially at a high significance level) provides only weak evidence in favor of the random walk hypothesis. We therefore applied a more powerful test to both measures of income dispersion, the DF-GLS test. In the DF-GLS test (Elliott et al., 1996), the null hypothesis is that the dependent variable follows a random walk, possibly with drift, while the alternative hypothesis is that the variable is stationary around a linear trend. It is the GLS regression in the first step of the DF-GLS test that improves the test s ability to discriminate between the null hypothesis and the alternative hypothesis, thereby increasing the power of the unit root test. Table 4 presents the results of the DF-GLS test, with one lag, using data for SD(LNPCPI) t and SD(LNAWPJ) t. In the case of both SD(LNPCPI) t and SD(LNAWPJ) t the value of the test statistic is unambiguously larger than the critical values at the various levels of significance; the hypothesis of a unit root process cannot be rejected. 6 6 The DF-GLS test is sensitive to the choice of lag length. The test fails to reject the null hypothesis even with up to five lags.

592 Drennan, Lobo, and Strumsky Table 4. Results of DF-GLS test (one lag) Variable t test statistic 1% critical value 5% critical value 10% critical value SD(PCPI) t 2.490 3.770 3.190 2.890 SD(LNAWPJ) t 1.211 3.770 3.190 2.890 Number of observations: 33 6. Discussion A visual and statistical examination of the time series for the standard deviation of the natural logarithm of metropolitan per capita personal income and average wage per job argue against the hypothesis of unconditional sigma convergence. Our statistical results suggest that the time series of the standard deviations can be characterized as random walks with positive drifts. From this we infer that dispersion among metropolitan economies with respect to per capita personal income and wages is not decreasing over time. Given that the period covered is 33 years, 1969 2001, we cannot characterize our result as an aberration. The careful analysis of sigma convergence for states from 1880 through 1988 by Barro and Sala-i-Martin (1991) does support sigma convergence over almost all of that long period. However, they do note that from the mid- 1970s through 1988, the state data exhibits divergence. Our metropolitan data goes through 2001, and it also exhibits divergence from the mid-1970s. Of course, future work using a longer time period for metropolitan areas than we use here would be desirable. A worthwhile extension of our work would be to assess the stationarity of metropolitan PCPI and AWPJ directly, using panel data unit root tests, and we hope to carry out this study in the near future. 7 Further research on the question of metropolitan income convergence may not yield a definitive answer as to whether divergence has replaced convergence, but that does not preclude posing the question: if metropolitan income is not converging, then why is it not converging? If we assume that the long-term convergence for states, which Barro and Sala-i-Martin established for 1880 to the mid-1970s, is also true for metropolitan areas, then why was it replaced by divergence in the past quarter century? The distinction between conditional versus unconditional convergence is of no help in answering this question. Assuming unconditional convergence in the century ending about 1975, namely, what Barro and Sala-i-Martin (1991) established for states, it can hardly be claimed that the post-1975 divergence is due to the failure to measure conditional sigma convergence, or to take into account regional price differences. If different initial conditions peculiar to a place and different price levels prevent our observation of unconditional sigma convergence after 1975, then why were such differences unimportant in the prior 100 years? We offer, purely as informed speculation, some possible sources of the observed metropolitan income divergence of the past 25 years or so. First, transportation technology did not improve significantly after the mid-1970s. In the prior 100 years, 7 For a discussion of the use of panel data in testing for unit roots, see Levin and Lin (1992), Im et al. (1997), and Rapach (2002).

Income convergence across US metropolitan areas 593 railroads, trucks, refrigeration cars, the interstate highway system, and jet air transportation all lowered costs and shortened the time needed to move goods and people. Those developments raised the mobility of labor and capital and commodities, acting to equalize returns and prices among places. But with no further dramatic improvements in transportation technology, different characteristics and different price levels became more important. That is, the strong push for convergence arising from improving transportation technology was no longer present. Second, as Mankiw et al. (1992) noted, the neoclassical growth model infers diminishing returns to labor and capital, which in turn implies income convergence among places as labor and capital seek their highest returns. If, however, human capital has, in both history and growth theory, become a major input, and if human capital is not subject to diminishing returns, then income convergence among places may be interrupted or even become divergence. In a recent article, Acemoglu (2002) argued that technical change in the 20 th century is skill-biased, and that there has been an acceleration in skill-bias in the past few decades. So, if human capital is not an input subject to diminishing returns, and if there has been an acceleration in skill-biased technical change, then human capital has become a relatively more important input, weakening the tendency of income convergence among places. Finally, the analysis of regional economic growth by Borts (1960) emphasizes that regional divergence is to be expected when strong secular demand for the export of some high- income region stimulates faster growth than in some low-income region not specialized in the favored export. A high-wage region may grow more rapidly than a low-wage region if the demand for its export is growin. (Borts, 1960 p.326). One characteristic of the post-1975 era in the United States is the far more rapid growth in producer service industries than in manufacturing and distribution industries. That shift favored some metropolitan areas but not others; the favored ones were characterized by specialization in producer services, large size, and high concentration of human capital (Drennan, 2002). Appendix 1 These are the 24 metropolitan areas for which the Bureau of Labor Statistics publishes a specific annual consumer price index: Chicago-Gary-Kenosha, IL-IN-WI; Los Angeles-Riverside-Orange County, CA; New York-Northern NJ-Long Island, NY-NJ-CT-PA; Atlanta, GA; Boston-Brockton-Nashua, MA-NH-ME-CT; Cleveland-Akron, OH; Dallas-Fort Worth, TX; Detroit-Ann Arbor-Flint, MI; Houston-Galveston-Brazoria, TX; Miami-Fort Lauderdale, FL; Philadelphia- Wilmington-Atlantic City, PA-NJ-DE-MD; San Francisco-Oakland-San Jose, CA; Seattle- Tacoma-Bremerton,WA;Washington-Baltimore,DC-MD-VA-WV;Cincinnati-Hamilton,OH-KY- IN; Denver-Boulder-Greeley, CO; Kansas City, MO-KS; Milwaukee-Racine, WI; Minneapolis- St, Paul, MN-WI; Pittsburgh, PA; Portland-Salem, OR-WA; St. Louis, MO-IL; San Diego, CA; and Tampa-St. Petersburg-Clearwater, FL. References Acemoglu, D. (2002) Technical change, inequality, and the labor market, Journal of Economic Literature, 40: 7 72. Ayat, L., Burridge, P. (2000) Unit root tests in the presence of uncertainty about the non-stochastic trend, Journal of Econometrics, 95: 71 96.

594 Drennan, Lobo, and Strumsky Banerjee, A., Dolado, J.J., Galbraith, J.W., Hendry, D.F. (1993) Cointegration, Error Correction, and the Econometric Analysis of Non-Stationary Data. Oxford: Oxford University Press. Barro, R.J., Sala-i-Martin, X. (1991) Convergence across states and regions. Brookings Papers on Economic Activity, 1: 107 182. Ben-David, D., Lumsdaine, R.L., Papell, D.H. (2003) Unit roots, postwar slowdowns and long-run growth: evidence from two structural breaks. Empirical Economics, 28: 303 319. Blanchard, O.J., Katz, L.F. (1992) Regional evolutions. Brookings Papers on Economic Activity, 1: 1 75. Borts, G.H. (1960) The equalization of returns and regional economic growth. American Economic Review, 50: 319 34. Borts, G.H., Stein, J.L. (1964) Economic Growth in a Free Market. New York: Columbia University Press. Brockwell, P.J., Davis, R.A. (1991) Time Series: Theory and Methods. New York: Springer-Verlag. Browne, L.E. (1989) Shifting regional fortunes: the wheel turns. New England Economic Review, Federal Reserve Bank of Boston, May/June, 27 40. Campbell, J.Y., Mankiw, N.G. (1987) Are output fluctuations transitory? Quarterly Journal of Economics, 102: 857 880. Carlino, G.A. (1992) Are regional per capita earning diverging? Business Review, Federal Reserve Bank of Philadelphia, March/April, 3 12. Cheung, Y., Chinn, M.D. (1996) Deterministic, stochastic, and segmented trends in aggregate output: a cross-country analysis. Oxford Economic Papers, 48: 134 162. Crihfield, C.J., Panggabean, M. (1995) Growth and convergence in US cities. Journal of Urban Economics, 38: 138 165. Dickey, D.A., Fuller, W.A. (1979) Distribution of the estimators for autoregressive time series with a unit root. Journal of the American Statistical Association, 74: 427 431. Dickey, D.A., Fuller, W.A. (1981) Likelihood ratio statistics for autoregressive time series with a unit root. Econometrica, 49: 1057 1072. Drennan, M.P. (2002) The Information Economy and American Cities. Baltimore: Johns Hopkins University Press. Drennan, M.P., Lobo, J. (1999) A simple test for convergence of metropolitan income in the United States. Journal of Urban Economics, 46: 350 359. Drennan, M.P., Tobier, E., Lewis, J. (1996) The interruption of income convergence and income growth in large cities in the 1980s. Urban Studies, 33: 63 82. Elliott, G., Rothenberg, T.J., Stock, J.H. (1996) Efficient tests for an autoregressive unit root. Econometrica, 64, 813 836. Friedman, M. (1992) Do old fallacies ever die? Journal of Economic Literature, 29: 2129 2132. Fuller, W.A. (1976) Introduction to Statistical Time Series. New York: Wiley. Gardner, G.W., Kimbrough, K.P. (1989) The behavior of US tariff rates. American Economic Review, 79: 211 218. Garnick, D.H. (1990) Accounting for regional differences in per capita personal income growth: an update and extension. Survey of Current Business, 70: 29 40. Ghysels, E., Perron, P. (1993) The effect of seasonal adjustment filters on test for a unit root. Journal of Econometrics. 55: 57 98. Glaeser, E.L., Scheinkman, J.A., Shleifer, A. (1995) Economic growth in a cross-section of cities. Journal of Monetary Economics, 36: 117 143. Gujarati, D.N. (1995) Basic Econometrics. New York: McGraw-Hill, Inc. Hamilton, J.D. (1994) Time Series Analysis. Princeton: Princeton University Press. Im, K.S., Pesaran, H.M., Shin, Y. (1997) Testing for unit roots in heterogeneous panels. Working paper, Department of Applied Economics, University of Cambridge. Levin, A., Lin, C. (1992) Unit root tests in panel data: asymptotic and finite sample properties. Discussion Paper No. 92 93, Department of Economics, University of California, San Diego.

Income convergence across US metropolitan areas 595 MacKinnon, J.G. (1991) Critical values for cointegration tests. In R.F. Engle, and C.W.J. Granger (eds) Long-run Economic Relationships: Readings in Cointegration. Oxford: Oxford University Press. MacKinnon, J.G. (1996) Numerical distribution functions for unit root and cointegration tests. Journal of Applied Econometrics, 11: 601 618. Mallick, R. (1993) Convergence of state per capita incomes: an examination of its sources. Growth and Change, Summer, 325 334. Mankiw, N.G., Romer, D., Weil, D.N. (1992) A contribution to the empirics of economic growth. Quarterly Journal of Economics, 107: 407 437. Nelson, C.R., Plosser, C.I. (1982) Trends and random walks in macroeconomic time series: some evidence and implications. Journal of Monetary Economics, 10: 139 162. Nucci, A., Long, L. (1995) Spatial and demographic dynamics of metropolitan and nonmetropolitan territory in the United States. International Journal of Population Geography, 1: 165 181. Perloff, H.S. (1963) How a Region Grows. New York: Committee for Economic Development. Pindyck, R.S., Rubinfeld, D.L. (1999) Econometric models and economic forecasts. Boston: Irwin McGraw-Hill. Quah, D. (1993) Galton s fallacy and tests of the convergence hypothesis. Scandinavian Journal of Economics, 93: 427 443. Rapach, D.E. (2002) Are real GDP levels nonstationary? Evidence from panel data tests. Southern Economic Journal, 68: 473 495. Said, E., Dickey, D.A. (1984) Testing for unit roots in autoregressive moving average models of unknown order. Biometrika, 71: 99 607. Sala-i-Martin, X. (1996) The classical approach to convergence analysis. Economic Journal, 106: 1019 1036. Shapiro, S.S., Wilk, M.B. (1965) An analysis of variance test for normality (complete samples). Biometrika, 52: 591 611. Shiller, R., Perron, P. (1985) Testing the random walk hypothesis: power versus frequency of observation. Technical Working Paper 045, NBER, Cambridge, MA. Smith, R.P. (1999) Unit roots and all that: the impact of time-series methods on macroeconomics. Journal of Economic Methodology, July, 239 258. Stock, J.H., Watson, M.W. (2003) Introduction to Econometrics. Boston: AddisonWesley. Stulz, R., Wasserfallen, W. (1985) Macroeconomic time-series, business cycles, and macroeconomic policies. Carnegie-Rochester Conference Series on Public Policy, 22: 9 54. Vohra, R. (1996) How fast do we grow? Growth and Change, Winter, 47 54. Wasserfallen, W. (1986) Non-stationarities in macro-economic time series further evidence and implications. Canadian Journal of Economics, 19: 498 510.