United States of U-Star*

Abstract:

1 Introduction

A growing body of research exploits regional heterogeneity to understand the cyclical dynamics of the macroeconomy and the effectiveness of macroeconomic policy.¹ In order to do so correctly, however, it is important to account for structural differences across regional economies. One key structural difference is labor-market functioning, as captured by a region-specific natural rate of unemployment. But despite its importance, this feature of regional heterogeneity is not well understood. This paper estimates the natural rate of unemployment for U.S. states (the “NRU” or “U-star,” henceforth). We explore the determinants of the natural rate and demonstrate that the presence of regional heterogeneity carries significant implications for empirical Phillips curves fit to state-level data.

We associate the U-star with the noncyclical component of the unemployment rate, as measured using a trend-cycle decomposition. As is well known, identifying and estimating the NRU at the aggregate level became increasingly challenging from the 1990s on because the link between inflation and labor-market slack weakened dramatically (a phenomenon often referred to as the flattening—or even the “death –of the Phillips curve). In response, researchers began using the noncyclical or “trend” components of unemployment stocks or flows to recover estimates of the NRU.²

In related work, Ahn, 2023 further decomposes unemployment outflows by the duration of unemployment and uses these variables’ trend components, along with low-frequency changes in unemployment inflows, to estimate the NRU. By exploiting information from changes in the unemployment–duration distribution, Ahn, 2023’s approach decomposes the NRU into short-term and long-term components, which can be interpreted as frictional and structural unemployment, respectively. In this paper, we estimate Ahn, 2023’s model using unemployment levels and distributions across all 50 states and the District of Columbia in order to recover an NRU, which we refer to as trend U-star. By combining microdata from the Local Area Unemployment Statistics (LAUS) and the Current Population Survey (CPS), our approach produces state-level trend U-star estimates that consistently aggregate to a national trend U-star estimate.

These state-level U-star estimates deepen our understanding of both local labor-market conditions and the cyclical position of the aggregate economy. The state-level estimates reflect regional heterogeneity in low-frequency movements in labor-market flows and in the distributions of unemployment durations. This regional heterogeneity provides additional information about labor-market functioning that helps us infer the aggregate NRU, which can be difficult to identify using aggregate data alone and especially when the NRU is time-varying. In addition, using a large number of states sharpens inference and helps make the NRU estimates more robust, given that idiosyncratic variation, including measurement errors, can wash out to some degree in the aggregate.³ Rich regional information embedded in the disaggregated estimates also helps evaluate the role of structural factors—such as demographics, migration, and changes in industrial structure—in determining the natural rate. Finally, the regional estimates can uncover pockets of economic slack that are not apparent in the national-level data, which permits national and place-based stabilization policies to be coordinated.

The state-level estimates reveal substantial cross-state variation in trend U-star. Over the period 1978:Q1–2025:Q1, average levels of trend U-star range from 3.3 percent in Nebraska and South Dakota to 7.5 percent in Alaska, with agricultural states in the Midwest disproportionately represented in the bottom quartile and states in the Rust Belt and Sun Belt in the top quartile. In addition, the state-level estimates exhibit significant heterogeneity in their contours. For example, unemployment inflows in Nebraska and Nevada remain relatively flat over the sample period, while those in many other states exhibit downward trends. Although a number of states experience increases in the portion of trend U-star that reflects long-term unemployment (defined as unemployment durations that are longer than 6 months), others—such as Iowa and North Dakota—exhibit little discernible change in their long-term components. As a result, in many states declining unemployment inflows and rising long-term components partially offset each other, producing net downward trajectories with varying slopes.

These findings suggest that the secular decline seen in trend U-star at the national level is not uniform across states. Approximately twenty states show more pronounced reductions, while twenty-five states exhibit a slower pace of decline. In four states, trend U-star does not change or even rises. In addition, structural forces commonly associated with reductions in the NRU—such as declining labor-market dynamism, population aging, and rising educational attainment—–have not been felt uniformly across regions. That said, cross-state dispersion in trend U-star has narrowed over time, largely because the components attributable to short-term unemployment spells have converged across states.

The duration composition of trend U-star also varies substantially across states, with the long-term share ranging from about 10 percent in Wyoming to roughly 25 percent in the District of Columbia. This heterogeneity indicates that the relative contributions of frictional and structural unemployment to U-star differ markedly across states. For example, the U-stars for Rust Belt states tend to reflect a relatively larger share of longer-term (structural) unemployment, suggesting the presence of hysteresis effects in which job losers remain unemployed for extended periods, become increasingly detached from the labor market, and face reduced reemployment prospects. By contrast, agricultural states generally display lower shares of structural unemployment in their trend U-star estimates, reflecting the relatively high worker churn that is associated with the nature of production in the agricultural sector.

We use the state-level estimates to examine how demographics, industry mix, labor-market institutions, and other economic fundamentals shape cross-state variation in trend U-star. Educational composition matters, with higher average educational attainment associated with a net decline in trend U-star. Population aging also reduces trend U-star, as younger workers are associated with higher frictional unemployment. However, we find little evidence that either the share of women in employment or a state’s industry structure matter for trend U-star. Likewise, labor-market rigidities and institutional factors such as unionization, tax burdens, and downward nominal wage rigidity have mixed effects on trend U-star, as do stronger fundamentals (in the form of higher labor force participation and labor productivity).

Finally, we use our estimates of U-star to revisit the results of well-cited studies of the Phillips curve that use state-level data (e.g., McLeay and Tenreyro, 2020; Hazell et al., 2020; and Fitzgerald et al., 2020).⁴ In these studies, time and state fixed effects are included in the empirical model to recover the slope of the Phillips curve under two maintained assumptions: first, that the state-level natural rate is either constant (and so captured by state fixed effects) or varies over time but is perfectly correlated with the aggregate trend (and so captured by time fixed effects that can also control for changes in monetary policy regimes and inflation expectations); and second, that aggregate demand shocks are uncorrelated with the NRU. Our state-level trend U-star estimates indicate that these assumptions are not empirically supported, which biases previous estimates of the Phillips curve’s slope obtained using state-level data.

In particular, Hazell et al., 2020 find a small but statistically significant Phillips curve slope based on a new-Keynesian specification where the slope is the coefficient on a discounted forward sum of future unemployment rates. Under their assumptions regarding state-level natural rates, the forward sum of the unemployment rate can be interpreted as the forward sum of the unemployment gap. However, our finding of substantial regional heterogeneity in the time paths of U-star calls their assumptions into question. Moreover, because the unemployment-rate gap is mean-zero and stationary, the forward sum of the unemployment rate will essentially be a forward sum of the natural rate, implying that the statistically significant association that they find between the forward sum of unemployment and inflation is instead a relationship between inflation and the natural rate. Taken together, these considerations suggest that Hazell et al., 2020’s model is likely to be misspecified, and when we repeat their estimation procedure using state-specific unemployment gaps, we no longer obtain a statistically significant Phillips curve slope.

Hazell et al., 2020 also consider a specification that uses the lagged unemployment rate in place of the discounted forward sum. We find that the estimated coefficient on the lagged unemployment rate increases in magnitude and remains statistically significant once state-level trend U-star is taken into account. This result suggests that controlling for idiosyncratic variation in state-level U-star values permits better identification of the margin of labor-market slack that is relevant for inflation, and also suggests that a traditional Phillips curve specification does a better job than the new-Keynesian Phillips curve in characterizing inflation dynamics.

Importantly, our paper contributes to the literature on the natural rate of unemployment and unemployment dynamics. As inflation lost power to identify the natural rate, economists have paid greater attention to the trend unemployment rate as a proxy for the natural rate (see for example, Fleischman and Roberts, 2011; Tasci, 2012; Barnichon and Mesters, 2018; Crump et al., 2019; Ahn and Rudd, 2024; Hornstein and Kudlyak, 2026).⁵ Our paper is related to this approach, but distinguishes itself from the previous literature in three ways. First, we estimate state-level trend unemployment rate, and aggregate them up to recover the aggregate natural rate, which none of the previous studies attempted to do so. Second, the paper makes a methodological contribution by integrating nonlinear state-space estimation, duration-structure accounting, and Bayesian trend–cycle decomposition into a unified framework for measuring regional trend unemployment. For this, the model allows unemployment exit probabilities to vary with unemployment duration—an important extension relative to the previous literature, which decomposes the natural rate by the duration of unemployment. Third, our paper connects to research on heterogeneity in cyclical labor-market dynamics Blanchard and Katz, 1992; Aaronson et al., 2019; Cajner et al., 2025 by examining regional differences in labor-market functioning.

In the following section, we discuss how we construct our dataset. Section 3 introduces the duration-structure model and describes our estimation procedure, and Section 4 reports estimation results for the model parameters across states. Section 5 presents a decomposition of changes in trend unemployment rates and discusses regional patterns and their implications for aggregate U-star, and Section 6 explores implications for the Phillips curve. Section 7 concludes.

2 Data

For each U.S. state and the District of Columbia, we use CPS microdata to construct the number of individuals unemployed for 1 month (less than 5 weeks), 2–3 months (5–14 weeks), 4–6 months (15–26 weeks), 7–12 months (27–52 weeks), and 13 months or more (53 weeks and over). The monthly counts are aggregated from individual-level CPS responses based on the reported duration of unemployment and the respondent’s state of residence. We also construct the number of individuals in the civilian labor force and total employment by state. All series are seasonally adjusted using the X-13-ARIMA procedure.⁶ The sample period spans January 1976 to March 2025.

To ensure consistency with official statistics, we scale CPS-based unemployment counts to match published state-level unemployment totals from the BLS Local Area Unemployment Statistics (LAUS). We also use the seasonally adjusted LAUS labor force to compute state-level unemployment rates. Because LAUS does not provide information on the distribution of unemployment duration at the local level, our adjustment preserves each state’s distribution of unemployment durations from the CPS while aligning aggregate unemployment and labor force levels with the LAUS data. As a result, for each state and in each month, the sum of unemployment across duration categories equals the total number of unemployed persons reported in the LAUS data.

All of the CPS-based counts are computed using the individual sampling weights to ensure that the state-level estimates are representative of the civilian noninstitutional population. For the early part of the sample, we harmonize state identifiers to ensure consistent geographic definitions over time. These harmonized, population-weighted series are used to estimate a dynamic accounting identity between unemployment duration and subsequent state-level trend unemployment rates. The data thus combine the detailed duration information from the CPS micro data with the comprehensive coverage and benchmarking of the LAUS series, ensuring both internal consistency and cross-state comparability.

3 Estimating State-Level Trend Unemployment

We use the methodology derived in Ahn, 2023 to estimate the trend unemployment rate of each state.⁷ Section 3.1 describes the dynamic accounting identity that we employ as our measurement equation, and Section 3.2 discusses how we model the duration structure of unemployment hazards. Section 3.3 describes the state-space representation of our model and discusses how we identify the time-varying parameters that we estimate; finally, Section 3.4 describes how we handle the Covid-19 pandemic period.

3.1 A dynamic accounting identity

The model characterizes changes in the number of unemployed individuals in state \(s\) using unemployment inflows and unemployment hazards by duration. For each state, we use the observed counts of unemployed workers whose unemployment duration falls into five bins: less than 5 weeks, 5–14 weeks, 15–26 weeks, 27–52 weeks, and longer than 1 year. Let \[y_{st} = [U_{st}^{1}, U_{st}^{2.3}, U_{st}^{4.6}, U_{st}^{7.12}, U_{st}^{13.+}]^{\prime}\] be the vector of these duration categories in month \(t\).

The number of newly unemployed individuals, or unemployment inflows, in month \(t\) is denoted by \(w_{st}\), which satisfies \[\begin{equation} U_{st}^{1} = w_{st}.\tag{3.1} \end{equation}\]

The notation \(P_{st}(\tau)\) denotes the fraction of individuals in state \(s\) who were unemployed for one month or less at time \(t-\tau\) and who are still unemployed and searching at time \(t\). For example, the number of unemployed individuals with unemployment durations between 5 and 14 weeks can be written as: \[\begin{equation} U_{st}^{2.3} = w_{s,t-1}P_{st}(1) + w_{s,t-2}P_{st}(2).\tag{3.2} \end{equation}\] Analogously, \[\begin{eqnarray} U_{st}^{4.6} &=& \sum_{\tau=3}^{5} w_{s,t-\tau}P_{st}(\tau),\tag{3.3} \\ U_{st}^{7.12} &=& \sum_{\tau=6}^{11} w_{s,t-\tau}P_{st}(\tau),\tag{3.4} \\ U_{st}^{13.+} &=& \sum_{\tau=12}^{47} w_{s,t-\tau}P_{st}(\tau).\tag{3.5} \end{eqnarray}\]

We assume a maximum unemployment duration of four years, consistent with Hornstein, 2012.

The survival probability \(P_{st}(\tau)\) can be expressed as: \[P_{st}(\tau) = \prod_{h=1}^{\tau} p_{s,t-\tau+h}(h),\] where \(p_{st}(\tau)\) is the monthly probability that an unemployed individual with duration \(\tau\) in state \(s\) remains unemployed one more month.

3.2 Duration structure of unemployment hazards

We express the time-varying parameter \(p_{st}(\tau)\) as \[\begin{equation} p_{st}(\tau) = \exp[-\exp(x_{st}(\tau))],\tag{3.6} \end{equation}\] where \(x_{st}(\tau)\) follows a Laguerre polynomial expansion in level, slope, and curvature components: \[\begin{equation} x_{st}(\tau) = \begin{bmatrix} \beta_{st}^{0} \\ \beta_{st}^{1} \\ \beta_{st}^{2} \end{bmatrix}^{\prime} \begin{bmatrix} 1 \\ e^{-(N-\tau)/\lambda_{s}} \\ ((N-\tau)/\lambda_{st})e^{-(N-\tau)/\lambda_{s}} \end{bmatrix}.\tag{3.7} \end{equation}\]

Each parameter captures a distinct feature of the duration profile: \(\beta_{st}^{0}\) governs the level (the exit probability for newly unemployed individuals), \(\beta_{st}^{1}\) the slope (the overall deterioration in unemployment hazards), and \(\beta_{st}^{2}\) the curvature (nonlinearity in hazards). The location parameter \(\lambda_{st}\) controls where in the duration profile the non-monotonicity arises.

For estimation, we assume the continuation probabilities flatten beyond 12 months of unemployment: \[x_{st}(\tau) = \begin{cases} \beta^{0}_{st} + \beta^{1}_{st} e^{-(12-\tau)/\lambda_{s}} + \beta^{2}_{st}((12-\tau)/\lambda_{s})e^{-(12-\tau)/\lambda_{s}}, & 1 \le \tau < 12,\\[6pt] \beta^{0}_{st} + \beta^{1}_{st}, & \tau \ge 12. \end{cases}\]

3.3 State-space representation and identification

Our goal is to estimate the four time-varying parameters \(w_{st}\), \(\beta^{0}_{st},\beta^{1}_{st}\), and \(\beta^{2}_{st}\), along with the constant \(\lambda_{s}\). Since the dynamic accounting identity is a nonlinear function of the four dynamic latent variables, our model is cast into nonlinear state-space form. The dynamic accounting identity serves as the basis for the measurement equation. Let \(\mathbf{H}(\cdot)\) denote the right-hand-side of equations (3.1) – (3.5) and let \(y_{st}\) denote the vector of five data points \([U_{st}^{1},U_{st}^{2.3},U_{st}^{4.6},U_{st}^{7.12},U_{st}^{13.+}]^{\prime}\). The measurement equation is: \[\begin{equation*} y_{st} = \mathbf{H}(\Xi_{st}) + r_{st}, \label{eq:meas} \end{equation*}\] where \(\Xi_{st}\) is the vector capturing the history of latent variables between \(t\) and \(t-47\), \[\begin{eqnarray*} \Xi_{st} &=& [\xi_{st}^{\prime},\xi_{s,t-1}^{\prime}, \cdots, \xi_{s,t-47}^{\prime}]^{\prime} , \\ \xi_{st} &=& [w_{st}, \beta_{st}^{0}, \beta_{st}^{1}, \beta_{st}^{2}]^{\prime}, \end{eqnarray*}\] and \(r_{st}=[r_{st}^{1},r_{st}^{2.3},r_{st}^{4.6},r_{st}^{7.12},r_{st}^{13.+}]'\) is the vector of measurement errors for each data point in \(y_{it}\). We assume that \(r_{st}\) is drawn independently from the normal distribution with a diagonal covariance matrix \(R_i\), \[\begin{equation*} r_{st}\sim N(0,R_s) . \end{equation*}\]

For the state equation, we assume \(\xi_{st}\) evolves according to an independent random walk: \[\begin{equation*} \underbrace{\xi_{st}}_{4\times 1}=\xi _{s,t-1}+\underbrace{\epsilon_{st}}_{4 \times 1} , \label{state} \end{equation*}\] where the vector innovation, \(\epsilon _{st}=[\epsilon_{st}^{w}\), \(\epsilon_{st}^{\beta _{0}}\), \(\epsilon _{st}^{\beta _{1}}\), \(\epsilon_{st}^{\beta _{2}}]^{\prime}\) is drawn from a normal distribution with a diagonal covariance matrix \(\Sigma_i\), \[\begin{equation*} \underbrace{\epsilon_{st}}_{4 \times 1}\sim N(\underbrace{0}_{4 \times 1},\underbrace{\Sigma_s}_{4 \times 4}) . \end{equation*}\] The full state vector \(\Xi_{st}\) evolves according to \[\begin{equation*} \Xi_{st} = F \, \Xi_{s,t-1} + \underline{\epsilon_{st}} , \end{equation*}\] where \(\underline{\epsilon_{st}}\) stacks the \(\epsilon_{st}\) vectors.

For each state the model has 10 parameters that need to be estimated: the diagonal terms in the variance matrices \(\Sigma_s\) and \(R_s\) along with \(\lambda_s\). Because the measurement equation is nonlinear in the latent variables, the extended Kalman filter is used to form the likelihood function for the observed data \(\{y_{s1},...,y_{sT}\}\) and to infer the unobserved latent variables \(\{\xi_{s1},...,\xi_{sT}\}\). For the rest of the paper, we report the full-sample smoothed estimates, \(\{\xi_{s1|T},\xi_{s2|T},...,\xi_{sT|T}\}\).

The intuition for how the latent variables are identified is as follows. Suppose the economy is in a steady state, with the number of unemployed individuals distributed across five duration categories: \([U_{s}^{1}, U_{s}^{2.3}, U_{s}^{4.6}, U_{s}^{7.12}, U_{s}^{13.+}]\). Given the dynamic accounting identity and assuming zero measurement error, we can identify five steady-state values of \([w_{s}, \beta^{0}_{s}, \beta^{1}_{s}, \beta^{2}_{s}, \lambda_{s}]\) that exactly match the five data points. In this setup, \(w_{s}\) is identified from \(U_{s}^{1}\), while the remaining four duration categories inform the average unemployment continuation probability for each group. The model fits these implied continuation probabilities by duration using a profile of unemployment-exit probabilities, characterized by Laguerre polynomials that capture the level, slope, and curvature and that are parameterized by the four coefficients \(\beta^{0}_{s}\), \(\beta^{1}_{s}\), \(\beta^{2}_{s}\), and \(\lambda_{s}\).

In a dynamic setting, changes in the five data points reflect shifts in \(w_{st}\) as well as in the implied average unemployment exit probabilities for each duration category. To capture these changes, four of the five parameters—\([w_{s}, \beta^{0}_{s}, \beta^{1}_{s}, \beta^{2}_{s}]\)—are assumed to evolve over time as a random walk. The overall difference in unemployment exit probabilities between the 2–3 month and 13+ month duration categories identifies \(\beta^{1}_{s}\), while the exit probability for the 2–3 month category identifies \(\beta^{0}_{s}\). The exit probabilities for the 4–6 month and 7–12 month categories pin down \(\beta^{2}_{s}\) and \(\lambda_{s}\), respectively. Note that we assume \(\lambda_{s}\) to be constant.

The assumption regarding the measurement errors implies that the model targets the smoothed component of the observed data, interpreting high-frequency fluctuations as largely transitory noise.

3.4 Covid-19 treatment

The data exhibit extraordinarily large swings during the pandemic, and fitting the model without special treatment of the Covid-19 outliers runs the risk of distorting the estimates. Following recent approaches used to deal with pandemic-period outliers (for example, Lenza and Primiceri, 2022 and Ng, 2021), we estimate the model parameters using data through December 2019 and filter the latent variables through March 2025 after excluding observations with unusually large swings. This approach prevents dramatic swings in the data from distorting the model parameters; without it, the model will allow the parameters to change abruptly so as to fit the pandemic outliers (see Lenza and Primiceri, 2022). However, this approach by itself cannot prevent the two-sided inference in a state-space model from changing significantly, as the model carries information from the pandemic observations backward and allows it to influence the smoothed estimates of the latent variables. Changes in the history of the latent variables that arise from large transitory outliers like these can be problematic if our goal is to estimate the trend components of the latent variables.

To mitigate this problem, we follow Ng, 2021 and exclude the period with extreme transitory swings from our dataset. Specifically, we pre-treat the data by removing observations between March 2020 and June 2022 and setting the missing observations to the average value of March 2020 and June 2022.⁸ This approach prevents the observations with extreme transitory swings from influencing the two-sided inference over history, thereby enhancing the robustness of the trend estimates.⁹

3.5 Estimated time-varying parameters by state

Appendix B displays the estimates of \(\hat{w}_{s,t|T}\), \(\hat{\beta}_{s,t|T}^{0}\), and \(\hat{\beta}_{s,t|T}^{2}\) for each of the 50 U.S. states and the District of Columbia.¹⁰ The estimates reveal common cyclical patterns as well as important heterogeneity across states. States with more cyclically sensitive labor markets (such as Nevada, Michigan, and Florida) exhibit sharper fluctuations in \(\hat{w}_{s,t|T}\), while states with larger shares of employment in the public sector or in the technology sector (for example, D.C., Washington state, and Massachusetts) display smoother inflows.

Notably, \(\hat{\beta}^{0}_{s,t|T}\)—which captures the exit probability of newly unemployed workers—has declined across nearly all states since the 1980s, with particularly pronounced declines in states undergoing structural employment shifts, such as the manufacturing-intensive states of the Midwest. In contrast, \(\hat{\beta}^{2}_{s,t|T}\)—which captures changes in the relative decline in unemployment exit hazards for short-term unemployed workers relative to the long-term unemployed—fell sharply after the Great Recession, reflecting a rise in long-term unemployment across regions.

4 State-Level Trend U-Star Estimates

In this section, we recover trend U-star estimates for each state based on the trend components of the parameter estimates described in the previous section. We start with a Bayesian trend-cycle decomposition of the four time-varying parameters. We then recover estimates of trend U-star by feeding these trend components into the dynamic accounting identity.

4.1 Trend-cycle decomposition

We decompose each state’s smoothed parameters into trend and cyclical components by applying a Bayesian trend-cycle decomposition to the quarterly averages of \(\hat{w}_{s,t|T}\), \(\hat{\beta}_{s,t|T}^{0}\), \(\hat{\beta}_{s,t|T}^{1}\), and \(\hat{\beta}_{s,t|T}^{2}\).¹¹ The model that is typically employed in the literature uses data at quarterly frequencies (see, for example, Stock and Watson, 1998 or Grant and Chan, 2017), so we first convert our monthly smoothed estimates into quarterly values by taking averages within the quarter. We use an integrated random walk to characterize the stochastic trend of each estimate as a smooth curve, which is an appropriate way to model time-varying trends in two-sided (smoothed) estimates.¹²

We denote \(\hat{w}_{sq|T}\) as the average of the smoothed estimates \(\hat{w}_{st|T}\) in quarter \(q\). The term \(\hat{w}_{sq|T}\) is composed of the trend, \(\psi_{sq}^{w}\), and a transitory deviation from the trend, \(c_{sq}^{w}\), \[\begin{equation} \hat{w}_{sq|T} = \psi_{sq}^{w} + c_{sq}^{w} .\tag{4.1} \end{equation}\] The cyclical component is modeled as a zero-mean stationary AR(2) process: \[\begin{equation} c_{sq}^{w} = \phi_{1}^w c_{s,q-1}^{w} + \phi_{2}^w c_{s,q-2}^{w} + \epsilon_{sq}^{cw},\tag{4.2} \end{equation}\] where \(\epsilon_{sq}^{cw}\sim N(0,(\sigma_{s}^{cw})^2)\).¹³ We express the trend, \(\psi_{sq}^{w}\), as \[\begin{eqnarray} \psi_{sq}^{w} &=& \mu_{sq}^{w} + \psi_{s,q-1}^{w} , \nonumber \\ \mu_{sq}^{w} &=& \mu_{s,q-1}^{w} + \epsilon_{sq}^{\psi w},\tag{4.3} \end{eqnarray}\] where \(\epsilon_{sq}^{\psi w} \sim N(0,(\sigma_{s}^{\psi w})^2)\). Note that equation (4.3) is equivalent to \[\begin{equation} \triangle \psi_{sq}^{w} = \triangle \psi_{s,q-1}^{w} + \epsilon_{sq}^{\psi w}.\tag{4.4} \end{equation}\] Similar specifications are used to estimate the trend components of \(\hat{\beta}_{iq|T}^{0},\hat{\beta}_{sq|T}^{1}\), and \(\hat{\beta}_{sq|T}^{2}\), which we denote by \({\psi}_{sq}^{\beta0}\), \({\psi}_{sq}^{\beta1}\), and \({\psi}_{sq}^{\beta2}\), respectively.

To recover the trend unemployment rate for each state we also estimate each state’s trend labor force, \(\psi_{sq}^{LF}\). We again use an integrated random walk model because it provides a good way to capture the I(2) dynamics of the civilian labor force. (As it turns out, the estimates of trend U-star are relatively insensitive to the priors used for the trend-cycle decomposition of the labor force.) Details on the priors and the derivation of the Gibbs sampler can be found in the appendix.

We then feed the relevant trend estimates back into the dynamic accounting identity with the measurement errors set equal to zero. To do this, we convert the quarterly trend estimates into monthly series, \(\hat{\psi}_{st|T}^{w}\), \(\hat{\psi}_{st|T}^{\beta0}\), \(\hat{\psi}_{st|T}^{\beta1}\), \(\hat{\psi}_{st|T}^{\beta2}\), and \(\hat{\psi}_{st|T}^{LF}\), under the assumption that the monthly values for a given quarter are the same.¹⁴ The trend U-star for state \(s\) in month \(t\) is therefore defined as: \[\begin{equation*} U^{*}_{st} = 100 \times \frac{{\mathbf{1}}^{\prime }h(\hat{\psi}_{st|T},\hat{\psi}_{s,t-1|T},\hat{\psi}_{s,t-2|T},...,\hat{\psi}_{s,t-47|T}, \hat{\lambda}_{s})}{\hat{\psi}_{st|T}^{LF}}, \end{equation*}\]where \({\mathbf{1}}\) is a \((5 \times 1)\) vector of ones, \(h(\cdot)\) denotes the system of equations (3.1)-(3.5) with measurement errors set to zero, and \(\hat{\psi}_{st|T}\) is the vector \([\hat{\psi}_{st|T}^{w}, \hat{\psi}_{st|T}^{\beta0}, \hat{\psi}_{st|T}^{\beta1}, \hat{\psi}_{st|T}^{\beta2}]'\). The quarterly trend U-star is computed by taking the quarterly average of the monthly estimates.

5 Key Features of State-Level Trend Unemployment Rates

5.1 Changes in the overall distribution

Figure 1: The Trend Unemployment Rate and Its Distribution by State and Duration
Panel A. Total Trend Rat
Panel B. Distribution by State
Panel C. Short-term Component Panel D. Long-term Component

Notes to figure: Panel A displays the aggregate unemployment rate (black line) and the trend U-star (blue line), with the 68% posterior interval shown as a light-blue shaded region. Panel B shows the total trend unemployment rates for the 50 states and the District of Columbia (thin lines), along with the labor-force-weighted mean (solid line) and median (dashed line); the grey shaded band represents the 90% cross-state distribution. Panels C and D present analogous figures for the short-term and long-term components—defined as unemployment spells of six months or less and longer than six months, respectively. The vertical shaded bars denote periods of recession as defined by the NBER. Source: Authors’ calculation.

Accessible version

Figure 1 presents estimates of trend U-star together with the portions of the trend that reflect short-term unemployment (defined as unemployment spells that last six months or less) and longer-term unemployment (more than six months). Panel A plots the aggregate estimate (solid blue line) and its 68% credible interval, together with the actual unemployment rate (black line). Overall, the aggregate trend exhibits a decline from the 1980s through the 1990s, flattens during the 2000s, and then resumes its downward movement after the Great Recession. This downward trajectory primarily reflects demographic forces—including population aging, rising educational attainment, and reduced labor-market dynamism—which result in declining inflows into unemployment and a steady reduction in the short-term component of trend U-star. Notably, the aggregate long-term component remains roughly flat through 2000, edges upward between 2000 and 2010, and then declines modestly thereafter; this increase in the long-term component is responsible for the 2000–2010 upturn in the aggregate U-star estimate. Apparently, then, the factors pushing down the short-term component had no effect on the long-term component (or other factors impeded a simultaneous decline in the long-term component).

Panels B through D also display the 51 state-level estimates of total, short-term, and long-term U-star, with each estimate shown as a thin light-blue line. The solid black lines in the panels represent the mean estimate weighted by population, and the dashed black lines represent the median estimate. Overall, states share a common downward trend in the total and short-term components, but both the pace of decline and the level of U-star vary substantially across states, as shown by the wide dispersion of cross-state estimates in Panels B and C. A similar pattern emerges for the long-term component (Panel D), though the dispersion of these estimates is smaller. These patterns highlight the pronounced cross-state heterogeneity in both the level and dynamics of trend U-star.

Figure 2: Overall changes in trend U-star by state


Notes to figure: This figure displays changes in trend U-star between the average values in 1978 and those in the five-quarter period 2024:Q1–2025:Q1. The red dashed line is the corresponding national average. Source: Authors’ calculation.

Accessible version

Figure 2 summarizes state-level changes in trend U-star between the 1978 and 2024:Q1–2025:Q1 averages. This figure highlights significant heterogeneity in secular declines in state-level unemployment rates. In particular, about twenty states show larger declines than the national-level estimate; twenty-five states exhibit slower declines; and four states’ trend U-star did not change or even rose.

Figure 3: Evolution of Cross-State Dispersion in Trend Unemployment
Panel A. Total
Panel B. Short-term Component
Panel C. Long-term Component

Notes to figure: The panels show the standard deviation (blue line) and interquartile range (IQR) of the state-level trend unemployment rate estimates at each point in time. Panel A uses the estimates of total trend unemployment, while Panels B and C use the short- and long-term components, respectively. Source: Authors’ calculation.

Accessible version

Figure 3 shows the evolution of cross-state dispersion, measured using both the interquartile range and the standard deviation of the state-level estimates. As Panel A illustrates, cross-state dispersion declines from the mid-1980s through the mid-2000s before then stabilizing. This pattern is driven primarily by the short-term component (Panel B), with the dispersion of the long-term component remaining relatively constant over time. (Note that the convergence of the estimates does not imply that state-level estimates are becoming more synchronized, only that they are moving in a narrower range.)

5.2 Features of state-level estimates

Figure 4: Average Level of Trend U-Star and the Relative Importance
of Its Long-Term Component Across States
Panel A: Average Level of Trend U-star (1978:Q1-2025:Q1, %)
Panel B: Long-term Share (1978:Q1-2025:Q1, %)

Note to figure: Panel A displays the average level of trend U-star for U.S. states (in %). Panel B displays the share of each state’s total trend U-star that is accounted for by its long-term component (also in %). The sample period is 1978:Q1-2025:Q1.
Source: Authors’ calculation.

Accessible version

Figure 4 presents the mean level of each state’s trend U-star (Panel A) and the share that is accounted for by the long-term component (Panel B). The trend U-stars are highly heterogeneous across states, ranging from about 3.25% in Nebraska to roughly 7.5% in Alaska. The long-term shares also vary substantially, from just above 10% in Wyoming to around 25% in the District of Columbia. Taken together, these observations suggest considerable cross-state heterogeneity in labor-market functioning, and particularly in the relative importance of structural versus frictional unemployment.

Figure 5: Trend U-Star Levels and Long-term Shares
Panel A: Average Level of Trend U-star
Panel B: Long-term Share

Note to figure: This figure displays U-star quartile groupings (Panel A) and the long-term share groupings (Panel B) across states. The left column shows states in the lowest quartile (“1st Q”); the center column shows the middle two quartiles, and the right figure shows the highest quartile (“4th Q”). Colors correspond to the scales shown on the right. The sample period is 1978:Q1-2025:Q1.
Source: Authors’ calculation.

Accessible version

Figure 5 shows the geography of trend U-star and its long-term share. Panel A displays states with the lowest trend U-star (the bottom 25%, left), those in the middle 50% of the trend U-star distribution (middle 50%), and states with the highest trend U-star (the top 25%, right). States in the bottom 25% are clustered in the Upper Midwest and the mountain plains, including Nebraska, North Dakota, South Dakota, Iowa, and Montana, which appear in dark purple (left figure in Panel A). These states have strong agricultural as well as manufacturing bases and have high labor force participation rates. In these states, the long-term share also tends to be low, as can be seen by comparing this map to the corresponding map in Panel B. The top 25% includes states from the industrial Midwest and deep South, as well as California. The higher U-star levels and larger long-term shares in this group imply that workers in these states are more likely to experience structural unemployment.

Figure 6: Trend U-Star and Its Long-term Share
Panel A. All States
Panel B. By Census Region

Notes to figure: The charts plot the share of U-star that is accounted for by its long-term component (\(y\)-axis) against the average level of U-star (\(x\)-axis) for each state. The black lines in panel B are OLS regression lines.
Source: Authors’ calculation.

Accessible version

The positive association between the level of trend U-star and its long-term share is confirmed by the scatterplots in Figure 6, as well as by the Spearman rank correlation between the level of U-star and its long-term share, which is 0.53.¹⁵ A positive association is also evident within each Census region (Panel B), and is quantitatively strongest in the Northeast and Midwest.

5.3 Drivers of trend U-star

This section investigates the role of state-level factors in shaping variation in trend U-star. Consider the following model: \[\begin{equation} U_{st}^{*} = a_s + \sum_{i}b_{i}X_{st}^{i} + \varepsilon_{st},\tag{5.1} \end{equation}\] where \(U_{st}^{*}\) denotes trend U-star for state \(s\) in quarter \(t\); \(a_s\) is a state fixed effect; \(X_{st}^{i}\) denotes attribute \(i\) of state \(s\) in quarter \(t\), which receives coefficient \(b_i\); and \(\varepsilon_{st}\) is the error term. We consider four broad categories of state-level attributes: demographics, industry structure, labor-market rigidity, and state-specific labor force participation and labor productivity. As demographic variables, we include the population shares of high-school dropouts and the share of college graduates out of population aged 25 and over; the shares of individuals aged 16–24 and 55-plus among those aged 16 and older; and men’s share of total state employment. Industry structure is captured by the employment shares of manufacturing, professional and business services, and finance, as well as with an indicator for oil-producing states. Measures of labor-market rigidity include the union membership share, the employment share of large firms (defined as firms with 500 or more employees), the tax-to-income ratio, the level of the minimum wage, and the share of nominal wage changes that are zero.¹⁶ Finally, we include state-specific labor force participation rates and average labor productivity to capture broad conditions of labor supply and labor demand.

The estimation period runs from 1980:Q1 to 2018:Q4 and excludes Washington, D.C. because of data constraints. (Excluding Washington, D.C., the downward nominal wage rigidity measure is available from 1978 until 2020, while labor productivity is available from 1976 until 2018.) New Mexico and South Dakota are omitted from 2002 onward because data on the financial employment shares in those states are unavailable. The standard errors are clustered by state.

Table 1: Coefficient Estimates from Equation [eq:state]

Note: This table reports the coefficient estimates from equation (5.1). Standard errors (in parentheses) are clustered by state. * p<0.10, ** p<0.05, *** p<0.01.

Source: Author’s calculation.

Table 1 reports the coefficient estimates. Starting with educational attainment, a higher share of high-school dropouts (row 1) has a statistically significant positive effect on trend U-star through its effect on the short-term component (column 2), possibly because less-educated workers tend to have higher job-separation rates. The overall effect of a higher share of college graduates (row 2) is smaller and insignificant, reflecting a modest negative effect on the long-term component (column 3) that appears to offset a counterintuitive (but barely significant) positive effect on the short-term component. Overall, because the boost to aggregate U-star is larger for high-school dropouts than for college graduates, rising educational attainment would appear to lower trend U-star on net.

As shown in rows 3 and 4, a larger share of young workers (aged 16–24) raises trend U-star, while the effect of a larger share of older workers (aged 55 or more) is negative but not statistically significant. The population share of young workers primarily raises the short-term component of U-star, while the share of older workers has a marginally significant negative effect on the long-term component, indicating that population aging puts downward pressure on U-star. Finally, the effect on U-star of a higher relative share of men in employment (row 14) is insignificant.

We find little evidence that industry structure plays an important role in determining trend U-star. In the aggregate, a higher employment share in professional and business services (row 6) does not have a statistically significant effect on aggregate U-star, but this is because it has opposite effects on the duration components. Specifically, a larger employment share for professional services lowers the short-term component of U-star, but raises its long-term component, with both effects statistically significant at the 1% level. Here, two opposing forces are likely at work: Job-specific human capital in this high-skilled sector enhances short-term employment stability, but displaced workers face longer unemployment spells, thereby raising the long-term component. This pattern could obtain because these workers are more selective about job offers, have more financial resources to engage in an extended job search, or maintain relatively high reservation wages. This result suggests that the increased employment share of professional and business services over time has likely raised the long-term share of U-star, but left its aggregate level little changed. Finally, a greater role of oil production in a state (row 12) tends to lower both the short-term and long-term components of U-star (columns 2 and 3), while the employment shares of the manufacturing and finance industries do not exhibit statistically significant effects.

Labor-market rigidities and institutional factors have mixed effects on U-star. A higher tax-to-income ratio (row 10) and greater downward nominal wage rigidity (row 13) exert upward pressure on U-star. While we would expect (and find) that a higher tax burden reduces labor demand and restrains job creation (all else equal), its effect on U-star comes through the short-term component (column 2), which is perhaps surprising. Greater downward nominal wage rigidity raises both duration components of U-star (row 13), which is intuitive, while minimum wages have distinct, heterogeneous effects across the duration components of U-star (row 11). A higher minimum wage significantly lowers both the aggregate and short-term components, a finding consistent with the view that higher minimum wages reduce voluntary separations and enhance job attachment. At the same time, however, a higher minimum wage significantly raises the long-term component (column 3) and the long-term share (column 4). This result suggests that higher minimum wages may discourage firms from hiring low-productivity job seekers, thereby prolonging these workers’ unemployment spells and increasing their risk of long-term unemployment. Finally, the effects of higher unionization rates (row 8) and a larger employment share in large firms (row 9) have intuitive signs, but are not statistically significant.

A higher labor force participation rate (LFPR) in a state (row 15) is associated with a reduction in U-star, mostly through its long-term component (including the long-term share, column 4). Because the LFPR reflects the interaction of labor supply and labor demand, a higher participation rate suggests the presence of robust labor demand that effectively draws individuals into the workforce. This alignment between demand and supply could in turn be associated with improved long-term labor-market matching and functioning, thereby reducing prolonged unemployment spells. In contrast, higher labor productivity (row 16) does not exhibit a statistically significant effect, aside from a positive association with the long-term share (column 4). This association might reflect the fact that higher productivity driven by restructuring or by the exit of less-productive sectors can lengthen search durations for displaced workers whose skills have become obsolete, thus raising the long-term share in U-star.

6 Implications for the Phillips Curve

It is now clear that state-level estimates of U-star differ substantially in terms of both their levels and their evolution over time. We next show that this heterogeneity has important implications for identifying the slope of the Phillips curve using state-level data. Previous research on the Phillips curve based on state-level data (e.g., Hazell et al., 2020; Fitzgerald et al., 2020; and McLeay and Tenreyro, 2020) employed time and state fixed effects in their empirical models. State fixed effects are designed to capture differences in the levels of state-specific natural rates and other time-invariant state-specific attributes. Conversely, time fixed effects account for national-level economic conditions, such as shifts in monetary policy regimes, movements in inflation expectations, and changes in the aggregate natural rate of unemployment. Implicit in this approach is the assumption that changes in state-level natural rates are perfectly correlated with the national aggregate. Finally, the last assumption made by these studies is that aggregate demand shocks are uncorrelated with the natural rate. The presence of significant and persistent heterogeneity in state-level U-stars calls each of these assumptions into question.

For example, in Hazell et al., 2020’s two-way fixed-effects specification, their instrument capturing heterogeneous state-level responses to changes in aggregate demand may not adequately isolate the structural relationship between the state-level unemployment-rate gap and (nontradables) price inflation—that is, the slope of the Phillips curve. Using our state-level U-star estimates, we revisit Hazell et al., 2020 and re-estimate the Phillips curve, replacing state-level unemployment rates with the corresponding state-level unemployment-rate gaps (while maintaining all the other features of their empirical framework).¹⁷

6.1 Estimating state-level Phillips curves

Hazell et al., 2020 estimate the following forward-looking new-Keynesian Phillips curve in order to recover the slope \(\kappa\) of the aggregate Phillips curve: \[\begin{equation} \pi^{N}_{it} = \alpha_i + \gamma_t - \kappa \sum_{j=0}^{T} \beta^{j} u_{i,t+j} - \lambda \sum_{j=0}^{T} \beta^{j} \hat{p}^{N}_{i,t+j} + \tilde{\omega}^{N}_{it} + \eta^{N}_{it},\tag{6.1} \end{equation}\] where \(\pi^{N}_{it}\) is the four-quarter percent change in nontradable goods prices for state \(i\) at time \(t\), \(\alpha_i\) is a state fixed effect, \(\gamma_t\) is a time fixed effect, \(u_{i,t+j}\) is the unemployment rate of state \(i\) at time \(t+j\), \(\hat{p}^{N}_{i,t+j}\) is the relative price of nontradables in state \(i\) at time \(t+j\), \(\tilde{\omega}^{N}_{it}\) is an error term that is related to supply shocks in the nontradable goods sector of state \(i\), and \(\eta^{N}_{it}\) is an error term that includes an expectation error that is orthogonal to information known at time \(t\) along with a truncation error that results from using a finite \(T\) in equation (6.1).¹⁸ The parameter \(\beta\) denotes a discount factor, which is set equal to 0.99.

Equation (6.1) can be estimated using standard GMM methods by instrumenting for the two forward sums. Hazell et al., 2020 consider two ways to do this. The first approach is to instrument for the two forward sums using four-quarter lagged unemployment, \(u_{i,t-4}\), and the four-quarter lagged relative price of nontradables, \(\hat{p}^N_{i,t-4}\). Assuming rational expectations, these lagged variables are uncorrelated with the expectations error in \(\eta^N_{i,t}\). The second approach involves constructing a Bartik instrument that captures the idea that demand for nontradables in state \(i\) will rise relatively more when there is a national increase in demand for the types of tradable goods that state \(i\) produces relatively more of. The instrument that captures tradable-demand spillovers is given by \[\begin{equation} \text{tradable Demand}_{i,t} = \sum_{k} \tilde{S}_{k,i} \times \Delta_{3Y} \log S_{-i,k,t} ,\tag{6.2} \end{equation}\] where \(\tilde{S}_{k,i}\) is the average employment share of industry \(k\) in state \(i\) over time, and \(\Delta_{3Y} \log S_{-i,k,t}\) is the three-year growth in national employment of industry \(k\) at time \(t\) excluding state \(i\).¹⁹

Hazell et al., 2020 set \(T = 20\) quarters in the forward sums in equation (6.1), which implies that five years’ worth of observations at the end of the sample are discarded. To minimize the effect of this reduction in sample size, they use a two-sample two-stage least squares (2SLS) regression where the first-stage regression is estimated on a sample that omits the most-recent five years and the second stage is estimated on the full sample. (Standard errors are clustered at the state level, and are corrected using the method developed by Chodorow-Reich and Wieland, 2020.)

If we are willing to assume that \(u_{it}\) and \(p^N_{it}\) follow AR(1) processes, then the time-\(t\) expected forward sums in equation (6.1) can be expressed in terms of the AR parameters, the discount factor \(\beta\), and the time-\(t\) values of \(u_{it}\) and \(p^N_{it}\). This motivates a second specification that Hazell et al., 2020 consider: \[\begin{equation} \pi^{N}_{it} = \alpha_i + \gamma_t - \psi u_{i,t} - \delta p^{N}_{i,t} + \varepsilon_{it},\tag{6.3} \end{equation}\] where \(\varepsilon_{it}\) is an error term.²⁰ Because inflation is defined as a four-quarter change, Hazell et al., 2020 actually estimate equation (6.3) using \(u_{i,t-4}\) and \(p^N_{i,t-4}\), \[\begin{equation} \pi^{N}_{it} = \alpha_i + \gamma_t - \psi u_{i,t-4} - \delta p^{N}_{i,t-4} + \varepsilon_{it},\tag{6.4} \end{equation}\] so that inflation over the next four quarters represents the outcome of the current-quarter values of \(u\) and \(p^N\). Equation (6.4) can be estimated by OLS; alternatively, we can instrument for the lagged unemployment rate using the tradable demand variable defined in equation (6.2).

The assumption that is being made about the natural rate with the two-way fixed effects is that the state-level natural rate is either constant over time—and therefore captured by state fixed effects—or co-moves perfectly with the aggregate natural rate, and so can be captured by time fixed effects. Under these assumptions, Hazell et al., 2020 interpret the forward sum of the unemployment rate as the forward sum of the unemployment-rate gap, and similarly interpret the lagged unemployment rate in equation (6.4) as the lagged unemployment-rate gap. However, if state-level trend unemployment is not constant or does not closely track the aggregate trend, both equations (6.1) and (6.4) are misspecified and their estimated coefficients are likely biased. As we have seen, our state-level estimates indicate that trend U-star is generally neither constant nor perfectly aligned with its aggregate counterpart. We would therefore expect the estimated Phillips curve coefficients to change when the state-level unemployment rate is replaced with the corresponding state-level unemployment-rate gap, defined as the difference between a state’s unemployment rate and its trend U-star.

The original sample period used in Hazell et al., 2020 is 1978–2018. In Hazell et al., 2020’s dataset, some states do not have unemployment rate data after 2017; we therefore restrict our sample period to 1978–2017, which implies that unemployment rates and U-star estimates are available for every state in each period. As we will show, doing so yields estimates that are quite close to Hazell et al., 2020’s original results.²¹

Table 2: Estimation Results

Note: This table presents estimates of \(\kappa\) and \(\psi\) from regression specifications (6.1) and (6.4), respectively. The outcome variable is the four-quarter change in nontradable prices (in percent). The estimates in the first three columns use the fourth lags of quarterly state unemployment and the relative price of nontradables as instruments (this corresponds to OLS for \(\psi\)). “No FE” denotes specifications with no fixed effects included; “No TE” denotes specifications that include state but not time fixed effects; “SE&TE” denotes specifications with both state and time fixed effects. In the fourth column, “Bartik (SE&TE),” we replace lagged unemployment in the instrument set with the tradeable-demand (Bartik) instrument and include both state and time fixed effects. Standard errors, clustered by state and corrected using the procedure from Chodorow-Reich and Wieland, 2020, are reported. Standard errors are in parentheses * p<0.10, ** p<0.05, *** p<0.01.

Source: Authors’ calculation.

1. Unempl. Rate (\(\kappa\))

2. Eq. (6.4) with UR (\(\psi\))

3. UR-gap (\(\kappa\))

4. Eq. (6.4) with UR-gap (\(\psi\))

Table 2 compares the estimation results that we obtain using either the unemployment rate or the unemployment-rate gap. Panels 1 and 2 use the unemployment rate to replicate the results in Hazell et al., 2020. As in Hazell et al., 2020, the slope of the Phillips curve (\(\kappa\) in equation 6.1) has the “wrong” sign when the fixed effects are omitted (panel 1, columns 1 and 2). Using both state and time fixed effects (columns 3 and 4) yields a Phillips curve slope that is statistically significant but extremely flat; this conclusion is robust when we instrument for the unemployment rate with the Bartik instrument (column 4). As shown in panel 2, if we estimate \(\psi\) from equation (6.4), the coefficients become much larger in absolute magnitude. Note that the role of the unemployment rate in equation (6.4) is fully consistent with a conventional Phillips curve, in which inflation is determined by excess supply or demand in labor and product markets, and where the unemployment rate serves as a proxy for the degree of resource utilization rather than as a predictor of future real marginal costs. Hence, these results do not rule out that the conventional interpretation of the unemployment–inflation relationship provides a better description of inflation dynamics than the new-Keynesian model.²² Leaving that aside, we would point out that all of the results we obtain are very similar to what Hazell et al., 2020 report.

Replacing the unemployment rate with its gap (panel 3) causes the results to change in several interesting ways. First, the slope estimate from the specification without fixed effects (column 1) receives the expected (positive) sign. Second, whether fixed effects are included has essentially no effect on the slope—the estimates with and without fixed effects (columns 1 to 3) are roughly equal in magnitude—but the slopes are no longer statistically significant. When the two-way fixed effects specification with the Bartik instrument is used (column 4), the coefficient on \(\kappa\) becomes statistically significant and is approximately twice as large as the other estimates that use the gap, and is also roughly twice as large as the corresponding estimate obtained using the unemployment rate. The results for \(\psi\)—the slope coefficient from equation (6.4)—are broadly similar (panel 4).

These findings suggest a different interpretation of the results in Hazell et al., 2020. First, the fact that we no longer obtain a statistically significant estimate of \(\kappa\) when we use the gap specification (column 3) likely reflects the fact that variations in state-level U-star have predictive power for state-level nontradables price inflation. By smoothing through any cyclical variation, the discounted forward sum of the unemployment rate will approximate the discounted forward sum of U-star—that is, the trend component will dominate the forward sum.²³ On average, both U-star and inflation trend lower over the sample period. Hence, rather than recovering a Phillips curve correlation, the specifications in panel 1 are capturing this low-frequency co-movement between inflation and the natural rate.

In addition, our finding that the gap-based estimate of \(\kappa\) increases when we use the Bartik instrument has two implications. First, in contrast to Hazell et al., 2020’s claim (p. 1331), inclusion of two-way fixed effects is apparently not sufficient to capture relevant supply shocks (such as shocks to the natural rate of unemployment).²⁴ Second, the larger value of \(\kappa\) that obtains in the gap-based specification when the Bartik instrument is used suggests that we are not recovering a structural estimate of the slope. As Hazell et al., 2020 note, the Bartik demand instrument is more persistent than the unemployment rate itself; they use this fact to explain why the “reduced-form” parameter \(\psi\) receives a larger coefficient when the Bartik instrument is used. Specifically, Hazell et al., 2020 argue that the size of \(\psi\) could be lower if (for example) a demand shock is attributable to a temporary policy change, while \(\kappa\) should be policy invariant, as it represents a structural parameter (p. 1332). (Put differently, \(\psi\) embeds both the structural parameter \(\kappa\) as well as the average persistence of unemployment rate movements, but this persistence can vary depending on the source and nature of the shock to unemployment.) However, under the gap-based specification we obtain an estimate of \(\kappa\) that is nearly twice as large when the Bartik instrument is used, suggesting that we are not in fact recovering a structural estimate of the slope.²⁵

Table 3: Estimation Results with Short- and Long-term Unemployment

Note: See Table 2. Source: Authors’ calculation.

1. UR \(<7\) months

2. Eqn. (6.4)

3. U-gap \(<7\) months

4. Eqn. (6.4)

5. UR \(>6\) months

6. Eqn. (6.4)

7. U-gap \(>6\) months

8. Eqn. (6.4)

Next, we assess whether the Phillips curve slope differs by unemployment duration by replacing the state-level unemployment rates and unemployment-rate gaps with their short-term and long-term components. Across many states, the trends in these duration components have moved in opposite directions since the 2000s; their cyclical dynamics also differ, which suggests it might be informative to estimate separate Phillips curves for short-term and long-term unemployment. Table 3, which mirrors the structure of Table 2, reports these estimation results.

The first noteworthy finding is that the slope coefficients from the models with two-way fixed effects (column 3) are all statistically insignificant when the unemployment gap is used. In contrast, both the short-term and long-term gaps receive statistically significant coefficients in the model that uses the Bartik instrument (panels 3 and 7, column 4), which is consistent with the aggregate results. However, the coefficient on each individual gap is larger than the coefficient on the aggregate gap (in the case of the short-term gap, the coefficient is more than 2.5 times larger than its aggregate counterpart). Similarly, the coefficients on the short- and long-term lagged unemployment rate gap (\(\psi\)) obtained using the Bartik instrument are significantly larger than the corresponding coefficient on the aggregate gap (compare column 4 of panels 4 and 8 with column 4 of panel 4 in Table 2). Together, these results suggest that differences in the cyclicality of the short-term and long-term unemployment components partially offset each other in the aggregate, resulting in a more-muted overall cyclical response.

The statistically significant coefficient on the long-term gap is also noteworthy inasmuch as it points to a possible role for hysteresis in inflation dynamics. The long-term unemployed are likely to include workers prone to scarring effects and labor force detachment, which are the core mechanisms of unemployment hysteresis. Changes in the long-term component of the gap that result from changes in local demand (as captured by the Bartik instrument) receive a significant coefficient in the Phillips curve, which in turn suggests a potential interaction between demand shocks and structural unemployment. Finally, comparing panel 1 of Table 3 with panel 1 of Table 2 suggests that the original results of Hazell et al., 2020 are largely driven by the short-term unemployment rate. Given that the overall trend in the unemployment rate was predominantly influenced by its short-term component during the sample period, this finding further supports the conclusion that those results reflect an association between U-star and inflation.

We next repeat Hazell et al., 2020’s exercise that looks at the stability of the slope coefficients before and after 1990. Table 4 shows results for the models with two-way fixed effects (columns 1 and 2 give the slope estimates when lagged unemployment is used as an instrument, and columns 3 and 4 give the estimates when the Bartik instrument is used). For the gap-based models, neither estimate of \(\kappa\) is statistically significant after 1990 (columns 2 and 4); in both subsamples, we again find the result that the gap-based estimates of \(\kappa\) increase noticeably (though not significantly) when the Bartik instrument is used.

It is worth noting that the flattening of the Phillips curve that is implied by the \(\psi\) estimates in the model with two-way fixed effects is similar in magnitude to what we recover from a conventional aggregate Phillips curve fit to quarterly data, so long as we account for the shift in the inflation process that occurred in the mid-1990s. Prior to the mid-1990s, inflation dynamics were essentially \(I(1)\), implying that inflation was characterized by an “accelerationist” specification during this period. Subsequently—and at least until the pandemic-related inflation surge—inflation became mean-reverting around a stable stochastic trend.²⁶ Simple quarterly Phillips curves that account for this shift in the inflation process imply a slope of 0.225 prior to the mid-1990s with a subsequent decline to 0.026.²⁷ These point estimates are very similar to the ones we obtain for \(\psi\) pre- and post-1990, which in turn suggests that the flattening of the Phillips curve documented in aggregate data is also being captured here. (Note that this interpretation differs from the one given by Hazell et al., 2020, who argue that the apparent flattening of the aggregate Phillips curve is an artifact of not controlling for inflation expectations.)

Table 4: Phillips Curve Slope Pre- and Post-1990

Note: All regressions include state fixed effects. Standard errors in parentheses. \(^\ast\) \(p<\)0.10, \(^{\ast \ast}\) \(p<\)0.05, \(^{\ast \ast \ast}\) \(p<\)0.01.

A. Unemployment rate

B. Unemployment gap

The preceding analysis has two important implications for the identification of the Phillips curve using state-level data. To begin with, because state-level U-stars exhibit meaningful time variation and substantial heterogeneity across states, the assumption that a state-level U-star is constant or closely aligned with an aggregate estimate is likely to be a poor one (by extension, two-way fixed effects will be unable to control for the heterogeneity in state-level U-stars over time or at a particular point in time). Second, the slope estimates from the forward-looking Phillips curve specification appear to reflect a correlation between state-specific natural rates and nontradables inflation rather than a relation between inflation and cyclical movements in unemployment. By extension, the large and statistically significant correlation between inflation and demand-driven variation in the lagged unemployment rate or unemployment gap is consistent with the standard description of the Phillips curve as a relation between labor-market slack and inflation, as opposed to a new-Keynesian interpretation in which the unemployment rate acts as a proxy for expectations of future costs.

6.2 Discussion of the structural model

The finding that the discounted forward sum of unemployment gaps in equation (6.1) plays little to no role in inflation determination is in line with earlier tests of the new-Keynesian Phillips curve using aggregate data. Rudd and Whelan, 2005 proposed testing the basic new-Keynesian Phillips curve, \[\begin{equation} \pi_t = \beta E_t \pi_{t+1} + \kappa x_t ,\tag{6.5} \end{equation}\] where \(x_t\) denotes a measure of excess demand or cost pressures such as an unemployment or output gap, by solving it forward \(K\) periods to yield \[\begin{equation} \pi_t = \kappa E_t \sum_{i=0}^{K} x_{t+i} + \beta^{K+1} E_t \pi_{t+K+1} .\tag{6.6} \end{equation}\] (Note that this solution is exact and does not suffer from truncation error, in contrast to the finite-sum approximation used by Hazell et al., 2020.) Equation (6.6) can then be estimated by GMM under the assumption that the realized future value of a variable will equal its time-\(t\) expectation plus an unforecastable (rational) expectational error; alternatively, VAR-based forecasts of \(x_{t+i}\) can be generated and used to assess the model’s ability to track inflation.²⁸ Tests of equation (6.6) using aggregate data typically found little support for the model, in that the forward sum had little to no ability to explain actual inflation.²⁹

The results that we obtain from state-level data can be viewed as confirming the new-Keynesian model’s inability to provide a good description of inflation dynamics. The estimates of \(\kappa\) that we found using unemployment-rate gaps as the driving variable \(x_t\) were not only small, but were consistently statistically insignificant, implying that the model’s prediction that forward-looking behavior should dominate price setting finds no support in the data.

Tests of equation (6.6) also found that the new-Keynesian model was unable to explain the significant role played by lagged inflation in empirical inflation equations prior to the mid-1990s. This latter fact is directly relevant to the main argument of Hazell et al., 2020’s paper. Hazell et al., 2020 document that their estimated model explains almost none of the variation in nonhousing inflation, and use their finding of a flat Phillips curve to argue that the Volcker disinflation of the 1980s was mostly the result of a shift in long-run inflation expectations as opposed to the rise in unemployment that was induced by tight monetary policy.³⁰ But the real reason that the model cannot fit the 1980s well is because the forward sum of equation (6.6) cannot account for the persistence in inflation that made the Volcker disinflation so costly in terms of unemployment.

We might also be surprised at Hazell et al., 2020’s claim that long-term inflation expectations matter in the context of the new-Keynesian Phillips curve. Equation (6.5) involves a short-term expectation, and while the closed-form solution of the model (6.6) involves a forward sum that goes to infinity as \(K\) does, the expectation of that forward sum is the (short-term) expectation of next period’s inflation rate. Moreover, allowing for nonzero trend inflation in a Calvo model significantly changes the nature of the price-setting problem (and the model’s solution) so that the resulting model looks very different from a simpler version like (6.5).³¹ More broadly, Werning, 2022 examined the role of short- versus long-term expectations in the context of a general class of sticky-price models (albeit models in which the effects of nonzero trend inflation aren’t considered). He argues that short-run inflation expectations are more important, in the sense that the influence of long-term expectations goes to zero as the long term gets longer; specifically, his result shows that the influence of \(E_t \pi_{t+i}\) goes to zero as \(i\) gets large.³²

These considerations suggest that finding a role for long-run expectations in canonical sticky-price models would be harder than one might think.³³ However, Hazell et al., 2020 claim that it’s actually a straightforward undertaking. They start with the basic new-Keynesian Phillips curve, \[\begin{equation} \pi_t = \beta E_t \pi_{t+1} - \kappa \left( u_t - u^n_t \right) + \nu_t ,\tag{6.7} \end{equation}\] where \((u_t - u^n_t)\) is the unemployment gap (\(u^n\) is the natural rate) and \(\nu_t\) captures other inflation shocks. The closed-form solution of the model can be written as \[\begin{equation} \pi_t = -\kappa E_t \sum \limits_{j=0}^{\infty} \beta^j \left( u_{t+j} - u^n_{t+j} \right) + v_t,\tag{6.8} \end{equation}\] where \(v_t\) collects the expected future discounted sum of the \(\nu\) terms, \[\begin{equation} v_t = E_t \sum \limits_{j=0}^\infty \beta^j \nu_{t+j}\tag{6.9} \end{equation}\] (typically values of \(\nu\) after time \(t\) are assumed to have an unconditional mean of zero, though they don’t have to). But Hazell et al., 2020 instead solve the model to yield the equivalent expression \[\begin{equation} \pi_t = -\kappa E_t \sum \limits_{j=0}^{\infty} \beta^j u_{t+j} + \omega_t ,\tag{6.10} \end{equation}\] where \(\omega_t\) collects the expected discounted forward sum of the \(u^n\) and \(\nu\) terms, \[\begin{equation} \omega_t = E_t \sum \limits_{j=0}^\infty \beta^j (\kappa u^n_{t+j} + \nu_{t+j}).\tag{6.11} \end{equation}\] They then define the “transitory’’ component of unemployment, \(\tilde u_{t}\), as \[\begin{equation} \tilde u_t = u_t - E_t u_{t+\infty}.\tag{6.12} \end{equation}\] Adding and subtracting \(\frac{\kappa}{1-\beta} E_t u_{t+\infty}\) on the right side of equation (6.10) and noting that \[\begin{equation} -\frac{\kappa}{1-\beta}E_t u_{t+\infty} = E_t \pi_{t+\infty} ,\tag{6.13} \end{equation}\] lets us rewrite the equation in terms of \(\tilde u\) as \[\begin{equation} \pi_t = -\kappa E_t \sum \limits_{j=0}^{\infty} \beta^j \tilde u_{t+j} + E_t \pi_{t+\infty} + \omega_t .\tag{6.14} \end{equation}\] Hazell et al., 2020 claim that the presence of the \(E_t \pi_{t+\infty}\) term in equation (6.14), which enters the equation with a coefficient of one, demonstrates the importance of long-run expectations.

This solution is puzzling when we compare it with equation (6.6). As that equation shows, if inflation is bounded, then the terminal inflation term \(\beta^{K+1} E_t \pi_{t+K+1}\) should go to zero as \(K\) goes to infinity; that’s the reason that a terminal inflation term isn’t present in equations (6.8) or (6.10) and it’s also the reason that a rational-expectations solution exists. So why does a \(E_t \pi_{t+\infty}\) term appear in equation (6.14)?

The puzzle is resolved when we see that the solution proposed by Hazell et al., 2020 is either vacuous or leads to inconsistencies. As was noted in the derivation to equation (6.14), all we are doing is adding and subtracting the same term \(E_t \pi_{t+\infty} = \frac{-\kappa}{1-\beta} E_t u_{t+\infty}\) on the right-hand side of the equation, and we assume that the public’s expectations respect this consistency relation (equation 6.13). Hence, if people’s expectation \(E_t \pi_{t+\infty}\) were to rise (say) by a percentage point, they should revise their expectation for \(E_t u_{t+\infty}\) by \(-(1-\beta)/\kappa\). But that implies no change to the right-hand side of equation (6.14), and so no change in inflation, for the same expected sequence of current and future actual unemployment rates \(E_t u_{t+j}\). And that in turn implies that inflation will not converge to the new \(E_t \pi_{t+\infty}\) and the unemployment rate will not converge to the new \(E_t u_{t+\infty}\), which violates the definition of these terms and also implies that the unconditional mean of \(\tilde u\) will not be zero. (This last point implies that the solution Hazell et al., 2020 obtain when they assume \(\beta=1\) is incorrect.)

We also run into problems if we assume that \(E_t \pi_{t+\infty}\) and \(E_t u_{t+\infty}\) are free to change independently, so that equation (6.13) doesn’t have to hold. If we raise \(u_{t+j}\) and \(E_t u_{t+\infty}\) by the same amount in every period, inflation will be unchanged according to equation (6.14). So we get the same inflation rate even though the sequence of unemployment gaps \(u_{t+j} - u^n_{t+j}\) is different in the two cases, which obviously violates equation (6.8).³⁴ And there is another, common-sense reason to be skeptical of the implications of Hazell et al., 2020’s expression (6.14) if \(E_t \pi_{t+\infty}\) is allowed to vary independently. Similar to Werning, 2022, \(E_t \pi_{t+\infty}\) is an expectation of far-future inflation rather than long-term inflation expectations as they are typically thought of (and that Hazell et al., 2020 use in their empirical work with aggregate data). It seems a priori implausible that a revision to the expectation of the rate of inflation in the very far future would have an immediate effect on current inflation, but that is what equation (6.14) implies.³⁵

7 Conclusion

This paper estimates state-level trend unemployment rates using a duration-structure model that exploits information in unemployment inflows and duration-dependent outflows. Applying the framework of Ahn, 2023 to all 50 U.S. states and the District of Columbia reveals substantial heterogeneity in trend U-stars across regions, particularly in the component of U-star that reflects long-term (or structural) unemployment. Agricultural states exhibit relatively small structural unemployment shares, while Rust Belt and several Sun Belt states display elevated long-term unemployment, highlighting persistent regional differences in labor-market functioning. We also find that the secular decline in trend U-star at the national level has not been broad-based, with U-stars in roughly half of the states showing little or no decline. Although cross-state dispersion has narrowed over time, this convergence is driven mainly by the short-term component, while heterogeneity in long-term unemployment remains pronounced.

Using these estimates, we revisit recent research that has used state-level data to estimate the slope of the Phillips curve. Once the heterogeneous behavior of trend U-star across states is accounted for and the unemployment rate is replaced by the unemployment gap, the slope of the forward-looking Phillips curve becomes statistically insignificant, even when state and time fixed effects are controlled for. This result suggests that previous state-level estimates that find a significant relationship between unemployment and inflation in the context of a new-Keynesian model largely reflect comovement between inflation and slow-moving components of trend unemployment, rather than a relationship between inflation and cyclical movements in labor-market slack. The results also imply that the forward-looking new-Keynesian Phillips curve provides as poor a characterization of state-level inflation dynamics as it does of aggregate inflation.

Overall, our findings underscore the importance of accounting for regional heterogeneity in the natural rate when measuring labor-market slack and estimating the Phillips curve on regional data, with obvious implications for both empirical modeling and the design of stabilization policies.

A Priors

A1 Trend–cycle model

For each series of length \(T\), let \(y_t\) denote the (quarterly, time-aggregated) observation. The model is the unobserved-components specification \[\begin{align} y_t &= \tau_t + c_t,\tag{A.1} \\ \Delta^{2}\tau_t &= \tau_t - 2\tau_{t-1} + \tau_{t-2} = u_t, \qquad u_t \sim \N(0,\sigma_\tau^{2}),\tag{A.2} \\ c_t &= \phi_1 c_{t-1} + \phi_2 c_{t-2} + \varepsilon_t, \qquad \varepsilon_t \sim \N(0,\sigma_c^{2}),\tag{A.3} \end{align}\] for \(t=1,\dots,T\), with \(u_t \perp \varepsilon_s\) for all \(t,s\). The trend \(\tau_t\) follows an I(2) process so that its first difference—the local growth rate—evolves as a random walk; this is the standard “smooth trend” specification of Harvey (1989) and underlies the Chan–Grant style sampler used here. The cycle \(c_t\) follows a stationary AR(2) process whose roots are restricted to lie inside the unit circle.

Stacking (A.2) for \(t=1,\dots,T\) yields \[H_2\,\boldsymbol{\tau} = \alpha_{\tau_0} + \boldsymbol{u}, \qquad \boldsymbol{u}\sim\N(\boldsymbol{0},\sigma_\tau^{2} I_T),\] where \(H_2\) is the second-difference matrix and \(\alpha_{\tau_0}\) collects the initial conditions \((\tau_0,\tau_{-1})\). Stacking (A.3) similarly yields \[H_\phi(y - \boldsymbol{\tau}) = \boldsymbol{\varepsilon}, \qquad \boldsymbol{\varepsilon}\sim\N(\boldsymbol{0},\sigma_c^{2} I_T),\] with \(H_\phi\) the lower-triangular AR(2) matrix built from \(\phi=(\phi_1,\phi_2)'\). The two unknown initial trend values \(\boldsymbol{\tau}_0 \equiv (\tau_0,\tau_{-1})'\) are treated as parameters and sampled from their full conditional distribution, exactly as in (A.8) below.

A2 Prior specification

The unknown parameters are \(\boldsymbol{\theta} = (\phi_1, \phi_2, \sigma_c^{2}, \sigma_\tau^{2}, \tau_0, \tau_{-1})\). Priors are independent across blocks: \[p(\boldsymbol{\theta}) \;=\; p(\phi)\, p(\sigma_c^{2})\, p(\sigma_\tau^{2})\, p(\boldsymbol{\tau}_0).\]

A2.1 AR(2) cycle coefficients \(\phi\)

The autoregressive coefficients are given a bivariate Gaussian prior truncated to the stationary region: \[\begin{equation} \phi \;\sim\; \N\!\bigl(\phi_0,\, V_\phi\bigr)\, \mathbf{1}\{\phi \in S\}.\tag{A.4} \end{equation}\] The prior mean \(\phi_0\) encodes the prior belief that the cyclical component exhibits the positively autocorrelated, hump-shaped impulse response typical of business-cycle dynamics in unemployment-related series. The value across states and series is \(\phi_0 = (1.3,-0.7)'\). The prior precision \(V_\phi^{-1}\) is set to \(I_2\) (unit precision) for most series; for \(j=1\) it is loosened to \(10^{-3}\, I_2\) in many states (a near-flat prior over \(\phi\)). The exact value of \(V_\phi^{-1}\) for each state and series is reported in Tables A.1–A.5.

A2.2 Stationarity restriction

The indicator \(\mathbf{1}\{\phi \in S\}\) in (A.4) restricts \(\phi\) to the AR(2) stationarity triangle. Operationally, the sampler accepts a candidate draw \(\phi^{c}\) only if \[\begin{equation} \phi_1^{c}+\phi_2^{c} < 0.99, \qquad \phi_2^{c}-\phi_1^{c} < 0.99, \qquad \phi_2^{c} > -0.99,\tag{A.5} \end{equation}\] which is the standard set of inequalities defining the interior of the stationarity region (with a small margin so that draws on the boundary are excluded). Rejected candidates leave \(\phi\) unchanged at its previous value. This restriction is identical across all states and all series.

A2.3 Cycle innovation variance \(\sigma_c^{2}\)

The cycle innovation variance is given a conjugate inverse-gamma prior: \[\begin{equation} \sigma_c^{2} \;\sim\; \IG\!\bigl(\nu_c,\, S_c\bigr),\tag{A.6} \end{equation}\] implying the full conditional \[\sigma_c^{2} \mid \cdot \;\sim\; \IG\!\Bigl(\nu_c + \tfrac{T}{2},\; S_c + \tfrac{1}{2}(c-X_\phi\phi)'(c-X_\phi\phi)\Bigr).\] Hyperparameters \((\nu_c,S_c)\) are calibrated separately for each series and each state; see Tables A.1–A.5. Series for which a tight prior is appropriate (notably the duration parameter \(\beta_0\) (\(j=2\)), where \((\nu_c,S_c)=(10^{-6},\,10^{-6})\) across all states) use very small values, yielding a near-improper prior on \(\sigma_c^{2}\). The labor force series uses larger values in those states for which the cycle component is non-negligible.

A2.4 Trend innovation variance \(\sigma_\tau^{2}\)

Following Chan and Grant (2017) and the bounded prior approach of Frisch–Waugh-style decompositions, the trend innovation variance is given a uniform prior on a bounded interval: \[\begin{equation} \sigma_\tau^{2} \;\sim\; \Uni\bigl(0,\, \overline{\sigma}_\tau^{2}\bigr),\tag{A.7} \end{equation}\] with the upper bound \(\overline{\sigma}_\tau^{2}\) chosen series by series and state by state. The full conditional density is \[p\!\bigl(\sigma_\tau^{2} \mid \cdot\bigr) \;\propto\; (\sigma_\tau^{2})^{-T/2} \exp\!\Bigl\{-\tfrac{1}{2\sigma_\tau^{2}} \sum_{t=2}^{T}(\Delta\tau_t-\Delta\tau_{t-1})^{2}\Bigr\}\, \mathbf{1}\bigl\{0<\sigma_\tau^{2}<\overline{\sigma}_\tau^{2}\bigr\},\] which is sampled via a discretized grid (500 grid points, with end-points jittered to avoid placing mass on the boundary). The tight upper bound plays the role of a smoothness restriction on the trend: It precludes large random shifts in trend growth and disciplines the trend–cycle decomposition in the absence of strong informational content from the data alone. The values of \(\overline{\sigma}_\tau^{2}\) used for each state and series are reported in Tables A.1–A.5.

A2.5 Initial trend conditions

The pair of pre-sample trend values \(\boldsymbol{\tau}_0 = (\tau_0,\tau_{-1})'\) is given a Gaussian prior \[\begin{equation} \boldsymbol{\tau}_0 \;\sim\; \N\!\bigl(a_0,\, B_0\bigr).\tag{A.8} \end{equation}\]

For the inflow rate and the three duration-distribution parameters (\(j=1,\dots,4\)), the prior mean is \[a_0 \;=\; \bigl(\bar y,\,\bar y\bigr)' \;+\; \delta_j\cdot\mathbf{1}_2,\] where \(\bar y\) is the sample mean of the series over the available pre-COVID sample and \(\delta_j\) is a small series- and state-specific shift used to nudge the initial trend toward economically plausible levels. For the labor force series, the prior mean is \(a_0 = (y_1, y_1)'\), the first observed value of the series. The prior covariance \(B_0\) is diagonal with magnitude \(10\), \(100\), or \(1{,}000\) depending on the state and series; see Tables A.1–A.5.

A2.6 Sampler initialization

Throughout, the Markov chain is initialized at \(\phi^{(0)}=\phi_0\), \(\sigma_c^{2(0)}=\nu_c\), \(\sigma_\tau^{2(0)}=\overline{\sigma}_\tau^{2}\), and \(\boldsymbol{\tau}_0^{(0)}=a_0\). The sampler is run for \(N_{\text{sim}}=10{,}000\) post-burn-in iterations after a burn-in of \(1{,}000\), and posterior summaries are based on the resulting draws.

A3 State-level hyperparameter values

Table A.1: Hyperparameters for the unemployment inflow rate (\(j=1\)) by state. \(\delta_j\) is the shift applied to the prior mean: \(a_0 = (\bar y, \bar y)' + \delta_j\cdot\mathbf{1}_2\). \(\nu_c, S_c\) are the inverse-gamma shape and scale; \(\overline{\sigma}^2_\tau\) is the upper bound of the uniform prior on the trend innovation variance.

Tables A.1–A.5 report the full set of hyperparameters used for each of the 51 states for each of the five series, namely the unemployment inflow rate (\(j=1\)), the three duration-distribution parameters \(\beta_0\), \(\beta_1\), \(\beta_2\) (\(j=2,3,4\)), and the labor force. The columns are: \(\delta_j\) (the shift in the prior mean of \(\boldsymbol{\tau}_0\) for that series), \(B_0\) (the prior covariance of \(\boldsymbol{\tau}_0\)), \(\phi_0\) (the prior mean of the AR(2) coefficients), \(V_\phi^{-1}\) (the prior precision of \(\phi\)), \(\nu_c\) and \(S_c\) (the inverse-gamma hyperparameters for \(\sigma_c^{2}\)), and \(\overline{\sigma}_\tau^{2}\) (the upper bound of the uniform prior on \(\sigma_\tau^{2}\)).

Table A.2: Hyperparameters for the duration-distribution parameter \(\beta_0\) (\(j=2\)) by state.

Table A.3: Hyperparameters for the duration-distribution parameter \(\beta_1\) (\(j=3\)) by state.

Table A.4: Hyperparameters for the duration-distribution parameter \(\beta_2\) (\(j=4\)) by state.

Table A.5: Hyperparameters for the labor force series by state. \(a_0=(y_1,y_1)'\) for all states.

B Factor Estimates

Figures B.1 – B.6 display each state’s factor estimates and their trends. For each state, the upper left panel shows the estimated inflows, the upper right panel \(\beta_0\), the lower left \(\beta_1\), and the lower right panel \(\beta_2\). The black lines are the factor estimates, the solid blue lines are the posterior means of trend estimates, and the dashed blue lines are the 68 percent posterior intervals.

Figure B.1: Factor estimates (1)
(a) Alaska
(b) Alabama
(c) Arkansas
(d) Arizona
(e) California
(f) Colorado
(g) Connecticut
(h) District of Columbia
(i) Delaware
(j) Florida
Notes to Figure B.1: For each state, the inflows are displayed in the upper left panel, \(\beta_0\) in the upper right panel, \(\beta_1\) in the lower left panel, \(\beta_2\) in the lower right panel. The black lines are the factor estimates, the solid blue lines are the posterior means of trend estimates, and the dashed blue lines are the 68 percent posterior intervals. Shaded areas are NBER recessions. Source: Authors’ calculation.

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Figure B.2: Factor estimates (2)
(a) Georgia
(b) Hawaii
(c) Iowa
(d) Idaho
(e) Illinois
(f) Indiana
(g) Kansas
(h) Kentucky
(i) Louisiana
(j) Massachusetts
Notes to Figure B.2: For each state, the inflows are displayed in the upper left panel, \(\beta_0\) in the upper right panel, \(\beta_1\) in the lower left panel, \(\beta_2\) in the lower right panel. The black lines are the factor estimates, the solid blue lines are the posterior means of trend estimates, and the dashed blue lines are the 68 percent posterior intervals. Shaded areas are NBER recessions. Source: Authors’ calculation.

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Figure B.3: Factor estimates (3)
(a) Maryland
(b) Maine
(c) Michigan
(d) Minnesota
(e) Missouri
(f) Mississippi
(g) Montana
(h) North Carolina
(i) North Dakota
(j) Nebraska
Notes to Figure B.3: For each state, the inflows are displayed in the upper left panel, \(\beta_0\) in the upper right panel, \(\beta_1\) in the lower left panel, \(\beta_2\) in the lower right panel. The black lines are the factor estimates, the solid blue lines are the posterior means of trend estimates, and the dashed blue lines are the 68 percent posterior intervals. Shaded areas are NBER recessions. Source: Authors’ calculation.

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Figure B.4: Factor estimates (4)
(a) New Hampshire
(b) New Jersey
(c) New Mexico
(d) Nevada
(e) New York
(f) Ohio
(g) Oklahoma
(h) Oregon
(i) Pennsylvania
(j) Rhode Island
Notes to Figure B.4: For each state, the inflows are displayed in the upper left panel, \(\beta_0\) in the upper right panel, \(\beta_1\) in the lower left panel, \(\beta_2\) in the lower right panel. The black lines are the factor estimates, the solid blue lines are the posterior means of trend estimates, and the dashed blue lines are the 68 percent posterior intervals. Shaded areas are NBER recessions. Source: Authors’ calculation.

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Figure B.5: Factor estimates (5)
(a) South Carolina
(b) South Dakota
(c) Tennessee
(d) Texas
(e) Utah
(f) Virginia
(g) Vermont
(h) Washington
(i) Wisconsin
(j) West Virginia
Notes to Figure B.5: For each state, the inflows are displayed in the upper left panel, \(\beta_0\) in the upper right panel, \(\beta_1\) in the lower left panel, \(\beta_2\) in the lower right panel. The black lines are the factor estimates, the solid blue lines are the posterior means of trend estimates, and the dashed blue lines are the 68 percent posterior intervals. Shaded areas are NBER recessions. Source: Authors’ calculation.

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Figure B.6: Factor estimates (6)
(a) Wyoming
Notes to Figure B.6: For each state, the inflows are displayed in the upper left panel, \(\beta_0\) in the upper right panel, \(\beta_1\) in the lower left panel, \(\beta_2\) in the lower right panel. The black lines are the factor estimates, the solid blue lines are the posterior means of trend estimates, and the dashed blue lines are the 68 percent posterior intervals. Shaded areas are NBER recessions. Source: Authors’ calculation.

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Footnotes