2.1. Background on recent changes to immigration policy

Eight major immigration policy changes are relevant to this paper. Among these are changes in the quota of various categories of immigration, level of border patrol enforcement, and amnesty for irregular immigrants. ^{Footnote 1} Changes to immigration policy are important in identifying the aggregate effects of immigration on the US economy. These changes affected the flow of immigration to the USA in several ways.

The Immigration and Nationality Act of 1965 (INA, also known as the Hart-Celler Act) first permitted non-Caucasian immigration, which had been prohibited since 1924. The Act had important long-run implications on the demographic composition of immigration, number of new immigrants, and migration decision. The enactment of the Refugee Act of 1980 and the Immigration Reform and Control Act of 1986 (IRCA) began an era of nonrestrictive immigration policies by creating new categories of permanent immigration and a nationwide amnesty of irregular immigration. Coupled with the American Homecoming and Immigration Act of 1990 (IMMACT), which increased immigration quota, these changes in immigration policy raised quarterly immigration influx by almost threefold, peaking at 1990Q1. Between the mid-1990s and the 2001 dot-com bubble, quotas for employment-based visa (and effectively the number of employment-based permanent residency applications) were raised three times in 1990, 1998, and 2000 up to 195,000, before it decreased to its pre-1990 level of 65,000 in 2004 through the H-1B Reform Act. The re-adjustment of quota generated kinks in at least the employment-based category of permanent immigration. Changes in these policies could also induce structural breaks in immigration. Given the history of US immigration policy during the sample, it is important to study the macroeconomic dynamics of immigration using an empirical model that can account for structural change in the US economy tied to changes in immigration policy.

2.2 Data

I use time series on the flow of new permanent residency as the measurement of immigration. I assume this measure of immigration proxies for the number of new immigrants eligible to work in the US labor market. ^{Footnote 2} The source is Yearbook of Immigration Statistics published by the DHS. This series contains annual flow of foreign-born civilian admittance to the USA by category but excludes temporary immigration admittance (nonimmigrant visa holders) such as foreign students or visitors. The sample covers 1953–2017. ^{Footnote 3}

A problem with the DHS immigration data is its low frequency at an annual rate. I interpolate the official annual series to quarterly from 1953Q1 to 2017Q4 using a regression-based approach developed by Silva and Cardoso (Reference Silva and Cardoso2001). ^{Footnote 4} The top panel of Figure 1 plots the interpolated quarterly immigration. Eight major immigration policy dates are labeled with arrows. Immigration begins to resemble a unit root process after the enactment of the 1965 INA. Until 1990, immigration displays a positive trend and the fluctuations around trend are small. There is a large peak in immigration from 1988 to 1991, which coincides with the enactment of the 1986 IRCA and the 1990 IMMACT. This peak subsides by 1993. Immigration reverts to the pre-1990 trend after the peak but exhibits greater volatility around the trend.

Figure 1. Data.

Notes: The sample size ranges from 1953Q1 to 2017Q4. Immigration policy reform dates are labeled with arrows. Gray shaded areas are NBER recession trough dates. X-axis: sample date; y-axis: $10^5$ persons (immigration), percentage points (ALP, consumption, term spread), log points (hours worked).

Source: Yearbook of Immigration, Bureau of Labor Statistics, Cociuba et al. (Reference Cociuba, Prescott and Ueberfeldt2018), and FRED.

I test for unit roots in the growth rate and log levels of immigration. Augmented Dickey–Fuller test rejects the existence of unit root at 1% when the series is in growth rate but not in log levels. As a result, the interpolated quarterly immigration series is treated as observationally equivalent to an I(1) series. However, I note that structural breaks can be responsible for creating random walk like behavior in immigration.

Figure 1 also contains four other macroeconomic indicators on which the SVARs are estimated. ALP is utilization-adjusted following Basu et al. (Reference Basu, Fernald and Kimball2006). Hours worked is log weekly hours per capita. Consumption is a Fisher ideal index of constant dollar services and nondurable goods per capita consumption expenditures. Treasury term spread is the difference between yield on 10-year constant maturity Treasury and yield on 90-day Treasury.

3.1 The TVP-SV-SVAR

The data vector $z_t = [\Delta{\log}(x_t),\, {\log}(n_t),\,\Delta{\log}(I_t), \,\Delta{\log}(c_t), \, s_t]$ collects ALP growth ( $\Delta{\log}(x_t)$ ), log weekly hours ( ${\log}(n_t)$ ), immigration growth ( $\Delta{\log}(I_t)$ ), consumption growth ( $\Delta{\log}(c_t)$ ), and the term spread ( $s_t$ ). ^{Footnote 5} I estimate

\begin{equation*}z_t = b_{t} + \sum_{l=1}^p{B}_{l,t}z_{t-l} + A_{0,t}^{-1}\Sigma_t^{\frac{1}{2}}\epsilon_t, \quad \epsilon_t \sim N(0, I),\end{equation*}

where $b_{t}$ denotes the $5\times1$ row vector of time-varying constants, ${B}_{l,t}$ are $5\times5$ matrices of the reduced-form TVPs for lags $l=1,...,p$ and $t=1, ..., T$ . The impact matrix $A_{0,t}$ has ones on the diagonal and can be non-recursive for the off-diagonal elements. The $5\times5$ diagonal matrix $\Sigma_t^{\frac{1}{2}} = diag\{\sigma_{k,t}\}$ collects the square roots of the SVs of the structural shocks $\epsilon_t$ at t along the diagonal with zeroes everywhere else. The concentrated form of the SVAR is

(1) \begin{equation}A_{0,t}(z_t - X_t'\mathbb{B}_t) = \Sigma_t^{\frac{1}{2}}\epsilon_t,\end{equation}

where $X_t' = I_{n}\otimes[z_{t-1}'...\,z_{t-p}'\,1]$ and $\mathbb{B}_t = [vec({B}_{1,t})'...\,vec({B}_{p,t})'\,b_t]'$ . The off-diagonal entries in $A_{0,t}$ represent the contemporaneous restrictions. I denote $a_{jk,t}$ the free elements in $A_{0,t}$ .

As is customary in the TVP-VAR literature, parameter blocks $(\mathbb{B}_t, \, A_{0,t}, \, \sigma_{k,t})$ are treated as latent variables that evolve as driftless random walks and geometric random walks:

(2) \begin{equation}p(\mathbb{B}_t|\mathbb{B}_{t-1},Q) = I(\mathbb{B}_t)f(\mathbb{B}_t|\mathbb{B}_{t-1},Q),\end{equation}

(3) \begin{equation}a_{jk,t} = a_{jk,t-1} + \zeta_{t},\end{equation}

(4) \begin{equation}log(\sigma_{k,t}) = log(\sigma_{k,t-1}) + \tau_{k,t},\end{equation}

where $I(\mathbb{B}_t)$ is an indicator function that rejects unstable draws of $\mathbb{B}_t$ , the law of motion $f(\mathbb{B}_t|\mathbb{B}_{t-1},Q)$ is given by $\mathbb{B}_t = \mathbb{B}_{t-1} + \eta_t$ . Let $\tau_t = [\tau_{1,t},...,\tau_{5,t}]$ , then $[\eta_t, \zeta_t, \tau_t]$ are mean zero, $i.i.d.$ disturbances with variance–covariance matrices [Q, S, W].

Putting the blocks of latent variables together yields

(5) \begin{equation}\mathcal{V} = Var\left(\begin{bmatrix} \epsilon_t \\ \eta_t \\ \zeta_t \\ \tau_t \\ \end{bmatrix}\right) = \begin{bmatrix} I_5 & 0 & 0 &0\\ 0 & Q & 0 &0\\ 0 & 0 & S &0\\ 0 & 0 & 0 &W\\ \end{bmatrix},\end{equation}

where Q, S, and W are full rank. Note that this setup accommodates SVARs lacking TVPs and/or SV. The estimation of a static-coefficient SVAR requires setting $Q = S = W =0$ . A TVP-SVAR with no SV sets $W =0$ . A static-coefficient SVAR with SV sets $Q = S = 0$ .

3.2. Identification

This section outlines the strategies I use to identify the SVARs. I employ a novel set of short- and long-run restrictions to identify a supply shock to immigration. I label the immigration shock a supply shock following Borjas (Reference Borjas2003). The definition of immigration in this paper is similar to Borjas’ notion of influx of immigrants to the USA that he refers to as immigration supply. This is also to clarify that I am not identifying an immigration policy shock.

The complication in identifying the immigration supply shock stems from the endogeneity of immigration. Forward-looking immigrants self-select into countries and labor markets in which they anticipate their job market prospects are best (Friedberg (Reference Friedberg2001), Borjas (Reference Borjas2003)). To identify an immigration supply shock in the macroeconomy, the permanent income hypothesis and news about future TFP shocks are used to tackle the forward-looking aspect of immigration.

The five identified shocks are TFP, labor demand, immigration, transitory consumption, and news about future TFP. ^{Footnote 6} This section concludes with a comparison of four identification schemes and specifications on TVP and/or SV. A Bayesian model selection exercise explores the preference the data have over the competing models.

3.2.1 Short-run restrictions

Following standard practice in the immigration literature (e.g., Pischke and Velling (Reference Pischke and Velling1997), Card (Reference Card2007), Dalbis et al. (Reference Dalbis, Boubtane and Coulibaly2016)), the first identifying assumption is migration decisions are made prior to the entry date. Partridge and Rickman (Reference Partridge and Rickman2009) document the delayed migration response to changes in economic conditions. Therefore, the response of immigration to macroeconomic incentives lags the date an immigrant enters the labor market. Imposing this restriction on $A_{0,t}$ yields the first set of short-run identifying restrictions:

\begin{equation*}\text{SR1}: \quad\quad a_{31,t} = a_{32,t} = a_{34,t} = a_{35,t} = 0.\end{equation*}

These short-run identifying restrictions force immigration to react only to its own shock at impact.

The news shock literature (Forni et al. (Reference Forni, Gambetti and Luca2014), Beaudry et al. (Reference Beaudry, Feve, Guay and Portier2019), among others) identify ALP to only respond to its own shock on impact. I use this as the second set of short-run restrictions which consists of

\begin{equation*}\text{SR2}: \quad\quad a_{12,t} = a_{13,t} = a_{14,t} = a_{15,t} = 0.\end{equation*}

Note the immigration supply shock affects ALP only with a lag. Stated differently, I assume immigration faces labor market rigidity of at least one quarter.

A neoclassical model of the labor market predicts immigration causes labor productivity to fall and hours worked to rise; see Card (Reference Card2001) and Borjas (Reference Borjas2006). Nonetheless, the microeconomics literature on immigration often reports conflicting empirical evidence about the impact of immigration on labor market outcomes, according to Kerr and Kerr (Reference Kerr and Kerr2011). Given no clear advice, I leave the impact response of hours worked to an immigration supply shock unrestricted.

Finally, an inflow of immigration affects consumption. There are studies documenting the effect immigration has on the receiving economy through a change in consumption expenditure. For instance, Coppel et al. (Reference Coppel, Dumont and Visco2001) and Hong and McLaren (Reference Hong and McLaren2016) find immigration increases consumption, because immigration enlarges an economy through an immediate increase in the number of households. However, since there is little evidence on whether this effect occurs at impact or with a lag, I consider two alternatives. The first option imposes

\begin{equation*}\text{SR3}: \quad\quad a_{43,t} = 0.\end{equation*}

Under SR3, consumption responds to the immigration supply shock with a lag. The alternative assumes that consumption reacts to the immigration supply shock at impact, which implies $a_{43,t}$ is estimated.

To sum up, I consider two short-run identifications:

(6) \begin{equation}A_{0,t}^I = \begin{bmatrix} 1 & 0 & 0 & 0 & 0\\ a_{21,t} & 1 & a_{23,t} & a_{24,t} & a_{25,t}\\ 0 & 0 & 1 & 0 & 0\\ a_{41,t} & a_{42,t} & a_{43,t} & 1 & a_{45,t}\\ a_{51,t} & a_{52,t} & a_{53,t} & a_{54,t} & 1\\ \end{bmatrix} \,\, \text{and}\,\, A_{0,t}^{II} = \begin{bmatrix} 1 & 0 & 0 & 0 & 0\\ a_{21,t} & 1 & a_{23,t} & a_{24,t} & a_{25,t}\\ 0 & 0 & 1 & 0 & 0\\ a_{41,t} & a_{42,t} & 0 & 1 & a_{45,t}\\ a_{51,t} & a_{52,t} & a_{53,t} & a_{54,t} & 1\\\end{bmatrix}.\end{equation}

The impact matrix $A_{0,t}^I$ only imposes SR1 and SR2, while $A_{0,t}^{II}$ imposes SR1, SR2, and SR3. For both alternatives, consumption is allowed to respond to the TFP shock on impact following the permanent income hypothesis. A TFP shock permanently raises the household income and therefore consumption expenditure. Because of the willingness to smooth consumption over time, the agents take advantage of a higher income by increasing consumption on impact.

3.2.2 Long-run restrictions

A long-run identification is imposed by restricting the elements of the cumulative response matrix, $D_t$ . First, compute the cumulative impact matrix:

(7) \begin{equation}D_t = J(I_{5} - \mathbb{B}_t)^{-1}J'A_{0,t}^{-1}\Sigma_t^{\frac{1}{2}},\end{equation}

where $J = [I_5 \dots 0_5]$ is a selection matrix. Denote $d_{jk,t}$ the elements in $D_t$ , setting $d_{jk,t} = 0$ imposes long-run neutrality of variable j to shock k.

Let the restricted long-run cumulative matrix be $\tilde{D}_t$ . Solve for the restricted structural intercepts and slope parameters matrix $\tilde{\mathbb{B}}_t$ by inverting equation (7). This step is important because the reduced-form parameters in $\mathbb{B}_t$ are no longer consistent with the restricted long-run matrix. Imposing long-run restrictions induces nonlinearities to the reduced-form VAR parameter space because of the inversion of the equation. Therefore, I also check the eigenvalues of $\tilde{\mathbb{B}}_t$ to ensure stationarity of the SVAR. The implementation of long-run restrictions to (7) requires additional sampling steps that I outline in the following section.

One incentive of immigration is an increase in expected lifetime earnings. For example, Coppel et al. (Reference Coppel, Dumont and Visco2001) and Damette and Fromentin (Reference Damette and Fromentin2013) argue the decision to migrate depends on the economic prosperity of the host country. I assume the anticipated long-run performance of the US economy matters for the immigration decision. Hence, the long-run restrictions rest on the permanent income potential immigrants expect when making the decision to move. Consumption captures these changes in expected lifetime earnings of immigrants. Since immigration is observationally equivalent to an I(1) series, I assume the level of immigration is independent of the transitory labor demand and transitory consumption shocks in the long run. These assumptions are reflected in the first collection of long-run restrictions:

\begin{equation*}\text{LR1}: \quad\quad d_{32,t} = d_{34,t} = 0.\end{equation*}

The cumulative impact of a TFP shock on immigration, $d_{31,t}$ , is left unrestricted. As a result, estimates of $d_{31,t}$ measure the incentive to immigrate at the aggregate level.

Another important factor that shapes migration decision is news about TFP. The news shock represents the expectations agents hold about TFP in the future. I identify a news shock to capture the forward-looking nature of migration decision. Following the recent news literature, the identifying restrictions of the news shock include (i) SR2 and (ii) ALP responds to the news shock in the long run. ^{Footnote 7}

Immigration may be important to ALP in the long run, but micro-level data show conflicting evidence about the long-run effect of immigration on ALP. Firm-level data show immigration could raise long-run productivity through complementing domestic labor force, technological transfers, or innovation engagement; see Feyrer (Reference Feyrer2007) and Hunt and Gauthier-Loiselle (Reference Hunt and Gauthier-Loiselle2010). On the other hand, Ortega and Peri (Reference Ortega and Peri2013) and Paserman (Reference Paserman2013) find no effect on long-run productivity that is produced by immigration. I consider both alternatives in restricting the long-run response of ALP to an immigration supply shock. Further, I restrict ALP to not respond to labor demand shocks in the long run following GalÍ (Reference GalÍ1999). This identifying restriction assumes transitory demand shocks do not have permanent effects on I(1) variables:

\begin{equation*}\text{LR2}: \quad\quad d_{12,t} = d_{14,t} = 0,\end{equation*}

and

\begin{equation*}\text{LR3}: \quad\quad d_{13,t} = 0.\end{equation*}

The former set of restrictions imposes LR3. The immigration supply shock has no long-run impact on ALP. The alternative is to allow immigration supply shock to have long-run effects on ALP. This alternative leaves $d_{13,t}$ unrestricted.

Next, consumption is assumed to only respond to TFP, immigration supply, and news shocks in the long run. These responses are embodied in the restrictions:

\begin{equation*}\text{LR4}: \quad\quad d_{42,t} = d_{44,t} = 0.\end{equation*}

Transitory shocks in labor demand and consumption do not permanently alter consumption. In other words, only I(1) shocks that permanently alter the expected lifetime earnings have a long-run impact on consumption.

In summary, I consider two alternative long-run identification schemes, which are labeled $\tilde{D}_t^I$ and $\tilde{D}_t^{II}$ ,

(8) \begin{equation}\tilde{D}_t^I = \begin{bmatrix} d_{11,t} & 0 & 0 & 0 &d_{15,t}\\ d_{21,t} & d_{22,t} & d_{23,t} & d_{24,t} & d_{25,t}\\ d_{31,t} & 0 & d_{33,t} & 0 & d_{35,t}\\ d_{41,t} & 0 & d_{43,t} & 0 & d_{45,t}\\ d_{51,t} & d_{52,t} & d_{53,t} & d_{53,t} & d_{55,t}\\ \end{bmatrix}\,\, \text{and}\,\, \tilde{D}_t^{II} = \begin{bmatrix} d_{11,t} & 0 & d_{13,t} & 0 &d_{15,t}\\ d_{21,t} & d_{22,t} & d_{23,t} & d_{24,t} & d_{25,t}\\ d_{31,t} & 0 & d_{33,t} & 0 & d_{35,t}\\ d_{41,t} & 0 & d_{43,t} & 0 & d_{45,t}\\ d_{51,t} & d_{52,t} & d_{53,t} & d_{53,t} & d_{55,t}\\ \end{bmatrix}.\end{equation}

The two long-run identification schemes differ in imposing LR3 on ALP. In $\tilde{D}_t^{I}$ , ALP is long-run neutral to an immigration supply shock. In $\tilde{D}_t^{II}$ , ALP responds to an immigration supply shock in the long run.

3.2.3 Competing models

Given the short- and long-run identifying restrictions, I list all SVAR specifications using different combinations of these restrictions in Table 1. The two short-run restrictions ( $A_{0,t}^{I}$ and $A_{0,t}^{II}$ ) and the two long-run restrictions ( $\tilde{D}_t^{I}$ and $\tilde{D}_t^{II}$ ) give four possible identification schemes. Models 1–4 estimate TVP-SV-SVARs with each identification. ^{Footnote 8}

Table 1. List of models

Notes: TVP denotes time-varying parameters; SV is stochastic volatility; FP denotes fixed-parameter.

I also examine the importance of time variation in the SVAR parameters and in the SVs of the structural errors. I estimate SVARs with four different identifications by turning the TVPs and SVs on and off. Models 5–8 estimate TVP-SVARs with constant SV. Models 9–12 have constant intercepts and slope parameters with SV. Models 13–16 estimate fixed-parameter SVARs. This yields a total of 16 models to be compared in the Bayesian model selection exercise.

3.3 Bayesian estimation

The TVP-SV-SVAR is estimated in state-space form with (1) as the system of observation equations and (2)–(4) as the state equations using a Bayesian MCMC sampler. The goal is to obtain the posterior distribution of the states $\mathbb{B}$ , $\Sigma$ , and $A_0$ using a Metropolis-within-Gibbs sampler developed by Canova and Perez-Forero (Reference Canova and Perez-Forero2015). The SVs are sampled with the 10-component mixture normal routine via a tracking indicator variable (s) À la Omori et al. (Reference Omori, Chib, Shephard and Nakajima2007). A sketch of the sampling steps follows. ^{Footnote 9}

Step 1 Set initial values $(\mathbb{B}^T_0, A^T_{0,0}, \Sigma^T_0, s^T_0, \mathcal{V}_0)$ and set $i=1$ ,

Step 2 Draw the reduced-form intercept and slope parameters $\mathbb{B}^T_i$ from $p(\mathbb{B}^T_i | z^T, s^T_{i-1}, \Sigma^T_{i-1}, \mathcal{V}_{i-1}) {\cdot} I_B(\mathbb{B}^T_i)$ using the Carter–Kohn algorithm, where $I_B(\mathbb{B}^T_i)$ truncates the posterior distribution to ensure the stability of the companion form. Impose long-run identifications for each draw of $\mathbb{B}^T_i$ to obtain the structural slope parameters $\mathbb{\tilde{B}}^T_i$ ,

Step 3 Draw $A^T_{0,i}$ from $p(A^T_{0,i} | z^T, \mathbb{\tilde{B}}^T_i, s^T_{i-1}, \Sigma^T_{i-1}, \mathcal{V}_{i-1})$ ,

Step 4 Draw $\Sigma^T_i$ through auxiliary variables $s^T_i$ ,

Step 5 Draw hyperparameters $\mathcal{V}_i$ given $(\mathbb{\tilde{B}}^T_i, A^T_{0,i}, \Sigma^T_i, z^T)$ ,

Step 6 Repeat step 2 through 4 M times. The last N ( $<M$ ) draws are engaged to construct the posterior of the SVARs.

I set M = 400,000. The burn-in uses the first 200,000 draws. I apply a thinning factor of 50 to the remaining N = 200,000 iterations to reduce the autocorrelation across draws. The baseline TVP-SV-SVAR model produces acceptance rates of the Metropolis step and the rate of stationary draws in the Gibbs step of 24.4% and 32.1%.